Relational inductive biases, deep learning, and graph networks

Peter W. Battaglia1∗, Jessica B. Hamrick1, Victor Bapst1,

Alvaro Sanchez-Gonzalez1, Vinicius Zambaldi1, Mateusz Malinowski1,

Andrea Tacchetti1, David Raposo1, Adam Santoro1, Ryan Faulkner1,

Caglar Gulcehre1, Francis Song1, Andrew Ballard1, Justin Gilmer2,

George Dahl2, Ashish Vaswani2, Kelsey Allen3, Charles Nash4,

Victoria Langston1, Chris Dyer1, Nicolas Heess1,

Daan Wierstra1, Pushmeet Kohli1, Matt Botvinick1,

Oriol Vinyals1, Yujia Li1, Razvan Pascanu1

1DeepMind; 2Google Brain; 3MIT; 4University of Edinburgh

Abstract

Artificial intelligence (AI) has undergone a renaissance recently, making major progress inkey domains such as vision, language, control, and decision-making. This has been due, inpart, to cheap data and cheap compute resources, which have fit the natural strengths of deeplearning. However, many defining characteristics of human intelligence, which developed undermuch different pressures, remain out of reach for current approaches. In particular, generalizingbeyond one’s experiences—a hallmark of human intelligence from infancy—remains a formidablechallenge for modern AI.

The following is part position paper, part review, and part unification. We argue thatcombinatorial generalization must be a top priority for AI to achieve human-like abilities, and thatstructured representations and computations are key to realizing this objective. Just as biologyuses nature and nurture cooperatively, we reject the false choice between “hand-engineering”and “end-to-end” learning, and instead advocate for an approach which benefits from theircomplementary strengths. We explore how using relational inductive biases within deep learningarchitectures can facilitate learning about entities, relations, and rules for composing them. Wepresent a new building block for the AI toolkit with a strong relational inductive bias—the graphnetwork—which generalizes and extends various approaches for neural networks that operateon graphs, and provides a straightforward interface for manipulating structured knowledge andproducing structured behaviors. We discuss how graph networks can support relational reasoningand combinatorial generalization, laying the foundation for more sophisticated, interpretable,and flexible patterns of reasoning. As a companion to this paper, we have also released anopen-source software library for building graph networks, with demonstrations of how to usethem in practice.

1 Introduction

A key signature of human intelligence is the ability to make “infinite use of finite means” (Humboldt,1836; Chomsky, 1965), in which a small set of elements (such as words) can be productivelycomposed in limitless ways (such as into new sentences). This reflects the principle of combinatorialgeneralization, that is, constructing new inferences, predictions, and behaviors from known buildingblocks. Here we explore how to improve modern AI’s capacity for combinatorial generalization by

∗Corresponding author: peterbattaglia@google.com

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biasing learning towards structured representations and computations, and in particular, systemsthat operate on graphs.

Humans’ capacity for combinatorial generalization depends critically on our cognitive mecha-nisms for representing structure and reasoning about relations. We represent complex systems ascompositions of entities and their interactions1 (Navon, 1977; McClelland and Rumelhart, 1981;Plaut et al., 1996; Marcus, 2001; Goodwin and Johnson-Laird, 2005; Kemp and Tenenbaum, 2008),such as judging whether a haphazard stack of objects is stable (Battaglia et al., 2013). We usehierarchies to abstract away from fine-grained differences, and capture more general commonalitiesbetween representations and behaviors (Botvinick, 2008; Tenenbaum et al., 2011), such as parts ofan object, objects in a scene, neighborhoods in a town, and towns in a country. We solve novelproblems by composing familiar skills and routines (Anderson, 1982), for example traveling to anew location by composing familiar procedures and objectives, such as “travel by airplane”, “toSan Diego”, “eat at”, and “an Indian restaurant”. We draw analogies by aligning the relationalstructure between two domains and drawing inferences about one based on corresponding knowledgeabout the other (Gentner and Markman, 1997; Hummel and Holyoak, 2003).

Kenneth Craik’s “The Nature of Explanation” (1943), connects the compositional structure ofthe world to how our internal mental models are organized:

...[a human mental model] has a similar relation-structure to that of the process itimitates. By ‘relation-structure’ I do not mean some obscure non-physical entity whichattends the model, but the fact that it is a working physical model which works in thesame way as the process it parallels... physical reality is built up, apparently, from a fewfundamental types of units whose properties determine many of the properties of themost complicated phenomena, and this seems to afford a sufficient explanation of theemergence of analogies between mechanisms and similarities of relation-structure amongthese combinations without the necessity of any theory of objective universals. (Craik,1943, page 51-55)

That is, the world is compositional, or at least, we understand it in compositional terms. Whenlearning, we either fit new knowledge into our existing structured representations, or adjust thestructure itself to better accommodate (and make use of) the new and the old (Tenenbaum et al.,2006; Griffiths et al., 2010; Ullman et al., 2017).

The question of how to build artificial systems which exhibit combinatorial generalization hasbeen at the heart of AI since its origins, and was central to many structured approaches, includinglogic, grammars, classic planning, graphical models, causal reasoning, Bayesian nonparametrics, andprobabilistic programming (Chomsky, 1957; Nilsson and Fikes, 1970; Pearl, 1986, 2009; Russell andNorvig, 2009; Hjort et al., 2010; Goodman et al., 2012; Ghahramani, 2015). Entire sub-fields havefocused on explicit entity- and relation-centric learning, such as relational reinforcement learning(Dzeroski et al., 2001) and statistical relational learning (Getoor and Taskar, 2007). A key reasonwhy structured approaches were so vital to machine learning in previous eras was, in part, becausedata and computing resources were expensive, and the improved sample complexity afforded bystructured approaches’ strong inductive biases was very valuable.

In contrast with past approaches in AI, modern deep learning methods (LeCun et al., 2015;Schmidhuber, 2015; Goodfellow et al., 2016) often follow an “end-to-end” design philosophy whichemphasizes minimal a priori representational and computational assumptions, and seeks to avoidexplicit structure and “hand-engineering”. This emphasis has fit well with—and has perhaps beenaffirmed by—the current abundance of cheap data and cheap computing resources, which make

1Whether this entails a “language of thought” (Fodor, 1975) is beyond the scope of this work.

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trading off sample efficiency for more flexible learning a rational choice. The remarkable and rapidadvances across many challenging domains, from image classification (Krizhevsky et al., 2012;Szegedy et al., 2017), to natural language processing (Sutskever et al., 2014; Bahdanau et al., 2015),to game play (Mnih et al., 2015; Silver et al., 2016; Moravcık et al., 2017), are a testament to thisminimalist principle. A prominent example is from language translation, where sequence-to-sequenceapproaches (Sutskever et al., 2014; Bahdanau et al., 2015) have proven very effective without usingexplicit parse trees or complex relationships between linguistic entities.

Despite deep learning’s successes, however, important critiques (Marcus, 2001; Shalev-Shwartzet al., 2017; Lake et al., 2017; Lake and Baroni, 2018; Marcus, 2018a,b; Pearl, 2018; Yuille andLiu, 2018) have highlighted key challenges it faces in complex language and scene understanding,reasoning about structured data, transferring learning beyond the training conditions, and learningfrom small amounts of experience. These challenges demand combinatorial generalization, and so itis perhaps not surprising that an approach which eschews compositionality and explicit structurestruggles to meet them.

When deep learning’s connectionist (Rumelhart et al., 1987) forebears were faced with analogouscritiques from structured, symbolic positions (Fodor and Pylyshyn, 1988; Pinker and Prince, 1988),there was a constructive effort (Bobrow and Hinton, 1990; Marcus, 2001) to address the challengesdirectly and carefully. A variety of innovative sub-symbolic approaches for representing and reasoningabout structured objects were developed in domains such as analogy-making, linguistic analysis,symbol manipulation, and other forms of relational reasoning (Smolensky, 1990; Hinton, 1990;Pollack, 1990; Elman, 1991; Plate, 1995; Eliasmith, 2013), as well as more integrative theories forhow the mind works (Marcus, 2001). Such work also helped cultivate more recent deep learningadvances which use distributed, vector representations to capture rich semantic content in text(Mikolov et al., 2013; Pennington et al., 2014), graphs (Narayanan et al., 2016, 2017), algebraic andlogical expressions (Allamanis et al., 2017; Evans et al., 2018), and programs (Devlin et al., 2017;Chen et al., 2018b).

We suggest that a key path forward for modern AI is to commit to combinatorial generalizationas a top priority, and we advocate for integrative approaches to realize this goal. Just as biology doesnot choose between nature versus nurture—it uses nature and nurture jointly, to build wholes whichare greater than the sums of their parts—we, too, reject the notion that structure and flexibility aresomehow at odds or incompatible, and embrace both with the aim of reaping their complementarystrengths. In the spirit of numerous recent examples of principled hybrids of structure-based methodsand deep learning (e.g., Reed and De Freitas, 2016; Garnelo et al., 2016; Ritchie et al., 2016; Wuet al., 2017; Denil et al., 2017; Hudson and Manning, 2018), we see great promise in synthesizingnew techniques by drawing on the full AI toolkit and marrying the best approaches from todaywith those which were essential during times when data and computation were at a premium.

Recently, a class of models has arisen at the intersection of deep learning and structuredapproaches, which focuses on approaches for reasoning about explicitly structured data, in particulargraphs (e.g. Scarselli et al., 2009b; Bronstein et al., 2017; Gilmer et al., 2017; Wang et al., 2018c; Liet al., 2018; Kipf et al., 2018; Gulcehre et al., 2018). What these approaches all have in commonis a capacity for performing computation over discrete entities and the relations between them.What sets them apart from classical approaches is how the representations and structure of theentities and relations—and the corresponding computations—can be learned, relieving the burdenof needing to specify them in advance. Crucially, these methods carry strong relational inductivebiases, in the form of specific architectural assumptions, which guide these approaches towardslearning about entities and relations (Mitchell, 1980), which we, joining many others (Spelke et al.,1992; Spelke and Kinzler, 2007; Marcus, 2001; Tenenbaum et al., 2011; Lake et al., 2017; Lake andBaroni, 2018; Marcus, 2018b), suggest are an essential ingredient for human-like intelligence.

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Box 1: Relational reasoning

We define structure as the product of composing a set of known building blocks. “Structuredrepresentations” capture this composition (i.e., the arrangement of the elements) and “structuredcomputations” operate over the elements and their composition as a whole. Relational reasoning,then, involves manipulating structured representations of entities and relations, using rulesfor how they can be composed. We use these terms to capture notions from cognitive science,theoretical computer science, and AI, as follows:

◦ An entity is an element with attributes, such as a physical object with a size and mass.◦ A relation is a property between entities. Relations between two objects might includesame size as, heavier than, and distance from. Relations can have attributes aswell. The relation more than X times heavier than takes an attribute, X, whichdetermines the relative weight threshold for the relation to be true vs. false. Relationscan also be sensitive to the global context. For a stone and a feather, the relation fallswith greater acceleration than depends on whether the context is in air vs. in avacuum. Here we focus on pairwise relations between entities.◦ A rule is a function (like a non-binary logical predicate) that maps entities and relations

to other entities and relations, such as a scale comparison like is entity X large? andis entity X heavier than entity Y?. Here we consider rules which take one or twoarguments (unary and binary), and return a unary property value.

As an illustrative example of relational reasoning in machine learning, graphical models (Pearl,1988; Koller and Friedman, 2009) can represent complex joint distributions by making explicitrandom conditional independences among random variables. Such models have been verysuccessful because they capture the sparse structure which underlies many real-world generativeprocesses and because they support efficient algorithms for learning and reasoning. For example,hidden Markov models constrain latent states to be conditionally independent of others giventhe state at the previous time step, and observations to be conditionally independent given thelatent state at the current time step, which are well-matched to the relational structure of manyreal-world causal processes. Explicitly expressing the sparse dependencies among variablesprovides for various efficient inference and reasoning algorithms, such as message-passing, whichapply a common information propagation procedure across localities within a graphical model,resulting in a composable, and partially parallelizable, reasoning procedure which can be appliedto graphical models of different sizes and shape.

In the remainder of the paper, we examine various deep learning methods through the lens oftheir relational inductive biases, showing that existing methods often carry relational assumptionswhich are not always explicit or immediately evident. We then present a general framework forentity- and relation-based reasoning—which we term graph networks—for unifying and extendingexisting methods which operate on graphs, and describe key design principles for building powerfularchitectures using graph networks as building blocks. We have also released an open-source libraryfor building graph networks, which can be found here: github.com/deepmind/graph nets.

2 Relational inductive biases

Many approaches in machine learning and AI which have a capacity for relational reasoning

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Box 2: Inductive biases

Learning is the process of apprehending useful knowledge by observing and interacting with theworld. It involves searching a space of solutions for one expected to provide a better explanationof the data or to achieve higher rewards. But in many cases, there are multiple solutions whichare equally good (Goodman, 1955). An inductive bias allows a learning algorithm to prioritizeone solution (or interpretation) over another, independent of the observed data (Mitchell,1980). In a Bayesian model, inductive biases are typically expressed through the choice andparameterization of the prior distribution (Griffiths et al., 2010). In other contexts, an inductivebias might be a regularization term (McClelland, 1994) added to avoid overfitting, or it mightbe encoded in the architecture of the algorithm itself. Inductive biases often trade flexibilityfor improved sample complexity and can be understood in terms of the bias-variance tradeoff(Geman et al., 1992). Ideally, inductive biases both improve the search for solutions withoutsubstantially diminishing performance, as well as help find solutions which generalize in adesirable way; however, mismatched inductive biases can also lead to suboptimal performanceby introducing constraints that are too strong.

Inductive biases can express assumptions about either the data-generating process or the spaceof solutions. For example, when fitting a 1D function to data, linear least squares followsthe constraint that the approximating function be a linear model, and approximation errorsshould be minimal under a quadratic penalty. This reflects an assumption that the data-generating process can be explained simply, as a line process corrupted by additive Gaussiannoise. Similarly, L2 regularization prioritizes solutions whose parameters have small values,and can induce unique solutions and global structure to otherwise ill-posed problems. This canbe interpreted as an assumption about the learning process: that searching for good solutionsis easier when there is less ambiguity among solutions. Note, these assumptions need not beexplicit—they reflect interpretations of how a model or algorithm interfaces with the world.

(Box 1) use a relational inductive bias. While not a precise, formal definition, we use this term torefer generally to inductive biases (Box 2) which impose constraints on relationships and interactionsamong entities in a learning process.

Creative new machine learning architectures have rapidly proliferated in recent years, with(perhaps not surprisingly given the thesis of this paper) practitioners often following a design patternof composing elementary building blocks to form more complex, deep2 computational hierarchies andgraphs3. Building blocks such as “fully connected” layers are stacked into “multilayer perceptrons”(MLPs), “convolutional layers” are stacked into “convolutional neural networks” (CNNs), and astandard recipe for an image processing network is, generally, some variety of CNN composed witha MLP. This composition of layers provides a particular type of relational inductive bias—thatof hierarchical processing—in which computations are performed in stages, typically resulting inincreasingly long range interactions among information in the input signal. As we explore below, thebuilding blocks themselves also carry various relational inductive biases (Table 1). Though beyondthe scope of this paper, various non-relational inductive biases are used in deep learning as well: forexample, activation non-linearities, weight decay, dropout (Srivastava et al., 2014), batch and layernormalization (Ioffe and Szegedy, 2015; Ba et al., 2016), data augmentation, training curricula, andoptimization algorithms all impose constraints on the trajectory and outcome of learning.

2This pattern of composition in depth is ubiquitous in deep learning, and is where the “deep” comes from.3Recent methods (Liu et al., 2018) even automate architecture construction via learned graph editing procedures.

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Component Entities Relations Rel. inductive bias Invariance

Fully connected Units All-to-all Weak -Convolutional Grid elements Local Locality Spatial translationRecurrent Timesteps Sequential Sequentiality Time translationGraph network Nodes Edges Arbitrary Node, edge permutations

Table 1: Various relational inductive biases in standard deep learning components. See also Section 2.

To explore the relational inductive biases expressed within various deep learning methods, wemust identify several key ingredients, analogous to those in Box 1: what are the entities, what arethe relations, and what are the rules for composing entities and relations, and computing theirimplications? In deep learning, the entities and relations are typically expressed as distributedrepresentations, and the rules as neural network function approximators; however, the precise formsof the entities, relations, and rules vary between architectures. To understand these differencesbetween architectures, we can further ask how each supports relational reasoning by probing:

◦ The arguments to the rule functions (e.g., which entities and relations are provided as input).◦ How the rule function is reused, or shared, across the computational graph (e.g., across different

entities and relations, across different time or processing steps, etc.).◦ How the architecture defines interactions versus isolation among representations (e.g., by

applying rules to draw conclusions about related entities, versus processing them separately).

2.1 Relational inductive biases in standard deep learning building blocks

2.1.1 Fully connected layers

Perhaps the most common building block is a fully connected layer (Rosenblatt, 1961). Typicallyimplemented as a non-linear vector-valued function of vector inputs, each element, or “unit”, ofthe output vector is the dot product between a weight vector, followed by an added bias term, andfinally a non-linearity such as a rectified linear unit (ReLU). As such, the entities are the units inthe network, the relations are all-to-all (all units in layer i are connected to all units in layer j),and the rules are specified by the weights and biases. The argument to the rule is the full inputsignal, there is no reuse, and there is no isolation of information (Figure 1a). The implicit relationalinductive bias in a fully connected layer is thus very weak: all input units can interact to determineany output unit’s value, independently across outputs (Table 1).

2.1.2 Convolutional layers

Another common building block is a convolutional layer (Fukushima, 1980; LeCun et al., 1989). It isimplemented by convolving an input vector or tensor with a kernel of the same rank, adding a biasterm, and applying a point-wise non-linearity. The entities here are still individual units (or gridelements, e.g. pixels), but the relations are sparser. The differences between a fully connected layerand a convolutional layer impose some important relational inductive biases: locality and translationinvariance (Figure 1b). Locality reflects that the arguments to the relational rule are those entities inclose proximity with one another in the input signal’s coordinate space, isolated from distal entities.Translation invariance reflects reuse of the same rule across localities in the input. These biasesare very effective for processing natural image data because there is high covariance within local

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(a) Fully connectedSh

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Figure 1: Reuse and sharing in common deep learning building blocks. (a) Fully connected layer,in which all weights are independent, and there is no sharing. (b) Convolutional layer, in whicha local kernel function is reused multiple times across the input. Shared weights are indicated byarrows with the same color. (c) Recurrent layer, in which the same function is reused across differentprocessing steps.

neighborhoods, which diminishes with distance, and because the statistics are mostly stationaryacross an image (Table 1).

2.1.3 Recurrent layers

A third common building block is a recurrent layer (Elman, 1990), which is implemented over asequence of steps. Here, we can view the inputs and hidden states at each processing step as theentities, and the Markov dependence of one step’s hidden state on the previous hidden state andthe current input, as the relations. The rule for combining the entities takes a step’s inputs andhidden state as arguments to update the hidden state. The rule is reused over each step (Figure 1c),which reflects the relational inductive bias of temporal invariance (similar to a CNN’s translationalinvariance in space). For example, the outcome of some physical sequence of events should notdepend on the time of day. RNNs also carry a bias for locality in the sequence via their Markovianstructure (Table 1).

2.2 Computations over sets and graphs

While the standard deep learning toolkit contains methods with various forms of relational inductivebiases, there is no “default” deep learning component which operates on arbitrary relational structure.We need models with explicit representations of entities and relations, and learning algorithms whichfind rules for computing their interactions, as well as ways of grounding them in data. Importantly,entities in the world (such as objects and agents) do not have a natural order; rather, orderingscan be defined by the properties of their relations. For example, the relations between the sizes ofa set of objects can potentially be used to order them, as can their masses, ages, toxicities, andprices. Invariance to ordering—except in the face of relations—is a property that should ideally bereflected by a deep learning component for relational reasoning.

Sets are a natural representation for systems which are described by entities whose order isundefined or irrelevant; in particular, their relational inductive bias does not come from the presenceof something, but rather from the absence. For illustration, consider the task of predicting the center

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Molecule Mass-Spring System

n-body System Rigid Body System

Sentence and Parse Tree Image and Fully-Connected Scene Graph

Figure 2: Different graph representations. (a) A molecule, in which each atom is represented as anode and edges correspond to bonds (e.g. Duvenaud et al., 2015). (b) A mass-spring system, inwhich the rope is defined by a sequence of masses which are represented as nodes in the graph (e.g.Battaglia et al., 2016; Chang et al., 2017). (c) A n-body system, in which the bodies are nodes andthe underlying graph is fully connected (e.g. Battaglia et al., 2016; Chang et al., 2017). (d) A rigidbody system, in which the balls and walls are nodes, and the underlying graph defines interactionsbetween the balls and between the balls and the walls (e.g. Battaglia et al., 2016; Chang et al., 2017).(e) A sentence, in which the words correspond to leaves in a tree, and the other nodes and edgescould be provided by a parser (e.g. Socher et al., 2013). Alternately, a fully connected graph couldbe used (e.g. Vaswani et al., 2017). (f) An image, which can be decomposed into image patchescorresponding to nodes in a fully connected graph (e.g. Santoro et al., 2017; Wang et al., 2018c).

of mass of a solar system comprised of n planets, whose attributes (e.g., mass, position, velocity,etc.) are denoted by {x1,x2, . . . ,xn}. For such a computation, the order in which we consider theplanets does not matter because the state can be described solely in terms of aggregated, averagedquantities. However, if we were to use a MLP for this task, having learned the prediction for aparticular input (x1,x2, . . . ,xn) would not necessarily transfer to making a prediction for the sameinputs under a different ordering (xn,x1, . . . ,x2). Since there are n! such possible permutations, inthe worst case, the MLP could consider each ordering as fundamentally different, and thus requirean exponential number of input/output training examples to learn an approximating function.A natural way to handle such combinatorial explosion is to only allow the prediction to dependon symmetric functions of the inputs’ attributes. This might mean computing shared per-object

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features {f(x1), . . . , f(xn)} which are then aggregated in a symmetric way (for example, by takingtheir mean). Such an approach is the essence of the Deep Sets and related models (Zaheer et al.,2017; Edwards and Storkey, 2016; Pevny and Somol, 2017), which we explore further in Section 4.2.3.

Of course, permutation invariance is not the only important form of underlying structure inmany problems. For example, each object in a set may be affected by pairwise interactions withthe other objects in the set (Hartford et al., 2018). In our planets scenario, consider now thetask of predicting each individual planet’s position after a time interval, ∆t. In this case, usingaggregated, averaged information is not enough because the movement of each planet depends onthe forces the other planets are exerting on it. Instead, we could compute the state of each objectas x′i = f(xi,

∑j g(xi,xj)), where g could compute the force induced by the j-th planet on the

i-th planet, and f could compute the future state of the i-th planet which results from the forcesand dynamics. The fact that we use the same g everywhere is again a consequence of the globalpermutation invariance of the system; however, it also supports a different relational structurebecause g now takes two arguments rather than one.4

The above solar system examples illustrate two relational structures: one in which there areno relations, and one which consists of all pairwise relations. Many real-world systems (such asin Figure 2) have a relational structure somewhere in between these two extremes, however, withsome pairs of entities possessing a relation and others lacking one. In our solar system example, ifthe system instead consists of the planets and their moons, one may be tempted to approximateit by neglecting the interactions between moons of different planets. In practice, this meanscomputing interactions only between some pairs of objects, i.e. x′i = f(xi,

∑j∈δ(i) g(xi,xj)), where

δ(i) ⊆ {1, . . . , n} is a neighborhood around node i. This corresponds to a graph, in that the i-thobject only interacts with a subset of the other objects, described by its neighborhood. Note, theupdated states still do not depend in the order in which we describe the neighborhood.5

Graphs, generally, are a representation which supports arbitrary (pairwise) relational struc-ture, and computations over graphs afford a strong relational inductive bias beyond that whichconvolutional and recurrent layers can provide.

3 Graph networks

Neural networks that operate on graphs, and structure their computations accordingly, have beendeveloped and explored extensively for more than a decade under the umbrella of “graph neuralnetworks” (Gori et al., 2005; Scarselli et al., 2005, 2009a; Li et al., 2016), but have grown rapidlyin scope and popularity in recent years. We survey the literature on these methods in the nextsub-section (3.1). Then in the remaining sub-sections, we present our graph networks framework,which generalizes and extends several lines of work in this area.

3.1 Background

Models in the graph neural network family (Gori et al., 2005; Scarselli et al., 2005, 2009a; Li et al.,2016) have been explored in a diverse range of problem domains, across supervised, semi-supervised,unsupervised, and reinforcement learning settings. They have been effective at tasks thought tohave rich relational structure, such as visual scene understanding tasks (Raposo et al., 2017; Santoro

4We could extend this same analysis to increasingly entangled structures that depend on relations among triplets(i.e., g(xi,xj ,xk)), quartets, and so on. We note that if we restrict these functions to only operate on subsets of xi

which are spatially close, then we end back up with something resembling CNNs. In the most entangled sense, wherethere is a single relation function g(x1, . . . ,xn), we end back up with a construction similar to a fully connected layer.

5The invariance which this model enforces is the invariance under isomorphism of the graph.

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et al., 2017) and few-shot learning (Garcia and Bruna, 2018). They have also been used to learnthe dynamics of physical systems (Battaglia et al., 2016; Chang et al., 2017; Watters et al., 2017;van Steenkiste et al., 2018; Sanchez-Gonzalez et al., 2018) and multi-agent systems (Sukhbaataret al., 2016; Hoshen, 2017; Kipf et al., 2018), to reason about knowledge graphs (Bordes et al., 2013;Onoro-Rubio et al., 2017; Hamaguchi et al., 2017), to predict the chemical properties of molecules(Duvenaud et al., 2015; Gilmer et al., 2017), to predict traffic on roads (Li et al., 2017; Cui et al.,2018), to classify and segment images and videos (Wang et al., 2018c; Hu et al., 2017) and 3Dmeshes and point clouds (Wang et al., 2018d), to classify regions in images (Chen et al., 2018a), toperform semi-supervised text classification (Kipf and Welling, 2017), and in machine translation(Vaswani et al., 2017; Shaw et al., 2018; Gulcehre et al., 2018). They have been used within bothmodel-free (Wang et al., 2018b) and model-based (Hamrick et al., 2017; Pascanu et al., 2017;Sanchez-Gonzalez et al., 2018) continuous control, for model-free reinforcement learning (Hamricket al., 2018; Zambaldi et al., 2018), and for more classical approaches to planning (Toyer et al.,2017).

Many traditional computer science problems, which involve reasoning about discrete entities andstructure, have also been explored with graph neural networks, such as combinatorial optimization(Bello et al., 2016; Nowak et al., 2017; Dai et al., 2017), boolean satisfiability (Selsam et al., 2018),program representation and verification (Allamanis et al., 2018; Li et al., 2016), modeling cellularautomata and Turing machines (Johnson, 2017), and performing inference in graphical models(Yoon et al., 2018). Recent work has also focused on building generative models of graphs (Li et al.,2018; De Cao and Kipf, 2018; You et al., 2018; Bojchevski et al., 2018), and unsupervised learning ofgraph embeddings (Perozzi et al., 2014; Tang et al., 2015; Grover and Leskovec, 2016; Garcıa-Duranand Niepert, 2017).

The works cited above are by no means an exhaustive list, but provide a representative cross-section of the breadth of domains for which graph neural networks have proven useful. We pointinterested readers to a number of existing reviews which examine the body of work on graph neuralnetworks in more depth. In particular, Scarselli et al. (2009a) provides an authoritative overview ofearly graph neural network approaches. Bronstein et al. (2017) provides an excellent survey of deeplearning on non-Euclidean data, and explores graph neural nets, graph convolution networks, andrelated spectral approaches. Recently, Gilmer et al. (2017) introduced the message-passing neuralnetwork (MPNN), which unified various graph neural network and graph convolutional networkapproaches (Monti et al., 2017; Bruna et al., 2014; Henaff et al., 2015; Defferrard et al., 2016;Niepert et al., 2016; Kipf and Welling, 2017; Bronstein et al., 2017) by analogy to message-passingin graphical models. In a similar vein, Wang et al. (2018c) introduced the non-local neural network(NLNN), which unified various “self-attention”-style methods (Vaswani et al., 2017; Hoshen, 2017;Velickovic et al., 2018) by analogy to methods from computer vision and graphical models forcapturing long range dependencies in signals.

3.2 Graph network (GN) block

We now present our graph networks (GN) framework, which defines a class of functions forrelational reasoning over graph-structured representations. Our GN framework generalizes andextends various graph neural network, MPNN, and NLNN approaches (Scarselli et al., 2009a; Gilmeret al., 2017; Wang et al., 2018c), and supports constructing complex architectures from simplebuilding blocks. Note, we avoided using the term “neural” in the “graph network” label to reflectthat they can be implemented with functions other than neural networks, though here our focus ison neural network implementations.

The main unit of computation in the GN framework is the GN block, a “graph-to-graph” module

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Box 3: Our definition of “graph”

Attributes

viek

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u

vsk vrk

u

vi

ek

Here we use “graph” to mean a directed, attributed multi-graph with a global attribute. In ourterminology, a node is denoted as vi, an edge as ek, and the global attributes as u. We also usesk and rk to indicate the indices of the sender and receiver nodes (see below), respectively, foredge k. To be more precise, we define these terms as:

Directed : one-way edges, from a “sender” node to a “receiver” node.Attribute : properties that can be encoded as a vector, set, or even another graph.Attributed : edges and vertices have attributes associated with them.Global attribute : a graph-level attribute.Multi-graph : there can be more than one edge between vertices, including self-edges.

Figure 2 shows a variety of different types of graphs corresponding to real data that we may beinterested in modeling, including physical systems, molecules, images, and text.

which takes a graph as input, performs computations over the structure, and returns a graph asoutput. As described in Box 3, entities are represented by the graph’s nodes, relations by the edges,and system-level properties by global attributes. The GN framework’s block organization emphasizescustomizability and synthesizing new architectures which express desired relational inductive biases.The key design principles are: Flexible representations (see Section 4.1); Configurable within-blockstructure (see Section 4.2); and Composable multi-block architectures (see Section 4.3).

We introduce a motivating example to help make the GN formalism more concrete. Considerpredicting the movements a set of rubber balls in an arbitrary gravitational field, which, instead ofbouncing against one another, each have one or more springs which connect them to some (or all) ofthe others. We will refer to this running example throughout the definitions below, to motivate thegraph representation and the computations operating over it. Figure 2 depicts some other commonscenarios that can be represented by graphs and reasoned over using graph networks.

3.2.1 Definition of “graph”

Within our GN framework, a graph is defined as a 3-tuple G = (u, V, E) (see Box 3 for details ofgraph representations). The u is a global attribute; for example, u might represent the gravitationalfield. The V = {vi}i=1:Nv is the set of nodes (of cardinality Nv), where each vi is a node’s attribute.For example, V might represent each ball, with attributes for position, velocity, and mass. TheE = {(ek, rk, sk)}k=1:Ne is the set of edges (of cardinality N e), where each ek is the edge’s attribute,rk is the index of the receiver node, and sk is the index of the sender node. For example, E mightrepresent the presence of springs between different balls, and their corresponding spring constants.

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Algorithm 1 Steps of computation in a full GN block.

function GraphNetwork(E, V , u)for k ∈ {1 . . . N e} do

e′k ← φe (ek,vrk ,vsk ,u) . 1. Compute updated edge attributesend forfor i ∈ {1 . . . Nn} do

let E′i = {(e′k, rk, sk)}rk=i, k=1:Ne

e′i ← ρe→v (E′i) . 2. Aggregate edge attributes per nodev′i ← φv (e′i,vi,u) . 3. Compute updated node attributes

end forlet V ′ = {v′}i=1:Nv

let E′ = {(e′k, rk, sk)}k=1:Ne

e′ ← ρe→u (E′) . 4. Aggregate edge attributes globallyv′ ← ρv→u (V ′) . 5. Aggregate node attributes globallyu′ ← φu (e′, v′,u) . 6. Compute updated global attributereturn (E′, V ′,u′)

end function

3.2.2 Internal structure of a GN block

A GN block contains three “update” functions, φ, and three “aggregation” functions, ρ,

e′k = φe (ek,vrk ,vsk ,u)

v′i = φv(e′i,vi,u

)u′ = φu

(e′, v′,u

)e′i = ρe→v

(E′i)

e′ = ρe→u(E′)

v′ = ρv→u(V ′) (1)

where E′i = {(e′k, rk, sk)}rk=i, k=1:Ne , V′ = {v′i}i=1:Nv , and E′ =

⋃iE′i = {(e′k, rk, sk)}k=1:Ne .

The φe is mapped across all edges to compute per-edge updates, the φv is mapped across allnodes to compute per-node updates, and the φu is applied once as the global update. The ρfunctions each take a set as input, and reduce it to a single element which represents the aggregatedinformation. Crucially, the ρ functions must be invariant to permutations of their inputs, and shouldtake variable numbers of arguments (e.g., elementwise summation, mean, maximum, etc.).

3.2.3 Computational steps within a GN block

When a graph, G, is provided as input to a GN block, the computations proceed from the edge, tothe node, to the global level. Figure 3 shows a depiction of which graph elements are involved ineach of these computations, and Figure 4a shows a full GN block, with its update and aggregationfunctions. Algorithm 1 shows the following steps of computation:

1. φe is applied per edge, with arguments (ek,vrk ,vsk ,u), and returns e′k. In our springs example,this might correspond to the forces or potential energies between two connected balls. Theset of resulting per-edge outputs for each node, i, is, E′i = {(e′k, rk, sk)}rk=i, k=1:Ne . And

E′ =⋃iE′i = {(e′k, rk, sk)}k=1:Ne is the set of all per-edge outputs.

2. ρe→v is applied to E′i, and aggregates the edge updates for edges that project to vertex i, intoe′i, which will be used in the next step’s node update. In our running example, this mightcorrespond to summing all the forces or potential energies acting on the ith ball.

12

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(a) Edge update

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(b) Node update

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(c) Global update

Figure 3: Updates in a GN block. Blue indicates the element that is being updated, and blackindicates other elements which are involved in the update (note that the pre-update value of theblue element is also used in the update). See Equation 1 for details on the notation.

3. φv is applied to each node i, to compute an updated node attribute, v′i. In our runningexample, φv may compute something analogous to the updated position, velocity, and kineticenergy of each ball. The set of resulting per-node outputs is, V ′ = {v′i}i=1:Nv .

4. ρe→u is applied to E′, and aggregates all edge updates, into e′, which will then be used in thenext step’s global update. In our running example, ρe→u may compute the summed forces(which should be zero, in this case, due to Newton’s third law) and the springs’ potentialenergies.

5. ρv→u is applied to V ′, and aggregates all node updates, into v′, which will then be used inthe next step’s global update. In our running example, ρv→u might compute the total kineticenergy of the system.

6. φu is applied once per graph, and computes an update for the global attribute, u′. In ourrunning example, φu might compute something analogous to the net forces and total energyof the physical system.

Note, though we assume this sequence of steps here, the order is not strictly enforced: it is possibleto reverse the update functions to proceed from global, to per-node, to per-edge updates, for example.Kearnes et al. (2016) computes edge updates from nodes in a similar manner.

3.2.4 Relational inductive biases in graph networks

Our GN framework imposes several strong relational inductive biases when used as components ina learning process. First, graphs can express arbitrary relationships among entities, which meansthe GN’s input determines how representations interact and are isolated, rather than those choicesbeing determined by the fixed architecture. For example, the assumption that two entities have arelationship—and thus should interact—is expressed by an edge between the entities’ correspondingnodes. Similarly, the absence of an edge expresses the assumption that the nodes have no relationshipand should not influence each other directly.

Second, graphs represent entities and their relations as sets, which are invariant to permutations.This means GNs are invariant to the order of these elements6, which is often desirable. For example,the objects in a scene do not have a natural ordering (see Sec. 2.2).

Third, a GN’s per-edge and per-node functions are reused across all edges and nodes, respectively.This means GNs automatically support a form of combinatorial generalization (see Section 5.1):because graphs are composed of edges, nodes, and global features, a single GN can operate ongraphs of different sizes (numbers of edges and nodes) and shapes (edge connectivity).

6Note, an ordering can be imposed by encoding the indices in the node or edge attributes, or via the edgesthemselves (e.g. by encoding a chain or partial ordering).

13

4 Design principles for graph network architectures

The GN framework can be used to implement a wide variety of architectures, in accordance withthe design principles listed above in Section 3.2, which also correspond to the sub-sections (4.1,4.2, and 4.3) below. In general, the framework is agnostic to specific attribute representations andfunctional forms. Here, however, we focus mainly on deep learning architectures, which allow GNsto act as learnable graph-to-graph function approximators.

4.1 Flexible representations

Graph networks support highly flexible graph representations in two ways: first, in terms of therepresentation of the attributes; and second, in terms of the structure of the graph itself.

4.1.1 Attributes

The global, node, and edge attributes of a GN block can use arbitrary representational formats. Indeep learning implementations, real-valued vectors and tensors are most common. However, otherdata structures such as sequences, sets, or even graphs could also be used.

The requirements of the problem will often determine what representations should be used forthe attributes. For example, when the input data is an image, the attributes might be representedas tensors of image patches; however, when the input data is a text document, the attributes mightbe sequences of words corresponding to sentences.

For each GN block within a broader architecture, the edge and node outputs typically correspondto lists of vectors or tensors, one per edge or node, and the global outputs correspond to a singlevector or tensor. This allows a GN’s output to be passed to other deep learning building blockssuch as MLPs, CNNs, and RNNs.

The output of a GN block can also be tailored to the demands of the task. In particular,

◦ An edge-focused GN uses the edges as output, for example to make decisions about interactionsamong entities (Kipf et al., 2018; Hamrick et al., 2018).◦ A node-focused GN uses the nodes as output, for example to reason about physical systems

(Battaglia et al., 2016; Chang et al., 2017; Wang et al., 2018b; Sanchez-Gonzalez et al., 2018).◦ A graph-focused GN uses the globals as output, for example to predict the potential energy of

a physical system (Battaglia et al., 2016), the properties of a molecule (Gilmer et al., 2017),or answers to questions about a visual scene (Santoro et al., 2017).

The nodes, edges, and global outputs can also be mixed-and-matched depending on the task. Forexample, Hamrick et al. (2018) used both the output edge and global attributes to compute a policyover actions.

4.1.2 Graph structure

When defining how the input data will be represented as a graph, there are generally two scenarios:first, the input explicitly specifies the relational structure; and second, the relational structure mustbe inferred or assumed. These are not hard distinctions, but extremes along a continuum.

Examples of data with more explicitly specified entities and relations include knowledge graphs,social networks, parse trees, optimization problems, chemical graphs, road networks, and physicalsystems with known interactions. Figures 2a-d illustrate how such data can be expressed as graphs.

Examples of data where the relational structure is not made explicit, and must be inferred orassumed, include visual scenes, text corpora, programming language source code, and multi-agent

14

systems. In these types of settings, the data may be formatted as a set of entities without relations,or even just a vector or tensor (e.g., an image). If the entities are not specified explicitly, they mightbe assumed, for instance, by treating each word in a sentence (Vaswani et al., 2017) or each localfeature vector in a CNN’s output feature map, as a node (Watters et al., 2017; Santoro et al., 2017;Wang et al., 2018c) (Figures 2e-f). Or, it might be possible to use a separate learned mechanism toinfer entities from an unstructured signal (Luong et al., 2015; Mnih et al., 2014; Eslami et al., 2016;van Steenkiste et al., 2018). If relations are not available, the simplest approach is to instantiate allpossible directed edges between entities (Figure 2f). This can be prohibitive for large numbers ofentities, however, because the number of possible edges grows quadratically with the number ofnodes. Thus developing more sophisticated ways of inferring sparse structure from unstructureddata (Kipf et al., 2018) is an important future direction.

4.2 Configurable within-block structure

The structure and functions within a GN block can be configured in different ways, which offersflexibility in what information is made available as inputs to its functions, as well as how output edge,node, and global updates are produced. In particular, each φ in Equation 1 must be implementedwith some function, f , where f ’s argument signature determines what information it requires asinput; in Figure 4, the incoming arrows to each φ depict whether u, V , and E are taken as inputs.Hamrick et al. (2018) and Sanchez-Gonzalez et al. (2018) used the full GN block shown in Figure 4a.Their φ implementations used neural networks (denoted NNe, NNv, and NNu below, to indicate thatthey are different functions with different parameters). Their ρ implementations used elementwisesummation, but averages and max/min could also be used,

φe (ek,vrk ,vsk ,u) := fe (ek,vrk ,vsk ,u) = NNe ([ek,vrk ,vsk ,u]) (2)

φv(e′i,vi,u

):= fv

(e′i,vi,u

)= NNv

([e′i,vi,u]

)φu(e′, v′,u

):= fu

(e′, v′,u

)= NNu

([e′, v′,u]

)ρe→v

(E′i)

:= =∑

{k: rk=i}

e′k

ρv→u(V ′)

:= =∑i

v′i

ρe→u(E′)

:= =∑k

e′k

where [x,y, z] indicates vector/tensor concatenation. For vector attributes, a MLP is often used forφ, while for tensors such as image feature maps, CNNs may be more suitable.

The φ functions can also use RNNs, which requires an additional hidden state as input andoutput. Figure 4b shows a very simple version of a GN block with RNNs as φ functions: there is nomessage-passing in this formulation, and this type of block might be used for recurrent smoothing ofsome dynamic graph states. Of course, RNNs as φ functions could also be used in a full GN block(Figure 4a).

A variety of other architectures can be expressed in the GN framework, often as differentfunction choices and within-block configurations. The remaining sub-sections explore how a GN’swithin-block structure can be configured in different ways, with examples of published work whichuses such configurations. See the Appendix for details.

15

(a) Full GN block

u0,u0hid

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E0, E0hid

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V 0, V 0hid

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Edge block Node block Global block

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(b) Independent recurrent block

Edge block Node block Global block

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(c) Message-passing neural network

Edge block Node block Global block

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(d) Non-local neural network

Edge block Node block Global block

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(e) Relation network

Edge block Node block Global block

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(f) Deep set

Figure 4: Different internal GN block configurations. See Section 3.2 for details on the notation,and Section 4 for details about each variant. (a) A full GN predicts node, edge, and global outputattributes based on incoming node, edge, and global attributes. (b) An independent, recurrentupdate block takes input and hidden graphs, and the φ functions are RNNs (Sanchez-Gonzalezet al., 2018). (c) An MPNN (Gilmer et al., 2017) predicts node, edge, and global output attributesbased on incoming node, edge, and global attributes. Note that the global prediction does notinclude aggregated edges. (d) A NLNN (Wang et al., 2018c) only predicts node output attributes.(e) A relation network (Raposo et al., 2017; Santoro et al., 2017) only uses the edge predictionsto predict global attributes. (f) A Deep Set (Zaheer et al., 2017) bypasses the edge update andpredicts updated global attributes.

4.2.1 Message-passing neural network (MPNN)

Gilmer et al. (2017)’s MPNN generalizes a number of previous architectures and can be translatednaturally into the GN formalism. Following the MPNN paper’s terminology (see Gilmer et al.(2017), pages 2-4):

◦ the message function, Mt, plays the role of the GN’s φe, but does not take u as input,◦ elementwise summation is used for the GN’s ρe→v,◦ the update function, Ut, plays the role of the GN’s φv,

16

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Figure 5: NLNNs as GNs. A schematic showing how NLNNs (Wang et al., 2018c) are implementedby the φe and ρe→v under the GN framework. Typically, NLNNs assume that different regions ofan image (or words in a sentence) correspond to nodes in a fully connected graph, and the attentionmechanism defines a weighted sum over nodes during the aggregation step.

◦ the readout function, R, plays the role of the GN’s φu, but does not take u or E′ as input,and thus an analog to the GN’s ρe→u is not required;◦ dmaster serves a roughly similar purpose to the GN’s u, but is defined as an extra node

connected to all others, and thus does not influence the edge and global updates directly. Itcan then be represented in the GN’s V .

Figure 4c shows how an MPNN is structured, according to the GN framework. For details andvarious MPNN architectures, see the Appendix.

4.2.2 Non-local neural networks (NLNN)

Wang et al. (2018c)’s NLNN, which unifies various “intra-/self-/vertex-/graph-attention” approaches(Lin et al., 2017; Vaswani et al., 2017; Hoshen, 2017; Velickovic et al., 2018; Shaw et al., 2018),can also be translated into the GN formalism. The label “attention” refers to how the nodes areupdated: each node update is based on a weighted sum of (some function of) the node attributes ofits neighbors, where the weight between a node and one of its neighbors is computed by a scalarpairwise function between their attributes (and then normalized across neighbors). The publishedNLNN formalism does not explicitly include edges, and instead computes pairwise attention weightsbetween all nodes. But various NLNN-compliant models, such as the vertex attention interactionnetwork (Hoshen, 2017) and graph attention network (Velickovic et al., 2018), are able to handleexplicit edges by effectively setting to zero the weights between nodes which do not share an edge.

As Figures 4d and 5 illustrate, the φe is factored into the scalar pairwise-interaction functionwhich returns the unnormalized attention term, denoted αe (vrk ,vsk) = a′k, and a vector-valuednon-pairwise term, denoted βe (vsk) = b′k. In the ρe→v aggregation, the a′k terms are normalizedacross each receiver’s edges, b′k, and elementwise summed:

φe (ek,vrk ,vsk ,u) := fe (vrk ,vsk) = (αe (vrk ,vsk) , βe (vsk)) = (a′k,b′k) = e′k

φv(e′i,vi,u

):= fv(e′i)

ρe→v(E′i)

:=1∑

{k: rk=i} a′k

∑{k: rk=i}

a′kb′k

In the NLNN paper’s terminology (see Wang et al. (2018c), pages 2-4):

◦ their f plays the role of the above α,

17

◦ their g plays the role of the above β.

This formulation may be helpful for focusing only on those interactions which are most relevant forthe downstream task, especially when the input entities were a set, from which a graph was formedby adding all possible edges between them.

Vaswani et al. (2017)’s multi-headed self-attention mechanism adds an interesting feature, wherethe φe and ρe→v are implemented by a parallel set of functions, whose results are concatenatedtogether as the final step of ρe→v. This can be interpreted as using typed edges, where the differenttypes index into different φe component functions, analogous to Li et al. (2016).

For details and various NLNN architectures, see the Appendix.

4.2.3 Other graph network variants

The full GN (Equation 2) can be used to predict a full graph, or any subset of (u′, V ′, E′), asoutlined in Section 4.1.1. For example, to predict a global property of a graph, V ′ and E′ can justbe ignored. Similarly, if global, node, or edge attributes are unspecified in the inputs, those vectorscan be zero-length, i.e., not taken as explicit input arguments. The same idea applies for other GNvariants which do not use the full set of mapping (φ) and reduction (ρ) functions. For instance,Interaction Networks (Battaglia et al., 2016; Watters et al., 2017) and the Neural Physics Engine(Chang et al., 2017) use a full GN but for the absence of the global to update the edge properties(see Appendix for details).

Various models, including CommNet (Sukhbaatar et al., 2016), structure2vec (Dai et al., 2016)(in the version of (Dai et al., 2017)), and Gated Graph Sequence Neural Networks (Li et al., 2016)have used a φe which does not directly compute pairwise interactions, but instead ignore the receivernode, operating only on the sender node and in some cases an edge attribute. This can be expressedby implementations of φe with the following signatures, such as:

φe (ek,vrk ,vsk ,u) := fe (vsk)

or φe (ek,vrk ,vsk ,u) := vsk + fe (ek)

or φe (ek,vrk ,vsk ,u) := fe (ek,vsk) .

See the Appendix for further details.Relation Networks (Raposo et al., 2017; Santoro et al., 2017) bypass the node update entirely

and predict the global output from pooled edge information directly (see also Figure 4e),

φe (ek,vrk ,vsk ,u) := fe (vrk ,vsk) = NNe ([vrk ,vsk ])

φu(e′, v′,u

):= fu

(e′)

= NNu

(e′)

ρe→u(E′)

:= =∑k

e′k

Deep Sets (Zaheer et al., 2017) bypass the edges update completely and predict the global outputfrom pooled nodes information directly (Figure 4f),

φv (ei,vi,u) := fv (vi,u) = NNv ([vi,u])

φu(e′, v′,u

):= fu

(v′)

= NNu

(v′)

ρv→u(V ′)

:= =∑i

v′i

PointNet (Qi et al., 2017) use similar update rule, with a max-aggregation for ρv→u and a two-stepnode update.

18

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GNcore<latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit><latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit><latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit><latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit>

⇥M<latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit><latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit><latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit><latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit>

(a) Composition of GN blocks

GNenc<latexit sha1_base64="KZY5NxgXVEC/q8QikcRvAbEAxBs=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclUQEXbgouNCVVLAPaEOYTKft0JlJmJkoJeZT3LhQxK1f4s6/cRKz0NYDA4dz7mXOPWHMqNKu+2UtLa+srq1XNqqbW9s7u3Ztr6OiRGLSxhGLZC9EijAqSFtTzUgvlgTxkJFuOL3M/e49kYpG4k7PYuJzNBZ0RDHSRgrs2oAjPZE8vbrJgpQInAV23W24BZxF4pWkDiVagf05GEY44URozJBSfc+NtZ8iqSlmJKsOEkVihKdoTPqGCsSJ8tMieuYcGWXojCJpntBOof7eSBFXasZDM5kHVfNeLv7n9RM9OvdTKuJE51cVH40S5ujIyXtwhlQSrNnMEIQlNVkdPEESYW3aqpoSvPmTF0nnpOG5De/2tN68KOuowAEcwjF4cAZNuIYWtAHDAzzBC7xaj9az9Wa9/4wuWeXOPvyB9fENyBqUTg==</latexit><latexit sha1_base64="KZY5NxgXVEC/q8QikcRvAbEAxBs=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclUQEXbgouNCVVLAPaEOYTKft0JlJmJkoJeZT3LhQxK1f4s6/cRKz0NYDA4dz7mXOPWHMqNKu+2UtLa+srq1XNqqbW9s7u3Ztr6OiRGLSxhGLZC9EijAqSFtTzUgvlgTxkJFuOL3M/e49kYpG4k7PYuJzNBZ0RDHSRgrs2oAjPZE8vbrJgpQInAV23W24BZxF4pWkDiVagf05GEY44URozJBSfc+NtZ8iqSlmJKsOEkVihKdoTPqGCsSJ8tMieuYcGWXojCJpntBOof7eSBFXasZDM5kHVfNeLv7n9RM9OvdTKuJE51cVH40S5ujIyXtwhlQSrNnMEIQlNVkdPEESYW3aqpoSvPmTF0nnpOG5De/2tN68KOuowAEcwjF4cAZNuIYWtAHDAzzBC7xaj9az9Wa9/4wuWeXOPvyB9fENyBqUTg==</latexit><latexit sha1_base64="KZY5NxgXVEC/q8QikcRvAbEAxBs=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclUQEXbgouNCVVLAPaEOYTKft0JlJmJkoJeZT3LhQxK1f4s6/cRKz0NYDA4dz7mXOPWHMqNKu+2UtLa+srq1XNqqbW9s7u3Ztr6OiRGLSxhGLZC9EijAqSFtTzUgvlgTxkJFuOL3M/e49kYpG4k7PYuJzNBZ0RDHSRgrs2oAjPZE8vbrJgpQInAV23W24BZxF4pWkDiVagf05GEY44URozJBSfc+NtZ8iqSlmJKsOEkVihKdoTPqGCsSJ8tMieuYcGWXojCJpntBOof7eSBFXasZDM5kHVfNeLv7n9RM9OvdTKuJE51cVH40S5ujIyXtwhlQSrNnMEIQlNVkdPEESYW3aqpoSvPmTF0nnpOG5De/2tN68KOuowAEcwjF4cAZNuIYWtAHDAzzBC7xaj9az9Wa9/4wuWeXOPvyB9fENyBqUTg==</latexit><latexit sha1_base64="KZY5NxgXVEC/q8QikcRvAbEAxBs=">AAAB+nicbVDLSsNAFL3xWesr1aWbYBFclUQEXbgouNCVVLAPaEOYTKft0JlJmJkoJeZT3LhQxK1f4s6/cRKz0NYDA4dz7mXOPWHMqNKu+2UtLa+srq1XNqqbW9s7u3Ztr6OiRGLSxhGLZC9EijAqSFtTzUgvlgTxkJFuOL3M/e49kYpG4k7PYuJzNBZ0RDHSRgrs2oAjPZE8vbrJgpQInAV23W24BZxF4pWkDiVagf05GEY44URozJBSfc+NtZ8iqSlmJKsOEkVihKdoTPqGCsSJ8tMieuYcGWXojCJpntBOof7eSBFXasZDM5kHVfNeLv7n9RM9OvdTKuJE51cVH40S5ujIyXtwhlQSrNnMEIQlNVkdPEESYW3aqpoSvPmTF0nnpOG5De/2tN68KOuowAEcwjF4cAZNuIYWtAHDAzzBC7xaj9az9Wa9/4wuWeXOPvyB9fENyBqUTg==</latexit>

GNdec<latexit sha1_base64="79QHnx4t/4kSfeuqQfRiz+zDMfI=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiQi6MJFwYWupIJ9QBvCZDJph85MwsxEKTGf4saFIm79Enf+jZM2C209MHA4517umRMkjCrtON9WZWV1bX2julnb2t7Z3bPr+10VpxKTDo5ZLPsBUoRRQTqaakb6iSSIB4z0gslV4fceiFQ0Fvd6mhCPo5GgEcVIG8m360OO9Fjy7Po297OQ4Ny3G07TmQEuE7ckDVCi7dtfwzDGKSdCY4aUGrhOor0MSU0xI3ltmCqSIDxBIzIwVCBOlJfNoufw2CghjGJpntBwpv7eyBBXasoDM1kEVYteIf7nDVIdXXgZFUmqicDzQ1HKoI5h0QMMqSRYs6khCEtqskI8RhJhbdqqmRLcxS8vk+5p03Wa7t1Zo3VZ1lEFh+AInAAXnIMWuAFt0AEYPIJn8ArerCfrxXq3PuajFavcOQB/YH3+ALjdlEQ=</latexit><latexit sha1_base64="79QHnx4t/4kSfeuqQfRiz+zDMfI=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiQi6MJFwYWupIJ9QBvCZDJph85MwsxEKTGf4saFIm79Enf+jZM2C209MHA4517umRMkjCrtON9WZWV1bX2julnb2t7Z3bPr+10VpxKTDo5ZLPsBUoRRQTqaakb6iSSIB4z0gslV4fceiFQ0Fvd6mhCPo5GgEcVIG8m360OO9Fjy7Po297OQ4Ny3G07TmQEuE7ckDVCi7dtfwzDGKSdCY4aUGrhOor0MSU0xI3ltmCqSIDxBIzIwVCBOlJfNoufw2CghjGJpntBwpv7eyBBXasoDM1kEVYteIf7nDVIdXXgZFUmqicDzQ1HKoI5h0QMMqSRYs6khCEtqskI8RhJhbdqqmRLcxS8vk+5p03Wa7t1Zo3VZ1lEFh+AInAAXnIMWuAFt0AEYPIJn8ArerCfrxXq3PuajFavcOQB/YH3+ALjdlEQ=</latexit><latexit sha1_base64="79QHnx4t/4kSfeuqQfRiz+zDMfI=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiQi6MJFwYWupIJ9QBvCZDJph85MwsxEKTGf4saFIm79Enf+jZM2C209MHA4517umRMkjCrtON9WZWV1bX2julnb2t7Z3bPr+10VpxKTDo5ZLPsBUoRRQTqaakb6iSSIB4z0gslV4fceiFQ0Fvd6mhCPo5GgEcVIG8m360OO9Fjy7Po297OQ4Ny3G07TmQEuE7ckDVCi7dtfwzDGKSdCY4aUGrhOor0MSU0xI3ltmCqSIDxBIzIwVCBOlJfNoufw2CghjGJpntBwpv7eyBBXasoDM1kEVYteIf7nDVIdXXgZFUmqicDzQ1HKoI5h0QMMqSRYs6khCEtqskI8RhJhbdqqmRLcxS8vk+5p03Wa7t1Zo3VZ1lEFh+AInAAXnIMWuAFt0AEYPIJn8ArerCfrxXq3PuajFavcOQB/YH3+ALjdlEQ=</latexit><latexit sha1_base64="79QHnx4t/4kSfeuqQfRiz+zDMfI=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiQi6MJFwYWupIJ9QBvCZDJph85MwsxEKTGf4saFIm79Enf+jZM2C209MHA4517umRMkjCrtON9WZWV1bX2julnb2t7Z3bPr+10VpxKTDo5ZLPsBUoRRQTqaakb6iSSIB4z0gslV4fceiFQ0Fvd6mhCPo5GgEcVIG8m360OO9Fjy7Po297OQ4Ny3G07TmQEuE7ckDVCi7dtfwzDGKSdCY4aUGrhOor0MSU0xI3ltmCqSIDxBIzIwVCBOlJfNoufw2CghjGJpntBwpv7eyBBXasoDM1kEVYteIf7nDVIdXXgZFUmqicDzQ1HKoI5h0QMMqSRYs6khCEtqskI8RhJhbdqqmRLcxS8vk+5p03Wa7t1Zo3VZ1lEFh+AInAAXnIMWuAFt0AEYPIJn8ArerCfrxXq3PuajFavcOQB/YH3+ALjdlEQ=</latexit>

GNcore<latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit><latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit><latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit><latexit sha1_base64="sfcetjjriA53KVhP8LRkSGs9KNA=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBFclUQEXbgouNCVVLAPaEOYTCft0HmEmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzJ0oY1cbzvp2V1bX1jc3KVnV7Z3dv3z2odbRMFSZtLJlUvQhpwqggbUMNI71EEcQjRrrR5Lrwu49EaSrFg5kmJOBoJGhMMTJWCt3agCMzVjy7ucvDDEtF8tCtew1vBrhM/JLUQYlW6H4NhhKnnAiDGdK673uJCTKkDMWM5NVBqkmC8ASNSN9SgTjRQTbLnsMTqwxhLJV9wsCZ+nsjQ1zrKY/sZJFUL3qF+J/XT018GWRUJKkhAs8PxSmDRsKiCDikimDDppYgrKjNCvEYKYSNratqS/AXv7xMOmcN32v49+f15lVZRwUcgWNwCnxwAZrgFrRAG2DwBJ7BK3hzcufFeXc+5qMrTrlzCP7A+fwBopiUyw==</latexit>

⇥M<latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit><latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit><latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit><latexit sha1_base64="xCEPSgjeJaAOppNxwTZXrwRukIg=">AAAB73icbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0MIiYGMjRDAfkBxhb7OXLNnbu+zOCeHIn7CxUMTWv2Pnv3GTXKGJDwYe780wMy9IpDDout9OYW19Y3OruF3a2d3bPygfHjVNnGrGGyyWsW4H1HApFG+gQMnbieY0CiRvBaPbmd964tqIWD3iJOF+RAdKhIJRtFK7iyLihtz3yhW36s5BVomXkwrkqPfKX91+zNKIK2SSGtPx3AT9jGoUTPJpqZsanlA2ogPesVRRu8bP5vdOyZlV+iSMtS2FZK7+nshoZMwkCmxnRHFolr2Z+J/XSTG89jOhkhS5YotFYSoJxmT2POkLzRnKiSWUaWFvJWxINWVoIyrZELzll1dJ86LquVXv4bJSu8njKMIJnMI5eHAFNbiDOjSAgYRneIU3Z+y8OO/Ox6K14OQzx/AHzucPqJqPrw==</latexit>

Gout<latexit sha1_base64="TKn8tu9S9/KYM9INqkcELhgYcuA=">AAAB+XicbVDLSsNAFJ3UV62vqEs3wSK4KokIunBRcKHLCvYBbQiT6aQdOo8wc1MooX/ixoUibv0Td/6NkzYLrR4YOJxzL/fMiVPODPj+l1NZW9/Y3Kpu13Z29/YP3MOjjlGZJrRNFFe6F2NDOZO0DQw47aWaYhFz2o0nt4XfnVJtmJKPMEtpKPBIsoQRDFaKXPcuygcCw1iLXGUwn0du3W/4C3h/SVCSOirRitzPwVCRTFAJhGNj+oGfQphjDYxwOq8NMkNTTCZ4RPuWSiyoCfNF8rl3ZpWhlyhtnwRvof7cyLEwZiZiO1mENKteIf7n9TNIrsOcyTQDKsnyUJJxD5RX1OANmaYE+MwSTDSzWT0yxhoTsGXVbAnB6pf/ks5FI/AbwcNlvXlT1lFFJ+gUnaMAXaEmukct1EYETdETekGvTu48O2/O+3K04pQ7x+gXnI9vWWaUGA==</latexit><latexit sha1_base64="TKn8tu9S9/KYM9INqkcELhgYcuA=">AAAB+XicbVDLSsNAFJ3UV62vqEs3wSK4KokIunBRcKHLCvYBbQiT6aQdOo8wc1MooX/ixoUibv0Td/6NkzYLrR4YOJxzL/fMiVPODPj+l1NZW9/Y3Kpu13Z29/YP3MOjjlGZJrRNFFe6F2NDOZO0DQw47aWaYhFz2o0nt4XfnVJtmJKPMEtpKPBIsoQRDFaKXPcuygcCw1iLXGUwn0du3W/4C3h/SVCSOirRitzPwVCRTFAJhGNj+oGfQphjDYxwOq8NMkNTTCZ4RPuWSiyoCfNF8rl3ZpWhlyhtnwRvof7cyLEwZiZiO1mENKteIf7n9TNIrsOcyTQDKsnyUJJxD5RX1OANmaYE+MwSTDSzWT0yxhoTsGXVbAnB6pf/ks5FI/AbwcNlvXlT1lFFJ+gUnaMAXaEmukct1EYETdETekGvTu48O2/O+3K04pQ7x+gXnI9vWWaUGA==</latexit><latexit sha1_base64="TKn8tu9S9/KYM9INqkcELhgYcuA=">AAAB+XicbVDLSsNAFJ3UV62vqEs3wSK4KokIunBRcKHLCvYBbQiT6aQdOo8wc1MooX/ixoUibv0Td/6NkzYLrR4YOJxzL/fMiVPODPj+l1NZW9/Y3Kpu13Z29/YP3MOjjlGZJrRNFFe6F2NDOZO0DQw47aWaYhFz2o0nt4XfnVJtmJKPMEtpKPBIsoQRDFaKXPcuygcCw1iLXGUwn0du3W/4C3h/SVCSOirRitzPwVCRTFAJhGNj+oGfQphjDYxwOq8NMkNTTCZ4RPuWSiyoCfNF8rl3ZpWhlyhtnwRvof7cyLEwZiZiO1mENKteIf7n9TNIrsOcyTQDKsnyUJJxD5RX1OANmaYE+MwSTDSzWT0yxhoTsGXVbAnB6pf/ks5FI/AbwcNlvXlT1lFFJ+gUnaMAXaEmukct1EYETdETekGvTu48O2/O+3K04pQ7x+gXnI9vWWaUGA==</latexit><latexit sha1_base64="TKn8tu9S9/KYM9INqkcELhgYcuA=">AAAB+XicbVDLSsNAFJ3UV62vqEs3wSK4KokIunBRcKHLCvYBbQiT6aQdOo8wc1MooX/ixoUibv0Td/6NkzYLrR4YOJxzL/fMiVPODPj+l1NZW9/Y3Kpu13Z29/YP3MOjjlGZJrRNFFe6F2NDOZO0DQw47aWaYhFz2o0nt4XfnVJtmJKPMEtpKPBIsoQRDFaKXPcuygcCw1iLXGUwn0du3W/4C3h/SVCSOirRitzPwVCRTFAJhGNj+oGfQphjDYxwOq8NMkNTTCZ4RPuWSiyoCfNF8rl3ZpWhlyhtnwRvof7cyLEwZiZiO1mENKteIf7n9TNIrsOcyTQDKsnyUJJxD5RX1OANmaYE+MwSTDSzWT0yxhoTsGXVbAnB6pf/ks5FI/AbwcNlvXlT1lFFJ+gUnaMAXaEmukct1EYETdETekGvTu48O2/O+3K04pQ7x+gXnI9vWWaUGA==</latexit>

Ginp<latexit sha1_base64="mvYjY6mgtt6w2Efm7YMP3XaOQ7I=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRQRcuCi50WcE+oB2GTJppQ5PMkGQKZeifuHGhiFv/xJ1/Y6adhVYPBA7n3EvOPVHKmTae9+VU1tY3Nreq27Wd3b39A/fwqKOTTBHaJglPVC/CmnImadsww2kvVRSLiNNuNLkt/O6UKs0S+WhmKQ0EHkkWM4KNlULXvQvzgcBmrETOZDqfh27da3gLoL/EL0kdSrRC93MwTEgmqDSEY637vpeaIMfKMMLpvDbINE0xmeAR7VsqsaA6yBfJ5+jMKkMUJ8o+adBC/bmRY6H1TER2sgipV71C/M/rZya+DoqLMkMlWX4UZxyZBBU1oCFTlBg+swQTxWxWRMZYYWJsWTVbgr968l/SuWj4XsN/uKw3b8o6qnACp3AOPlxBE+6hBW0gMIUneIFXJ3eenTfnfTlaccqdY/gF5+MbP22UBw==</latexit><latexit sha1_base64="mvYjY6mgtt6w2Efm7YMP3XaOQ7I=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRQRcuCi50WcE+oB2GTJppQ5PMkGQKZeifuHGhiFv/xJ1/Y6adhVYPBA7n3EvOPVHKmTae9+VU1tY3Nreq27Wd3b39A/fwqKOTTBHaJglPVC/CmnImadsww2kvVRSLiNNuNLkt/O6UKs0S+WhmKQ0EHkkWM4KNlULXvQvzgcBmrETOZDqfh27da3gLoL/EL0kdSrRC93MwTEgmqDSEY637vpeaIMfKMMLpvDbINE0xmeAR7VsqsaA6yBfJ5+jMKkMUJ8o+adBC/bmRY6H1TER2sgipV71C/M/rZya+DoqLMkMlWX4UZxyZBBU1oCFTlBg+swQTxWxWRMZYYWJsWTVbgr968l/SuWj4XsN/uKw3b8o6qnACp3AOPlxBE+6hBW0gMIUneIFXJ3eenTfnfTlaccqdY/gF5+MbP22UBw==</latexit><latexit sha1_base64="mvYjY6mgtt6w2Efm7YMP3XaOQ7I=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRQRcuCi50WcE+oB2GTJppQ5PMkGQKZeifuHGhiFv/xJ1/Y6adhVYPBA7n3EvOPVHKmTae9+VU1tY3Nreq27Wd3b39A/fwqKOTTBHaJglPVC/CmnImadsww2kvVRSLiNNuNLkt/O6UKs0S+WhmKQ0EHkkWM4KNlULXvQvzgcBmrETOZDqfh27da3gLoL/EL0kdSrRC93MwTEgmqDSEY637vpeaIMfKMMLpvDbINE0xmeAR7VsqsaA6yBfJ5+jMKkMUJ8o+adBC/bmRY6H1TER2sgipV71C/M/rZya+DoqLMkMlWX4UZxyZBBU1oCFTlBg+swQTxWxWRMZYYWJsWTVbgr968l/SuWj4XsN/uKw3b8o6qnACp3AOPlxBE+6hBW0gMIUneIFXJ3eenTfnfTlaccqdY/gF5+MbP22UBw==</latexit><latexit sha1_base64="mvYjY6mgtt6w2Efm7YMP3XaOQ7I=">AAAB+XicbVDLSgMxFL1TX7W+Rl26CRbBVZkRQRcuCi50WcE+oB2GTJppQ5PMkGQKZeifuHGhiFv/xJ1/Y6adhVYPBA7n3EvOPVHKmTae9+VU1tY3Nreq27Wd3b39A/fwqKOTTBHaJglPVC/CmnImadsww2kvVRSLiNNuNLkt/O6UKs0S+WhmKQ0EHkkWM4KNlULXvQvzgcBmrETOZDqfh27da3gLoL/EL0kdSrRC93MwTEgmqDSEY637vpeaIMfKMMLpvDbINE0xmeAR7VsqsaA6yBfJ5+jMKkMUJ8o+adBC/bmRY6H1TER2sgipV71C/M/rZya+DoqLMkMlWX4UZxyZBBU1oCFTlBg+swQTxWxWRMZYYWJsWTVbgr968l/SuWj4XsN/uKw3b8o6qnACp3AOPlxBE+6hBW0gMIUneIFXJ3eenTfnfTlaccqdY/gF5+MbP22UBw==</latexit>

(b) Encode-process-decode

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Gthid

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Gt�1hid

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Gtout

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Gtinp

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(c) Recurrent GN architecture

Figure 6: (a) An example composing multiple GN blocks in sequence to form a GN “core”. Here,the GN blocks can use shared weights, or they could be independent. (b) The encode-process-decodearchitecture, which is a common choice for composing GN blocks (see Section 4.3). Here, a GNencodes an input graph, which is then processed by a GN core. The output of the core is decodedby a third GN block into an output graph, whose nodes, edges, and/or global attributes would beused for task-specific purposes. (c) The encode-process-decode architecture applied in a sequentialsetting in which the core is also unrolled over time (potentially using a GRU or LSTM architecture),in addition to being repeated within each time step. Here, merged lines indicate concatenation, andsplit lines indicate copying.

4.3 Composable multi-block architectures

A key design principle of graph networks is constructing complex architectures by composing GNblocks. We defined a GN block as always taking a graph comprised of edge, node, and globalelements as input, and returning a graph with the same constituent elements as output (simplypassing through the input elements to the output when those elements are not explicitly updated).This graph-to-graph input/output interface ensures that the output of one GN block can be passedas input to another, even if their internal configurations are different, similar to the tensor-to-tensorinterface of the standard deep learning toolkit. In the most basic form, two GN blocks, GN1 andGN2, can be composed as GN1 ◦ GN2 by passing the output of the first as input to the second:G′ = GN2(GN1(G)).

Arbitrary numbers of GN blocks can be composed, as show in Figure 6a. The blocks canbe unshared (different functions and/or parameters, analogous to layers of a CNN), GN1 6=GN2 6= · · · 6= GNM , or shared (reused functions and parameters, analogous to an unrolled RNN),GN1 = GN2 = · · · = GNM . The white box around the GNcore in Figure 6a represents M repeatedinternal processing sub-steps, with either shared or unshared GN blocks. Shared configurationsare analogous to message-passing (Gilmer et al., 2017), where the same local update procedure isapplied iteratively to propagate information across the structure (Figure 7). If we exclude the globalu (which aggregates information from across the nodes and edges), the information that a nodehas access to after m steps of propagation is determined by the set of nodes and edges that are atmost m hops away. This can be interpreted as breaking down a complex computation into smallerelementary steps. The steps can also be used to capture sequentiality in time. In our ball-springexample, if each propagation step predicts the physical dynamics over one time step of duration ∆t,then the M propagation steps result in a total simulation time of, M ·∆t.

A common architecture design is what we call the encode-process-decode configuration (Hamricket al. (2018); also see Figure 6ba): an input graph, Ginp is transformed into a latent representation,G0, by an encoder, GNenc; a shared core block, GNcore, is applied M times to return GM ; andfinally an output graph, Gout, is decoded by GNdec. For example, in our running example, theencoder might compute the initial forces and interaction energies between the balls, the core might

19

m = 0<latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit><latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit><latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit><latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit>

m = 1<latexit sha1_base64="Zv5ybfKo/H8QvXQaOWqQVyOFtg4=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ73JkJnZZWZWCCGf4MWDIl79Im/+jZNkDxotaCiquunuilLBjfX9L6+wtr6xuVXcLu3s7u0flA+PWibJNMMmS0SiOxE1KLjCpuVWYCfVSGUksB2Nb+Z++xG14Yl6sJMUQ0mHisecUeuke3kd9MsVv+YvQP6SICcVyNHolz97g4RlEpVlghrTDfzUhlOqLWcCZ6VeZjClbEyH2HVUUYkmnC5OnZEzpwxInGhXypKF+nNiSqUxExm5TkntyKx6c/E/r5vZ+CqccpVmFhVbLoozQWxC5n+TAdfIrJg4Qpnm7lbCRlRTZl06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NUT3rP35r0vWwtePnMMv+B9fAO/Y41Z</latexit><latexit 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m = 2<latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit><latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit><latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit><latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit>

m = 3<latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit><latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit><latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit><latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit>

m = 0<latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit><latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit><latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit><latexit sha1_base64="ic6WezPV7TWar890N1QnpVOPjNA=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ7PJkJnZZaZXCCGf4MWDIl79Im/+jZNkDxotaCiquunuilIpLPr+l1dYW9/Y3Cpul3Z29/YPyodHLZtkhvEmS2RiOhG1XArNmyhQ8k5qOFWR5O1ofDP324/cWJHoB5ykPFR0qEUsGEUn3atrv1+u+DV/AfKXBDmpQI5Gv/zZGyQsU1wjk9TabuCnGE6pQcEkn5V6meUpZWM65F1HNVXchtPFqTNy5pQBiRPjSiNZqD8nplRZO1GR61QUR3bVm4v/ed0M46twKnSaIddsuSjOJMGEzP8mA2E4QzlxhDIj3K2EjaihDF06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NWT3rP35r0vWwtePnMMv+B9fAO9341Y</latexit>

m = 1<latexit sha1_base64="Zv5ybfKo/H8QvXQaOWqQVyOFtg4=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQS9CwIvHiOYByRJmJ73JkJnZZWZWCCGf4MWDIl79Im/+jZNkDxotaCiquunuilLBjfX9L6+wtr6xuVXcLu3s7u0flA+PWibJNMMmS0SiOxE1KLjCpuVWYCfVSGUksB2Nb+Z++xG14Yl6sJMUQ0mHisecUeuke3kd9MsVv+YvQP6SICcVyNHolz97g4RlEpVlghrTDfzUhlOqLWcCZ6VeZjClbEyH2HVUUYkmnC5OnZEzpwxInGhXypKF+nNiSqUxExm5TkntyKx6c/E/r5vZ+CqccpVmFhVbLoozQWxC5n+TAdfIrJg4Qpnm7lbCRlRTZl06JRdCsPryX9I6rwV+Lbi7qNSreRxFOIFTqEIAl1CHW2hAExgM4Qle4NUT3rP35r0vWwtePnMMv+B9fAO/Y41Z</latexit><latexit 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m = 2<latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit><latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit><latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit><latexit sha1_base64="oVNcNejQAmVb9Nn6VIeeV7f50ls=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9ktgl6EghePFe0HtEvJptk2NMkuSVYoS3+CFw+KePUXefPfmLZ70NYHA4/3ZpiZFyaCG+t536iwsbm1vVPcLe3tHxwelY9P2iZONWUtGotYd0NimOCKtSy3gnUTzYgMBeuEk9u533li2vBYPdppwgJJRopHnBLrpAd5Ux+UK17NWwCvEz8nFcjRHJS/+sOYppIpSwUxpud7iQ0yoi2ngs1K/dSwhNAJGbGeo4pIZoJsceoMXzhliKNYu1IWL9TfExmRxkxl6DolsWOz6s3F/7xeaqPrIOMqSS1TdLkoSgW2MZ7/jYdcM2rF1BFCNXe3YjommlDr0im5EPzVl9dJu17zvZp/f1lpVPM4inAG51AFH66gAXfQhBZQGMEzvMIbEugFvaOPZWsB5TOn8Afo8wfA541a</latexit>

m = 3<latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit><latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit><latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit><latexit sha1_base64="+jOsYdJ9SNJymWDVPaXJffNRSPM=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvQU9lVQS9CwYvHivYD2qVk02wbmmSXJCuUpT/BiwdFvPqLvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHu0kYYEkQ8UjTol10oO8ueiXK17NmwOvEj8nFcjR6Je/eoOYppIpSwUxput7iQ0yoi2ngk1LvdSwhNAxGbKuo4pIZoJsfuoUnzllgKNYu1IWz9XfExmRxkxk6DolsSOz7M3E/7xuaqPrIOMqSS1TdLEoSgW2MZ79jQdcM2rFxBFCNXe3YjoimlDr0im5EPzll1dJ67zmezX//rJSr+ZxFOEETqEKPlxBHe6gAU2gMIRneIU3JNALekcfi9YCymeO4Q/Q5w/Ca41b</latexit>

Figure 7: Example of message passing. Each row highlights the information that diffuses throughthe graph starting from a particular node. In the top row, the node of interest is in the upperright; in the bottom row, the node of interest is in the bottom right. Shaded nodes indicate how farinformation from the original node can travel in m steps of message passing; bolded edges indicatewhich edges that information has the potential to travel across. Note that during the full messagepassing procedure, this propagation of information happens simultaneously for all nodes and edgesin the graph (not just the two shown here).

apply an elementary dynamics update, and the decoder might read out the final positions from theupdated graph state.

Similar to the encode-process-decode design, recurrent GN-based architectures can be built bymaintaining a hidden graph, Gthid, taking as input an observed graph, Gtinp, and returning an outputgraph, Gtout, on each step (see Figure 6c). This type of architecture can be particularly useful forpredicting sequences of graphs, such as predicting the trajectory of a dynamical system over time(e.g. Sanchez-Gonzalez et al., 2018). The encoded graph, output by GNenc, must have the samestructure as Gthid, and they can be easily combined by concatenating their corresponding ek, vi,and u vectors (where the upward arrow merges into the left-hand horizontal arrow in Figure 6c),before being passed to GNcore. For the output, the Gthid is copied (where the right-hand horizontalarrow splits into the downward arrow in Figure 6c) and decoded by GNdec. This design reuses GNblocks in several ways: GNenc, GNdec, and GNcore are shared across each step, t; and within eachstep, GNcore may perform multiple shared sub-steps.

Various other techniques for designing GN-based architectures can be useful. Graph skipconnections, for example, would concatenate a GN block’s input graph, Gm, with its output graph,Gm+1, before proceeding to further computations. Merging and smoothing input and hidden graphinformation, as in Figure 6c, can use LSTM- or GRU-style gating schemes, instead of simpleconcatenation (Li et al., 2016). Or distinct, recurrent GN blocks (e.g. Figure 4b) can be composedbefore and/or after other GN blocks, to improve stability in the representations over multiplepropagation steps (Sanchez-Gonzalez et al., 2018).

4.4 Implementing graph networks in code

Similar to CNNs (see Figure 1), which are naturally parallelizable (e.g. on GPUs), GNs have anatural parallel structure: since the φe and φv functions in Equation 1 are shared over the edgesand nodes, respectively, they can be computed in parallel. In practice, this means that with respect

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Box 4: Graph Nets open-source software library: github.com/deepmind/graph nets

We have released an open-source library for building GNs in Tensorflow/Sonnet. It includesdemos of how to create, manipulate, and train GNs to reason about graph-structured data, ona shortest path-finding task, a sorting task, and a physical prediction task. Each demo uses thesame GN architecture, which highlights the flexibility of the approach.

Shortest path demo: tinyurl.com/gn-shortest-path-demo

This demo creates random graphs, and trains a GN to label the nodes and edges on the shortestpath between any two nodes. Over a sequence of message-passing steps (as depicted by eachstep’s plot), the model refines its prediction of the shortest path.

Sort demo: tinyurl.com/gn-sort-demo

This demo creates lists of random numbers, and trains a GN to sort the list. After a sequenceof message-passing steps, the model makes an accurate prediction of which elements (columnsin the figure) come next after each other (rows).

Physics demo: tinyurl.com/gn-physics-demo

This demo creates random mass-spring physical systems, and trains a GN to predict the state ofthe system on the next timestep. The model’s next-step predictions can be fed back in as inputto create a rollout of a future trajectory. Each subplot below shows the true and predictedmass-spring system states over 50 timesteps. This is similar to the model and experiments in(Battaglia et al., 2016)’s “interaction networks”.

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to φe and φv, the nodes and edges can be treated like the batch dimension in typical mini-batchtraining regimes. Moreover, several graphs can be naturally batched together by treating them asdisjoint components of a larger graph. With some additional bookkeeping, this allows batchingtogether the computations made on several independent graphs.

Reusing φe and φv also improves GNs’ sample efficiency. Again, analogous to a convolutionalkernel, the number of samples which are used to optimize a GN’s φe and φv functions is the numberof edges and nodes, respectively, across all training graphs. For example, in the balls examplefrom Sec. 3.2, a scene with four balls which are all connected by springs will provide twelve (4× 3)examples of the contact interaction between them.

We have released an open-source software library for building GNs, which can be found here:github.com/deepmind/graph nets. See Box 4 for an overview.

4.5 Summary

In this section, we have discussed the design principles behind graph networks: flexible representa-tions, configurable within-block structure, and composable multi-block architectures. These threedesign principles combine in our framework which is extremely flexible and applicable to a wide rangeof domains ranging from perception, language, and symbolic reasoning. And, as we will see in theremainder of this paper, the strong relational inductive bias possessed by graph networks supportscombinatorial generalization, thus making it a powerful tool both in terms of implementation andtheory.

5 Discussion

In this paper, we analyzed the extent to which relational inductive bias exists in deep learningarchitectures like MLPs, CNNs, and RNNs, and concluded that while CNNs and RNNs do containrelational inductive biases, they cannot naturally handle more structured representations such assets or graphs. We advocated for building stronger relational inductive biases into deep learningarchitectures by highlighting an underused deep learning building block called a graph network,which performs computations over graph-structured data. Our graph network framework unifiesexisting approaches that also operate over graphs, and provides a straightforward interface forassembling graph networks into complex, sophisticated architectures.

5.1 Combinatorial generalization in graph networks

The structure of GNs naturally supports combinatorial generalization because they do not performcomputations strictly at the system level, but also apply shared computations across the entities andacross the relations as well. This allows never-before-seen systems to be reasoned about, becausethey are built from familiar components, in a way that reflects von Humboldt’s “infinite use of finitemeans” (Humboldt, 1836; Chomsky, 1965).

A number of studies have explored GNs’ capacity for combinatorial generalization. Battagliaet al. (2016) found that GNs trained to make one-step physical state predictions could simulatethousands of future time steps, and also exhibit accurate zero-shot transfer to physical systemswith double, or half, the number of entities experienced during training. Sanchez-Gonzalez et al.(2018) found similar results in more complex physical control settings, including that GNs trained asforward models on simulated multi-joint agents could generalize to agents with new numbers of joints.Hamrick et al. (2018) and Wang et al. (2018b) each found that GN-based decision-making policiescould transfer to novel numbers of entities as well. In combinatorial optimization problems, Bello

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et al. (2016); Nowak et al. (2017); Dai et al. (2017); Kool and Welling (2018) showed that GNs couldgeneralize well to problems of much different sizes than they had been trained on. Similarly, Toyeret al. (2017) showed generalization to different sizes of planning problems, and Hamilton et al. (2017)showed generalization to producing useful node embeddings for previously unseen data. On booleanSAT problems, Selsam et al. (2018) demonstrated generalization both to different problem sizes andacross problem distributions: their model retained good performance upon strongly modifying thedistribution of the input graphs and its typical local structure.

These striking examples of combinatorial generalization are not entirely surprising, given GNs’entity- and relation-centric organization, but nonetheless provide important support for the viewthat embracing explicit structure and flexible learning is a viable approach toward realizing bettersample efficiency and generalization in modern AI.

5.2 Limitations of graph networks

One limitation of GNs’ and MPNNs’ form of learned message-passing (Shervashidze et al., 2011)is that it cannot be guaranteed to solve some classes of problems, such as discriminating betweencertain non-isomorphic graphs. Kondor et al. (2018) suggested that covariance7 (Cohen and Welling,2016; Kondor and Trivedi, 2018), rather than invariance to permutations of the nodes and edgesis preferable, and proposed “covariant compositional networks” which can preserve structuralinformation, and allow it to be ignored only if desired.

More generally, while graphs are a powerful way of representing structure information, theyhave limits. For example, notions like recursion, control flow, and conditional iteration are notstraightforward to represent with graphs, and, minimally, require additional assumptions (e.g., ininterpreting abstract syntax trees). Programs and more “computer-like” processing can offer greaterrepresentational and computational expressivity with respect to these notions, and some have arguedthey are an important component of human cognition (Tenenbaum et al., 2011; Lake et al., 2015;Goodman et al., 2015).

5.3 Open questions

Although we are excited about the potential impacts that graph networks can have, we caution thatthese models are only one step forward. Realizing the full potential of graph networks will likely befar more challenging than organizing their behavior under one framework, and indeed, there are anumber of unanswered questions regarding the best ways to use graph networks.

One pressing question is: where do the graphs come from that graph networks operate over?One of the hallmarks of deep learning has been its ability to perform complex computations overraw sensory data, such as images and text, yet it is unclear the best ways to convert sensory datainto more structured representations like graphs. One approach (which we have already discussed)assumes a fully connected graph structure between spatial or linguistic entities, such as in theliterature on self-attention (Vaswani et al., 2017; Wang et al., 2018c). However, such representationsmay not correspond exactly to the “true” entities (e.g., convolutional features do not directlycorrespond to objects in a scene). Moreover, many underlying graph structures are much moresparse than a fully connected graph, and it is an open question how to induce this sparsity. Severallines of active research are exploring these issues (Watters et al., 2017; van Steenkiste et al., 2018;Li et al., 2018; Kipf et al., 2018) but as of yet there is no single method which can reliably extractdiscrete entities from sensory data. Developing such a method is an exciting challenge for future

7Covariance means, roughly, that the activations vary in a predictable way as a function of the ordering of theincoming edges.

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research, and once solved will likely open the door for much more powerful and flexible reasoningalgorithms.

A related question is how to adaptively modify graph structures during the course of computation.For example, if an object fractures into multiple pieces, a node representing that object also oughtto split into multiple nodes. Similarly, it might be useful to only represent edges between objectsthat are in contact, thus requiring the ability to add or remove edges depending on context. Thequestion of how to support this type of adaptivity is also actively being researched, and in particular,some of the methods used for identifying the underlying structure of a graph may be applicable (e.g.Li et al., 2018; Kipf et al., 2018).

Human cognition makes the strong assumption that the world is composed of objects andrelations (Spelke and Kinzler, 2007), and because GNs make a similar assumption, their behaviortends to be more interpretable. The entities and relations that GNs operate over often correspondto things that humans understand (such as physical objects), thus supporting more interpretableanalysis and visualization (e.g., as in Selsam et al., 2018). An interesting direction for future workis to further explore the interpretability of the behavior of graph networks.

5.4 Integrative approaches for learning and structure

While our focus here has been on graphs, one takeaway from this paper is less about graphsthemselves and more about the approach of blending powerful deep learning approaches withstructured representations. We are excited by related approaches which have explored this idea forother types of structured representations and computations, such as linguistic trees (Socher et al.,2011a,b, 2012, 2013; Tai et al., 2015; Andreas et al., 2016), partial tree traversals in a state-actiongraph (Guez et al., 2018; Farquhar et al., 2018), hierarchical action policies (Andreas et al., 2017),multi-agent communication channels (Foerster et al., 2016), “capsules” (Sabour et al., 2017), andprograms (Parisotto et al., 2017). Other methods have attempted to capture different types ofstructure by mimicking key hardware and software components in computers and how they transferinformation between each other, such as persistent slotted storage, registers, memory I/O controllers,stacks, and queues (e.g. Dyer et al., 2015; Grefenstette et al., 2015; Joulin and Mikolov, 2015;Sukhbaatar et al., 2015; Kurach et al., 2016; Graves et al., 2016).

5.5 Conclusion

Recent advances in AI, propelled by deep learning, have been transformative across many importantdomains. Despite this, a vast gap between human and machine intelligence remains, especially withrespect to efficient, generalizable learning. We argue for making combinatorial generalization a toppriority for AI, and advocate for embracing integrative approaches which draw on ideas from humancognition, traditional computer science, standard engineering practice, and modern deep learning.Here we explored flexible learning-based approaches which implement strong relational inductivebiases to capitalize on explicitly structured representations and computations, and presented aframework called graph networks, which generalize and extend various recent approaches for neuralnetworks applied to graphs. Graph networks are designed to promote building complex architecturesusing customizable graph-to-graph building blocks, and their relational inductive biases promotecombinatorial generalization and improved sample efficiency over other standard machine learningbuilding blocks.

Despite their benefits and potential, however, learnable models which operate on graphs areonly a stepping stone on the path toward human-like intelligence. We are optimistic about anumber of other relevant, and perhaps underappreciated, research directions, including marrying

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learning-based approaches with programs (Ritchie et al., 2016; Andreas et al., 2016; Gaunt et al.,2016; Evans and Grefenstette, 2018; Evans et al., 2018), developing model-based approaches with anemphasis on abstraction (Kansky et al., 2017; Konidaris et al., 2018; Zhang et al., 2018; Hay et al.,2018), investing more heavily in meta-learning (Wang et al., 2016, 2018a; Finn et al., 2017), andexploring multi-agent learning and interaction as a key catalyst for advanced intelligence (Nowak,2006; Ohtsuki et al., 2006). These directions each involve rich notions of entities, relations, andcombinatorial generalization, and can potentially benefit, and benefit from, greater interaction withapproaches for learning relational reasoning over explicitly structured representations.

Acknowledgements

We thank Tobias Pfaff, Danilo Rezende, Nando de Freitas, Murray Shanahan, Thore Graepel, JohnJumper, Demis Hassabis, and the broader DeepMind and Google communities for valuable feedbackand support.

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Appendix: Formulations of additional models

In this appendix we give more examples of how published networks can fit in the frame defined byEquation 1.

Interaction networks

Interaction Networks (Battaglia et al., 2016; Watters et al., 2017) and the Neural Physics EngineChang et al. (2017) use a full GN but for the absence of the global to update the edge properties:

φe (ek,vrk ,vsk ,u) := fe (ek,vrk ,vsk) = NNe ([ek,vrk ,vsk ])

φv(e′i,vi,u

):= fv

(e′i,vi,u

)= NNv

([e′i,vi,u]

)ρe→v

(E′i)

:= =∑

{k: rk=i}

e′k

That work also included an extension to the above formulation which output global, rather thanper-node, predictions:

φe (ek,vrk ,vsk ,u) := fe (ek,vrk ,vsk) = NNe ([ek,vrk ,vsk ])

φv(e′i,vi,u

):= fv

(e′i,vi,u

)= NNv

([e′i,vi,u]

)φu(e′, v′,u

):= fu

(v′,u

)= NNu

([v′,u]

)ρv→g

(V ′)

:= =∑i

v′i

Non-pairwise interactions

Gated Graph Sequence Neural Networks (GGS-NN) (Li et al., 2016) use a slightly generalizedformulation where each edge has an attached type tk ∈ {1, .., T}, and the updates are:

φe ((ek, tk) ,vrk ,vsk ,u) := fe (ek,vsk) = NNe,tk (vsk)

φv(e′i,vi,u

):= fv

(e′i,vi

)= NNv

([e′i,vi]

)ρe→v

(E′i)

:= =∑

{k: rk=i}

e′k

These updates are applied recurrently (the NNv is a GRU (Cho et al., 2014)), followed by a globaldecoder which computes a weighted sum of embedded final node states. Here each NNe,tk is a neuralnetwork with specific parameters.

CommNet (Sukhbaatar et al., 2016) (in the slightly more general form described by (Hoshen,2017)) uses:

φe (ek,vrk ,vsk ,u) := fe (vsk) = NNe (vsk)

φv(e′i,vi,u

):= fv

(e′i,vi

)= NNv

([e′i,NNv′ (vi)]

)ρe→v

(E′i)

:= =1

|E′i|∑

{k: rk=i}

e′k

38

Attention-based approaches

The various attention-based approaches use a φe which is factored into a scalar pairwise-interactionfunction which returns the unnormalized attention term, denoted αe (vrk ,vsk) = a′k, and a vector-valued non-pairwise term, denoted βe (vsk) = b′k,

φe (ek,vrk ,vsk ,u) := fe (vrk ,vsk) = (αe (vrk ,vsk) , βe (vsk)) = (a′k,b′k) = e′k

The single-headed self-attention (SA) in the Transformer architecture (Vaswani et al., 2017),implements the non-local formulation as:

αe (vrk ,vsk) = exp (NNαquery (vrk)ᵀ ·NNαkey (vsk))

βe (vsk) = NNβ (vsk)

φv(e′i,vi,u

):= fv

(e′i)

= NNv

(e′i)

where NNαquery , NNαkey , and NNβ are again neural network functions with different parameters andpossibly different architectures. They also use a multi-headed version which computes Nh parallele′hi using different NNαquery

h, NN

αkeyh

, NNβh , where h indexes the different parameters. These are

passed to fv and concatenated:

fv({e′hi }h=1...Nh

)= NNv

([e′1i , . . . , e

′Nhi ]

)Vertex Attention Interaction Networks (Hoshen, 2017) are very similar to single-headed SA,

but use Euclidean distance for the attentional similarity metric, with shared parameters across theattention inputs’ embeddings, and also use the input node feature in the node update function,

αe (vrk ,vsk) = exp(−‖NNα (vrk)−NNα (vsk) ‖2

)βe (vsk) = NNβ (vsk)

φv(e′i,vi,u

):= fv

(e′i)

= NNv

([e′i,vi]

)Graph Attention Networks (Velickovic et al., 2018) are also similar to multi-headed SA, but use

a neural network as the attentional similarity metric, with shared parameters across the attentioninputs’ embeddings:

αe (vrk ,vsk) = exp (NNα′ ([NNα (vrk) ,NNα (vsk)))

βe (vsk) = NNβ (vsk)

φv(e′i,vi,u

):= fv

({e′hi }h=1...Nh

)= NNv

([e′1i , . . . , e

′Nhi ]

)Stretching beyond the specific non-local formulation, Shaw et al. (2018) extended multi-headed

SA with relative position encodings. “Relative” refers to an encoding of the spatial distance betweennodes in a sequence or other signal in a metric space. This can be expressed in GN language as anedge attribute ek, and replacing the βe (vsk) from multi-headed SA above with:

βe (ek,vsk) = NNe (vsk) + ek

39

Belief Propagation embeddings

Finally, we briefly summarize how the general “structure2vec” algorithm of Dai et al. (2016) can fitinto our framework. In order to do so, we need to slightly modify our main Equation 1, i.e.:

εk = ρ

({el}sl=rk

rl 6=sk

):=∑rl=sksl 6=rk

el

e′k = φe (εk) := f(εk) = NN(εk)

e′i = ρ({e′k}rk=i

):=

∑{k: rk=i}

ek

v′i = φv(e′i)

:= f(e′i) = NN(e′i)

Edges’ features now takes the meaning of “message” between their receiver and sender; note thatthere is only one set of parameters to learn for both the edges and nodes updates.

40