Complex NetworksStructure and Dynamics

Ying-Cheng LaiDepartment of Mathematics and Statistics

Department of Electrical EngineeringArizona State University

Collaborators Adilson E. Motter, now at Max-Planck

Institute for Physics of Complex Systems, Dresden, Germany

Takashi Nishikawa, now at Department of Mathematics, Southern Methodist University

Complex Networks

Structures composed of a large number of elements linked together in an apparently fairly sophisticated fashion.

Examples:- Social networks- Internet and WWW (world-wide web)- Power grids- Brain and other neural networks

- Metabolic networks

Characteristics: - Large, sparse, and continuously evolving.

Social networks

Contacts and Influences – Poll & Kochen (1958)

– How great is the chance that two people chosen at random from the population will have a friend in common?

– How far are people aware of the available lines of contact?

The Small-World Problem – Milgram (1967)

– How many intermediaries are needed to move a letter from person A to person B through a chain of acquaintances?– Letter-sending experiment: starting in Nebraska/Kansas,

with a target person in Boston.

“Six degrees of separation”

Random graphs – Erdos & Renyi (1960)

Start with N nodes and for each pair of nodes, with probability p, add a link between them.

For large N, there is a giant connected component if the average connectivity (number of links per node) is larger than 1.

The average path length L in the giant component scales as L ln

N.

Minimal number of links one needs to follow to go from one node to another, on average.

Small-world networks – Watts & Strogatz (1998)

Start with a regular lattice and for each link, with probability p, rewire one extreme of the link at random.

fraction p of the links is converted into shortcuts

LC

Clustering coefficient C is the probability that two nodes are connected to each other,

given that they are both connected to a common node.

p

regular sw random

Scale-free networks – Barabasi & Albert (1999)

Growth: Start with few nodes and, at each time step, a new node with m links is added.

Preferential attachment: Each link connects with a node in the network according to a probability i proportional to the connectivity ki of the node: i ki .

The result is a network with an algebraic (scale-free) connectivity distribution: P(k) k -, where =3.

Questions

Which are the generic structural properties of real-world networks?

What sort of dynamical processes govern the emergence of these properties?

How does individual behavior aggregate to collective behavior?

Questions

Structure

Which are the generic structural properties of real-world networks?

Dynamics of the network

What sort of dynamical processes govern the emergence of these properties?

Dynamics on the network

How does individual behavior aggregate to collective behavior?

Network of word associationMotter, de Moura, Lai, & Dasgupta (2002)

Words correspond to nodes of the network; a link exists between two words if they express similar concepts.

• Motivation: structure and evolution of language, cognitive science.

Word association is a small-world network

N <k> C L

Actual configuration 30244* 59.9 0.53 3.16

Random configuration 30244 59.9 0.002 2.5

*Source: online Gutenberg Thesaurus dictionary

Featured in Nature Science Update, New Scientist, Wissenschaft-online, etc.

“Three degrees of separation for English words”

Word association as a growing network

Preferential and random attachments [Liu et al (2002)]:

i (1- p) k i + p, 0 p 1 Scaling for the connectivity distribution:

P(k) [k + p/(1- p) ] - , = 3 + m-1 p/(1- p)

P(k): exponential for small k, algebraic for large k

= 3.5

Small-world phenomenon in scale-free networksMotter, Nishikawa, & Lai (2002)

The range R(Lij) of a link Lij connecting nodes i. and j is the length of the shortest path between i. and j in the absence of Lij.

Watts-Strogatz model: short average path length is due to long-range links

(shortcuts).

Scale-free networks also present very short L.

Are long-range links responsible for the short average path length of scale-free networks?

j

iR(Lij) = 3

Range-based attack

Short-range attack: links with shorter range are removed first.

Long-range attack: links with longer range are removed first.

Average of the inverse path length

Efficiency [Latora & Marchiori (2001)]

ijLNNE

1

)1(

2

Range-based attack on scale-free networks

Results for semirandom scale-free networks: P(k) k -

The connectivity distribution is more heterogeneous for smaller .

fraction of removed links

norm

aliz

ed

effi

cien

cy

N=5000

Newman, Strogatz, & Watts (2001)

Load of a link Lij is the number of shortest paths passing through Lij.

Links between highly connected nodes are more likely to have high load and small range.

Heterogeneity versus homogeneity

Results for growing networks with aging: i i- ki

Short average path length in scale-free networks

is mainly due to short-range links.

Other scale-free models

fraction of removed links

norm

aliz

ed

effi

cien

cy

<k>=6, N=5000

Dorogovtsev & Mendes (2000)

Cascade-based attacks on complex networksMotter & Lai (2002)

Statically: – L increases significantly in scale-free networks when highly

connected nodes are removed [Albert et al (2000)]; – the existence of a giant connected component does not

depend on the presence of these nodes [Broder et al (2000)]. Dynamically, if 1. the flow of a physical quantity, as characterized by load on

nodes, is important, and 2. the load can redistribute among other nodes when a node is

removed, intentional attacks may trigger a global cascade of overload

failures in heterogeneous networks.

Flow: at each time step, one unit of the relevant quantity is exchanged between every pair of nodes along the shortest path.

Capacity is proportional to the initial load:

Cj = (1 + ) lj (0), ( j=1,2, … N, 0).

Cascade: a node fails whenever the updated load exceeds the capacity, i.e., node j is removed at step n if lj (n) > Cj.

Simple model for cascading failure

load on a node = number of shortest paths passing through that node

Simulations

( =3, N 5000)

– random (squares) – connectivity (stars) – load (circles)

G: relative number of nodes in the largest connected component

Simulations

( =3, N 5000) (Western U.S. power grid, N=4941)

Simulations

Featured in Newsletter (Editorial), Equality: Better for network security;

NewsFactor, Cascading failures could crash the global Internet;

The Guardian, Electronic Pearl Harbor; etc.

Networks with heterogeneous distribution of load: “robust-yet-fragile”

( =3, N 5000) (Western U.S. power grid, N=4941)

Revisiting the original small-world problemMotter, Nishikawa, & Lai (2003)

After talking to a strange for a few minutes, you and the stranger often realize that you are linked through a mutual friend or through a short chain of acquaintances.

discovery of short paths existence of short paths

We want to model this phenomenon and find a criterion for plausible models of social networks.

Model for the identification of mutual acquaintances

People are naturally inclined to look for social connections that can identify them with a newly introduced person.

We assume that a person knows another person when this person knows the social coordinates of the other.

We also assume that when two people are introduced:

1. they exchange information defining their own social coordinates;

2. they exchange information defining the social coordinates of acquaintances that are socially close to the other

person.

Network model

Hierarchy of social structure: individuals are organized into groups, which in turn belong to groups of groups

and so on [Watts, Dodds, & Newman (2002)].

The distance along the tree structure defines a social distance between individuals in a hierarchy.

The society is organized into different but correlated hierarchies.

The network is built by connecting with higher probability pairs of closer individuals.

Social coordinates set of positions a person occupies in the hierarchies.

Trade-off between short paths and high correlations

Probability of discovering mutual acquaintances, acquaintances in the same social group, and acquaintances who know each other, after citing m=1, 2, and 20 acquaintances.

Scaling with system size: P N -1

N=106, n=250, H=2, g=100, b=10, =

: correlation between hierarchies : correlation between distribution of social ties and social distance

Discovery versus existence

The probability of finding a short chain of acquaintances between two people does not scale with typical distances in the underlying network of social ties.

Random networks are usually “smaller” than small-world networks, and because of that they are sometimes called themselves small-world networks.

But a random society would not allow people to find easily that “It is a small world!”

Word association is a small-world network, with a crossover from exponential to algebraic distribution of connectivity.

The short average path length observed in scale-free network is mainly due to short-range links.

Networks with skewed distribution of load may undergo cascades of overload failures.

The “small-world phenomenon” results from a trade-off between short paths and high correlations in the network of social ties.

Conclusions

Word association is a small-world network, with a crossover from exponential to algebraic distribution of connectivity.

The short average path length observed in scale-free network is mainly due to short-range links.

Networks with skewed distribution of load may undergo cascades of overload failures.

The “small-world phenomenon” results from a trade-off between short paths and high correlations in the network of social ties.

Conclusions

Recent developments in complex networks offer a framework to approach new and old problems in various disciplines.