The Structure of Information Pathways in a Social

Communication Network

Presented By: Under the guidance of: Tingting Xu Augustin Chainterau

Paper Objective

Study the temporal dynamics of communication using on-line data

Give temporal notion of ‘distance’ and ‘vector – clocks’ to formulate a temporal measure which will provide structural insights

Define the network backbone to be the sub-graph consisting of edges on which information has the potential to flow the quickest

Why Construct New Model

Discrete communication distributed non-uniformly over time

Direct and indirect flow of information

Discussion about recent research - has studied communication of an event-driven nature

The properties of systemic communication arguably determine much about the rate at which people in the network remain up-to-date on information about each other

The Present Work

Systemic communication and information pathways

Propose a framework for analyzing systemic communication based on inferring structural measures from the potential for information to flow between different nodes

Out-of-date information

Indirect paths – triangle-inequality violation

The Present Work

Data used here have complete histories of communication events over long periods of time

Main datasets - complete set of anonymized e-mail logs among all faculty and staff at a large university over two years

Enron e-mail corpus The complete set of user-talk communications among admins

and high-volume editors on Wikipedia

Vector clocks introduced by Lamport and refined by Mattern

Network backbone

Vector Clocks and Latency

Communication skeleton G

The latest view that v has of u at time t is denoted by

Define for all v and t

, refer as the vector clock of v at time t

Information latency is denoted by t -

An algorithm to compute the vector clocks for all nodes at all time in [0, T]

Latencies in Social Network Data

Consider only messages with at most c (ranging from 1 and 5) recipients

Focus on q-fraction of active e-mal users (Here q = 0.20)

For a time difference τ , we define the ball of radius τ around node v at time t, denoted Bτ (v, t), to be the set of all nodes whose latency with respect to v at time t is ≤ τ days.

For fixed t, the distribution of ball-sizes over nodes can be studied using a function ft(τ ), defined as the median value of |Bτ (v, t)| over all v

Open Worlds vs. Closed Worlds

Boundary specification problem – value of q-fraction [0, 1]

Quantifying the Strength of Weak Ties

The range of an edge , defined to be the unweighted shortest-path distance in the social network between and if were deleted

Edges of range greater than two are generally weak ties

Vector-clock analysis can provide evidence for the phenomenon that weak ties are the sources of important information to their endpoints

Define advance in ’s clock to be the sum of coordinatewise differences between before the update from and after the update from

Backbone Structures

Instantaneous Backbones

Define the backbone Ht at time t to be the graph on whose edge set is the collection of edges from G that are essential at time t.

An edge is essential if ’s most up-to-date view of is the result of direct communication from

Here the backbones Ht at fixed times t as instantaneous backbones, by contrast with the aggregate backbone which is based on an aggregate construction that takes all times into account.

Backbone Structures

An aggregate Backbone

For each edge in the communication skeleton G such that has sent ρv, w > 0 messages to over the full time interval [0, T], define the delay δv, w of the edge to be T/ ρv, w

The weighted graph Gδ obtained from the communication skeleton G by assigning a weight of δv, w to each edge

An edge in Gδ is essential if it forms the minimum-delay path between its two endpoints

Define the aggregate backbone H* to be the sub-graph of Gδ consisting only of essential edges

Backbone Structures

How to construct the aggregate Backbone H*

Compute a weighted shortest-paths tree rooted at each node of Gδ , using the delays as weights

The union of the edges in all these trees will be H*, by the following proposition

PROPOSITION An edge belongs to H* if and only if it lies on the minimum-delay path between some pair of nodes and

PROOF

Backbone Structures

How to construct the aggregate Backbone H*

Compute a weighted shortest-paths tree rooted at each node of Gδ , using the delays as weights

The union of the edges in all these trees will be H*, by the following proposition

PROPOSITION An edge belongs to H* if and only if it lies on the minimum-delay path between some pair of nodes and

Backbone Structures

Density and node degrees of the backbone

The backbone Ht and the aggregate backbone H* are surprisingly sparse related to a fairly dense communication skeleton G

This in other words, from the point of view of potential information flow, a significant majority of all edges in the social network are bypassed by faster indirected paths

Backbone Structures

Density and node degrees of the backbone

Considering the backbone also sheds further light on the role of high-degree nodes in the social network

High-degree nodes in the full communication skeleton G indeed have many incident edges in the aggregate backbone

However, the fraction of a node’s edges that are declared essential strictly decreases with degree.

Backbone Structures

Structure of the backbone

The backbone is trying to balance two competing objectives

Representing long range edges (recall definition of ‘range’)

Representing edges have high embeddedness and transmit information at short ranges over quick time scales

Define embeddedness of an edge to be the fraction of its endpoints’ neighbors that are common to both

For an edge , let and denote the sets of neighbors of the endpoints and respectively. Define the embeddedness of to be / | |

The backbone balances between two qualitatively different kinds of information flow

Varying Speed of Communication

Study what happens to information latencies (i.e. t - ) when each node varies the relative rates of its communication

Given a directed graph G, with a total rate for each node

Given a target set of nodes in G Each node chooses a rate at which to communicate to each of its

neighbors , subject to the constraint that

Define delays , where T is value of the time interval Question here is that: for a given bound , can we choose rates for each

node so that the median shortest-path delay between pairs in in the aggregate backbone is at most

Varying Speed of Communication

THEOREM The delay minimization problem defined above is NP – complete

Sketch of the proof of this theorem is in the paper

Consider simple local rules by which individuals in a network might vary rates of communication so as to influence the potential for information flow

Load-leveling vs. Load-concentrating

For accelerating potential information flow

Talk even more actively to one’s most frequent contacts Load-concentrating with > 1

or balance things out by increasing communication with the less frequent contacts? Load-leveling with < 1

Rescaling exponent , changing the communication rate to and then normalizing all rates from to keep its total outgoing message volume the same

Load-leveling vs. Load-concentrating

Extend the notion of delay to node-dependent delays which will have also a fixed delay of at each node

Total delay on a path becomes the sum of edges and node delays

As increases, there is a larger penalty for more-hop paths

The value of at which network latency is optimized decreases with * = 1 at days

The backbone becomes denser and the importance of quick indirect paths diminishes

Conclusions (I)

Make integral use of information about how nodes communicate over time

Develop structural measures based on the potential for information to flow

The sparse sub-graph of edges most essential to keeping people up-to-date – the backbone of the network – provides important structural insights that relate to embeddedness, the role of high-degree(i.e. hubs), and the strength of weak ties

Studied the effects on information flow as nodes vary the rate at which they communicate with others in the network using different strategies

Conclusions (II)

Discussions in other two datasets

The situations in sparsity of the aggregate and instantaneous backbones and the variation in node degrees are similar

Difference - the ‘core’ of active communicators is much smaller in both the Enron corpus and in Wikipedia, this makes the range of an edge in the unweighted communication skeleton harder to interpret and to correlate with other measures

Further investigation the principles that govern the dynamics of different types of information how these principles interact with the directed, weighted nature of social

communication networks

Thank You