Entropy and Heterogeneity Measures

for Directed Graphs

Cheng Ye, Richard C. Wilson, Cesar H. Comin, Luciano da F.

Costa, and Edwin R. Hancock

Outline

Introduction & Related Literature

Von Neumann Entropy of Directed Graphs

Heterogeneity Index & Commute Time

Experiments & Evaluations

Conclusions

Complex Networks

Young and active area of scientific research

inspired largely by the empirical study of real-

world networks

Airline network & social network

Complex Networks

Why do we study?

Play a significant role in modelling large-scale

systems

What do we study?

Structural complexity is one of the salient

properties

How do we study?

Graph-based theoretic methods are often used

Related Literature

Estrada has proposed

an index which can

quantify the

heterogeneity

characteristics of

undirected graphs

Depends on vertex

degree statistics and

graph size

Related Literature

Lower bound: 0

– regular graph

Upper bound: 1

– star graph

Related Literature

Han et al. have developed simplified expressions

of von Neumann entropy on undirected graphs

They replace the Shannon entropy by its

quadratic counterpart and simplify and

approximate the calculation

Undirected to Directed Networks

Directed networks are

more common in real-

world, e.g. World Wide

Web

Little work aimed at

characterizing directed

graphs’ structure

Our aim - extend

characterizations

developed for

undirected graphs to

the domain of directed

graphs

Von Neumann Entropy of

Undirected Graphs

Shannon entropy

associated with

normalized Laplacian

eigenvalues (Passeri &

Severini, 2008)

Replace it by the

quadratic entropy to

obtain the

approximate entropy (Han et al., 2012)

Repeat the computation for directed cases to extend von

Neumann entropy from undirected to directed graphs

Von Neumann Entropy of

Directed Graphs

Given a directed graph

G(V, E), the normalized

Laplacian matrix (Chung, 2005)

is the transition matrix

unique left eigenvector of

transition matrix

Notice the following

approximate equation

Compute the traces

Von Neumann Entropy of

Directed Graphs

Divide all edges into two types: one-direction

edges and bidirectional edges such that

and

Von Neumann entropy of directed graphs

or equivalently,

Von Neumann Entropy of Directed

Graphs

For weakly directed graphs, i.e.

For strongly directed graphs, i.e.

Both contain two terms: graph size and in/out-degree statistics

Symmetric/asymmetric & flow of information on network

Heterogeneity Index of Directed

Graphs

Follow Estrada, define

the single link irregularity

of a directed graph (measures difference in

out/in-degrees)

Sum single link

irregularity over all

edges

Lower bound: 0

– regular graph

Upper bound: 1

– star graph

Normalized

heterogeneity index

Commute Time of Directed

Graphs

Hitting time and commute time (resistance

distance) on a directed graph (Boley et al., 2011)

is the fundamental matrix

Commute time can be approximated by

Relationship between HI & CT

On undirected graphs

On directed graphs

Entropy for Weakly & Strongly

Directed Graphs

Weakly & strongly directed forms of von

Neumann entropy

Directed/Undirected Graph

Characterization

Von Neumann entropy has a better performance

on directed graphs

Heterogeneity Index & Commute

Time

Linear relationship between HI & CT

Directed Graph Characterization

using Heterogeneity Index

Different kinds of directed graphs can be well

separated

Conclusions

Develop von Neumann entropy for both weakly &

strongly directed graphs

Extend heterogeneity index to directed graphs

Explore the relationship between heterogeneity

index and commute time

Future work - testing our methods on more real-

world network data

End

Thank you very much!

Question & Answer