Soon-Hyung Yook, Sungmin Lee, Yup KimKyung Hee University

NSPCS 08

Unified centrality measure of complex networks: a dynamical

approach to a topological property

Overview• introduction– centrality measure– interplay between dynamical process and underlying

topology

• biased random walk centrality– analytic results– compare the analytic expectations with well known

centrality by numerical simulations• special example: shortest path betweenness centrality

• first systematic study on the edge centrality

• summary and discussion

Introduction

• Many properties of dynamical systems on complex networks are different from those expected by simple mean-field theory

– due to the heterogeneity of the underlying topology.

– scale-free networks: P(k)~k-

• Is it possible to use such dynamical properties to characterize the underlying topology of given networks?

Underlying topology & dynamics

• The dynamical properties of random walk provide some efficient methods to uncover the topological properties of underlying networks

Using the finite-size scaling of <Ree>One can estimate the scaling behavior of diameter

Lee, SHY, Kim Physica A 387, 3033 (2008)

Underlying topology & dynamics• Diffusive capture process (lamb-lion problem)– Related to the first passage properties of random walker

Nodes of large degrees plays a important role. exists some important components[Lee, SHY, Kim PRE 74 046118 (2006)]

Centrality• Centrality: importance of a vertex and an edge

Shortest path betweenness centrality (SPBC)• bi: fraction of shortest path between pairs of vertices in a network that pass through vertex i.

• h (j): starting (targeting) vertex• Total amount of traffic that pass through a vertex

The simplest one: degree (degree centrality), ki

Node and edge importance based on adjacency matrix eigenvalue[Restrepo, Ott, Hund PRL 97, 094102]

Closeness centrality:

Random walk centrality (RWC)

Essential or lethal proteins in protein-protein interaction networks

Various centrality and degree– node importance

• Node (or vertex) importance: – defined by eigenvalue of adjacency matrix

[Restrepo, Ott, Hund PRL 97, 094102]

PIN email

AS

Various centrality and degree– closeness centrality

[Kurdia et al. Engineering in Medicine and Biology Workshop, 2007]

PIN

Nodes having high degree

High closeness

Various centrality and degree– lithality

[Jeong et al. Nature 411, 41 (2007)]

Shortest Path Betweenness Centrality (SPBC)

for a vertex• SPBC distribution:

[Goh et al. PRL 87, 278701 (2001)]

SPBC and RWC

• SPBC and RWC [Newman, Social Networks 27, 39 (2005)]

Random Walk Centrality• RWC can find some vertices which do not lie on many shortest paths [Newman, Social Networks 27, 39 (2005)]

Motivation

Centrality of each node Related to degree of each node

Dynamical property(random walks) Related to degree of each node

Any relationship between them?

If yes, then is it possible to use a certain dynamical property in the investigation of topological properties, especially important component?

Unified and efficient framework to measure the centrality?

Biased Random Walk Centrality (BRWC)

• Generalize the RWC by biased random walker

• Count the number of traverse, NT, of vertices having degree k or edges connecting two vertices of degree k and k’

• NT: the basic measure of BRWC

• Note that both RWC and SPC depend on k

• In stationary state

Relationship between BRWC and SPBC for vertices

• For scale free network whose degree distribution satisfies a power-law P(k)~k-

NT(k) also scales as

• Average number of traverse a vertex i having degree k

• Nv(k): number of vertices having degree k

The probability to find a walker at one of the nodes of degree k

Thus

• SPBC; bv(k)

Relationship between BRWC and SPBC for vertices

thus,

But in the numerical simulations, we find that this relation holds for >3

Relationship between BRWC and SPBC for vertices

=1.0

=2.0=5/3

=0.7

=1.0=1.3

Relationship between BRWC and SPBC for vertices

Relationship between BRWC and SPBC for edges

• for uncorrelated network

number of edges connecting nodes of degree k and k’

thus

• By assuming that

Relationship between BRWC and SPBC for edges

3.04.3

0.66

0.77

Relationship between BRWC and SPBC for edges

Relationship between BRWC and SPBC for edges

Protein-Protein Interaction Network

Slight deviation of +1= and =/

Summary and Discussion

• We introduce a biased random walk centrality as a unified and efficient frame work for centrality.

• We show that the edge centrality satisfies a power-law.• In uncorrelated networks, the analytic expectations agree very well with the numerical

results. ,

• In real networks, numerical simulations show slight deviations from the analytic expectations.• This might come from the fact that the centrality affected by the other topological

properties of a network, such as degree-degree correlation.• The results are reminiscent of multifractal.

• D(q): generalized dimension• q=0: box counting dimension• q=1: information dimension• q=2: correlation dimension …

• In our BC measure

• for =0: simple RWBC is recovered

• If ; hubs have large BC

• If - ; dangling ends have large BC