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

NSPCS 08

Unified centrality measure of complex networks

Overview

• Introduction– interplay between dynamical process and underlying topology– centrality measure

• shortest path betweenness centrality• random walk centrality

• biased random walk betweenness centrality– analytic results– numerical simulations

• special example: shortest path betweenness centrality

• First systematic study on the edge centrality

• summary and discussion

Underlying topology & dynamics

• 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.

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

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

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

Underlying topology & dynamics

• Diffusive capture process– Related to the first passage properties of random walker

Nodes of large degrees plays a impor-tant 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 impor-tance

• 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]

PINNodes having high degree

High closeness

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?

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 degrees k and k’

• NT: the basic measure of BRWC• Note that both RWC and SPC depend on k

• In the limit t

Relationship between BRWC and SPBC for vertices

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

NT(k) also scales as

• Average number of traverse a vertex having degree k

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

The probability to find a walker at 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 rela-tion holds for g>3

Relationship between BRWC and SPBC for vertices

n=1.0

n=2.0n=5/3

b=0.7

b=1.0b=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 a+1=n and b=n/h=a/h

Summary and Discussion• We introduce a biased random walk 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 expec-tations.• 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 a=0: simple RWBC is recovered• If a; hubs have large BC• If a- ; dangling ends have large BC

Thank you for your attention!!

• Kwon et al. PRE 77, 066105 (2008)

Relationship between BRWC and SPBC for vertices

• Mapping to the weight network with weight

• Therefore, NT(k) also scales as

• Average number of traverse a vertex having degree k

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