Soon-Hyung Yook, Sungmin Lee, Yup Kim

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Unified centrality measure of complex networks a
dynamical approach to a topological property
NSPCS 08

• Soon-Hyung Yook, Sungmin Lee, Yup Kim
• Kyung Hee University

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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

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

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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 ltReegt One can
estimate the scaling behavior of diameter
Lee, SHY, Kim Physica A 387, 3033 (2008)
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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)
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Centrality

• Centrality importance of a vertex and an edge

• The simplest one degree (degree centrality), ki

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

• Random walk centrality (RWC)

• Essential or lethal proteins in protein-protein
  interaction networks

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Various centrality and degree node importance

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

PIN
email
AS
Restrepo, Ott, Hund PRL 97, 094102
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Various centrality and degree closeness
centrality
PIN
Kurdia et al. Engineering in Medicine and
Biology Workshop, 2007
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Various centrality and degree lithality
Jeong et al. Nature 411, 41 (2007)
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Shortest Path Betweenness Centrality (SPBC) for
a vertex

• SPBC distribution

Goh et al. PRL 87, 278701 (2001)
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SPBC and RWC

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

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Random Walk Centrality

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

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Motivation

• 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?

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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

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Relationship between BRWC and SPBC for vertices

• In stationary state

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

• 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 i having
  degree k

• Nv(k) number of vertices having degree k

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Relationship between BRWC and SPBC for vertices

• SPBC bv(k)

thus,
But in the numerical simulations, we find that
this relation holds for ggt3
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Relationship between BRWC and SPBC for vertices
b1.3
b1.0
n5/3
n2.0
b0.7
n1.0
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Relationship between BRWC and SPBC for vertices
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Relationship between BRWC and SPBC for edges

• for uncorrelated network

number of edges connecting nodes of degree k and
k
thus

• By assuming that

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Relationship between BRWC and SPBC for edges
0.77
3.0
0.66
4.3
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Relationship between BRWC and SPBC for edges
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Relationship between BRWC and SPBC for edges
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Protein-Protein Interaction Network
Slight deviation of a1n and bn/ha/h
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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
• q0 box counting dimension
• q1 information dimension
• q2 correlation dimension
• In our BC measure
• for a0 simple RWBC is recovered
• If a?? hubs have large BC
• If a?-? dangling ends have large BC

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Thank you !!