Unified centrality measure of complex networks
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Soon-Hyung Yook, Sungmin Lee, Yup KimKyung Hee University
NSPCS 08
Unified centrality measure of complex networks
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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
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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)
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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)]
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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
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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
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Various centrality and degree– closeness centrality
[Kurdia et al. Engineering in Medicine and Biology Workshop, 2007]
PINNodes having high degree
High closeness
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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
Centrality of each node Related to degree of each node
Dynamical property(random walks) Related to degree of each node
Any relationship between them?
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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 degrees k and k’
• NT: the basic measure of BRWC• Note that both RWC and SPC depend on k
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• 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
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• 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
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Relationship between BRWC and SPBC for vertices
n=1.0
n=2.0n=5/3
b=0.7
b=1.0b=1.3
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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
3.04.3 0.66
0.77
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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 a+1=n and b=n/h=a/h
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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
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Thank you for your attention!!
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• 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