NSPCS 08 Unified centrality measure of complex networks

NSPCS 08 Unified centrality measure of complex networks: a dynamical approach to a topological property Soon-Hyung Yook, Sungmin Lee, Yup Kim Kyung Hee University

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-g • 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 ü The simplest one: degree (degree centrality), ki ü Node and edge importance based on adjacency matrix eigenvalue [Restrepo, Ott, Hund PRL 97, 094102] ü Closeness centrality: ü 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 ü Random walk centrality (RWC) ü 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 PIN email AS [Restrepo, Ott, Hund PRL 97, 094102]

Various centrality and degree– closeness centrality PIN Nodes having high degree High closeness [Kurdia et al. Engineering in Medicine and Biology Workshop, 2007]

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 Any relationship between them? Dynamical property (random walks) Related to degree of each node Ø If yes, then is it possible to use a certain dynamical property in the investigation of topological properties, especially important component? Ø 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

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

Relationship between BRWC and SPBC for vertices • SPBC; bv(k) thus, But in the numerical simulations, we find that this relation holds for g>3

Relationship between BRWC and SPBC for vertices b=1. 3 b=1. 0 n=2. 0 n=5/3 n=1. 0 b=0. 7

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 0. 77 4. 3 3. 0 0. 66

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 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 a=0: simple RWBC is recovered • If a ; hubs have large BC • If a - ; dangling ends have large BC

Thank you !!