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

1. 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

2. 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

3. 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?

4. 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)

5. 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)]

6. Centrality • The simplest one: degree (degree centrality), ki • Centrality: importance of a vertex and an edge • Node and edge importance based on adjacency matrix eigenvalue • [Restrepo, Ott, Hund PRL 97, 094102] • Closeness centrality: • 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 • Random walk centrality (RWC) • Essential or lethal proteins in protein-protein interaction networks

7. 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]

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

9. Various centrality and degree– lithality • [Jeong et al. Nature 411, 41 (2007)]

10. Shortest Path Betweenness Centrality (SPBC) for a vertex • SPBC distribution: [Goh et al. PRL 87, 278701 (2001)]

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

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

13. 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 • 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?

14. 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

15. Relationship between BRWC and SPBC for vertices The probability to find a walker at one of the nodes of degree k • In stationary state 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

16. 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

17. Relationship between BRWC and SPBC for vertices b=1.3 b=1.0 n=5/3 n=2.0 b=0.7 n=1.0

18. Relationship between BRWC and SPBC for vertices

19. Relationship between BRWC and SPBC for edges • for uncorrelated network number of edges connecting nodes of degree k and k’ thus • By assuming that

20. Relationship between BRWC and SPBC for edges 0.77 3.0 0.66 4.3

21. Relationship between BRWC and SPBC for edges

22. Relationship between BRWC and SPBC for edges

23. Protein-Protein Interaction Network Slight deviation of a+1=n and b=n/h=a/h

24. 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

25. Thank you !!