1. Soon-Hyung Yook, Sungmin Lee, Yup Kim Kyung Hee University NSPCS 08 Unified centrality measure of complex networks

2. 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

3. 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 uncover the topological properties of underlying networks Using the finite-size scaling of One can estimate the scaling behavior of diameter Lee, SHY, Kim Physica A 387, 3033 (2008)

4. Underlying topology & dynamics Diffusive capture process – 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)]

5. Centrality Centrality: importance of a vertex and an edge Shortest path betweenness centrality (SPBC) b i : 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), k i 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

6. Various centrality and degree– node importance Node (or vertex) importance: – defined by eigenvalue of adjacency matrix [Restrepo, Ott, Hund PRL 97, 094102] PINemail AS

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

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

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

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

11. Motivation Centrality of each nodeRelated to degree of each nodeDynamical property (random walks) Related to degree of each node Any relationship between them?

12. Biased Random Walk Centrality (BRWC) Generalize the RWC by biased random walker Count the number of traverse, N T, of vertices having degree k or edges connecting two vertices of degrees k and k’ N T : the basic measure of BRWC Note that both RWC and SPC depend on k

13. 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 -  N T (k) also scales as Average number of traverse a vertex having degree k N v (k): number of vertices having degree k The probability to find a walker at nodes of degree k Thus

14. SPBC; b v (k) Relationship between BRWC and SPBC for vertices thus, But in the numerical simulations, we find that this relation holds for  >3

15. Relationship between BRWC and SPBC for vertices =1.0 =2.0 =5/3  =0.7  =1.0  =1.3

16. Relationship between BRWC and SPBC for vertices

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

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

19. Relationship between BRWC and SPBC for edges

21. Protein-Protein Interaction Network Slight deviation of  +1= and  = / 

22. 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 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  =0: simple RWBC is recovered If    ; hubs have large BC If   -  ; dangling ends have large BC

24. Kwon et al. PRE 77, 066105 (2008) Relationship between BRWC and SPBC for vertices Mapping to the weight network with weight Therefore, N T (k) also scales as Average number of traverse a vertex having degree k N v (k): number of vertices having degree k