# In the following we will only consider undirected networks.

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1 Roles in Networks

2 Roles in Networks Motivation for work: Let topology define network roles. Work by Kleinberg on directed graphs, used topology to define two types of roles: authorities and hubs. (Each node gets a score for each role) In the following we will only consider undirected networks. In this case authority and hub scores become the same: eigenvector centrality (EVC) scores Our goals: Use topology (via EVC) to define natural clusterings in a graph, and boundaries between the clusters Roles will include heads of clusters, other cluster members, and boundary nodes and links Telenor R&D 2

3 Our starting point: Eigenvector centrality (EVC) How central is a node? It s not just how many nodes you know (number of neighbors), but also how central they are, that determines how central you are: e i j = nn e ( i ) j or e i = λ 1 j A ij e j e i EVC is thus defined as the i th component of the principal eigenvector of the graph adjacency matrix A. Telenor R&D 3

4 A property of EVC Eigenvector centrality is smooth over the graph. Hence, the eigenvector centrality can be viewed as a (smoothly varying) height measure on each network node. Topographic view: Telenor R&D 4

5 Network role: Centers A node is defined as a Center when it is a local maximum in the eigenvector centrality. C Telenor R&D 5

6 Network role: Member of a Region Idea - Consider the topographic view of EVC: Need an unambiguous rule to place nodes in Regions, which are associated with Centers. Example rules: Distance from Centers: closest Center determines your Region. Topography: follow steepest-ascent path to find your Center. Telenor R&D 6

7 Border between regions Topographic picture: The Border consists of nodes for which the unambiguous rule gives more than one answer. Example rules: Distance: Border set is defined as nodes at equal distance from two or more centers. (So far we have only tried this one.) Topography: Border nodes have more than one steepest path, leading to more than one Center (a rare event) NB: in either case, border links can be defined: they connect two regions Telenor R&D 7

8 Other network roles: Bridges and danglers Using the distance rule to define the border we often get the following: Based on this we can define two additional network roles: Bridge = Border nodes lying on at least one non-self-retracing path between Centers. Danglers = Border nodes Bridges. These definitions also work for the topographic rule, but these roles are then rare With either rule, parallel bridges can occur. Telenor R&D 8

9 Example network C D B D C Telenor R&D 9

10 Gnutella visualization using these ideas: Telenor R&D 10

11 Some thoughts on centrality and well-connectedness Conjecture: a well-connected network will have relatively few Centers Conjecture: in a well-connected network, there will be many bridges (links and nodes) connecting the few Centers To test these conjectures, we need an independent definition of well-connectedness Power-law graphs are well-connected, in the sense that they are robust (do not fragment) against the loss of randomly chosen nodes Our Gnutella example is a power-law graph, and conforms to our two conjectures above Room for more work here! Telenor R&D 11

12 Thoughts, in pictures Telenor R&D 12

13 Summary and future directions We have a natural way for decomposing an undirected graph into separate clusters (Regions) We have found four distinct network roles in such graphs: Centers, Region members, Bridges, Danglers We can also define edges of Regions (nodes touching bridge links, and nodes linked to bridge nodes) We have ideas for generalizing this approach to directed graphs What s the use? Helps to understand the structure of a network Eg, gives us measures of well-connectedness robustness Serves as a guide for visualizing the network in 2D And there must be even more. Telenor R&D 13

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