Overlapping Communities, Edge Partitions & Line Graphs
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1 Warwick 22 nd September 2009 ECCS09 Overlapping Communities, Edge Partitions & Line Graphs Tim Evans, IMS and Theoretical Physics, Imperial College London Page 1
2 Our Node Centric Viewpoint Line Graphs Application to Community Detection Conclusions Page 2
3 Node Centric Viewpoint A network is 1. a set of nodes AND 2. a set of vertices We tend to have a very NODE centred viewpoint Page 3
4 Node Centric - Scale Free Networks Power law degree distributions of NODES Scale Free Random walk on NODES produces scale free graphs [TSE, Saramäki 2004] Exponential Page 4
5 Node Centric Node Partitions Community detection via partition of NODE set Page 5 Node partition of Karate club graph with optimal modularity [Agarwal & Kempe 2007]
6 Edge Centric Viewpoint? A graph is both a set of nodes AND a set of edges However we naturally focus on nodes not edges Can we find a trick to help us shift our viewpoint from nodes to edges? Page 6
7 Our Node Centric Viewpoint Line Graphs Application to Community Detection Conclusions Page 7
8 Nodes of a Line Graph 1. For every edge a in original graph G create a node a in the line graph L(G) a a g b g b G L(G) Page 8
9 Edges of a Line Graph 2. Connect the nodes a and b in the line graph L(G) if the corresponding edges in original graph G were coincident a a i i i g b g i b Page 9 G L(G)
10 Properties of a Line Graph Not usually a duality transformation L(L(G)) G (Almost) always reversible [Whitney 1932] L(G) G G L(G) Page 10
11 The Problem with a Standard Line Graph High degree nodes in original graph G over represented by edges in Line Graph L(G). Page 11 G Degree k node L(G) k(k-1)/2 edges
12 Solution Weighted Line Graphs [Evans & Lambiotte 2009] Original graph node of degree k produces line graph edges of weight 1/(k-1) Strength 1 G Degree k node Weight 1/(k-1) WL(G) k(k-1)/2 edges, Total weight k Page 12
13 Our Node Centric Viewpoint Line Graphs Application to Community Detection Conclusions Page 13
14 Community Detection - Node Centric version Partition nodes into communities Perform random walk on nodes Compare number of random walkers which stay within community after one step against number which remain within communities after infinite number of steps = Optimisation of Modularity [Girvan & Newman 2002; Lambiotte, Delvenne & Barahona 2008] Page 14
15 Community Detection - Edge Centric version Partition edges into communities Perform random walk on edges = Random walk on line graph vertices Compare number of random walkers which stay within community after one step against number which remain within communities after infinite number of steps = Optimisation of Modularity of Weighted Line Graph [Evans & Lambiotte 2009] Page 15
16 Karate Club Edge partition means individuals nodes can be members of more than one community Page 16 Ford-Fulkerson partition of Zachary
17 Karate Club Analysis Nodes in One Edge Community # k Fraction k In Green C Mr Hi (the Instructor) bridges several groups % % % % % 0 (Mr_Hi) 16 25% Page 17
18 Karate Club Edge Partition Nodes can be members of many communities An overlapping community structure for nodes Name Community Total k k in C 0 Mr Hi John A Page 18
19 South Florida Word Association Data Only showing nodes which have 90% of edges in one edge community except for BRIGHT
20 Edge Partition of Word in Paper Titles Some words have all edges in one partition they define these communities e.g. cassini Other words have edges in several communities stop words e.g. signature Page 20 Stem Total k k in C interplanetari cassini heliospher magnetopaus spacecraft signatur solitari radar 21 7 mhd 18 6
21 Our Node Centric Viewpoint Line Graphs Application to Community Detection Conclusions Page 21
22 Conclusions Line graphs move focus from nodes to edges with minimal effort Weighted line graphs avoid problem of over representation of high degree nodes Community detection on line graph produces overlapping node communities for original graph Evans & Lambiotte, Phys.Rev.E 80 (2009) Page 22
23 Additional Material
24 Zachary Karate Club Node Partition A Line Graph C Weighted Line graph D Weighted Line Graph E1
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