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- Title
MODULARITY BASED COMMUNITY DETECTION IN HETEROGENEOUS NETWORKS.
- Authors
Jingfei Zhang; Yuguo Chen
- Abstract
Consider simple bipartite graphs with m edges and degree sequence d = (d1;:::;dn 1) for one type of nodes, referred to as type-[1] nodes, and d 0 = (d 0 1;:::;d 0 n 2) for the other type of nodes, referred to type-[2] nodes. We then calculate the normalized mutual information (NMI) between 10 J. ZHANG AND Y. CHEN 0.04 0.06 0.08 0.10 0.12 0.0 0.2 0.4 0.6 0.8 1.0 Type 1 nodes r3 normalized mutual information 0.04 0.06 0.08 0.10 0.12 0.0 0.2 0.4 0.6 0.8 1.0 Type 2 nodes r3 normalized mutual information Figure 1: Average NMI between the true community membership and the community membership obtained from the proposed method (dashed line) and the homogenous model withK = 4 (solid line). the obtained community detection results and the true community membership. Statistica Sinica: Supplement 1 MODULARITY BASED COMMUNITY DETECTION IN HETEROGENEOUS NETWORKS Jingfei Zhang and Yuguo Chen University of Miami and University of Illinois at Urbana-Champaign Supplementary Material S1 Proof of Theorem 1 First, we state a theorem from McKay (2010) on the asymptotic number of simple graphs with forbidden edges. 1, E(A [12] ij) = d [12] i d [21] j m [12] (1 +o(1)): First, it is easy to derive that E(A [12] ij) = j d [12];d [21] jA [12] ij =1 j j d [12];d [21]j; wherej d [12];d [21] jA [12] ij =1 j is the total number of bipartite graphs with degree sequences d [12] for type-[1] nodes, d [21] for type-[2] nodes and a link between the ith node of type-[1] and the jth node of type-[2].
- Subjects
BERNOULLI equation; COMMUNITIES; BIPARTITE graphs
- Publication
Statistica Sinica, 2020, p1
- ISSN
1017-0405
- Publication type
Article
- DOI
10.5705/ss.202017.0399