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Searching for Network Modules

  • Giovanni RossiEmail author
Conference paper
First Online:
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 70)

Abstract

When analyzing complex networks, a key target is to uncover their modular structure, which means searching for a family of node subsets spanning each an exceptionally dense subnetwork. Objective function-based graph clustering procedures such as modularity maximization output a partition of nodes, i.e. a family of pair-wise disjoint subsets, although single nodes are likely to be included in multiple or overlapping modules. Thus in fuzzy clustering approaches each node may be included in different modules with different [0, 1]-ranged memberships. This work proposes a novel type of objective function for graph clustering, in the form of a multilinear polynomial extension whose coefficients are determined by network topology. It may be seen as a potential, taking values on fuzzy clusterings or families of fuzzy subsets of nodes over which every node distributes a unit membership. If suitably parameterized, this potential attains its maximum when every node concentrates its all unit membership on some module. Maximizers thus remain partitions, while the original discrete optimization problem is turned into a continuous version allowing to conceive alternative search strategies. The instance of the problem being a pseudo-Boolean function assigning real-valued cluster scores to node subsets, modularity maximization is employed to exemplify a so-called quadratic form, in that the scores of singletons and pairs also fully determine the scores of larger clusters, while the resulting multilinear polynomial potential function has degree 2. After considering further quadratic instances, different from modularity and obtained by interpreting network topology in alternative manners, a greedy local-search strategy for the continuous framework is analytically compared with an existing greedy agglomerative procedure for the discrete case. Overlapping is finally discussed in terms of multiple runs, i.e. several local searches with different initializations.

Keywords

Modularity Fuzzy clustering Pseudo-Boolean function 
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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering DISIUniversity of BolognaBolognaItaly

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