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Searching for Network Modules

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Advances in Information and Communication (FICC 2019)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 70))

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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.

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Notes

  1. 1.

    \(\mathcal B_k=\sum _{1\le l\le k}\mathcal S_{k,l}\) is the (Bell) number of partitions of a k-set, while \(\mathcal S_{k,l}\) is the Stirling number of the second kind, i.e. the number of partitions of a k-set into l blocks [3, 16, 33].

  2. 2.

    Modularity \(\mathcal Q\) is meant to evaluate modular structures in complex networks, while the modular elements of \((\mathcal P^N,\wedge ,\vee )\) are those partitions \(\hat{P}\) realizing, for all \(Q\in \mathcal P^N\), equality \(r(\hat{P}\wedge Q)+r(\hat{P}\vee Q)=r(\hat{P})+r(Q)\), where \(r(P)=n-|P|\) is the rank (see [3] on modular lattices/lattice functions).

  3. 3.

    The terminology and notation used here are standard in graph theory [11].

  4. 4.

    As usual colon ‘:’ stands for ‘such that’.

  5. 5.

    \(\varOmega (\mathcal F)\) is a generaliziation of the field \(2^P\) of subsets generated by partitions P, where \(2^{P_{\bot }}=2^N\), while \(2^{P^{\top }}=\{\emptyset ,N\}\).

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering DISI, University of Bologna, 40126, Bologna, Italy

    Giovanni Rossi

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  1. Giovanni Rossi
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Correspondence to Giovanni Rossi .

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Editors and Affiliations

  1. Faculty of Science and Engineering, Saga University, Saga, Japan

    Kohei Arai

  2. The Science and Information (SAI) Organization, Bradford, UK

    Rahul Bhatia

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Rossi, G. (2020). Searching for Network Modules. In: Arai, K., Bhatia, R. (eds) Advances in Information and Communication. FICC 2019. Lecture Notes in Networks and Systems, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-030-12385-7_42

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