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Hierarchical modeling of graphs using modular decomposition

Miguel Méndez1, Carenne Ludeña2* and Nicolás Bolívar1, 3
  • 1 Universidad Antonio Nariño, Facultad de Ciencias, Colombia
  • 2 Universidad de Bogotá Jorge Tadeo Lozano, Facultad de Ciencias Naturales e Ingeniería, Colombia
  • 3 Universidad Antonio Nariño, Centro de Investigación en Ciencias Básicas y Aplicadas, Colombia

For a variety of discrete structures such as graphs, boolean functions or set systems it is possible under certain conditions, to define a decomposition based on substitution of structures in a nested schema: the complete structure can be decomposed as a simpler outer structure each of whose components represent in turn internal structures (see Möhring, 1985; Möhring & Rademacher, 1984. Also, for a thorough review in the context of Operads, see Méndez, 2015, and the references therein). In the case of undirected graphs G = (V, E), such a decomposition yields a hierarchical structure which can be thought of as a tree. The internal vertex are subsets of V, starting with the root V, satisfying the property of being modules and having an associated type-tag. We will refer to it as the Modular decomposition (MD) of a graph. A module M ⊂ V satisfies for each v ∈ V − M that either v is connected to all b ∈ M or is not connected to any. Children of any given module M will be in turn be modules of M. Type-tags indicate the type of relations among these modules and can be of three types: complete, empty and prime. Complete relations indicate all children are related among them, empty relations indicate no children are related and prime relations occur in any other case. Classical well known results (see Méndez, 2015 for a review), establish that the MD of an undirected, simple graph is unique, so that the hierarchical structure is well defined. In a sense, MD can be thought of as a generalization of connected components of a graph, including the possibility of modeling other homogeneous relationships such as bipartite structures. This is interesting because much of classical graph models study mostly modular type behavior, defined by dense relationships among vertices within modules and sparce relations among modules, but this leaves out many interesting functional-like relationships such as affiliation networks. The MD of a graph is especially appealing because in many applications, ranging from social networks to protein-protein (Gagneur, Krause, Bouwmeester & Casari, 2004), genomic, proteonomic or brain networks (Meunier, Lambiotte, & Bullmore, 2010; Bassett & Sporns, 2017), there is a natural hierarchical structure induced by functional, spatial or other types of relationships that is not readily obtained by most popular graph models such as the classical Erdös-Rényi (ER) random graph models (Erdos & Renyi, 1959; Estrada & Knight, 2015), power degree distribution models (Barabasi & Albert, 1999; Bollobás & Riordan, 2003; Estrada & Knight, 2015), Exponential Random Graph Models (ERGMs) (Holland & Leinhardt, 1981), Hierarchical ERGMs (Schweinberger & Handcock, 2015) or stochastic Block models (Holland, Laskey & Leinhardt, 1983), although the latter is based on a similar module-like decomposition principle. Additionally, the MD can identify logical relations among the members of the modules, e.g. exclusive alternatives for the rest of the network (Gagneur, Krause, Bouwmeester & Casari, 2004), as well as interesting structures within the network, especially for brain networks, such as cliques (Sizemore et al, 2018), which are complete (serial) components within a module. Finally, the MD is known to be achievable in linear-time (Habib, de Montgolfier & Paul, 2004), which is desirable for large-scale networks, such as those from the brain. In this work we propose a novel hierarchical graph model based on MDs. Our approach simulates simple outer structures at each level of the tree-like decomposition, selecting with a given probability the type-tag at each step. Vertices are then associated to each module using a preferential attachment procedure. We present simulations of the proposed procedure and characteristics of the MD of biological networks.

Acknowledgements

This work was partially supported by Banco de la República, Colombia, Project 3.991: Aplicación de la descomposición modular de grafos para el análisis de grafos multiniveles.

References

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Keywords: random graph models, modular decomposition of graphs, biological networks, complex systems, brain networks

Conference: 2nd International Neuroergonomics Conference, Philadelphia, PA, United States, 27 Jun - 29 Jun, 2018.

Presentation Type: Oral Presentation

Topic: Neuroergonomics

Citation: Méndez M, Ludeña C and Bolívar N (2019). Hierarchical modeling of graphs using modular decomposition. Conference Abstract: 2nd International Neuroergonomics Conference. doi: 10.3389/conf.fnhum.2018.227.00006

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Received: 02 Apr 2018; Published Online: 27 Sep 2019.

* Correspondence: Dr. Carenne Ludeña, Universidad de Bogotá Jorge Tadeo Lozano, Facultad de Ciencias Naturales e Ingeniería, Bogotá, Colombia, carennec.ludenac@utadeo.edu.co