Event Abstract Back to Event 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 Barabasi, A.-L. & Albert, R. (1999). Emergence of Scaling in Random Networks. Science, 286, 509-512. doi: 10.1126/science.286.5439.509. Bassett, D. S. & Sporns, O. (2017). Network neuroscience. Nature Neuroscience, 20(3), 353-364. doi: 10.1038/nn.4502. Bollobás, B. & Riordan, O. M. (2003). Mathematical results on scale‐free random graphs. In S. Bornholdt & H. G. Schuster (Eds), Handbook of Graphs and Networks (pp. 1-32). doi:10.1002/3527602755.ch1. Erdos, P. & Renyi, A. (1959). On Random Graphs I. Publ. Math. Debrecen, 6, 290. Estrada, E., & Knight, P. A. (2015). A First Course in Network Theory. Oxford. Gagneur, J., Krause, R., Bouwmeester, T., & Casari, G. (2004). Modular decomposition of protein-protein interaction networks. Genome Biology, 5(8), R57. doi: 10.1186/gb-2004-5-8-r57. Habib, M., De Montgolfier, F. & Paul, C. (2004). A Simple Linear-Time Modular Decomposition Algorithm. Algorithm Theory - SWAT 2004, 187-198. doi: 10.1007/b98413. Holland, P. & Leinhardt, S. (1981). An Exponential Family of Probability Distributions for Directed Graphs. Journal of the American Statistical Association, 76(373), 33-50. doi: 10.2307/2287037. Holland, P., Laskey, K. & Leinhardt, S. (1983). Stochastic Blockmodels: First Steps. Social Networks, 5, 109-137. doi: 10.1016/0378-8733(83)90021-7. Méndez, M. (2015). Set Operads in Combinatorics and Computer Science. Springer Briefs in Mathematics. Springer. Meunier, D., Lambiotte, R. & Bullmore, E. (2010). Modular and hierarchically modular organization of brain networks. Frontiers in Neuroscience. doi: 10.3389/fnins.2010.00200. Möhring, R.H. & Rademacher, F.J. (1984). Substitution decomposition for discrete structures and connections with combinatorial optimization. Annals of Discrete Mathematics, 19, 257-356. doi: 10.1016/S0304-0208(08)72966-9. Möhring, R.H. (1985). Algorithmic aspects of the substitution decomposition in optimization over relations, set systems and boolean functions. Annals of Operations Research, 4(1), 195 -225. doi: 10.1007/BF02022041. Schweinberger, M. & Handcock, M.S. (2015). Local dependence in random graphs: characterization, properties, and statistical inference. Journal of the Royal Statistical Society: Series B, 77, 647-676. doi: 10.1111/rssb.12081. Sizemore, A.E., Giusti, C., Kuhn, A., Vettel, J.M., Betzel, R.F. & Bassett, D.S. (2018). Cliques and cavities in the human connectome. Journal of Computational Neuroscience, 44(1), 115-145. doi: 10.1007/s10827-017-0672-6. 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 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. 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 Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Miguel Méndez Carenne Ludeña Nicolás Bolívar Google Miguel Méndez Carenne Ludeña Nicolás Bolívar Google Scholar Miguel Méndez Carenne Ludeña Nicolás Bolívar PubMed Miguel Méndez Carenne Ludeña Nicolás Bolívar Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Full Text

Published Version
Open DOI Link

Get access to 115M+ research papers

Discover from 40M+ Open access, 2M+ Pre-prints, 9.5M Topics and 32K+ Journals.

Sign Up Now! It's FREE

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call