Abstract
Methods that generate networks sharing a given degree distribution and global clustering can induce changes in structural properties other than that controlled for. Diversity in structural properties, in turn, can affect the outcomes of dynamical processes operating on those networks. Since exhaustive sampling is not possible, we propose a novel evolutionary framework for mapping this structural diversity. The three main features of this framework are: (a) subgraph-based encoding of networks, (b) exact mutations based on solving systems of Diophantine equations, and (c) heuristic diversity-driven mechanism to drive resolution changes in the MapElite algorithm. We show that our framework can elicit networks with diversity in their higher-order structure and that this diversity affects the behaviour of the complex contagion model. Through a comparison with state of the art clustered network generation methods, we demonstrate that our approach can uncover a comparably diverse range of networks without needing computationally unfeasible mixing times. Further, we suggest that the subgraph-based encoding provides greater confidence in the diversity of higher-order network structure for low numbers of samples and is the basis for explaining our results with complex contagion model. We believe that this framework could be applied to other complex landscapes that cannot be practically mapped via exhaustive sampling.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anand, K., Bianconi, G.: Entropy measures for networks: toward an information theory of complex topologies. Phys. Rev. E 80(4), 045,102 (2009)
Betzel, R.F., Bassett, D.S.: Generative models for network neuroscience: prospects and promise. J. R. Soc. Interface 14, 20170,623 (2017)
Courtney, O.T., Bianconi, G.: Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E 93(6), 062,311 (2016)
Cully, A., Demiris, Y.: Quality and diversity optimization: a unifying modular framework. IEEE Trans. Evol. Comput. 22(2), 245–259 (2018)
Del Genio, C.I., Kim, H., Toroczkai, Z., Bassler, K.E.: Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PLoS ONE 5(4), 1–7 (2010)
Eames, K.T.: Modelling disease spread through random and regular contacts in clustered populations. Theor. Popul. Biol. 73(1), 104–111 (2008)
Eiben, Á.E., Hinterding, R., Michalewicz, Z.: Parameter control in evolutionary algorithms. IEEE Trans. Evol. Comput. 3(2), 124–141 (1999)
Erdos, P., Gallai, T.: Gráfok eloırt fokszámú pontokkal. Matematikai Lapok 11, 264–274 (1960)
Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 35–41 (1977)
Giagkiozis, I., Purshouse, R.C., Fleming, P.J.: An overview of population-based algorithms for multi-objective optimisation. Int. J. Syst. Sci. 46(9), 1572–1599 (2015)
Green, D.M., Kiss, I.Z.: Large-scale properties of clustered networks: implications for disease dynamics. J. Biol. Dyn. 4(5), 431–445 (2010)
Havas, G., Majewski, B.S., Matthews, K.R.: Extended GCD and hermite normal form algorithms via lattice basis reduction. Exp. Math. 7(2), 125–136 (1998)
House, T., Davies, G., Danon, L., Keeling, M.J.: A motif-based approach to network epidemics. Bull. Math. Biol. 71(7), 1693–1706 (2009)
House, T., Keeling, M.J.: The impact of contact tracing in clustered populations. PLoS Comput. Biol. 6(3), e1000,721 (2010)
Karrer, B., Newman, M.E.: Random graphs containing arbitrary distributions of subgraphs. Phys. Rev. E 82(6), 066,118 (2010)
Keeling, M.J., et al.: Networks and the epidemiology of infectious disease. Interdiscip. Perspect. Infect. Dis. (2011)
Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks. Springer (2017)
Liu, H.L., Chen, L., Deb, K., Goodman, E.D.: Investigating the effect of imbalance between convergence and diversity in evolutionary multiobjective algorithms. IEEE Trans. Evol. Comput. 21(3), 408–425 (2017)
Lovász, L.: Large networks and graph limits. Am. Math. Soc. 60 (2012)
Miller, J.C.: Complex contagions and hybrid phase transitions. J. Complex Netw. 4(2), 201–223 (2015)
Mouret, J.B., Clune, J.: Illuminating search spaces by mapping elites. arXiv:1504.04909 v1 (2015)
Newman, M.E.: Spread of epidemic disease on networks. Phys. Rev. E 66(1), 016,128 (2002)
Nordmoen, J., Samuelsen, E., Ellefsen, K.O., Glette, K.: Dynamic mutation in map-elites for robotic repertoire generation. In: Artificial Life Conference Proceedings, pp. 598–605. MIT Press (2018)
Orsini, C., et al.: Quantifying randomness in real networks. Nat. Commun. 6, 8627 (2015)
Overbury, P., Kiss, I.Z., Berthouze, L.: A genetic algorithm-based approach to mapping the diversity of networks sharing a given degree distribution and global clustering. In: Complex Networks & Their Applications V, vol. 693, pp. 223–233 (2016)
Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925 (2015)
Pugh, J.K., Soros, L.B., Stanley, K.O.: Quality diversity: a new frontier for evolutionary computation. Front Robot. AI 3, 40 (2016)
Read, J.M., Keeling, M.J.: Disease evolution on networks: the role of contact structure. Proc. R. Soc. B 270(1516), 699–708 (2003)
Ritchie, M., Berthouze, L., House, T., Kiss, I.Z.: Higher-order structure and epidemic dynamics in clustered networks. J. Theor. Biol. 348, 21–32 (2014)
Ritchie, M., Berthouze, L., Kiss, I.Z.: Generation and analysis of networks with a prescribed degree sequence and subgraph family: higher-order structure matters. J. Complex Netw. 5(1), 1–31 (2017)
Robins, G., Pattison, P., Kalish, Y., Lusher, D.: An introduction to exponential random graph (p*) models for social networks. Soc. Netw. 29(2), 173–191 (2007)
Vassiliades, V., Chatzilygeroudis, K., Mouret, J.B.: A comparison of illumination algorithms in unbounded spaces. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, pp. 1578–1581. ACM (2017)
Vassiliades, V., Chatzilygeroudis, K., Mouret, J.B.: Using centroidal Voronoi tessellations to scale up the multi-dimensional archive of phenotypic elites algorithm. IEEE Trans. Evol. Comput. (2017)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440 (1998)
Zhou, A., Qu, B.Y., Li, H., Zhao, S.Z., Suganthan, P.N., Zhang, Q.: Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol. Comput. 1(1), 32–49 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Overbury, P., Kiss, I.Z., Berthouze, L. (2019). Mapping Structural Diversity in Networks Sharing a Given Degree Distribution and Global Clustering: Adaptive Resolution Grid Search Evolution with Diophantine Equation-Based Mutations. In: Aiello, L., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L. (eds) Complex Networks and Their Applications VII. COMPLEX NETWORKS 2018. Studies in Computational Intelligence, vol 812. Springer, Cham. https://doi.org/10.1007/978-3-030-05411-3_57
Download citation
DOI: https://doi.org/10.1007/978-3-030-05411-3_57
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05410-6
Online ISBN: 978-3-030-05411-3
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)