International Science Index


10007536

Generalization of Clustering Coefficient on Lattice Networks Applied to Criminal Networks

Abstract:A lattice network is a special type of network in which all nodes have the same number of links, and its boundary conditions are periodic. The most basic lattice network is the ring, a one-dimensional network with periodic border conditions. In contrast, the Cartesian product of d rings forms a d-dimensional lattice network. An analytical expression currently exists for the clustering coefficient in this type of network, but the theoretical value is valid only up to certain connectivity value; in other words, the analytical expression is incomplete. Here we obtain analytically the clustering coefficient expression in d-dimensional lattice networks for any link density. Our analytical results show that the clustering coefficient for a lattice network with density of links that tend to 1, leads to the value of the clustering coefficient of a fully connected network. We developed a model on criminology in which the generalized clustering coefficient expression is applied. The model states that delinquents learn the know-how of crime business by sharing knowledge, directly or indirectly, with their friends of the gang. This generalization shed light on the network properties, which is important to develop new models in different fields where network structure plays an important role in the system dynamic, such as criminology, evolutionary game theory, econophysics, among others.
References:
[1] W. Li, A. Bashan, S. V. Buldyrev, H. E. Stanley, and S. Havlin, “Cascading failures in interdependent lattice networks: The critical role of the length of dependency links,” Phys. Rev. Lett., vol. 108, p. 228702, May 2012. (Online). Available: http://link.aps.org/doi/10.1103/PhysRevLett.108.228702.
[2] M. N. Kuperman and S. Risau-Gusman, “Relationship between clustering coefficient and the success of cooperation in networks,” Phys. Rev. E, vol. 86, p. 016104, Jul 2012. (Online). Available: http://link.aps.org/doi/10.1103/PhysRevE.86.016104.
[3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, “Complex networks: Structure and dynamics,” Physics Reports, vol. 424, no. 4-5, pp. 175 – 308, 2006. (Online). Available: http://www.sciencedirect.com/science/article/pii/S037015730500462X.
[4] R. Albert and A.-L. Barab´asi, “Statistical mechanics of complex networks,” Rev. Mod. Phys., vol. 74, pp. 47–97, Jan 2002. (Online). Available: http://link.aps.org/doi/10.1103/RevModPhys.74.47.
[5] F. Vega-Redondo, Complex Social Networks. Cambridge University Press, 2007.
[6] D. J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness. Princenton University Press, 1999.
[7] C. Gros, Complex and Adaptive Dynamical Systems: A Primer. Springer-Verlag, 2008.
[8] A. Calv´o-Armengoi and Y. Zenou, “Social networks and crime decisions: The role of social structure in facilitating delinquent behavior,” International Economic Review, vol. 45, no. 3, pp. 939–958, 2004. (Online). Available: http://www.jstor.org/stable/3663642.
[9] M. Carlie, “Into the abyss: A personal journey into the world of street gangs.” (Online). Available: http://people.missouristate.edu/MichaelCarlie/site map.htm.
[10] C. Moukarzel, S. Gonc¸alves, J. Iglesias, M. Rodr´ıguez-Achach, and R. Huerta-Quintanilla, “Wealth condensation in a multiplicative random asset exchange model,” Eur. Phys. J. Special Topics, vol. 143, pp. 75–79, 2007. (Online). Available: http://dx.doi.org/10.1140/epjst/e2007-00073-3.
[11] C. H. S. Monta˜na, R. Huerta-Quintanilla, and M. Rodr´ıguez-Achach, “Class formation in a social network with asset exchange,” Physica A Statistical Mechanics and its Applications, vol. 390, pp. 320–340, 2011. (Online). Available: http://www.sciencedirect.com/science/article/B6TVG-517J27V-4/2/3181 04b3a1d77c8a308251bdb8c5a1e6.
[12] G. Szab´o and G. F´ath, “Evolutionary games on graphs,” Physics Reports, vol. 446, no. 4?6, pp. 97 – 216, 2007. (Online). Available: http://www.sciencedirect.com/science/article/pii/S0370157307001810.
[13] C. P. Roca, J. A. Cuesta, and A. S´anchez, “Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics,” Physics of Life Reviews, vol. 6, no. 4, pp. 208 – 249, 2009. (Online). Available: http://www.sciencedirect.com/science/article/pii/S1571064509000256.
[14] L. G. Moyano and A. S´anchez, “Evolving learning rules and emergence of cooperation in spatial prisoner’s dilemma,” Journal of Theoretical Biology, vol. 259, no. 1, pp. 84 – 95, 2009. (Online). Available: http://www.sciencedirect.com/science/article/pii/S0022519309000988.