Use of Enumerative Combinatorics for Proving the Applicability of an Asymptotic Stability Result on Discrete-Time SIS Epidemics in Complex Networks

Abstract

In this paper, we justify by the use of Enumerative Combinatorics, the applicability of an asymptotic stability result on Discrete-Time Epidemics in Complex Networks, where the complex dynamics of an epidemic model to identify the nodes that contribute the most to the propagation process are analyzed, and, because of that, are good candidates to be controlled in the network in order to stabilize the network to reach the extinction state. The epidemic model analyzed was proposed and published in 2011 by of Gómez et al. The asymptotic stability result obtained in the present article imply that it is not necessary to control all nodes, but only a minimal set of nodes if the topology of the network is not regular. This result could be important in the spirit of considering policies of isolation or quarantine of those nodes to be controlled. Simulation results using a refined version of the asymptotic stability result were presented in another paper of the second author for large free-scale and regular networks that corroborate the theoretical findings. In the present article, we justify the applicability of the controllability result obtained in the mentioned paper in almost all the cases by means of the use of Combinatorics.

Combinatorial Enumeration of Graphs

“…In [30] I used the combinatorial enumeration methods and probability for showing the applicability of the selection node criteria in a virus spreading control problem in complex networks. The main purpose of this section is to illustrate how the enumerative combinatorics in combination with probability theory can be used for demonstrating mathematically a result in an application field.…”

“…We can mention that the case of homogeneity in the behavior of the nodes and their interaction cannot be discarded given what has been observed in the reaction of the agents in the context of social networks is that they try to minimize the conflict. Many successful models can, for example [30][31][32][33], base their predicting effectiveness on the homogeneity of the behavior of the nodes and their interaction. By the other side, in [30] we have obtained a criteria for selecting the nodes to be controlled, but such criteria fail if we have homogeneity in the behavior of the nodes and, at the same time, the topology of the network is regular.…”

“…Many successful models can, for example [30][31][32][33], base their predicting effectiveness on the homogeneity of the behavior of the nodes and their interaction. By the other side, in [30] we have obtained a criteria for selecting the nodes to be controlled, but such criteria fail if we have homogeneity in the behavior of the nodes and, at the same time, the topology of the network is regular. Then, what we want to do here is to justify the applicability of node selection criteria, keeping the homogeneity of the nodes and trying to compare the number of regular graphs with n vertices with the total of graphs that can be constructed with n vertices.…”

“…From the results of 1, it can be observed that the number of possible connected graphs C p grows very fast in terms of the number p of vertices. It should be mentioned that (29) and (30) are recurrence relations instead of a closed formula. The recurrences (29) or (30) can be used for the calculation of C p with a computer program.…”

“…It should be mentioned that (29) and (30) are recurrence relations instead of a closed formula. The recurrences (29) or (30) can be used for the calculation of C p with a computer program. The generating functions can be used to solve recurrences and obtain a closed mathematical expression for the nth term of the succession associated with the recurrence.…”