Output-Feedback Control for Discrete-Time Spreading Models in Complex Networks

Abstract

Abstract:The problem of stabilizing the spreading process to a prescribed probability distribution over a complex network is considered, where the dynamics of the nodes in the network is given by discrete-time Markov-chain processes. Conditions for the positioning and identification of actuators and sensors are provided, and sufficient conditions for the exponential stability of the desired distribution are derived. Simulations results for a network of N = 10 6 corroborate our theoretical findings.

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

“…In fact, this previous model was presented as an application example of the results obtained in a section of [1]. The criteria obtained in [1] as well as the one obtained in the mentioned previous model, can be considered as a good option when the number of nodes to be controlled with respect to the total number of nodes is small. The number of nodes to be controlled depends on the topology of the complex network as well as on the values of the state transition parameters in each node.…”

“…Due to this drawback, we decided to show through an enumerative combinatorial argument that, even in the case of homogeneous behavior of the nodes, the criterion is applicable in most cases because the probability that a randomly generated graph will be regular tends to zero as the number of nodes grows arbitrarily. Simultaneously, the second author, in collaboration with other colleagues, refined the selection criterion of nodes to be controlled obtained in Section 5 of the present article and tested by simulation that the criterion worked under the hypothesis of non-homogeneity, for scale-free type topologies in the article [1].…”

“…The bifurcation analysis that will be presented in Section 4 as well as the criterion for selection of control nodes that will be presented in Section 5 of the present article were unpublished results prior to the more general and refined criterion obtained in the article [1]. During the development of the criterion of selection of nodes to be controlled, presented in Section 5 of the present article, we faced the problem that in the case that the behavior transition and interaction of the nodes were homogenous and the network had a topology of regular type, that is to say that the nodes had the same degree, we would be forced to have to control all the nodes since we could not distinguish them.…”

“…The second author of the present article, who is also the first author of [1], in a previous result that is less general than the model of the virus spreading on complex networks presented in [1], was interested in establishing the conditions allowing for detecting those nodes of a complex network that should be controlled such that the system can be steered to a stable virus extinction state. In fact, this previous model was presented as an application example of the results obtained in a section of [1]. The criteria obtained in [1] as well as the one obtained in the mentioned previous model, can be considered as a good option when the number of nodes to be controlled with respect to the total number of nodes is small.…”

Research Frontier in Chaos Theory and Complex Networks

“…In the paper "Output-feedback control for discrete-time spreading models in complex networks", Alarcón et al [8] analyzes a Markov chain-based model for Susceptible-Infected-Susceptible (SIS) spreading dynamics over complex networks, and proposes a control mechanism to stabilize the epidemic extinction state. They derive conditions for positioning and identification of actuators and sensors, as well as sufficient conditions for the exponential stability of a desired distribution.…”