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.
Recent work on information survival in sensor and human P2P networks try to study the datum preservation or the virus spreading in a network under the dynamical system approach. Some interesting solutions propose to use non-linear dynamical systems and fixed point stability theorems, providing closed form formulas that depend on the largest eigenvalue of the dynamic system matrix. Given that in the Web there can be messages from one place to another, and that these messages can be, with some probability, new unclassified virus warning messages as well as worms or other kind of viruses, the sites can be infected very fast. The question to answer is how and when a network infection can become global and how it can be controlled or at least how to stabilize the spreading in such a way that it becomes confined below a fixed portion of the network. In this paper, we try to be a step ahead in this direction and apply classic results of the dynamical systems theory to model the behavior of a network where warning messages and viruses spread.
341Advs. Complex Syst. 2011.14:341-358. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/09/15. For personal use only.342 C. R. Lucatero and R. B. Jaquez sensor networks, social nets or wireless networks, where information is to be stored, generated and retrieved. So under these new environments, it is very important to study and model how the information is spread or how to keep the spreading of a virus under control in such a way that the information is still useful under these vulnerable circumstances. In [8] the problem of information survival threshold in sensor and P2P networks, modeling the problem as a nonlinear dynamical system and using fixed point stability theorems, and obtaining a closed form solution that depends on an additional parameter, the largest eigenvalue of the dynamical system matrix is being studied. In the sensor networks for instance, the nodes can loss their communication links and the nodes can stop working because of a system failure produced by a virus infection and quarantine process or a system maintenance procedure. Under such conditions they try to answer the following question: Under what conditions can a datum survive in a sensor network? Given that the nodes as well as the links can fail with some probability the obvious model can be a Markov Chain, but such a model can grow in complexity very quickly because the number of possible states becomes 3 N where N is the number of states. To avoid this mathematical problem, one alternative is to model the system as a nonlinear dynamical system. Recently, there have appeared in the conferences and journal articles some very interesting and relevant research articles about the virus spread behavior in a P2P network or in scale free nets such as the Web. In [10] the authors study the communication mechanisms for gossip based protocols. Another very recent and interesting work on how to distribute antidotes for controlling the epidemics spread is presented in [3]. In this research t...
The dynamics of decisions in complex networks is studied within a Markov process framework using numerical simulations combined with mathematical insight into the process mechanisms. A mathematical discrete-time model is derived based on a set of basic assumptions on the convincing mechanisms associated to two opinions. The model is analyzed with respect to multiplicity of critical points, illustrating in this way the main behavior to be expected in the network. Particular interest is focussed on the effect of social network and exogenous mass media-based influences on the decision behavior. A set of numerical simulation results is provided illustrating how these mechanisms impact the final decision results. The analysis reveals (i) the presence of fixed-point multiplicity (with a maximum of four different fixed points), multistability, and sensitivity with respect to process parameters, and (ii) that mass media have a strong impact on the decision behavior.
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