This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total @math possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of relating dynamical properties of Boolean networks to graphical properties of the underlying interactions.
Virus Spreading . Virus spreading across a computer network can be modeled as a Boolean network, where @math and @math represent infected and healthy computers, respectively @cite_23 . The proposed Boolean gossip process may characterize more possibilities for two computers during an interaction: two computers, infected or not, are both infected ( @math ); two computers, infected or not, are both cured ( @math ), etc.
@cite_23
1987511126 2398538542 2124486211 2025392770
The effect of virus spreading in a telecommunication network, where a certain curing strategy is deployed, can be captured by epidemic models. In the N-intertwined model proposed and studied in [1], [2], the probability of each node to be infected depends on the curing and infection rate of its neighbors. In this paper, we consider the case where all infection rates are equal and different values of curing rates can be deployed within a given budget, in order to minimize the overall infection of the network. We investigate this difficult optimization together with a related problem where the curing budget must be minimized within a given level of network infection. Some properties of these problems are derived and several solution algorithms are proposed. These algorithms are compared on two real world network instances, while Erdos-Renyi graphs and some special graphs such as the cycle, the star, the wheel and the complete bipartite graph are also addressed. The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations. The strong analogy between biological viruses and their computational counterparts has motivated the authors to adapt the techniques of mathematical epidemiology to the study of computer virus propagation. In order to allow for the most general patterns of program sharing, a standard epidemiological model is extended by placing it on a directed graph and a combination of analysis and simulation is used to study its behavior. The conditions under which epidemics are likely to occur are determined, and, in cases where they do, the dynamics of the expected number of infected individuals are examined as a function of time. It is concluded that an imperfect defense against computer viruses can still be highly effective in preventing their widespread proliferation, provided that the infection rate does not exceed a well-defined critical epidemic threshold. > We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log1.5 n).