Spreading Processes over Socio-Technical Networks with Phase-Type Transmissions

Most theoretical tools available for the analysis of spreading processes over networks assume exponentially distributed transmission and recovery times. In practice, the empirical distribution of transmission times for many real spreading processes, such as the spread of web content through the Internet, are far from exponential. To bridge this gap between theory and practice, we propose a methodology to model and analyze spreading processes with arbitrary transmission times using phase-type distributions. Phase-type distributions are a family of distributions that is dense in the set of positive-valued distributions and can be used to approximate any given distributions. To illustrate our methodology, we focus on a popular model of spreading over networks: the susceptible-infected-susceptible (SIS) networked model. In the standard version of this model, individuals informed about a piece of information transmit this piece to its neighbors at an exponential rate. In this paper, we extend this model to the case of transmission rates following a phase-type distribution. Using this extended model, we analyze the dynamics of the spread based on a vectorial representations of phase-type distributions. We illustrate our results by analyzing spreading processes over networks with transmission and recovery rates following a Weibull distribution

Worst-Case Scenarios for Greedy, Centrality-Based Network Protection Strategies

The task of allocating preventative resources to a computer network in order to protect against the spread of viruses is addressed. Virus spreading dynamics are described by a linearized SIS model and protection is framed by an optimization problem which maximizes the rate at which a virus in the network is contained given finite resources. One approach to problems of this type involve greedy heuristics which allocate all resources to the nodes with large centrality measures. We address the worst case performance of such greedy algorithms be constructing networks for which these greedy allocations are arbitrarily inefficient. An example application is presented in which such a worst case network might arise naturally and our results are verified numerically by leveraging recent results which allow the exact optimal solution to be computed via geometric programming

Disease spread over randomly switched large-scale networks

In this paper we study disease spread over a randomly switched network, which is modeled by a stochastic switched differential equation based on the so called $N$-intertwined model for disease spread over static networks. Assuming that all the edges of the network are independently switched, we present sufficient conditions for the convergence of infection probability to zero. Though the stability theory for switched linear systems can naively derive a necessary and sufficient condition for the convergence, the condition cannot be used for large-scale networks because, for a network with $n$ agents, it requires computing the maximum real eigenvalue of a matrix of size exponential in $n$. On the other hand, our conditions that are based also on the spectral theory of random matrices can be checked by computing the maximum real eigenvalue of a matrix of size exactly $n$