Masaki Ogura

Stability of Spreading Processes over Time-Varying Large-Scale Networks

In this paper, we analyze the dynamics of spreading processes taking place over time-varying networks. A common approach to model time-varying networks is via Markovian random graph processes. This modeling approach presents the following limitation: Markovian random graphs can only replicate switching patterns with exponential inter-switching times, while in real applications these times are usually far from exponential. In this paper, we introduce a flexible and tractable extended family of processes able to replicate, with arbitrary accuracy, any distribution of inter-switching times. We then study the stability of spreading processes in this extended family. We first show that a direct analysis based on Itôs formula provides stability conditions in terms of the eigenvalues of a matrix whose size grows exponentially with the number of edges. To overcome this limitation, we derive alternative stability conditions involving the eigenvalues of a matrix whose size grows linearly with the number of nodes. Based on our results, we also show that heuristics based on aggregated static networks approximate the epidemic threshold more accurately as the number of nodes grows, or the temporal volatility of the random graph process is reduced. Finally, we illustrate our findings via numerical simulations.

Stability of switching systems and generalized joint spectral radius

Abstract This paper extends the notion of generalized joint spectral radius with exponents, originally defined for a finite set of matrices, to probability distributions. We show that, under a certain invariance condition, the radius is calculated as the spectral radius of a matrix that can be easily computed, extending the classical counterpart. Using this result we investigate the mean stability of switching systems. In particular we establish the equivalence of mean square stability, simultaneous contractibility in square mean, and the existence of a quadratic Lyapunov function. Also the stabilization of positive switching systems is studied. Numerical examples are given to illustrate the results.

Stability Analysis of Positive Semi-Markovian Jump Linear Systems with State Resets

This paper studies the mean stability of positive semi-Markovian jump linear systems. We show that their mean stability is characterized by the spectral radius of a matrix that is easy to compute. In deriving the condition we use a certain discretization of a semi-Markovian jump linear system that preserves stability. Also we show a characterization for the exponential mean stability of continuous-time positive Markovian jump linear systems. Numerical examples are given to illustrate the results.

Epidemic Processes over Adaptive State-Dependent Networks

In this paper we study the dynamics of epidemic processes taking place in adaptive networks of arbitrary topology. We focus our study on the adaptive susceptible-infected-susceptible (ASIS) model, where healthy individuals are allowed to temporarily cut edges connecting them to infected nodes in order to prevent the spread of the infection. In this paper we derive a closed-form expression for a lower bound on the epidemic threshold of the ASIS model in arbitrary networks with heterogeneous node and edge dynamics. For networks with homogeneous node and edge dynamics, we show that the resulting lower bound is proportional to the epidemic threshold of the standard SIS model over static networks, with a proportionality constant that depends on the adaptation rates. Furthermore, based on our results, we propose an efficient algorithm to optimally tune the adaptation rates in order to eradicate epidemic outbreaks in arbitrary networks. We confirm the tightness of the proposed lower bounds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures.

Optimal Design of Switched Networks of Positive Linear Systems via Geometric Programming

In this paper, we propose an optimization framework to design a network of positive linear systems whose structure switches according to a Markov process. The optimization framework herein proposed allows the network designer to optimize the coupling elements of a directed network, as well as the dynamics of the nodes in order to maximize the stabilization rate of the network and/or the disturbance rejection against an exogenous input. The cost of implementing a particular network is modeled using polynomial cost functions, which allow for a wide variety of modeling options. In this context, we show that the cost-optimal network design can be efficiently found using geometric programming in polynomial time. We illustrate our results with a practical problem in network epidemiology, namely, the cost-optimal stabilization of the spread of a disease over a time-varying contact network.

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 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 n.

A Limit Formula for Joint Spectral Radius with p-radius of Probability Distributions

Abstract In this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the p -radius of an associated probability distribution when p tends to ∞. Allowing the set to have infinitely many matrices, the obtained formula extends the results in the literature. Based on the formula, we then present a novel characterization of the stability of switched linear systems for an arbitrary switching signal via the existence of stochastic Lyapunov functions of any higher degrees. Numerical examples are presented to illustrate the results.

Efficient method for computing lower bounds on the p-radius of switched linear systems

Abstract This paper proposes lower bounds on a quantity called L p -norm joint spectral radius, or in short, p -radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models, algorithms for computing the p -radius are only available in a very limited number of particular cases. The proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via Kronecker products and do not place any requirements on the set of matrices. We show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds. A Markovian extension of the proposed lower bounds is also presented.

Stability of Markov regenerative switched linear systems

In this paper, we give a necessary and sufficient condition for mean stability of switched linear systems having a Markov regenerative process as its switching signal. This class of switched linear systems, which we call Markov regenerative switched linear systems, contains Markov jump linear systems and semi-Markov jump linear systems as special cases. We show that a Markov regenerative switched linear system is m th mean stable if and only if a particular matrix is Schur stable, under the assumption that either m is even or the system is positive.

A general class of spreading processes with non-Markovian dynamics

In this paper we propose a general class of models for spreading processes we call the SI*V * model. Unlike many works that consider a fixed number of compartmental states, we allow an arbitrary number of states on arbitrary graphs with heterogeneous parameters for all nodes and edges. As a result, this generalizes an extremely large number of models studied in the literature including the MSEIV, MSEIR, MSEIS, SEIV, SEIR, SEIS, SIV, SIRS, SIR, and SIS models. Furthermore, we show how the SI*V * model allows us to model non-Poisson spreading processes letting us capture much more complicated dynamics than existing works such as information spreading through social networks or the delayed incubation period of a disease like Ebola. This is in contrast to the overwhelming majority of works in the literature that only consider dynamics that can be captured by Markov processes. After developing the stochastic model, we analyze its deterministic mean-field approximation and provide conditions for when the disease-free equilibrium is stable. Simulations illustrate our results.