In previous work, we developed the scaled SIS process, which models the dynamics of SIS epidemics over networks. We derived for the scaled SIS process a closed-form expression for the time-asymptotic probability distribution of the configurations of all the agents in the network, which explicitly exhibits the underlying network topology through its adjacency matrix. This is accomplished for networks that are finite-size and of arbitrary topology. This paper determines which network configuration is the most probable. We prove that, for a range of epidemics parameters, this combinatorial problem leads to a submodular optimization problem, which is \emph{exactly} solvable in polynomial time. We relate the most-probable configuration to the network structure, in particular, to the existence of high density subgraphs. Depending on the model parameters, subset of agents may be more likely to be infected than others; these more-vulnerable agents form subgraphs that are denser than the overall network. We illustrate our results with a 193 node social network and the 4941 node Western US power grid under different model parameters.

The interest in distributed control methods for power systems is motivated by the need for scalable solutions to handle the coordination of an increasing number of distributed resources. This paper presents a fully distributed multilevel method to solve the DC Optimal Power Flow problem (DC-OPF). Our proposed approach constitutes a distributed iterative mechanism to solve the first order optimality conditions of the DC-OPF problem using the fact that optimality conditions involve local variable couplings. The proposed distributed structure requires each bus to update a few local variables and exchange information with neighboring buses. Our multilevel distributed approach distributes the computation at several levels, i.e., nodes, subareas and areas. It allows for synchronous information exchanges, i.e., after each iteration, at the nodal level and asynchronous communication, i.e., after multiple iterations, between subareas and areas. To define meaningful subareas, we are using a graph theoretic partitioning method derived from an epidemics model. We compare the performance of the proposed partitioning method over a random partitioning method using the IEEE 118-bus system.

To study signals on networks, to detect epidemics, or to predict blackouts, we need to understand network topology and its impact on the behavior of network processes. The high dimensionality of large networks presents significant analytical and computational challenges; only specific network structures have been studied without approximation. We consider the impact of network topology on the limiting behavior of a dynamical process obeying the stochastic rules of SIS (susceptible-infected-susceptible) epidemics using the scaled SIS process. We introduce the network effect ratio, which captures the preference of individual agents versus the preference of society (i.e., network) and investigate its effects.

Peer influence and interactions between agents in a population give rise to complex, nonlinear behaviors. This paper adopts the SIS (susceptible-infected-susceptible) framework from epidemiology to analytically study how network topology affects the diffusion of ideas/opinions/beliefs/innovations in social networks. We introduce the \emph{scaled SIS process}, which models peer influence as neighbor-to-neighbor infections. We model the scaled SIS process as a continuous-time Markov process and derive for this process its closed form equilibrium distribution. The adjacency matrix that describes the underlying social network is explicitly reflected in this distribution. The paper shows that interesting population asymptotic behaviors occur for scenarios where the individual tendencies of each agent oppose peer influences. Specifically, we determine how the most probable configuration of agent states (i.e., the population configuration with maximum equilibrium distribution) depends on both model parameters and network topology. We show that, for certain regions of the parameter space, this and related issues reduce to standard graph questions like the maximum independent set problem.

Bond percolation is a network process that traditionally addresses the question when there is a path between two sites or two clusters in a network. This has been used to study the circulation of goods or flows in networked structures as well as network resilience. This paper proposes and analyzes a \emph{dynamic} bond percolation model where bonds (i.e., edges) open (i.e., form) or close (i.e., terminate) according to random micro interactions. We model the edge dynamics through topology independent and topology-dependent processes\textemdash spontaneous formation or termination of edges and the formation of edge \{a, b\} between two sites $a$ and $b$ that depends on the current number of existing edges at $a$ and $b$. We show that the resulting network process is Markov and reversible, and we determine analytically its equilibrium distribution, avoiding having to solve a eigenvalue-eigenvector problem that quickly becomes intractable for even moderate sized networks.

We model a SIS (susceptible-infected-susceptible) epidemics over a static, finite-sized network as a continuous-time Markov process using the scaled SIS epidemics model. In our previous work, we derived the closed form description of the equilibrium distribution that explicitly accounts for the network topology and showed that the most probable equilibrium state demonstrates threshold behavior. In this paper, we will show how subgraph structures in the network topology impact the most probable state of the long run behavior of a SIS epidemics (i.e., stochastic diffusion process) over any static, finite-sized, network.

Current research is interested in identifying how topology impacts epidemics in networks. In this paper, we model SIS (susceptible-infected-susceptible) epidemics as a continuous-time Markov process and for which we can obtain a closed form description of the equilibrium distribution. Such distribution describes the long-run behavior of the epidemics. The adjacency matrix of the network topology is reflected explicitly in the formulation of the equilibrium distribution. Secondly, we are interested in analyzing the model in the regime where the topology dependent infection process opposes the topology independent healing process. Specifically, how will network topology affect the most probable long-run network state? We show that for $k$-regular graph topologies, the most probable network state transitions from the state where everyone is healthy to one where everyone is infected at a threshold that depends on $k$ but not on the size of the graph.

We are interested in investigating the spread of contagion in a network, $G$, which describes the interactions between the agents in the system. The topology of this network is often neglected due to the assumption that each agent is connected with every other agents; this means that the network topology is a complete graph. While this allows for certain simplifications in the analysis, we fail to gain insight on the diffusion process for non-complete network topology. In this paper, we offer a continuous-time Markov chain infection model that explicitly accounts for the network topology, be it complete or non-complete. Although we characterize our process using parameters from epidemiology, our approach can be applied to many application domains. We will show how to generate the infinitesimal matrix that describes the evolution of this process for any topology. We also develop a general methodology to solve for the equilibrium distribution by considering symmetries in $G$. Our results show that network topologies have dramatic effect on the spread of infections.