Dynamic message-passing equations for models with unidirectional dynamics

Understanding and quantifying the dynamics of disordered out-of-equilibrium models is an important problem in many branches of science. Using the dynamic cavity method on time trajectories, we construct a general procedure for deriving the dynamic message-passing equations for a large class of models with unidirectional dynamics, which includes the zero-temperature random field Ising model, the susceptible-infected-recovered model, and rumor spreading models. We show that unidirectionality of the dynamics is the key ingredient that makes the problem solvable. These equations are applicable to single instances of the corresponding problems with arbitrary initial conditions, and are asymptotically exact for problems defined on locally tree-like graphs. When applied to real-world networks, they generically provide a good analytic approximation of the real dynamics.Comment: Final versio

Dynamic message-passing approach for kinetic spin models with reversible dynamics

A method to approximately close the dynamic cavity equations for synchronous reversible dynamics on a locally tree-like topology is presented. The method builds on $(a)$ a graph expansion to eliminate loops from the normalizations of each step in the dynamics, and $(b)$ an assumption that a set of auxilary probability distributions on histories of pairs of spins mainly have dependencies that are local in time. The closure is then effectuated by projecting these probability distributions on $n$-step Markov processes. The method is shown in detail on the level of ordinary Markov processes ($n=1$), and outlined for higher-order approximations ($n>1$). Numerical validations of the technique are provided for the reconstruction of the transient and equilibrium dynamics of the kinetic Ising model on a random graph with arbitrary connectivity symmetry.Comment: 6 pages, 4 figure

A matrix product algorithm for stochastic dynamics on networks, applied to non-equilibrium Glauber dynamics

We introduce and apply a novel efficient method for the precise simulation of stochastic dynamical processes on locally tree-like graphs. Networks with cycles are treated in the framework of the cavity method. Such models correspond, for example, to spin-glass systems, Boolean networks, neural networks, or other technological, biological, and social networks. Building upon ideas from quantum many-body theory, the new approach is based on a matrix product approximation of the so-called edge messages -- conditional probabilities of vertex variable trajectories. Computation costs and accuracy can be tuned by controlling the matrix dimensions of the matrix product edge messages (MPEM) in truncations. In contrast to Monte Carlo simulations, the algorithm has a better error scaling and works for both, single instances as well as the thermodynamic limit. We employ it to examine prototypical non-equilibrium Glauber dynamics in the kinetic Ising model. Because of the absence of cancellation effects, observables with small expectation values can be evaluated accurately, allowing for the study of decay processes and temporal correlations.Comment: 5 pages, 3 figures; minor improvements, published versio

Contagion in an interacting economy

We investigate the credit risk model defined in Hatchett & K\"{u}hn under more general assumptions, in particular using a general degree distribution for sparse graphs. Expanding upon earlier results, we show that the model is exactly solvable in the $N\rightarrow \infty$ limit and demonstrate that the exact solution is described by the message-passing approach outlined by Karrer and Newman, generalized to include heterogeneous agents and couplings. We provide comparisons with simulations of graph ensembles with power-law degree distributions.Comment: 21 pages, 6 figure

Network infection source identification under the SIRI model

We study the problem of identifying a single infection source in a network under the susceptible-infected-recovered-infected (SIRI) model. We describe the infection model via a state-space model, and utilizing a state propagation approach, we derive an algorithm known as the heterogeneous infection spreading source (HISS) estimator, to infer the infection source. The HISS estimator uses the observations of node states at a particular time, where the elapsed time from the start of the infection is unknown. It is able to incorporate side information (if any) of the observed states of a subset of nodes at different times, and of the prior probability of each infected or recovered node to be the infection source. Simulation results suggest that the HISS estimator outperforms the dynamic message pass- ing and Jordan center estimators over a wide range of infection and reinfection rates.Comment: 5 pages, 3 figures; to present in ICASSP 201

Message-Passing Methods for Complex Contagions

Message-passing methods provide a powerful approach for calculating the expected size of cascades either on random networks (e.g., drawn from a configuration-model ensemble or its generalizations) asymptotically as the number $N$ of nodes becomes infinite or on specific finite-size networks. We review the message-passing approach and show how to derive it for configuration-model networks using the methods of (Dhar et al., 1997) and (Gleeson, 2008). Using this approach, we explain for such networks how to determine an analytical expression for a "cascade condition", which determines whether a global cascade will occur. We extend this approach to the message-passing methods for specific finite-size networks (Shrestha and Moore, 2014; Lokhov et al., 2015), and we derive a generalized cascade condition. Throughout this chapter, we illustrate these ideas using the Watts threshold model.Comment: 14 pages, 3 figure

Recurrent Dynamic Message Passing with Loops for Epidemics on Networks

Several theoretical methods have been developed to approximate prevalence and threshold of epidemics on networks. Among them, the recurrent dynamic message-passing (rDMP) theory offers a state-of-the-art performance by preventing the echo chamber effect in network edges. However, the rDMP theory was derived in an intuitive ad-hoc way, lacking a solid theoretical foundation and resulting in a probabilistic inconsistency flaw. Furthermore, real-world networks are clustered and full of local loops like triangles, whereas rDMP is based on the assumption of a locally tree-like network structure, which makes rDMP potentially inefficient on real applications. In this work, for the recurrent-state epidemics, we first demonstrate that the echo chamber effect exits not only in edges but also in local loops, which rDMP-like method can not avoid. We then correct the deficiency of rDMP in a principled manner, leading to the natural introduction of new higher-order dynamic messages, extending rDMP to handle local loops. By linearizing the extended message-passing equations, a new epidemic threshold estimation is given by the inverse of the leading eigenvalue of a matrix named triangular non-backtracking matrix. Numerical experiments conducted on synthetic and real-world networks to evaluate our method, the efficacy of which is validated in epidemic prevalence and threshold prediction tasks. In addition, our method has the potential to speed up the solution of the immunization, influence maximization, and robustness optimization problems in the networks.Comment: Submitted, 14 pages, 7 figure