Matrix product algorithm for stochastic dynamics on networks applied to nonequilibrium Glauber dynamics

Dynamical phase transitions in graph cellular automata

Discrete dynamical systems can exhibit complex behavior from the iterative application of straightforward local rules. A famous class of examples comes from cellular automata whose global dynamics are notoriously challenging to analyze. To address this, we relax the regular connectivity grid of cellular automata to a random graph, which gives the class of graph cellular automata. Using the dynamical cavity method and its backtracking version, we show that this relaxation allows us to derive asymptotically exact analytical results on the global dynamics of these systems on sparse random graphs. Concretely, we showcase the results on a specific subclass of graph cellular automata with "conforming nonconformist" update rules, which exhibit dynamics akin to opinion formation. Such rules update a node's state according to the majority within their own neighborhood. In cases where the majority leads only by a small margin over the minority, nodes may exhibit nonconformist behavior. Instead of following the majority, they either maintain their own state, switch it, or follow the minority. For configurations with different initial biases towards one state we identify sharp dynamical phase transitions in terms of the convergence speed and attractor types. From the perspective of opinion dynamics this answers when consensus will emerge and when two opinions coexist almost indefinitely.

Matrix Product Belief Propagation for reweighted stochastic dynamics over graphs

Stochastic processes on graphs can describe a great variety of phenomena ranging from neural activity to epidemic spreading. While many existing methods can accurately describe typical realizations of such processes, computing properties of extremely rare events is a hard task, particularly so in the case of recurrent models, in which variables may return to a previously visited state. Here, we build on the matrix product cavity method, extending it fundamentally in two directions: First, we show how it can be applied to Markov processes biased by arbitrary reweighting factors that concentrate most of the probability mass on rare events. Second, we introduce an efficient scheme to reduce the computational cost of a single node update from exponential to polynomial in the node degree. Two applications are considered: inference of infection probabilities from sparse observations within the SIRS epidemic model and the computation of both typical observables and large deviations of several kinetic Ising models.

Theory of Nonequilibrium Local Search on Random Satisfaction Problems

We study local search algorithms to solve instances of the random k-satisfiability problem, equivalent to finding (if they exist) zero-energy ground states of statistical models with disorder on random hypergraphs. It is well known that the best such algorithms are akin to nonequilibrium processes in a high-dimensional space. In particular, algorithms known as focused, and which do not obey detailed balance, outperform simulated annealing and related methods in the task of finding the solution to a complex satisfiability problem, that is to find (exactly or approximately) the minimum in a complex energy landscape. A physical question of interest is if the dynamics of these processes can be well predicted by the well-developed theory of equilibrium Gibbs states. While it has been known empirically for some time that this is not the case, an alternative systematic theory that does so has been lacking. In this Letter we introduce such a theory based on the recently developed technique of cavity master equations and test it on the paradigmatic random 3-satisfiability problem. Our theory predicts the qualitative form of the phase boundary between the satisfiable (SAT) and unsatisfiable (UNSAT) region of the phase diagram where the numerics of a focused Metropolis search and cavity master equation cannot be distinguished.