Matrix-Product Edge-Message Toolkit

		"The matrix product approximation for the dynamic cavity method" T. Barthel arXiv:1904.03312, pdf Stochastic dynamics of classical degrees of freedom, defined on vertices of locally tree-like graphs, can be studied in the framework of the dynamic cavity method which is exact for tree graphs. Such models correspond for example to spin-glass systems, Boolean networks, neural networks, and other technical, biological, and social networks. The central objects in the cavity method are edge messages - conditional probabilities of two vertex variable trajectories. In this paper, we discuss a rather pedagogical derivation for the dynamic cavity method, give a detailed account of the novel matrix product edge message (MPEM) algorithm for the solution of the dynamic cavity equation as introduced in Phys. Rev. E 97, 010104(R) (2018), and present optimizations and extensions. Matrix product approximations of the edge messages are constructed recursively in an iteration over time. Computation costs and precision can be tuned by controlling the matrix dimensions of the MPEM in truncations. Without truncations, the dynamics is exact. In contrast to Monte Carlo simulations, the approach has a better error scaling. Hence, it gives for example access to low probability events and decaying observables like temporal correlations. We discuss optimized truncation schemes and an extension that allows to capture models which have a continuum time limit.
		"A matrix product algorithm for stochastic dynamics on networks, applied to non-equilibrium Glauber dynamics" T. Barthel, C. De Bacco, and S. Franz Phys. Rev. E 97, 010104(R) (2018) 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.