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We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially-structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model) and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics whilst holding the degree distribution constant. In particular, we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to. We further show that there is a complex interplay between degree distribution and asymmetry when they may co-vary; and that the results are robust to correlations in the network or the initial condition.
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We introduce tensor network contraction algorithms for counting satisfying assignments of constraint satisfaction problems (#CSPs). We represent each arbitrary #CSP formula as a tensor network, whose full contraction yields the number of satisfying assignments of that formula, and use graph theoretical methods to determine favorable orders of contraction. We employ our heuristics for the solution of #P-hard counting boolean satisfiability (#SAT) problems, namely monotone #1-in-3SAT and #Cubic-Vertex-Cover, and find that they outperform state-of-the-art solvers by a significant margin.
An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable network activations within a limited critical range. In this range, the activity of neural populations in the network persists between the extremes of quickly dying out, or activating the whole network. The nerve fiber network of the mammalian cerebral cortex possesses a modular organization extending across several levels of organization. Using a basic spreading model without inhibition, we investigated how functional activations of nodes propagate through such a hierarchically clustered network. The simulations demonstrated that persistent and scalable activation could be produced in clustered networks, but not in random networks of the same size. Moreover, the parameter range yielding critical activations was substantially larger in hierarchical cluster networks than in small-world networks of the same size. These findings indicate that a hierarchical cluster architecture may provide the structural basis for the stable and diverse functional patterns observed in cortical networks.
Empirical evidence shows that the rate of irregular usage of English verbs exhibits discontinuity as a function of their frequency: the most frequent verbs tend to be totally irregular. We aim to qualitatively understand the origin of this feature by studying simple agent--based models of language dynamics, where each agent adopts an inflectional state for a verb and may change it upon interaction with other agents. At the same time, agents are replaced at some rate by new agents adopting the regular form. In models with only two inflectional states (regular and irregular), we observe that either all verbs regularize irrespective of their frequency, or a continuous transition occurs between a low frequency state where the lemma becomes fully regular, and a high frequency one where both forms coexist. Introducing a third (mixed) state, wherein agents may use either form, we find that a third, qualitatively different behavior may emerge, namely, a discontinuous transition in frequency. We introduce and solve analytically a very general class of three--state models that allows us to fully understand these behaviors in a unified framework. Realistic sets of interaction rules, including the well-known Naming Game (NG) model, result in a discontinuous transition, in agreement with recent empirical findings. We also point out that the distinction between speaker and hearer in the interaction has no effect on the collective behavior. The results for the general three--state model, although discussed in terms of language dynamics, are widely applicable.
It has been recently discovered that the measles virus can wipe out the adaptive immune system, destroying B lymphocytes and reducing the diversity of non-specific B cells of the infected host. In particular, this implies that previously acquired immunization from vaccination or direct exposition to other pathogens could be erased in a phenomenon named immune amnesia, whose effects can become particularly worrisome given the actual rise of anti-vaccination movements. Here we present the first attempt to incorporate immune amnesia into standard models of epidemic spreading. In particular, we analyze diverse variants of a model that describes the spreading of two concurrent pathogens causing measles and another generic disease: the SIR-IA model. Analytical and computational studies confirm that immune amnesia can indeed have important consequences for epidemic spreading, significantly altering the vaccination coverage required to reach herd-immunity for concurring infectious diseases. More specifically, we uncover the existence of novel propagating and endemic phases which are induced by immune amnesia, that appear both in fully-connected and more structured networks, such as random networks and power-law degree-distributed ones. In particular, the transitions from a quiescent state into these novel phases can become rather abrupt in some cases that we specifically analyze. Furthermore, we discuss the meaning and consequences of our results and their relation with, e.g., immunization strategies, together with the possibility that explosive types of transitions may emerge, making immune-amnesia effects particularly dramatic. This work opens the door to further developments and analyses of immune amnesia effects, contributing, more generally, to the theory of interacting epidemics on complex networks.
We discuss a non-reversible Markov chain Monte Carlo (MCMC) algorithm for particle systems, in which the direction of motion evolves deterministically. This sequential direction-sweep MCMC generalizes the widely spread MCMC sweep methods for particle or spin indices. The sequential direction-sweep MCMC can be applied to a wide range of original reversible or non-reversible Markov chains, such as the Metropolis algorithm or the event-chain Monte Carlo algorithm. For a simplified two-dimensional dipole model, we show rigorously that sequential MCMC leaves the stationary probability distribution unchanged, yet it profoundly modifies the Markov-chain trajectory. Long excursions, with persistent rotation in one direction, alternate with long sequences of rapid zigzags resulting in persistent rotation in the opposite direction. We show that sequential MCMC can have shorter mixing times than the algorithms with random updates of directions. We point out possible applications of sequential MCMC in polymer physics and in molecular simulation.
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