Damián Horacio Zanette

Demographic stochasticity in predator-prey (or activator-inhibitor) systems can generate persistent oscillations driven by intrinsic fluctuations, even when the corresponding deterministic mean-field descriptions exhibit only damped oscillations that vanish asymptotically as the system approaches a stable fixed point. Here we demonstrate that sustained stochastic oscillations may also emerge when the prey population responds deterministically to fluctuations in the predator population. In this framework, predator dynamics is governed by a stochastic process, whereas prey dynamics obeys an equation that is continuous in both time and population density, as expected in the limit of a large, homogeneous prey population. We compare the behavior of this hybrid stochastic-deterministic system with that of its mean-field counterpart and quantitatively characterize the resulting stochastic oscillatory regime, with particular emphasis on its dependence on the predator population size and the prey response time.

We study an ecology-inspired model for a population of bounded size, whose dynamics is governed by random birth, death, and immigration events. Stochastic fluctuations in the number of individuals give rise to a succession of alternating active and vacant periods, where the population is, respectively, extant and extinct. Using both analytical and numerical techniques, we characterize the statistics of the two kinds of period, quantifying their duration and frequency, and the typical population sizes in active periods. In sharp contrast to the deterministic mean-field behavior, governed by logistic dynamics, active periods may exhibit pronounced bimodality: either short durations with very small populations, or much longer durations with population sizes close to the maximum. We also investigate how these results change when the population evolves on random networks of three classes: Erdős-Rényi, regular, and geographic. The main effect of the network structure is to induce population clustering, with individuals aggregated into localized groups. This, in turn, limits population growth and increases the frequency of vacant periods.

We study a variety of stochastic contact processes—directly related to models of rumor and disease spreading—from the viewpoint of their constants of motion, either exact or approximated. As in deterministic systems, constants of motion in stochastic dynamics make it possible to reduce the number of relevant variables, thereby confining the set of accessible states and facilitating analytical treatment. For processes of rumor propagation based on the Maki–Thompson model, we show how to construct exact constants of motion as linear combinations of conserved quantities in each elementary contact event, and how they relate to the constants of motion of the corresponding mean-field equations, which are obtained as the continuous-time, large-size limit of the stochastic process. For SIR epidemic models, both in homogeneous systems and on heterogeneous networks, we find that a similar procedure produces approximate constants of motion, whose average value is preserved along the evolution. We also give examples of exact and approximate constants of motion built as nonlinear combinations of the relevant variables, whose expressions are suggested by their mean-field counterparts.

We study a variety of stochastic contact processes --directly related to models of rumor and disease spreading-- from the viewpoint of their constants of motion, either exact or approximated. Much as in deterministic systems, constants of motion in stochastic dynamics make it possible to reduce the number of relevant variables, confining the set of accessible states, and thus facilitating their analytical treatment. For processes of rumor propagation based on the Maki-Thompson model, we show how to construct exact constants of motion as linear combinations of conserved quantities in each elementary contact event, and how they relate to the constants of motion of the corresponding mean-field equations, which are obtained as the continuous-time, large-size limit of the stochastic process. For SIR epidemic models, both in homogeneous systems and on heterogeneous networks, we find that a similar procedure produces approximate constants of motion, whose average value is preserved along the evolution. We also give examples of exact and approximate constants of motion built as nonlinear combinations of the relevant variables, whose expressions are suggested by their mean-field counterparts.

We study the properties of discrete-time random walks on networks formed by randomly interconnected cliques, namely, random networks of cliques. Our purpose is to derive the parameters that define the network structure—specifically, the distribution of clique size and the abundance of inter-clique links—from the observation of selected statistical features along the random walk. To this end, we apply a Bayesian approach based on recording the times spent by the walker inside successively visited cliques. The procedure is illustrated with some numerical examples of diverse complexity, where the relevant structural parameters are successfully recovered.

We study the properties of discrete-time random walks on networks formed by randomly interconnected cliques, namely, random networks of cliques. Our purpose is to derive the parameters that define the network structure -- specifically, the distribution of clique size and the abundance of inter-clique links -- from the observation of selected statistical features along the random walk. To this end, we apply a Bayesian approach based on recording the times spent by the walker inside successively visited cliques. The procedure is illustrated with some numerical examples of diverse complexity, where the relevant structural parameters are successfully recovered.

We study the structural properties of networks formed by random sets of bit strings—namely the ordered arrays of binary variables representing, for instance, genetic information or cultural profiles. Two bit strings are connected by a network link when they are sufficiently similar to each other, i.e., when their Hamming distance is below a certain threshold. Using both analytical and numerical techniques, we determine the degree distribution and the conditions for the existence of a giant component in this kind of network. In addition, we analyze their clustering, assortativity, and mean geodesic distance. We show that these properties combine features specific to random networks with characteristics that derive from the Hamming metrics implicit in the definition of similarity between bit strings.

We study the structural properties of networks formed by random sets of bit strings –namely, ordered arrays of binary variables representing, for instance, genetic information or cultural profiles. Two bit strings are connected by a network link when they are sufficiently similar to each other, i. e. when their Hamming distance is below a certain threshold. Using both analytical and numerical techniques, we determine the degree distribution and the conditions for the existence of a giant component in this kind of network. In addition, we analyze their clustering, assortativity, and mean geodesic distance. We show that these properties combine features specific to random networks with characteristics that derive from the Hamming metrics implicit in the definition of similarity between bit strings.

We disclose a critical phenomenon induced by structural properties of the contact pattern in a stylized model of rumor propagation over a population of agents. The contact pattern is given by a random network of cliques, formed by fully interconnected groups of nodes of identical size with randomly distributed connections between groups. As demonstrated numerically using finite-size scaling analysis, the process exhibits a critical transition between a regime where the rumor remains confined to a negligible part of the population and a regime where it attains a finite portion of the system. We determine the critical point and the critical exponent of the transition for different clique sizes. The phenomenon is analogous to that observed for the same kind of process in Watts-Strogatz small-world networks, and is likely due to the combination of large clustering and short mean geodesic distances that also characterizes random networks of cliques.