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Register a new userMultivariate Generating Functions for Information Spread on Multi-Type Random Graphs
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We study the spread of information on multi-type directed random graphs. In such graphs the vertices are partitioned into distinct types (communities) that have different transmission rates between themselves and with other types. We construct multivariate generating functions and use multi-type branching processes to derive an equation for the size of the large out-components in multi-type random graphs with a general class of degree distributions. We use our methods to analyse the spread of epidemics and verify the results with population based simulations.
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Working in the multi-type Galton-Watson branching-process framework we analyse the spread of a pandemic via a most general random contact graph. Our model consists of several communities, and takes an input parameters that outline the contacts between individuals in different communities. Given these parameters, we determine whether there will be a pandemic outbreak and if yes, we calculate the size of the giant--connected-component of the graph, thereby, determining the fraction of the population of each type that would contract the disease before it ends. We show that the disease spread has a natural evolution direction given by the Perron-Frobenius eigenvector of a matrix whose entries encode the average number of individuals of one type expected to be infected by an individual of another type. The corresponding eigenvalue is the basic reproduction number of the pandemic. We perform numerical simulations that compare homogeneous and heterogeneous disease spread graphs and quantify the difference between the pandemics. We elaborate on the difference between herd immunity and the end of the pandemics and the effect of countermeasures on the fraction of infected population.
We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can be also useful for understanding spatial learning by living organisms.
A self-organized model with social percolation process is proposed to describe the propagations of information for different trading ways across a social system and the automatic formation of various groups within market traders. Based on the market structure of this model, some stylized observations of real market can be reproduced, including the slow decay of volatility correlations, and the fat tail distribution of price returns which is found to cross over to an exponential-type asymptotic decay in different dimensional systems.
Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random walk model with long range memory for which not only the mean square displacement (MSD) can be obtained exactly in the asymptotic limit, but also the propagator. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anti-correlated Levy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.
Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the Central Limit Theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.
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