Effects of degree-biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks.

We introduce a generalized rumor spreading model and analytically investigate the spreading of rumors on scale-free (SF) networks. In the standard rumor spreading model, each node has an infectivity equal to its degree, and connectivity is uniform across all links. To generalize this model, we introduce an infectivity function that determines the number of simultaneous contacts that a given node (individual) may establish with its connected neighbors and a connectivity strength function (CSF) for the direct link between two connected nodes. These lead to a degree-biased propagation of rumors. For nonlinear functions, this generalization is reflected in the infectivity's exponent α and the CSF's exponent ß. We show that, by adjusting exponents α and ß, the epidemic threshold can be controlled. This feature is absent in the standard rumor spreading model. In addition, we obtain a critical threshold. We show that the critical threshold for our generalized model is greater than that of the standard model on a finite SF network. Theoretically, we show that ß=-1 leads to a maximum spreading of rumors, and computation results on different networks verify our theoretical prediction. Also, we show that a smaller α leads to a larger spreading of rumors. Our results are interesting since we obtain these results regardless of the network topology and configuration.

Static and dynamic phase transitions in multidimensional voting models on continua.

A voting model (or a generalization of the Glauber model at zero temperature) on a multidimensional lattice is defined as a system composed of a lattice, each site of which is either empty or occupied by a single particle. The reactions of the system are such that two adjacent sites, one empty, the other occupied, may evolve to a state where both of these sites are either empty or occupied. The continuum version of this model in a D-dimensional region with a boundary is studied, and two general behaviors of such systems are investigated, the stationary behavior of the system, and the dominant way of relaxation of the system toward its stationary state. Based on the first behavior, a static phase transition (discontinuous changes in the stationary profiles of the system) is studied. Based on the second behavior, a dynamical phase transition (discontinuous changes in the relaxation times of the system) is studied. It is shown that the static phase transition is induced by the bulk reactions only, while the dynamical phase transition is a result of both bulk reactions and boundary conditions.

Solvable multispecies reaction-diffusion processes.

A family of one-dimensional multispecies reaction-diffusion processes on a lattice is introduced. It is shown that these processes are exactly solvable, provided a nonspectral matrix equation is satisfied. Some general remarks on the solutions to this equation, and some special solutions are given. The large-time behavior of the conditional probabilities of such systems is also investigated.

Asymmetric one-dimensional exclusion processes: a two-parameter exactly solvable example.

We consider a two-parameter family of asymmetric exclusion processes for particles living on a continuous one-dimensional space. Using the Bethe ansatz, the exact solution to the master equation, and from that the drift and the diffusion rate in the two particle sector, are obtained.