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 β. We show that, by adjusting exponents α and β, 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 β=-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.

By considering the master equation of the partially asymmetric diffusion process on a one-dimensional lattice, the most general boundary condition (i.e. interactions) for the multi-species reaction-diffusion processes is considered. Resulting system has various interactions including diffusion to left and right, two-particle interactions AαA β → AγA δ and the extended n-particle drop-push interactions to left and right. We obtain three distinct new models. The conditions on reaction rates to ensure the solvability of the resulting models are obtained. The two-particle conditional probabilities are calculated exactly. * alimohmd@ut.ac.ir

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In this study, we have obtained the exact solutions of the Schrödinger equation for a multi-layered quantum antidot (MLQAD) within the effective mass approximation and dielectric continuum model for the spherical symmetry. The MLQAD is nano-structured semiconductor system that consists of a spherical core (e.g. Ga 1−x Al x As) and a coated spherical shell (e.g. Ga 1−y Al y As) as the whole anti-dot is embedded inside a bulk material (e.g. GaAs). The dependence of the electron energy spectrum and its radial probability density on nano-system radius are studied. The numeric calculations and analysis of oscillator strength of intersubband quantum transition from the ground state into two first allowed excited states at the varying radius, for both the finite and infinite confining potential (CP) as well as constant shell thickness, are performed. It is shown that, in particular, the binding energy and the oscillator strength of the hydrogenic impurity of a MLQAD behave differently from that of a single-layered quantum antidot (SLQAD). For a MLQAD with finite core and shell CPs, the state energies and the oscillator strengths of the impurity are found to be dependent on the shell thickness. At the large core radius and very small shell thickness, our results are closer to respective values for a SLQAD that previously reported.

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We introduce the generalized rumor spreading model and investigate some properties of this model on different complex social networks. Despite pervious rumor models that both the spreader-spreader (SS) and the spreader-stifler (SR) interactions have the same rate α, we define α (1) and α (2) for SS and SR interactions, respectively. The effect of variation of α (1) and α (2) on the final density of stiflers is investigated. Furthermore, the influence of the topological structure of the network in rumor spreading is studied by analyzing the behavior of several global parameters such as reliability and efficiency. Our results show that while networks with homogeneous connectivity patterns reach a higher reliability, scale-free topologies need a less time to reach a steady state with respect the rumor.