Intertwined Hamiltonians in Two Dimensional Curved Spaces

The problem of intertwined Hamiltonians in two dimensional curved spaces is investigated. Explicit results are obtained for Euclidean plane,Minkowski plane, Poincar{\' e} half plane ($AdS_2$), de Sitter Plane ($dS_2$), sphere, and torus. It is shown that the intertwining operator is related to the Killing vector fields and the isometry group of corresponding space. It is shown that the intertwined potentials are closely connected to the integral curves of the Killing vector fields. Two problems of considered as applications of the formalism presented in the paper. The first one is the problem of Hamiltonians with equispaced energy levels and the second one is the problem of Hamiltonians whose spectrum are like the spectrum of a free particle.Comment: To appear in Annals of Physic

Fixation time in evolutionary graphs: A mean-field approach

Using an analytical method we calculate average conditional fixation time of mutants in a general graph-structured population of two types of species. The method is based on Markov chains and uses a mean-field approximation to calculate the corresponding transition matrix. Analytical results are compared with the results of simulation of the Moran process on a number of network structures

Community detection based on "clumpiness" matrix in complex networks

The "clumpiness" matrix of a network is used to develop a method to identify its community structure. A "projection space" is constructed from the eigenvectors of the clumpiness matrix and a border line is defined using some kind of angular distance in this space. The community structure of the network is identified using this borderline and/or hierarchical clustering methods. The performance of our algorithm is tested on some computer-generated and real-world networks. The accuracy of the results is checked using normalized mutual information. The effect of community size heterogeneity on the accuracy of the method is also discussed.Comment: 18 pages and 13 figure

Phase synchronization on scale-free and random networks in the presence of noise

In this work we investigate the stability of synchronized states for the Kuramoto model on scale-free and random networks in the presence of white noise forcing. We show that for a fixed coupling constant, the robustness of the globally synchronized state against the noise is dependent on the noise intensity on both kinds of networks. At low noise intensities the random networks are more robust against losing the coherency but upon increasing the noise, at a specific noise strength the synchronization among the population vanishes suddenly. In contrast, on scale-free networks the global synchronization disappears continuously at a much larger critical noise intensity respect to the random networks

Path to fixation of evolutionary processes in graph-structured populations

We study the spreading of a single mutant in graph-structured populations with a birth-death update rule. We use a mean-field approach and a Markov chain dynamics to investigate the effect of network topology on the path to fixation. We obtain approximate analytical formulas for average time versus the number of mutants in the fixation process starting with a single mutant for several network structures, namely, cycle, complete graph, two- and three-dimensional lattices, random graph, regular graph, Watts–Strogatz network, and Barabási–Albert network. In the case of the cycle and complete graph, the results are accurate and in line with the results obtained by other methods. In the case of two- and three-dimensional lattice structures, some efforts are made in other studies to provide an analytical justification for simulation results of the evolutionary process, but they can explain just the onset of the fixation process, not the whole process. The results of the analytical approach of the present paper are well fitted to the simulation results throughout the whole fixation process. Moreover, we analyze the dynamics of evolution for a number of complex structures, and in all cases, we obtain analytical results which are in good agreement with simulations. Our results may shed some light on the process of fixation during the whole path to fixation