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We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-Renyi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The results are in agreement with numerical simulations for a broad range of network sizes and connectivities. The average and standard deviation of the distribution are also obtained. In the case that the mean degree scales as $N^{alpha}$ with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance $d=lfloor 1/alpha rfloor$ from each other. The distribution of shortest path lengths between nodes of degree $m$ and the rest of the network is calculated. Its average is shown to be a monotonically decreasing function of $m$, providing an interesting relation between a local property and a global property of the network. The methodology presented here can be applied to more general classes of networks.
We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths:
We consider an Erdos-Renyi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N->infinity one obtains a graph of finite mean degree c. In this regim
We reconsider the problem of the critical behavior of a three-dimensional $O(m)$ symmetric magnetic system in the presence of random anisotropy disorder with a generic trimodal random axis distribution. By introducing $n$ replicas to average over dis
We study a model for a random walk of two classes of particles (A and B). Where both species are present in the same site, the motion of As takes precedence over that of Bs. The model was originally proposed and analyzed in Maragakis et al., Phys. Re
We calculate analytically the critical connectivity $K_c$ of Random Threshold Networks (RTN) for homogeneous and inhomogeneous thresholds, and confirm the results by numerical simulations. We find a super-linear increase of $K_c$ with the (average) a