Critical Points of Correlated Percolation in a Gravitational Link-adding Network Model

Motivated by the importance of geometric information in real systems, a new model for long-range correlated percolation in link-adding networks is proposed with the connecting probability decaying with a power-law of the distance on the two-dimensional(2D) plane. By overlapping it with Achlioptas process, it serves as a gravity model which can be tuned to facilitate or inhibit the network percolation in a generic view, cover a broad range of thresholds. Moreover, it yields a set of new scaling relations. In the present work, we develop an approach to determine critical points for them by simulating the temporal evolutions of type-I, type-II and type-III links(chosen from both inter-cluster links, an intra-cluster link compared with an inter-cluster one, and both intra-cluster ones, respectively) and corresponding average lengths. Numerical results have revealed objective competition between fractions, average lengths of three types of links, verified the balance happened at critical points. The variation of decay exponents $a$ or transmission radius $R$ always shifts the temporal pace of the evolution, while the steady average lengths and the fractions of links always keep unchanged just as the values in Achlioptas process. Strategy with maximum gravity can keep steady average length, while that with minimum one can surpass it. Without the confinement of transmission range, $bar{l} to infty$ in thermodynamic limit, while $bar{l}$ does not when with it. However, both mechanisms support critical points. In two-dimensional free space, the relevance of correlated percolation in link-adding process is verified by validation of new scaling relations with various exponent $a$, which violates the scaling law of Weinribs.

Scaling of critical connectivity of mobile ad hoc communication networks

In this paper, critical global connectivity of mobile ad hoc communication networks (MAHCN) is investigated. We model the two-dimensional plane on which nodes move randomly with a triangular lattice. Demanding the best communication of the network, we account the global connectivity $eta$ as a function of occupancy $sigma$ of sites in the lattice by mobile nodes. Critical phenomena of the connectivity for different transmission ranges $r$ are revealed by numerical simulations, and these results fit well to the analysis based on the assumption of homogeneous mixing . Scaling behavior of the connectivity is found as $eta sim f(R^{beta}sigma)$, where $R=(r-r_{0})/r_{0}$, $r_{0}$ is the length unit of the triangular lattice and $beta$ is the scaling index in the universal function $f(x)$. The model serves as a sort of site percolation on dynamic complex networks relative to geometric distance. Moreover, near each critical $sigma_c(r)$ corresponding to certain transmission range $r$, there exists a cut-off degree $k_c$ below which the clustering coefficient of such self-organized networks keeps a constant while the averaged nearest neighbor degree exhibits a unique linear variation with the degree k, which may be useful to the designation of real MAHCN.