Scale-free network on a vertical plane

Limit theorems for random spatial drainage networks

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of R d , d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1) d . The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

Limit theorems for random spatial drainage networks

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of Rd, d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1)d. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

Impacts of preference and geography on epidemic spreading

We investigate the standard susceptible-infected-susceptible model on a random network to study the effects of preference and geography on diseases spreading. The network grows by introducing one random node with m links on a Euclidean space at unit time. The probability of a new node i linking to a node j with degree k(j) at distance d(ij) from node i is proportional to k(j)(A)/d(ij)(B), where A and B are positive constants governing preferential attachment and the cost of the node-node distance. In the case of A=0 , we recover the usual epidemic behavior with a critical threshold below which diseases eventually die out. Whereas for B=0 , the critical behavior is absent only in the condition A=1. While both ingredients are proposed simultaneously, the network becomes robust to infection for larger A and smaller B.

Spatially embedded random networks

Many real-world networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domain-specific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real-world networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of small-world spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples.

Numerical evaluation of the upper critical dimension of percolation in scale-free networks

We propose numerical methods to evaluate the upper critical dimension d(c) of random percolation clusters in Erdös-Rényi networks and in scale-free networks with degree distribution P(k) approximately k(-lambda), where k is the degree of a node and lambda is the broadness of the degree distribution. Our results support the theoretical prediction, d(c) = 2(lambda - 1)(lambda - 3) for scale-free networks with 3 < lambda < 4 and d(c) = 6 for Erdös-Rényi networks and scale-free networks with lambda > 4 . When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d(c) = 6 for all lambda > 2 . Our method also yields a better numerical evaluation of the critical percolation threshold p(c) for scale-free networks. Our results suggest that the finite size effects increases when lambda approaches 3 from above.

Geographical networks evolving with an optimal policy

In this article we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a node, having randomly assigned coordinates in a 1x1 square, is added and connected to a previously existing node i, which minimizes the quantity ri2/kialpha, where ri is the geographical distance, ki the degree, and alpha a free parameter. The degree distribution obeys a power-law form when alpha=1, and an exponential form when alpha=0. When alpha is in the interval (0, 1), the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge length will sharply increase when alpha exceeds the critical value alphac=1, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.

Phase transitions in an Ising model on a Euclidean network

A one-dimensional network on which there are long-range bonds at lattice distances l>1 with the probability P(l) proportional to l(-delta) has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbors apart from their nearest neighbors for 0<or=delta<2. It is observed that there is a finite temperature phase transition in the entire range. For 0<or=delta<1, finite-size scaling behavior of various quantities are consistent with mean-field exponents while for 1<or=delta<or=2, the exponents depend on delta. The results are discussed in the context of earlier observations on the topology of the underlying network.