Weighted networks as randomly reinforced urn processes

Abstract

We analyze weighted networks as randomly reinforced urn processes, in which the edge-total weights are determined by a reinforcement mechanism. We develop a statistical test and a procedure based on it to study the evolution of networks over time, detecting the "dominance" of some edges with respect to the others and then assessing if a given instance of the network is taken at its steady state or not. Distance from the steady state can be considered as a measure of the relevance of the observed properties of the network. Our results are quite general, in the sense that they are not based on a particular probability distribution or functional form of the random weights. Moreover, the proposed tool can be applied also to dense networks, which have received little attention by the network community so far, since they are often problematic. We apply our procedure in the context of the International Trade Network, determining a core of "dominant edges. [10], and others, which can be efficiently described by a network structure, where the nodes are the system entities and the edges represent the relations between them. All the models that produce complex networks are based either on preferential attachment (or copying mechanism) or on a fitness (hidden variables) microscopic mechanism. Unfortunately, no statistical method has been developed in order to assess the relevance of both experimental data and model simulations. In this paper, we present a model of network evolution based on randomly reinforced urn (RRU) processes [11][12][13][14]. In our model we map the weight associated to a given edge with the number of balls of a given color, which are added in an urn so that at a given time step the probability of picking an edge (color) depends on the total weight associated with it until that time. At each time step we first extract an edge (color) with probability proportional to its total weight and then we associate to it a random weight (number of added balls) which increases its total weight. This results in a preferential attachment (PA) rule for edges with random weights. Hence, although our model can be considered as a particular refinement in the class of the PA mechanisms, its novelty is in the connection that we can establish between complex networks and RRU models [15]. RRU theory allows us to develop a procedure for the detection of the "dominant edges" in the evolution of a weighted network and for an evaluation of the distance from the steady state of the network, in the sense that we can assess if the structure observed at a given time is what we can expect at the steady state or not. The novelty of our methodology is also related to its applicability to dense, weighted networks (a situation often problematic both for modeling and for randomization) [16].We consider a system with N vertices and Lpotential edges (directed or not). Hereafter we indicate the various edges by the index (with ∈ [1,L]). Our model defines a weighted adjacency matrix W t for every time step t, where the generic element w t =...

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