Measuring the variability of local characteristics in complex networks: Empirical and analytical analysis

We examine the dynamics for the average degree of a node's neighbors in complex networks. It is a Markov stochastic process, and at each moment of time, this quantity takes on its values in accordance with some probability distribution. We are interested in some characteristics of this distribution: its expectation and its variance, as well as its coefficient of variation. First, we look at several real communities to understand how these values change over time in social networks. The empirical analysis of the behavior of these quantities for real networks shows that the coefficient of variation remains at high level as the network grows. This means that the standard deviation and the mean degree of the neighbors are comparable. Then, we examine the evolution of these three quantities over time for networks obtained as simulations of one of the well-known varieties of the Barabási-Albert model, the growth model with nonlinear preferential attachment (NPA) and a fixed number of attached links at each iteration. We analytically show that the coefficient of variation for the average degree of a node's neighbors tends to zero in such networks (albeit very slowly). Thus, we establish that the behavior of the average degree of neighbors in Barabási-Albert networks differs from its behavior in real networks. In this regard, we propose a model based on the NPA mechanism with the rule of random number of edges added at each iteration in which the dynamics of the average degree of neighbors is comparable to its dynamics in real networks.