Temporal Behavior of Local Characteristics in Complex Networks with Preferential Attachment-Based Growth

Abstract

The study of temporal behavior of local characteristics in complex growing networks makes it possible to more accurately understand the processes caused by the development of interconnections and links between parts of the complex system that occur as a result of its growth. The spatial position of an element of the system, determined on the basis of connections with its other elements, is constantly changing as the result of these dynamic processes. In this paper, we examine two non-stationary Markov stochastic processes related to the evolution of Barabási–Albert networks: the first describes the dynamics of the degree of a fixed node in the network, and the second is related to the dynamics of the total degree of its neighbors. We evaluate the temporal behavior of some characteristics of the distributions of these two random variables, which are associated with higher-order moments, including their variation, skewness, and kurtosis. The analysis shows that both distributions have a variation coefficient close to 1, positive skewness, and a kurtosis greater than 3. This means that both distributions have huge standard deviations that are of the same order of magnitude as the expected values. Moreover, they are asymmetric with fat right-hand tails.

Keywords

• network analysis
• complex networks
• Markov process
• stochastic equation
• preferential attachment model
• network evolution