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.
Highlights
Many technological, biological, and social systems can be represented by underlying complex networks
Numerous examples of new successful technology companies show that the temporal behavior of network elements is very diverse: elements that appeared much later can take a more dominant position in the complex system than elements that appeared in the early stages of the system development
We study complex systems whose growth is based on the use of the preferential attachment mechanism and show that while the “first-moving” effect is performed, on average, in relation to the node degree, the temporal behavior of this quantity has an important feature: its coefficient of variation is close to
Summary
Biological, and social systems can be represented by underlying complex networks. The higher-order moment-related quantities, such as variation, the coefficient of asymmetry (skewness), and kurtosis, allow one to clearer understand the dynamic behavior of the degree of a vertex and to more definitely characterize the underlying stochastic process Another local characteristic of a node (in addition to its degree), which is of interest, is the sum of the degrees of all its neighbors. The recent paper [26] studies the behavior in the limit of the degree of an individual node in the Barabási–Albert model, and it shows that after some scaling procedures, this stochastic process converges to a Yule process (in distribution) Based on this findings, the paper examines why the limit degree distribution of a node picked uniformly at random (as the network grows to infinity) matches the limit distribution of the number of species chosen randomly in a Yule model (as time goes to infinity). We expand the study with the analysis of the total degree of neighbors of node
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