Two-walks degree assortativity in graphs and networks

Abstract

Abstract. Degree ssortativity is the tendency for nodes of high degree (resp. low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is usually quantified by the Pearson correlation coefficient of the degree-degree correlation. Here we extend this concept to account for the effect of second neighbours to a given node in a graph. That is, we consider the two-walks degree of a node as the sum of all the degrees of its adjacent nodes. The two-walks degree assortativity of a graph is then the Pearson correlation coefficient of the two-walks degree-degree correlation. We found here analytical expression for this two-walks degree assortativity index as a function of contributing subgraphs. We then study all the 261,000 connected graphs with 9 nodes and observe the existence of assortative-assortative and disassortative-disassortative graphs according to degree and two-walks degree, respectively. More surprinsingly, we observe a class of graphs which are degree disassortative and two-walks degree assortative. We explain the existence of some of these graphs due to the presence of certain topological features, such as a node of low-degree connected to high-degree ones. More importantly, we study a series of 49 real-world networks, where we observe the existence of the disassortative-assortative class in several of them. In particular, all biological networks studied here were in this class. We also conclude that no graphs/networks are possible with assortative-disassortative structure.

General formulation of long-range degree correlations in complex networks

“…The average lth neighbor degree k l (k) is the average degree of nodes separated by l from a node of degree k. They found that social networks exhibit disassortative degree correlations on long-range scales, while nonsocial networks do not indicate such a tendency. The two-walks degree assortativity proposed by Allen-Perkins et al [50] is another type of assortativity measure be-yond nearest neighbors. This quantity is defined as the Pearson correlation coefficient of the sum of the nearestneighbor degrees of adjacent nodes, which reflects second neighbor degree correlations.…”

Conformity: A Path-Aware Homophily Measure for Node-Attributed Networks

“…Since the only direct neighborhood (or egonetwork) of nodes can not be taken into consideration due to its limited expressive power (inherited in the scale-free-like degree distribution of complex networks), the issue is to define connectivity boundednesses able to circumscribe those nodes whose importance is fundamental in the assortative attitude measurement of a target one. While some lines of research focused on degree assortativity [13] (extended to cope with higher-order notions of node neighborhood such as a two-walks degree correlation [14] or transsortativity [15]), the node-attributed counterpart of the problem has not received much of attention.…”

Modelos de redes adaptativas críticamente auto-organizadas : aplicación al estudio de modelos de población

“….102 6.1. Asortatividad de grado, r k , y asortatividad de dos-pasos, rk, calculadas para las redes del modelo de [APGP16], empleando la metodología de [APPE17], cuando el sistema se halla en un régimen estacionario estadístico. Los datos se han obtenido al simular el sistema durante t = 5000 unidades de macro-tiempo, con C = 3. .…”

“…A continuación se reproducen algunos de sus principales hallazgos, ya que éstos constituyen el punto de partida de [APPE17], presentado en el capítulo 4.…”

“…Finalmente, atendiendo a los anteriores resultados, el estudio de [APPE17] revela dos características importantes de las redes del modelo de [APGP16]:…”