Influence of reciprocal arcs on the degree distribution and degree correlations

Reciprocal arcs represent the lowest order cycle possible to find in directed graphs without self-loops. Representing also a measure of feed-back between vertices, it is interesting to understand how reciprocal arcs influence other properties of complex networks. In this paper we focus on influence of reciprocal arcs on vertex degree distribution and degree correlations. We show that there is a fundamental difference between properties observed on the static network compared to the properties of networks which are obtained by simple evolution mechanism driven by reciprocity. We also present a way to statistically infer the portion of reciprocal arcs which can be explained as a consequence of feed-back process on the static network. In the rest of the paper the influence of reciprocal arcs on a model of growing network is also presented. It is shown that our model of growing network nicely interpolates between BA model for undirected and the BA model for directed networks.

💡 Research Summary

The paper investigates how reciprocal arcs—pairs of opposite‑direction edges between two vertices—affect two fundamental structural properties of directed networks: the degree distribution and degree–degree correlations. Reciprocal arcs are the simplest possible directed cycles (aside from self‑loops) and can be interpreted as a quantitative measure of feedback between nodes. The authors distinguish between static networks, where the set of reciprocal arcs is already present, and evolving networks, where reciprocal arcs are generated by a growth mechanism that explicitly incorporates feedback.

Static‑network analysis
Using the adjacency matrix (A) of a directed graph, the authors define a reciprocal‑arc matrix (S = A \circ A^{\top}) (element‑wise product) and the reciprocal‑arc fraction
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