Reciprocity in directed networks

TL;DR

This paper analyzes the statistical properties of reciprocity in directed networks using concepts from statistical physics, exploring entropy and free energy across different ensembles and extending to complex reciprocal structures.

Contribution

It introduces a comprehensive statistical physics framework for modeling reciprocity in directed networks, including new ensemble analyses and extensions to complex reciprocal motifs.

Findings

Derived limiting entropy and free energy densities for various ensembles.

Analyzed sparse network cases in the grand canonical ensemble.

Discussed extensions to reciprocal triangles and star densities.

Abstract

Reciprocity is an important characteristic of directed networks and has been widely used in the modeling of World Wide Web, email, social, and other complex networks. In this paper, we take a statistical physics point of view and study the limiting entropy and free energy densities from the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble whose sufficient statistics are given by edge and reciprocal densities. The sparse case is also studied for the grand canonical ensemble. Extensions to more general reciprocal models including reciprocal triangle and star densities will likewise be discussed.