Statistical Mechanics of Directed Networks

Abstract

Directed networks are essential for representing complex systems, capturing the asymmetry of interactions in fields such as neuroscience, transportation, and social networks. Directionality reveals how influence, information, or resources flow within a network, fundamentally shaping the behavior of dynamical processes and distinguishing directed networks from their undirected counterparts. Robust null models are crucial for identifying meaningful patterns in these representations, yet designing models that preserve key features remains a significant challenge. One such critical feature is reciprocity, which reflects the balance of bidirectional interactions in directed networks and provides insights into the underlying structural and dynamical principles that shape their connectivity. This paper introduces a statistical mechanics framework for directed networks, modeling them as ensembles of interacting fermions. By controlling the reciprocity and other network properties, our formalism offers a principled approach to analyzing directed network structures and dynamics, introducing new perspectives and models and analytical tools for empirical studies.

Keywords: Fermi statistics; complex networks; directed networks; maximum entropy; reciprocity.

Figure 1

Possible fermionic states between a pair of nodes i and j, and their associated energies. The solid arrow indicates the presence of a directed link and the dashed arrow an empty state. When the two fermions simultaneously occupy the two states, $i\to j$ and $j\to i$, the total energy includes a correction $\Delta {\epsilon }_{ij}$ added to the sum of the energies of the partially occupied states.

Figure 2

Reciprocity of the interacting directed ${\mathbb{S}}^d$ model for fully correlated in- and out-energies, as a function of $\Delta \epsilon$. Different curves correspond to different temperatures ${\beta }^{-1}$.