Subgraphs and network motifs in geometric networks

TL;DR

This paper analytically investigates the occurrence of subgraphs and network motifs in geometric networks, revealing which motifs arise from geometric constraints and which are likely due to other biological or social factors.

Contribution

The study provides analytical solutions for subgraph counts in geometric networks, including models with arbitrary degree sequences and directional biases, and identifies invariant motif ratios.

Findings

Analytical formulas for 3 and 4-node subgraphs in geometric networks.

Invariant measures such as feedback to feed-forward loop ratios.

Many real-world motifs are not explained solely by geometric constraints.

Abstract

Many real-world networks describe systems in which interactions decay with the distance between nodes. Examples include systems constrained in real space such as transportation and communication networks, as well as systems constrained in abstract spaces such as multivariate biological or economic datasets and models of social networks. These networks often display network motifs: subgraphs that recur in the network much more often than in randomized networks. To understand the origin of the network motifs in these networks, it is important to study the subgraphs and network motifs that arise solely from geometric constraints. To address this, we analyze geometric network models, in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes. We present analytical solutions for the numbers of all 3 and 4-node subgraphs, in both directed and non-directed geometric networks. We also analyze geometric networks with arbitrary degree sequences, and models with a field that biases for directed edges in one direction. Scaling rules for scaling of subgraph numbers with system size, lattice dimension and interaction range are given. Several invariant measures are found, such as the ratio of feedback and feed-forward loops, which do not depend on system size, dimension or connectivity function. We find that network motifs in many real-world networks, including social networks and neuronal networks, are not captured solely by these geometric models. This is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information-processing functions.