Motifs in Triadic Random Graphs based on Steiner Triple Systems

TL;DR

This paper introduces a new class of generative network models based on Steiner Triple Systems and exponential random graphs, enabling the study of triadic motifs and their correlations in complex networks.

Contribution

It combines Steiner Triple Systems with ERGMs to create models that generate networks with specific triadic motif profiles, addressing limitations of previous models.

Findings

Models produce networks with non-trivial triadic Z-score profiles.

Identifies inherent correlations between triad pattern abundances.

Provides analytical degree distributions for the proposed models.

Abstract

Conventionally, pairwise relationships between nodes are considered to be the fundamental building blocks of complex networks. However, over the last decade the overabundance of certain sub-network patterns, so called motifs, has attracted high attention. It has been hypothesized, these motifs, instead of links, serve as the building blocks of network structures. Although the relation between a network's topology and the general properties of the system, such as its function, its robustness against perturbations, or its efficiency in spreading information is the central theme of network science, there is still a lack of sound generative models needed for testing the functional role of subgraph motifs. Our work aims to overcome this limitation. We employ the framework of exponential random graphs (ERGMs) to define novel models based on triadic substructures. The fact that only a small portion of triads can actually be set independently poses a challenge for the formulation of such models. To overcome this obstacle we use Steiner Triple Systems (STS). These are partitions of sets of nodes into pair-disjoint triads, which thus can be specified independently. Combining the concepts of ERGMs and STS, we suggest novel generative models capable of generating ensembles of networks with non-trivial triadic Z-score profiles. Further, we discover inevitable correlations between the abundance of triad patterns, which occur solely for statistical reasons and need to be taken into account when discussing the functional implications of motif statistics. Moreover, we calculate the degree distributions of our triadic random graphs analytically.