Motifs in Triadic Random Graphs based on Steiner Triple Systems
arXiv:1304.2924 · doi:10.1103/PhysRevE.88.022805
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.
13 pages, 10 figures
References in corpus (15)
- Emergence of scaling in random networks
- Statistical mechanics of complex networks
- The structure and function of complex networks
- Stochastic blockmodels and community structure in networks
- The Yeast Cell-Cycle Network Is Robustly Designed
- Growing Scale-Free Networks with Tunable Clustering
- The complex network of global cargo ship movements
- Random graphs with clustering
- Highly clustered scale-free networks
- Tuning clustering in random networks with arbitrary degree distributions
- Solution for the properties of a clustered network
- Evolving Clustered Random Networks
- Synchronization Properties of Network Motifs
- The interplay of microscopic and mesoscopic structure in complex networks
- Network motifs come in sets: correlations in the randomization process