  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.
