Growing scale-free simplices.

The past two decades have seen significant successes in our understanding of complex networked systems, from the mapping of real-world social, biological and technological networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions, captured by dyadic links, and provide limited insight into higher-order structure, in which a group of several components represents the basic interaction unit. Such multi-component interactions can only be grasped through simplicial complexes, which have recently found applications in social and biological contexts, as well as in engineering and brain science. What, then, are the generative models recovering the patterns observed in real-world simplicial complexes? Here we introduce, study, and characterize a model to grow simplicial complexes of order two, i.e. nodes, links and triangles, that yields a highly flexible range of empirically relevant simplicial network ensembles. Specifically, through a combination of preferential and/or non preferential attachment mechanisms, the model constructs networks with a scale-free degree distribution and an either bounded or scale-free generalized degree distribution - the latter accounting for the number of triads surrounding each link. Allowing to analytically control the scaling exponents we arrive at a highly general scheme by which to construct ensembles of synthetic complexes displaying desired statistical properties.