Growing scale-free simplices

Abstract

The past two decades have seen significant successes in our understanding of networked systems, from the mapping of real-world networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions and provide limited insight into higher-order structures. Such multi-component interactions can only be grasped through simplicial complexes, which have recently found applications in social, technological, and biological contexts. Here we introduce a model to grow simplicial complexes of order two, i.e., nodes, links, and triangles, that can be straightforwardly extended to structures containing hyperedges of larger order. Specifically, through a combination of preferential and/or nonpreferential attachment mechanisms, the model constructs networks with a scale-free degree distribution and an either bounded or scale-free generalized degree distribution. We arrive at a highly general scheme with analytical control of the scaling exponents to construct ensembles of synthetic complexes displaying desired statistical properties.

Full reconstruction of simplicial complexes from binary time-series data

“…High-order interactions are nonetheless ubiquitous in complex networks and its importance has been gradually recognized with an accumulating interest, eventually generating an explosive growth of research recently [18][19][20][21][22][23][24]. The structure of networks with high-order connections, also known as simplicial complexes, are represented by tensors of high orders.…”

“…With the initial conditions for P 0 j→i and ε i , the values of P 0 j→i and ε i can be obtained by iterating Eqs. (21)(22)(23)(24) until convergence is achieved. It is worth noting that P 0 j→i is a probability and we need to determine the "approximate" neighbors of the node under reconstruction.…”

“…(f) The values of P 0 j→13 for node 13 obtained by iterating Eqs. (21)(22)(23)(24), and only nonzero values are shown. (g) Compressed data matrix that records only the columns in the data matrix S giving P 0 j→13 > 0.…”

“…For example, the drive-response and adaptive synchronization methods use data from continuous-time nonlinear coupled systems [2,3,5,11], while the maximum likelihood estimation method is suitable for data from discrete-time dynamics [13,14,16]. In this paper, motivated by the fact that high-order networks have become a stateof-the-art subfield of research in network science [18][19][20][21][22][23][24], we develop a reconstruction framework for finding from time-series data network topology with high-order interactions.…”

Higher-Order Networks

“…In Ref. [107] a further variation of the simplicial NGF model is considered, allowing new d-dimensional simplices to be attached to a face ˛chosen according to uniform attachment, (i.e. according to … OE0 ˛) with probability q and according to generalized preferential attachment (i.e.…”