We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an existing simplex is uniform among simplices of dimension $<m$, the number $S_d$ of simplices of any admissible dimension $d\leq m$ is an asymptotically self-averaging random variable. This feature allows us to determine the asymptotic growth law of the average of $S_d$ when the number of vertices diverges. We also probe the degree distribution, examine the probabilities of various extreme outcomes, and analyze the characteristics of the first vertex.
The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique number of geometric graphs or scan statistics. We extend existing results in the...
We introduce an incidence-based random walk on the edges of a random two-dimensional simplicial complex with a complete $1$-skeleton and independently retained triangular faces. The dynamics combine two transport channels, one mediated by vertices and the other by triangular faces, through an effective transition opera...
C. T. Martínez-Martínez, Francisco J Sevilla· 0 citations
A graphic arrangement $\A_G$ associated with a simple graph $G$ is a classical and well-studied object in the theory of hyperplane arrangements. In this note, we show that, for a connected graph $G$, a slight modification of the logarithmic vector field $D(\A_G)$ of $\A_G$ is isomorphic to the face ring of a certain si...
This work provides a perturbative approximation scheme recovering the modulation parameter between pure random walk and teleportation mode acting as the expansion parameter for simplicial complexes: a specific type of geometric and algebraic structures that encapsulates higher-order interactions.
Diego Febbe, Duccio Fanelli, Gianluca Peri et al.· 0 citations
We analyse topology of random simplicial complexes in the medial regime. We show that these complexes are highly connected and have homotopy type of iterated suspensions. One of our main tools is a new combinatorial criterion for high connectivity of simplicial complexes, which is more flexible than conicity. We show t...
J. Barmak, M. Farber· 0 citations
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