The Linial-Meshulam complex model is a natural higher dimensional analog of the Erdős-Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present article aims to extend that result to more general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled, we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.

We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higherdimensional counterpart of Frieze's ζ(3)-limit theorem for the Erdős-Rényi graph process. The main results include solutions to questions posed in an earlier study by Hiraoka and Shirai about the Linial-Meshulam complex process and the random clique complex process. One of the key elements of the arguments is a new upper bound on the Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links. This bound can be regarded as a quantitative version of the cohomology vanishing theorem.2010 Mathematics Subject Classification. Primary 05C80, 60D05; Secondary 55U10, 05E45, 60C05. Key Words and Phrases. Linial-Meshulam complex process, random clique complex process, multiparameter random simplicial complex, lifetime sum, Betti number. K n (t) decreases, and β 0 (K n (t)) denotes the zeroth (reduced) Betti number of K n (t), that is, the number of connected components of K n (t) minus one. This type of relation holds for a general increasing family of graphs. Applying this formula and analyzing β 0 (K n (t)) in detail, Frieze [7] obtained the following significant result about the behavior of W n .and for any ε > 0,Recently, there has been a growing interest in studying random simplicial complexes as a higher-dimensional generalization of random graphs. Since an Erdős-Rényi graph can be regarded as a one-dimensional random simplicial complex, and graph connectivity can be equivalently described as the vanishing of the zeroth (reduced) homology, it is natural to seek a higher-dimensional analogue to the theory of Erdős-Rényi's G(n, p) model. The d-Linial-Meshulam model [14] and the random clique complex model [12] are typical models of this type. The d-Linial-Meshulam model Y d (n, p) is defined as the distribution of d-dimensional random simplicial complexes with n vertices and the complete (d−1)-dimensional skeleton such that each d-simplex is placed with independent probability p. The random clique complex model C(n, p) is defined as the distribution of the clique complex of the Erdős-Rényi graph that follows G(n, p). Here, given a graph G, its clique complex Cl(G) is defined as the maximal simplicial complex among those for which the one-dimensional skeletons are equal to G. Linial, Meshulam, and Wallach [14,16] exhibited the threshold for the vanishing of the (d − 1)-th homology for the d-Linial-Meshulam model, which is analogous to the connectivity threshold of the Erdős-Rényi graph. Later, Kahle [13] obtained similar results for the random clique complex model.Along another line, Hiraoka and Shirai [10] obtained a higher-dimensional analogue of (1.1) in the context of the theory of persistent homology. Persistent homologies can describe the topological features of a filtration (i.e., an increasing family of simplicial complexes; see, e.g., [3,17]). In par...

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respetively, for finite metric spaces, first introduced by Leinster and Hepworth-Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behavior is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.