相关论文: Structure of large random hypergraphs
Hypergraphs provide a fundamental framework for representing complex systems involving interactions among three or more entities. As empirical hypergraphs grow in size, characterizing their structural properties becomes increasingly…
A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct…
We use a tensor unfolding technique to prove a new identifiability result for discrete bipartite graphical models, which have a bipartite graph between an observed and a latent layer. This model family includes popular models such as…
Linear structural equation models, which relate random variables via linear interdependencies and Gaussian noise, are a popular tool for modeling multivariate joint distributions. These models correspond to mixed graphs that include both…
Sampling from combinatorial families can be difficult. However, complicated families can often be embedded within larger, simpler ones, for which easy sampling algorithms are known. We take advantage of such a relationship to describe a…
We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong…
An $(\alpha,\beta)$-ruling set of a graph $G=(V,E)$ is a set $R\subseteq V$ such that for any node $v\in V$ there is a node $u\in R$ in distance at most $\beta$ from $v$ and such that any two nodes in $R$ are at distance at least $\alpha$…
We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within…
Structural properties of large random maps and lambda-terms may be gleaned by studying the limit distributions of various parameters of interest. In our work we focus on restricted classes of maps and their counterparts in the…
A hypergraph $G=(V,E)$ is $(k,\ell)$-sparse if no subset $V'\subset V$ spans more than $k|V'|-\ell$ hyperedges. We characterize $(k,\ell)$-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
Hypergraphs are a generalization of graphs in which edges can connect any number of vertices. They allow the modeling of complex networks with higher-order interactions, and their spectral theory studies the qualitative properties that can…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…
Given a family of hypergraphs $\mathcal{H}$, we say that a hypergraph $\Gamma$ is $\mathcal{H}$-universal if it contains every $H \in \mathcal{H}$ as a subgraph. For $D, r \in \mathbb{N}$, we construct an $r$-uniform hypergraph with…
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $\alpha_c(\Delta)$ and provide (i) for $\alpha <…
We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance…
In recent decades, hypergraphs and their analysis through Topological Data Analysis (TDA) have emerged as powerful tools for understanding complex data structures. Various methods have been developed to construct hypergraphs -- referred to…
To learn (statistical) dependencies among random variables requires exponentially large sample size in the number of observed random variables if any arbitrary joint probability distribution can occur. We consider the case that sparse data…
We propose a generic framework to describe classical Ising-like models defined on arbitrary graphs. The energy spectrum is shown to be the Hadamard transform of a suitably defined sparse "coding" vector associated with the graph. We expect…