Related papers: The correlation decay (CD) tree and strong spatial…
We investigate hierarchical structure in various complex systems according to Minimum Spanning Tree methods. Firstly, we investigate stock markets where the graphis obtained from the matrix of correlations coefficient computed between all…
We study the minimum spanning tree distribution on the space of spanning trees of the $n$-by-$n$ grid for large $n$. We establish bounds on the decay rates of the probability of the most and the least probable spanning trees as…
Strong spatial mixing (SSM) is an important quantitative notion of correlation decay for Gibbs distributions arising in statistical physics, probability theory, and theoretical computer science. A longstanding conjecture is that the uniform…
The collection of all the strongly connected components in a directed graph, among each cluster of which any node has a path to another node, is a typical example of the intertwining structure and dynamics in complex networks, as its…
In a recent paper the last author proved that absence of complex zeros of the partition function of the hard-core model near a parameter $\lambda>0$ implies a form of correlation decay called strong spacial mixing. In this paper we…
We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve…
Real-world networks often exhibit strong transitivity with nontrivial local clustering spectra and degree correlations. Such features are not easily modeled in tractable network models, creating an obstacle to the theoretical understanding…
The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth. In this paper we survey results related to this concept…
Random intersection graphs model networks with communities, assuming an underlying bipartite structure of groups and individuals, where these groups may overlap. Group memberships are generated through the bipartite configuration model.…
Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their…
Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local…
Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging…
We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…
We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…
The goal of this paper is to provide a general purpose result for the coupling of exploration processes of random graphs, both undirected and directed, with their local weak limits when this limit is a marked Galton-Watson process. This…
Probabilistic graphical models with frustration exhibit rugged energy landscapes that trap iterative optimization dynamics. These landscapes are shaped not only by local interactions, but crucially also by the global loop structure of the…
Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize…
We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing…
Recent work has established that large informatics graphs such as social and information networks have non-trivial tree-like structure when viewed at moderate size scales. Here, we present results from the first detailed empirical…
We consider correlation decay in the hard-core model with fugacity $\lambda$ on a rooted tree $T$ in which the arity of each vertex is independently Poisson distributed with mean $d$. Specifically, we investigate the question of which…