Related papers: A Characterization of Level-k Realizability for Cl…

Due to the increasing discovery and implementation of networks within all disciplines of life, the study of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a…

In the graph clustering problem with a planted solution, the input is a graph on $n$ vertices partitioned into $k$ clusters, and the task is to infer the clusters from graph structure. A standard assumption is that clusters induce…

The concept of nestedness, in particular for ecological and economical networks, has been introduced as a structural characteristic of real interacting systems. We suggest that the nestedness is in fact another way to express a mesoscale…

Grouping the nodes of a graph into clusters is a standard technique for studying networks. We study a problem where we are given a directed network and are asked to partition the graph into a sequence of coherent groups. We assume that…

In the present paper we discuss the clustering procedure in the case where instead of a single metric we have a family of metrics. In this case we can obtain a partially ordered graph of clusters which is not necessarily a tree. We discuss…

Mader conjectured in 1979 that an average degree of at least $3k-1$ in a graph is sufficient for the existence of a $(k+1)$-connected subgraph. The following minimum degree analogue holds: Every graph with minimum degree at least $3k-1$…

An intense activity is nowadays devoted to the definition of models capturing the properties of complex networks. Among the most promising approaches, it has been proposed to model these graphs via their clique incidence bipartite graphs.…

We propose a novel method to cluster gene networks. Based on a dissimilarity built using correlation structures, we consider networks that connect all the genes based on the strength of their dissimilarity. The large number of genes require…

Despite the huge recent breakthroughs in neural networks (NNs) for artificial intelligence (specifically deep convolutional networks) such NNs do not achieve human-level performance: they can be hacked by images that would fool no human and…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

Extensive studies have been done to understand the principles behind architectures of real networks. Recently, evidences for hierarchical organization in many real networks have also been reported. Here, we present a new hierarchical model…

A wiring diagram is a labeled directed graph that represents an abstract concept such as a temporal process. In this article, we introduce the notion of a quasi-skeleton wiring diagram graph, and prove that quasi-skeleton wiring diagram…

Core-periphery (CP) structure is an important meso-scale network property where nodes group into a small, densely interconnected {core} and a sparse {periphery} whose members primarily connect to the core rather than to each other. While…

The objective of clustering is to discover natural groups in datasets and to identify geometrical structures which might reside there, without assuming any prior knowledge on the characteristics of the data. The problem can be seen as…

A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there is a subgroup $K\leq G$ such that $HK\unlhd G$ and $H\cap K$ is contained in $H_G$, the core of $H$ in $G$. We characterize the solvability of finite groups $G$ with…

Community structure is a commonly observed feature of real networks. The term refers to the presence in a network of groups of nodes (communities) that feature high internal connectivity, but are poorly connected between each other. Whereas…

A class of groups C is root in a sense of K. W. Gruenberg if it is closed under taking subgroups and satisfies the Gruenberg condition: for any group X and for any subnormal sequence Z \leqslant Y \leqslant X with factors in C, there exists…

Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are…

Network data sets are often constructed by some kind of thresholding procedure. The resulting networks frequently possess properties such as heavy-tailed degree distributions, clustering, large connected components and short average…

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…