Related papers: Modularity of regular and treelike graphs
Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a…
We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound…
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…
We study large-scale, distributed graph clustering. Given an undirected graph, our objective is to partition the nodes into disjoint sets called clusters. A cluster should contain many internal edges while being sparsely connected to other…
The study of the sub-structure of complex networks is of major importance to relate topology and functionality. Many efforts have been devoted to the analysis of the modular structure of networks using the quality function known as…
Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases -…
Modularity was introduced as a measure of goodness for the community structure induced by a partition of the set of vertices in a graph. Then, it also became an objective function used to find good partitions, with high success.…
Traditionally, graph quality metrics focus on readability, but recent studies show the need for metrics which are more specific to the discovery of patterns in graphs. Cluster analysis is a popular task within graph analysis, yet there is…
Modularity is a parameter indicating the presence of community structure in the graph. Nowadays it lies at the core of widely used clustering algorithms. We study the modularity of the most classical random graph, binomial $G(n,p)$. In 2020…
The study of the topological structure of complex networks has fascinated researchers for several decades, and today we have a fairly good understanding of the types and reoccurring characteristics of many different complex networks.…
Real-data networks often appear to have strong modularity, or network-of-networks structure, in which subgraphs of various size and consistency occur. Finding the respective subgraph structure is of great importance, in particular for…
Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science.…
We prove that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, modularity tends to 1 as the number of edges tends to infinity.
Modularity is one of the most widely used quality measures for graph clusterings. Maximizing modularity is NP-hard, and the runtime of exact algorithms is prohibitive for large graphs. A simple and effective class of heuristics coarsens the…
A basic question in network community detection is how modular a given network is. This is usually addressed by evaluating the quality of partitions detected in the network. The Girvan-Newman (GN) modularity function is the standard way to…
Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured…
We consider Gallai's graph Modular Decomposition theory for network analytics. On the one hand, by arguing that this is a choice tool for understanding structural and functional similarities among nodes in a network. On the other, by…
We consider a large class of random geometric graphs constructed from samples $\mathcal{X}_n = \{X_1,X_2,\ldots,X_n\}$ of independent, identically distributed observations of an underlying probability measure $\nu$ on a bounded domain…
We present a compact matrix formulation of the modularity, a commonly used quality measure for the community division in a network. Using this formulation we calculate the density of modularities, a statistical measure of the probability of…
Many computational problems admit fast algorithms on special inputs, however, the required properties might be quite restrictive. E.g., many graph problems can be solved much faster on interval or cographs, or on graphs of small…