Related papers: Modularity and Graph Expansion

Much effort has gone into understanding the modular nature of complex networks. Communities, also known as clusters or modules, are typically considered to be densely interconnected groups of nodes that are only sparsely connected to other…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

Several algorithms have been proposed to compute partitions of networks into communities that score high on a graph clustering index called modularity. While publications on these algorithms typically contain experimental evaluations to…

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry,…

This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional…

The \emph{$r$-neighbourhood complexity} of a graph $G$ is the function counting, for a given integer $k$, the largest possible number, over all vertex-subsets $A$ of size $k$, of subsets of $A$ realized as the intersection between the…

Community detection, which involves partitioning nodes within a network, has widespread applications across computational sciences. Modularity-based algorithms identify communities by attempting to maximize the modularity function across…

Community detection algorithms have been widely used to study the organization of complex systems like the brain. A principal appeal of these techniques is their ability to identify a partition of brain regions (or nodes) into communities,…

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications…

Community detection in networks is the process of identifying unusually well-connected sub-networks and is a central component of many applied network analyses. The paradigm of modularity optimization stipulates a partition of the network's…

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…

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$…

We define, for any graph $G=(V,E)$, a boundary $\partial G \subseteq V$. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the…

Network detection is an important capability in many areas of applied research in which data can be represented as a graph of entities and relationships. Oftentimes the object of interest is a relatively small subgraph in an enormous,…

There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if…

Temporal graphs provide a useful model for many real-world networks. Unfortunately the majority of algorithmic problems we might consider on such graphs are intractable. There has been recent progress in defining structural parameters which…

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r,…

Statistical analysis and node clustering in hypergraphs constitute an emerging topic suffering from a lack of standardization. In contrast to the case of graphs, the concept of nodes' community in hypergraphs is not unique and encompasses…

Submodular width is a central structural measure governing the complexity of conjunctive query evaluation. In this paper we recast submodular width in geometric terms. We how that submodular width can be approximated, up to a factor $3/2$,…

This article introduces a concept and measure of graph compartmentalization. This new measure allows for principled comparison between graphs of arbitrary structure, unlike existing measures such as graph modularity. The proposed measure is…