Related papers: ({\alpha}, {\beta})-Modules in Graphs
Cluster deletion is an NP-hard graph clustering objective with applications in computational biology and social network analysis, where the goal is to delete a minimum number of edges to partition a graph into cliques. We first provide a…
Scaling model capacity has been vital in the success of deep learning. For a typical network, necessary compute resources and training time grow dramatically with model size. Conditional computation is a promising way to increase the number…
A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple…
Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph…
One of the most useful measures of cluster quality is the modularity of a partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of such edges in a random…
Submodular functions describe a variety of discrete problems in machine learning, signal processing, and computer vision. However, minimizing submodular functions poses a number of algorithmic challenges. Recent work introduced an…
We show here that the problem of maximizing a family of quantitative functions, encompassing both the modularity (Q-measure) and modularity density (D-measure), for community detection can be uniformly understood as a combinatoric…
Given a graph $G$, a subgraph $H$ is isometric if $d_H(u,v) = d_G(u,v)$ for every pair $u,v\in V(H)$, where $d$ is the distance function. A graph $G$ is distance preserving (dp) if it has an isometric subgraph of every possible order. A…
When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a…
This paper presents the foundation for a decomposition theory for Boolean networks, a type of discrete dynamical system that has found a wide range of applications in the life sciences, engineering, and physics. Given a Boolean network…
We consider the cell decomposition of the moduli space of real genus two curves with a marked point on the only real oval. The cells are enumerated by certain graphs with their weights describing the complex structure on a curve. We show…
Modularity is a general principle present in many fields. It offers attractive advantages, including, among others, ease of conceptualization, interpretability, scalability, module combinability, and module reusability. The deep learning…
A $(\beta,\delta,\Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $\Delta$ such that for every vertex $v\in V$, the probability that…
The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$…
Graph convolution is a recent scalable method for performing deep feature learning on attributed graphs by aggregating local node information over multiple layers. Such layers only consider attribute information of node neighbors in the…
Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition.…
Mining dense subgraphs where vertices connect closely with each other is a common task when analyzing graphs. A very popular notion in subgraph analysis is core decomposition. Recently, Esfahani et al. presented a probabilistic core…
In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$…
It is known that complete graphs and complete multipartite graphs have modularity zero. We show that the least number of edges we may delete from the complete graph $K_n$ to obtain a graph with non-zero modularity is $\lfloor n/2\rfloor…
Identifying clusters of vertices in graphs continues to be an important problem, and modularity continues to be used as a tool for solving the problem. Modularity, which measures the quality of a division of the vertices into clusters,…