Dense Subgraphs in Random Graphs

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then $\omega_{\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\alpha(\gamma,p)=\gamma\log\frac{\gamma}{p}+(1-\gamma)\log\frac{1-\gamma}{1-p}$, we show that $\omega_{\gamma}(G_{n,p})$ is one of the two integers closest to $\frac{2}{\alpha(\gamma,p)}\big(\log n-\log\log n+\log\frac{e\alpha(\gamma,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these "quasi-cliques" may overlap

Graph-based exploration and clustering analysis of semantic spaces

The goal of this study is to demonstrate how network science and graph theory tools and concepts can be effectively used for exploring and comparing semantic spaces of word embeddings and lexical databases. Specifically, we construct semantic networks based on word2vec representation of words, which is “learnt” from large text corpora (Google news, Amazon reviews), and “human built” word networks derived from the well-known lexical databases: WordNet and Moby Thesaurus. We compare “global” (e.g., degrees, distances, clustering coefficients) and “local” (e.g., most central nodes and community-type dense clusters) characteristics of considered networks. Our observations suggest that human built networks possess more intuitive global connectivity patterns, whereas local characteristics (in particular, dense clusters) of the machine built networks provide much richer information on the contextual usage and perceived meanings of words, which reveals interesting structural differences between human built and machine built semantic networks. To our knowledge, this is the first study that uses graph theory and network science in the considered context; therefore, we also provide interesting examples and discuss potential research directions that may motivate further research on the synthesis of lexicographic and machine learning based tools and lead to new insights in this area.peerReviewe

The Network Of Causal Relationships In The U.S. Stock Market

We propose a network-based framework to study causal relationships in financial markets and demonstrate the proposed approach by applying it to the entire U.S. stock market. Directed networks (referred to as causal market graphs) are constructed based on stock return time series data during 2001–2017 using Granger causality as a measure of pairwise causal relationships between all stocks. We consider the dynamics of structural properties of the constructed network snapshots, group stocks into network-based clusters, as well as identify the most “influen-tial” stocks via a PageRank algorithm. The proposed approaches offer a new angle for analyzing global characteristics and trends of the stock market using network-based techniques

On the maximum quasi-clique problem

AbstractGiven a simple undirected graph G=(V,E) and a constant γ∈(0,1), a subset of vertices is called a γ-quasi-clique or, simply, a γ-clique if it induces a subgraph with the edge density of at least γ. The maximum γ-clique problem consists in finding a γ-clique of largest cardinality in the graph. Despite numerous practical applications, this problem has not been rigorously studied from the mathematical perspective, and no exact solution methods have been proposed in the literature. This paper, for the first time, establishes some fundamental properties of the maximum γ-clique problem, including the NP-completeness of its decision version for any fixed γ satisfying 0<γ<1, the quasi-heredity property, and analytical upper bounds on the size of a maximum γ-clique. Moreover, mathematical programming formulations of the problem are proposed and results of preliminary numerical experiments using a state-of-the-art optimization solver to find exact solutions are presented

Network-based indices of individual and collective advising impacts in mathematics

Advising and mentoring Ph.D. students is an increasingly important aspect of the academic profession. We define and interpret a family of metrics (collectively referred to as “a-indices”) that can potentially be applied to “ranking academic advisors” using the academic genealogical records of scientists, with the emphasis on taking into account not only the number of students advised by an individual, but also subsequent academic advising records of those students. We also define and calculate the extensions of the proposed indices that account for student co-advising (referred to as “adjusted a-indices”). In addition, we extend some of the proposed metrics to ranking universities and countries with respect to their “collective” advising impacts, as well as track the evolution of these metrics over the past several decades. To illustrate the proposed metrics, we consider the social network of over 200,000 mathematicians (as of July 2018) constructed using the Mathematics Genealogy Project data.peerReviewe

On Maximum Degree-Based Γ-Quasi-Clique Problem: Complexity And Exact Approaches

We consider the problem of finding a degree-based γ-quasi-clique of maximum cardinality in a given graph for some fixed γ ∈ (0,1]. A degree-based γ-quasi-clique (often referred to as simply a quasi-clique) is a subgraph, where the degree of each vertex is at least γ times the maximum possible degree of a vertex in the subgraph. A degree-based γ-quasi-clique is a relative clique relaxation model, where the case of γ=1 corresponds to the well-known concept of a clique. In this article, we first prove that the problem is NP-hard for any fixed γ ∈ (0,1], which addresses one of the open questions in the literature. More importantly, we also develop new exact solution methods for solving the problem and demonstrate their advantages and limitations in extensive computational experiments with both random and real-world networks. Finally, we outline promising directions of future research including possible functional generalizations of the considered clique relaxation model. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(2), 136–152 2018