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

New Analytical Lower Bounds On The Clique Number Of A Graph

This paper proposes three new analytical lower bounds on the clique number of a graph and compares these bounds with those previously established in the literature. Two proposed bounds are derived from the well-known Motzkin–Straus quadratic programming formulation for the maximum clique problem. Theoretical results on the comparison of various bounds are established. Computational experiments are performed on random graph models such as the Erdös-Rényi model for uniform graphs and the generalized random graph model for power-law graphs that simulate graphs with different densities and assortativity coefficients. Computational results suggest that the proposed new analytical bounds improve the existing ones on many graph instances

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