Mathematical Foundations and Algorithms for Clique Relaxations in Networks

This dissertation establishes mathematical foundations for the properties exhibited by generalizations of cliques, as well as algorithms to find such objects in a network. Cliques are a model of an ideal group with roots in social network analysis. They have since found applications as a part of grouping mechanisms in computer vision, coding theory, experimental design, genomics, economics, and telecommunications among other fields. Because only groups with ideal properties form a clique, they are often too restrictive for identifying groups in many real-world networks. This motivated the introduction of clique relaxations that preserve some of the various defining properties of cliques in relaxed form. There are six clique relaxations that are the focus of this dissertation: s-clique, s-club, s-plex, k-core, quasi-clique, and k-connected subgraphs. Since cliques have found applications in so many fields, research into these clique relaxations has the potential to steer the course of much future research. The focus of this dissertation is on bringing organization and rigorous methodology to the formation and application of clique relaxations. We provide the first taxonomy focused on how the various clique relaxations relate on key structural properties demonstrated by groups. We also give a framework for how clique relaxations can be formed. This equips researchers with the ability to choose the appropriate clique relaxation for an application based on its structural properties, or, if an appropriate clique relaxation does not exist, form a new one. In addition to identifying the structural properties of the various clique relaxations, we identify properties and prove propositions that are important computationally. These assist in creating algorithms to find a clique relaxation quickly as it is immersed in a network. We give the first ever analysis of the computational complexity of finding the maximum quasi-clique in a graph. Such analysis identifies for researchers the appropriate set of computational tools to solve the maximum quasiclique problem. We further create a polynomial time algorithm for identifying large 2-cliques within unit disk graphs, a special class of graphs often arising in communication networks. We prove the algorithm to have a guaranteed 1=2-approximation ratio and finish with computational results

Provably and Efficiently Approximating Near-cliques using the Tur\'an Shadow: PEANUTS

Clique and near-clique counts are important graph properties with applications in graph generation, graph modeling, graph analytics, community detection among others. They are the archetypal examples of dense subgraphs. While there are several different definitions of near-cliques, most of them share the attribute that they are cliques that are missing a small number of edges. Clique counting is itself considered a challenging problem. Counting near-cliques is significantly harder more so since the search space for near-cliques is orders of magnitude larger than that of cliques. We give a formulation of a near-clique as a clique that is missing a constant number of edges. We exploit the fact that a near-clique contains a smaller clique, and use techniques for clique sampling to count near-cliques. This method allows us to count near-cliques with 1 or 2 missing edges, in graphs with tens of millions of edges. To the best of our knowledge, there was no known efficient method for this problem, and we obtain a 10x - 100x speedup over existing algorithms for counting near-cliques. Our main technique is a space-efficient adaptation of the Tur\'an Shadow sampling approach, recently introduced by Jain and Seshadhri (WWW 2017). This approach constructs a large recursion tree (called the Tur\'an Shadow) that represents cliques in a graph. We design a novel algorithm that builds an estimator for near-cliques, using an online, compact construction of the Tur\'an Shadow.Comment: The Web Conference, 2020 (WWW

Placenta-specific methylation of the vitamin D 24-hydroxylase gene: implications for feedback autoregulation of active vitamin D levels at the fetomaternal interface

Plasma concentrations of biologically active vitamin D (1,25- (OH)2D) are tightly controlled via feedback regulation of renal 1-hydroxylase (CYP27B1; positive) and 24-hydroxylase (CYP24A1; catabolic) enzymes. In pregnancy, this regulation is uncoupled, and 1,25-(OH)2D levels are significantly elevated, suggesting a role in pregnancy progression. Epigenetic regulation of CYP27B1 and CYP24A1 has previously been described in cell and animal models, and despite emerging evidence for a critical role of epigenetics in placentation generally, little is known about the regulation of enzymes modulating vitamin D homeostasis at the fetomaternal interface. In this study, we investigated the methylation status of genes regulating vitamin D bioavailability and activity in the placenta. No methylation of the VDR (vitamin D receptor) and CYP27B1 genes was found in any placental tissues. In contrast, the CYP24A1 gene is methylated in human placenta, purified cytotrophoblasts, and primary and cultured chorionic villus sampling tissue. No methylation was detected in any somatic human tissue tested. Methylation was also evident in marmoset and mouse placental tissue. All three genes were hypermethylated in choriocarcinoma cell lines, highlighting the role of vitaminDderegulation in this cancer. Gene expression analysis confirmed a reduced capacity for CYP24A1 induction with promoter methylation in primary cells and in vitro reporter analysis demonstrated that promoter methylation directly down-regulates basal promoter activity and abolishes vitamin D-mediated feedback activation. This study strongly suggests that epigenetic decoupling of vitamin D feedback catabolism plays an important role in maximizing active vitamin D bioavailability at the fetomaternal interface

The Violence of the Canon: Revisiting Contemporary Notions of Canonical Forms

The historical conditions surrounding the processes of forming a canon are rarely examined directly, yet it is these processes which govern over the realm of religious representations and identity constructions. In light of recent critical scholarship, it is imperative to address theologically the role that the canon plays within a religious tradition. This essay demonstrates the cultural necessity of canonical forms despite their “monotheistic tendency” to subdivide the world into binary oppositions. By utilizing a scale of violence to determine the impact of the canonical form upon culture this essay offers an account of canons and their role in forming religious identities over and beyond the violence they are said to provoke. Through this clarification, an alternative perspective of canons can emerge that reveals the violence at the core of cultural-canonical norms, thus providing a valuable distinction between differing (violence-concealing or violence-revealing) canonical forms

A meta-analysis of the investment-uncertainty relationship

In this article we use meta-analysis to investigate the investment-uncertainty relationship. We focus on the direction and statistical significance of empirical estimates. Specifically, we estimate an ordered probit model and transform the estimated coefficients into marginal effects to reflect the changes in the probability of finding a significantly negative estimate, an insignificant estimate, or a significantly positive estimate. Exploratory data analysis shows that there is little empirical evidence for a positive relationship. The regression results suggest that the source of uncertainty, the level of data aggregation, the underlying model specification, and differences between short- and long-run effects are important sources of variation in study outcomes. These findings are, by and large, robust to the introduction of a trend variable to capture publication trends in the literature. The probability of finding a significantly negative relationship is higher in more recently published studies. JEL Classification: D21, D80, E22 1

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