Bootstrap Percolation on Random Geometric Graphs

Bootstrap Percolation is a discrete-time process that models the spread of information or disease across the vertex set of a graph. We consider the following version of this process: Initially, each vertex of the graph is set active with probability p or inactive otherwise. Then, at each time step, every inactive vertex with at least k active neighbors becomes active. Active vertices will always remain active. The process ends when it reaches a stationary state. If all the vertices eventually become active, then we say we achieve percolation. This process has been widely studied on many families of graphs, deterministic and random. We analyze the Bootstrap Percolation process on a Random Geometric Graph. A Random Geometric Graph is obtained by choosing n vertices uniformly at random from the unit d-dimensional cube or torus, and joining any two vertices by an edge if they are within a certain distance, r, of each other. We obtain very precise results that characterize the final state of the Bootstrap Percolation process in terms of the parameters p and r with high probability as the number n of vertices tends to infinity. Adviser: Xavier Pérez Giméne

Prime Vertex Labelings of Families of Unicyclic Graphs

A simple n-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers 1 through n such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs, all of which are special cases of Seoud and Youssef\u27s conjecture that all unicyclic graphs have a prime labeling

Assessment of variation in immunosuppressive pathway genes reveals TGFBR2 to be associated with risk of clear cell ovarian cancer.

BACKGROUND: Regulatory T (Treg) cells, a subset of CD4+ T lymphocytes, are mediators of immunosuppression in cancer, and, thus, variants in genes encoding Treg cell immune molecules could be associated with ovarian cancer. METHODS: In a population of 15,596 epithelial ovarian cancer (EOC) cases and 23,236 controls, we measured genetic associations of 1,351 SNPs in Treg cell pathway genes with odds of ovarian cancer and tested pathway and gene-level associations, overall and by histotype, for the 25 genes, using the admixture likelihood (AML) method. The most significant single SNP associations were tested for correlation with expression levels in 44 ovarian cancer patients. RESULTS: The most significant global associations for all genes in the pathway were seen in endometrioid ( p = 0.082) and clear cell ( p = 0.083), with the most significant gene level association seen with TGFBR2 ( p = 0.001) and clear cell EOC. Gene associations with histotypes at p < 0.05 included: IL12 ( p = 0.005 and p = 0.008, serous and high-grade serous, respectively), IL8RA ( p = 0.035, endometrioid and mucinous), LGALS1 ( p = 0.03, mucinous), STAT5B ( p = 0.022, clear cell), TGFBR1 ( p = 0.021 endometrioid) and TGFBR2 ( p = 0.017 and p = 0.025, endometrioid and mucinous, respectively). CONCLUSIONS: Common inherited gene variation in Treg cell pathways shows some evidence of germline genetic contribution to odds of EOC that varies by histologic subtype and may be associated with mRNA expression of immune-complex receptor in EOC patients

Bootstrap Percolation on Random Geometric Graphs

Bootstrap Percolation is a discrete-time process that models the spread of information or disease across the vertex set of a graph. It was introduced in 1979 by Chalupa, Leith and Reich as a simple model of dynamics of ferromagnetism. We consider the following version of this process:Initially, each vertex of the graph is set active with probability p or inactive otherwise. At each time step, every inactive vertex with at least k active neighbors becomes active. Active vertices will always remain active. The process ends when it reaches a stationary state. If all the vertices eventually become active, then we say we achieve percolation. We analyze the Bootstrap Percolation process on a Random Geometric Graph. Random Geometric Graphs provide a simplified abstract model of spatial networks, and are particularly suitable to describe wireless ad-hoc networks. More precisely, a Random Geometric Graph is obtained by choosing n vertices uniformly at random from the unit d-dimensional cube or torus, and joining any two vertices by an edge if they are within a certain distance, r, from each other. Until now, very little was known about Bootstrap Percolation on Random Geometric Graphs, other than some initial results in a paper by Bradonjić and Saniee (2012). We obtain precise results that characterize the final state of the Bootstrap Percolation process in terms of the parameters p and r asymptotically almost surely as the number n of vertices tends to infinity.We show that, a.a.s., the process is either stationary from the very beginning (i.e. no inactive vertex ever changes to active) or almost all vertices eventually become active. Moreover, we prove that in the latter case the only obstacle to achieve full percolation is the presence of vertices of degree less than k. Indeed, as soon as r is large enough to guarantee that the minimum degree is at least k, a.a.s., the process is either stationary or attains percolation.Finally, we study a version of the model with a restricted focus of infection (i.e. active points can initially occur in a small region of the torus), and obtain analogous results for that case

No Evidence That Genetic Variation in the Myeloid-Derived Suppressor Cell Pathway Influences Ovarian Cancer Survival.

Background: The precise mechanism by which the immune system is adversely affected in cancer patients remains poorly understood, but the accumulation of immunosuppressive/protumorigenic myeloid-derived suppressor cells (MDSCs) is thought to be a prominent mechanism contributing to immunologic tolerance of malignant cells in epithelial ovarian cancer (EOC). To this end, we hypothesized genetic variation in MDSC pathway genes would be associated with survival after EOC diagnoses.Methods: We measured the hazard of death due to EOC within 10 years of diagnosis, overall and by invasive subtype, attributable to SNPs in 24 genes relevant in the MDSC pathway in 10,751 women diagnosed with invasive EOC. Versatile Gene-based Association Study and the admixture likelihood method were used to test gene and pathway associations with survival.Results: We did not identify individual SNPs that were significantly associated with survival after correction for multiple testing (P &lt; 3.5 × 10-5), nor did we identify significant associations between the MDSC pathway overall, or the 24 individual genes and EOC survival.Conclusions: In this well-powered analysis, we observed no evidence that inherited variations in MDSC-associated SNPs, individual genes, or the collective genetic pathway contributed to EOC survival outcomes.Impact: Common inherited variation in genes relevant to MDSCs was not associated with survival in women diagnosed with invasive EOC. Cancer Epidemiol Biomarkers Prev; 26(3); 420-4. ©2016 AACR

No evidence that genetic variation in the myeloid-derived suppressor cell pathway influences ovarian cancer survival

The precise mechanism by which the immune system is adversely affected in cancer patients remains poorly understood, but the accumulation of immune suppressive/pro-tumorigenic myeloid-derived suppressor cells (MDSCs) is thought to be one prominent mechanism contributing to immunologic tolerance of malignant cells in epithelial ovarian cancer (EOC). To this end, we hypothesized genetic variation in MDSC pathway genes would be associated with survival after EOC diagnoses.status: publishe