Do Metabolic Networks Follow a Power Law? A PSAMM Analysis

Inspired by the landmark paper “Emergence of Scaling in Random Networks” by Barabási and Albert, the field of network science has focused heavily on the power law distribution in recent years. This distribution has been used to model everything from the popularity of sites on the World Wide Web to the number of citations received on a scientific paper. The feature of this distribution is highlighted by the fact that many nodes (websites or papers) have few connections (internet links or citations) while few “hubs” are connected to many nodes. These properties lead to two very important observed effects: the so-called small world property and robustness to random attacks. The small world property holds that although the size of e.g. the World Wide Web is massive, the distance between any two sites is small (19 clicks separate any two websites). The robustness feature is a result of the networks having proportionally few hubs; a random attack (taking down a random website) is not likely to hit a hub like Facebook or Twitter. In biology, metabolic networks are constructed by linking metabolite nodes to one another if there is a biochemical reaction that converts one to another. The degrees of each node in these networks have been suggested to follow a power law distribution. In this project, we will use the PSAMM software to analyze metabolic networks in a collection of curated metabolic models from the literature with an aim to determine whether the degrees of nodes in each metabolic network follow a power law distribution. We will investigate how the different representations of the networks may influence the power law fitting. We will also compare the power law fittings between models across different species and note how similar or different the structure of the species’ networks are based on this specific feature

On powers of tight Hamilton cycles in randomly perturbed hypergraphs

We show that for $k \geq 3$, $r\geq 2$ and $\alpha> 0$, there exists $\varepsilon > 0$ such that if $p=p(n)\geq n^{-{\binom{k+r-2}{k-1}}^{-1}-\varepsilon}$ and $H$ is a $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\alpha n$, then asymptotically almost surely the union $H\cup G^{(k)}(n,p)$ contains the $r^{th}$ power of a tight Hamilton cycle. The bound on $p$ is optimal up to the value of $\varepsilon$ and this answers a question of Bedenknecht, Han, Kohayakawa and Mota

Lower bounds for integral functionals generated by bipartite graphs

We study lower estimates for integral fuctionals for which the structure of the integrand is defined by a graph, in particular, by a bipartite graph. Functionals of such kind appear in statistical mechanics and quantum chemistry in the context of Mayer’s transformation and Mayer’s cluster integrals. Integral functionals generated by graphs play an important role in the theory of graph limits. Specific kind of functionals generated by bipartite graphs are at the center of the famous and much studied Sidorenko’s conjecture, where a certain lower bound is conjectured to hold for every bipartite graph. In the present paper we work with functionals more general and lower bounds significantly sharper than those in Sidorenko’s conjecture. In his 1991 seminal paper, Sidorenko proved such sharper bounds for several classes of bipartite graphs. To obtain his result he used a certain way of “gluing” graphs. We prove his inequality for a new class of bipartite graphs by defining a different type of gluing

Bipartite subgraphs and quasi-randomness

We say that a family of graphs script G sign = {Gn : n ≥ 1} is p-quasi-random, 0 \u3c p \u3c 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by script Q signw(p) the class of all graphs H for which e(Gn) ≥ (1 + o(1))p(2n) and the number of not necessarily induced labeled copies of H in Gn is at most (1 + o(1))pe(H)n v(H) imply that script G sign is p-quasi-random. In this note, we show that all complete bipartite graphs Ka,b, a, b ≥ 2, belong to script Q signw(p) for all 0 \u3c p \u3c 1. © Springer-Verlag 2004

On the Size of Set Systems on [n] Not Containing Weak (r, ∆)-Systems

Let r 3 be an integer. A weak (r; ∆)-system is a family of r sets such that all pairwise intersections among the members have the same cardinality. We show that for n large enough, there exists a family F of subsets of [n] such that F does not contain a weak (r; ∆)-system and jF j 2 1 3 \Deltan 1=5 log 4=5 (r\Gamma1) : This improves an earlier result of P. Erdős and E. Szemerédi [ES 78] (cf. [E 90])

Bipartite Subgraphs and Quasi-randomness

Abstract. We say that a family of graphs G = {Gn: n ≥ 1} is p-quasi-random, 0 < p < 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by Q w � (p) the class of all graphs H for which e(Gn) ≥ (1 + and the number of not necessarily induced labeled copies of H in Gn is at o(1))p �n 2 most (1 + o(1))p e(H) n v(H) imply that G is p-quasi-random. In this note, we show that all complete bipartite graphs Ka,b, a, b ≥ 2, belong to Q w (p) for all 0 < p < 1. 1. Notation We start with fixing notation. For positive integers k, n and a real number x, we set [n] = {1,..., n} and (x)k = x(x − 1) × · · · × (x − k + 1). Given a graph G with vertex set V (G) and edge set E(G), v(G) stands for |V (G) | and e(G) for |E(G)|. Furthermore, for a subset X of V (G), G[X] denotes the subgraph induced by the vertices of X, and e(X) denotes the number of edges of G[X]. Given a vertex x ∈ V (G), NG(x) is the set of all vertices adjacent to x and, similarly, for a subset X of V (G), NG(X) denotes the set of all vertices adjacent to every vertex in X. Clearly, NG(X) = � x∈X NG(x). We also pu

Implementing Online Programs in Gateway Mathematics Courses for Students with Prerequisite Deficiencies

The percentage of college students receiving unproductive grades in introductory mathematics courses is a concern for post-secondary institutions across the country. Many factors can be targeted as potential explanations for this lack of success, yet none of these issues are more noteworthy than the fact that many students enter college mathematics courses with significant gaps in their fundamental mathematical background. In this paper, the authors discuss a way to implement an online remedial program to help students overcome their deficiencies following a “just-in-time teaching” model. Relevant data are presented supporting a positive outcome

Implementing Online Programs in Gateway Mathematics Courses for Students with Prerequisite Deficiencies

The percentage of college students receiving unproductive grades in introductory mathematics courses is a concern for post-secondary institutions across the country. Many factors can be targeted as potential explanations for this lack of success, yet none of these issues are more noteworthy than the fact that many students enter college mathematics courses with significant gaps in their fundamental mathematical background. In this paper, the authors discuss a way to implement an online remedial program to help students overcome their deficiencies following a “just-in-time teaching” model. Relevant data are presented supporting a positive outcome