Quasirandomness in hypergraphs

An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others. In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.Comment: 19 page

Quasi-random oriented graphs

We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of unoriented graphs, and by Chung and Graham in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G the main result of Chung and Graham which corresponds to the case that G is complete. One interesting aspect of these results is that exactly two of the four orientations of a four-cycle can be used for a quasi-randomness condition, i.e., if the number of appearances they make in D is close to the expected number in a random orientation of the same underlying graph, then the same is true for every small oriented graph HComment: 11 page

More on quasi-random graphs, subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases. The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

DENSE GRAPH LIMITS AND APPLICATIONS

In recent years, there has been a growing need to understand large networks and to devise effective strategies to analyze them. In this dissertation, our main objectives are to understand various structural properties of large networks under suitable general framework and develop general techniques to analyze important network models arising from applied fields of study. In the first part of this dissertation, we investigate properties of large networks that satisfy certain local conditions. In particular, we show that if the number of neighbors of each vertex and co-neighbors of each pair of vertices satises certain conditions then the number of copies of moderately large subgraphs is approximately same as that of an Erdos-Renyi random graph with appropriate edge density. We apply our results to different graph ensembles including exponential random graph models (ERGMs), thresholded graphs from high-dimensional correlation networks, Erdos-Renyi random graphs conditioned on large cliques and random d-regular graphs. In the second part of this dissertation, we study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. We derive limiting results for the structure of these models as the number of nodes goes to infinity. We also derive sucient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally, we study site percolation on a class of non-regular graphs satisfying some mild assumptions on the number of neighbors of each vertex and co-neighbors of each pair of vertices. We show that there is a sharp phase transition (in site percolation) for the class of graphs under consideration and that in the supercritical regime the giant component is unique.Doctor of Philosoph