Nathan Lemons

3-uniform hypergraphs avoiding a given odd cycle

We give upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In particular, we show that a 3-uniform hypergraph containing no cycle of length 2k+1 has less than 4k4n1+1/k+O(n) edges. Constructions show that these bounds are best possible (up to constant factor) for k=1,2,3, 5.

Hypergraph Extensions of the Erdős-Gallai Theorem

Abstract The Erdős-Gallai Theorem gives the maximum number of edges in a graph without a path of length k. We extend this result for Berge paths in r-uniform hypergraphs. We also find the extremal hypergraphs avoiding t-tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.

Hierarchical graphs for rule-based modeling of biochemical systems

BackgroundIn rule-based modeling, graphs are used to represent molecules: a colored vertex represents a component of a molecule, a vertex attribute represents the internal state of a component, and an edge represents a bond between components. Components of a molecule share the same color. Furthermore, graph-rewriting rules are used to represent molecular interactions. A rule that specifies addition (removal) of an edge represents a class of association (dissociation) reactions, and a rule that specifies a change of a vertex attribute represents a class of reactions that affect the internal state of a molecular component. A set of rules comprises an executable model that can be used to determine, through various means, the system-level dynamics of molecular interactions in a biochemical system.ResultsFor purposes of model annotation, we propose the use of hierarchical graphs to represent structural relationships among components and subcomponents of molecules. We illustrate how hierarchical graphs can be used to naturally document the structural organization of the functional components and subcomponents of two proteins: the protein tyrosine kinase Lck and the T cell receptor (TCR) complex. We also show that computational methods developed for regular graphs can be applied to hierarchical graphs. In particular, we describe a generalization of Nauty, a graph isomorphism and canonical labeling algorithm. The generalized version of the Nauty procedure, which we call HNauty, can be used to assign canonical labels to hierarchical graphs or more generally to graphs with multiple edge types. The difference between the Nauty and HNauty procedures is minor, but for completeness, we provide an explanation of the entire HNauty algorithm.ConclusionsHierarchical graphs provide more intuitive formal representations of proteins and other structured molecules with multiple functional components than do the regular graphs of current languages for specifying rule-based models, such as the BioNetGen language (BNGL). Thus, the proposed use of hierarchical graphs should promote clarity and better understanding of rule-based models.

Connected Components and Credential Hopping in Authentication Graphs

Modern enterprise computer networks rely on centrally managed authentication schemes that allow users to easily communicate access credentials to many computer systems and applications. The authentication events typically consist of a user connecting to a computer with an authorized credential. These credentials are often cached on the application servers which creates a risk that they may be stolen and used to hop between computers in the network. We examine computer network risk associated with credential hopping by creating and studying the structure of the authentication graph, a bipartite graph built from authentication events. We assume that an authentication graph with many short paths between computers represents a network that is more vulnerable to such attacks. Under this natural assumption, we use a measure of graph connectivity, namely the size of the largest connected component, to give a quantitative indicator of the networks susceptibility to such attacks. Motivated by graph theoretical results for component sizes in random intersection graphs, we propose a mitigation strategy, and perform experiments simulating an implementation using data from a large enterprise network. The results lead to realistic, actionable risk reduction strategies. To facilitate continued research opportunities we are also providing our authentication bipartite graph data set spanning 9 months and 708 million time-series edge records.

Hypergraph extensions of the Erdźs-Gallai Theorem

We extend the Erdźs-Gallai Theorem for Berge paths in r -uniform hypergraphs. We also find the extremal hypergraphs avoiding t -tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we fully characterize the degeneracy of random intersection graphs, and prove that the model asymptotically almost surely produces graphs with hyperbolicity at least

Fast Generation of Sparse Random Kernel Graphs.

\log {n}

Polychromatic colorings of arbitrary rectangular partitions

logn. Further, we prove that when degenerate, the graphs generated by this model belong to a bounded-expansion graph class with high probability, a property particularly suitable for the design of linear time algorithms.