Almost all triple systems with independent neighborhoods are semi-bipartite

The neighborhood of a pair of vertices $u,v$ in a triple system is the set of vertices $w$ such that $uvw$ is an edge. A triple system \HH is semi-bipartite if its vertex set contains a vertex subset $X$ such that every edge of \HH intersects $X$ in exactly two points. It is easy to see that if \HH is semi-bipartite, then the neighborhood of every pair of vertices in \HH is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets $[n]$ and independent neighborhoods are semi-bipartite. Our result can be viewed as an extension of the Erd\H os-Kleitman-Rothschild theorem to triple systems. The proof uses the Frankl-R\"odl hypergraph regularity lemma, and stability theorems. Similar results have recently been proved for hypergraphs with various other local constraints

Navigability is a Robust Property

The Small World phenomenon has inspired researchers across a number of fields. A breakthrough in its understanding was made by Kleinberg who introduced Rank Based Augmentation (RBA): add to each vertex independently an arc to a random destination selected from a carefully crafted probability distribution. Kleinberg proved that RBA makes many networks navigable, i.e., it allows greedy routing to successfully deliver messages between any two vertices in a polylogarithmic number of steps. We prove that navigability is an inherent property of many random networks, arising without coordination, or even independence assumptions

Partition function of periodic isoradial dimer models

Isoradial dimer models were introduced in \cite{Kenyon3} - they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of \cite{Kenyon3}, namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case \cite{KOS}.Comment: 12 pages, 2 figure

Incompatibility boundaries for properties of community partitions

We prove the incompatibility of certain desirable properties of community partition quality functions. Our results generalize the impossibility result of [Kleinberg 2003] by considering sets of weaker properties. In particular, we use an alternative notion to solve the central issue of the consistency property. (The latter means that modifying the graph in a way consistent with a partition should not have counterintuitive effects). Our results clearly show that community partition methods should not be expected to perfectly satisfy all ideally desired properties. We then proceed to show that this incompatibility no longer holds when slightly relaxed versions of the properties are considered, and we provide in fact examples of simple quality functions satisfying these relaxed properties. An experimental study of these quality functions shows a behavior comparable to established methods in some situations, but more debatable results in others. This suggests that defining a notion of good partition in communities probably requires imposing additional properties.Comment: 17 pages, 3 figure

Structural Analysis: Shape Information via Points-To Computation

This paper introduces a new hybrid memory analysis, Structural Analysis, which combines an expressive shape analysis style abstract domain with efficient and simple points-to style transfer functions. Using data from empirical studies on the runtime heap structures and the programmatic idioms used in modern object-oriented languages we construct a heap analysis with the following characteristics: (1) it can express a rich set of structural, shape, and sharing properties which are not provided by a classic points-to analysis and that are useful for optimization and error detection applications (2) it uses efficient, weakly-updating, set-based transfer functions which enable the analysis to be more robust and scalable than a shape analysis and (3) it can be used as the basis for a scalable interprocedural analysis that produces precise results in practice. The analysis has been implemented for .Net bytecode and using this implementation we evaluate both the runtime cost and the precision of the results on a number of well known benchmarks and real world programs. Our experimental evaluations show that the domain defined in this paper is capable of precisely expressing the majority of the connectivity, shape, and sharing properties that occur in practice and, despite the use of weak updates, the static analysis is able to precisely approximate the ideal results. The analysis is capable of analyzing large real-world programs (over 30K bytecodes) in less than 65 seconds and using less than 130MB of memory. In summary this work presents a new type of memory analysis that advances the state of the art with respect to expressive power, precision, and scalability and represents a new area of study on the relationships between and combination of concepts from shape and points-to analyses