The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.
Abstract-This paper describes an extension to stochastic Kronecker graphs that provides the special structure required for searchability, by defining a "distance"-dependent Kronecker operator. We show how this extension of Kronecker graphs can generate several existing social network models, such as the Watts-Strogatz small-world model and Kleinberg's latticebased model. We focus on a specific example of an expanding hypercube, reminiscent of recently proposed social network models based on a hidden hyperbolic metric space, and prove that a greedy forwarding algorithm can find very short paths of length O((log log n)2 ) for graphs with n nodes.I. INTRODUCTION There exists a large body of work exploring the various structural properties of social networks -small diameter, high clustering, heavy-tailed degree distributions, and searchability; see [1], [2], and [3] for surveys of this area. Many generative models have been proposed that capture some of these properties with varying levels of complexity, but the challenge remains to develop a simple, quantitative model that can exhibit all of these properties. For example, the simple Erdös-Rényi random graph maintains a small diameter, but fails to capture many of the other properties [4], [5]. The combination of small diameter and high clustering is often called the "small-world effect," and Watts and Strogatz (see section 2.2) generated much interest on the topic when they proposed a model that maintains these two characteristics simultaneously [6]. Several models were then proposed to explain the heavy-tailed degree distributions and densification of complex networks; these include the preferential attachment model [7], the forest-fire model [8], [9], Kronecker graphs [10] [11], and many others [1]. As demonstrated by Milgram's 1967 experiment using real people, individuals can discover and use short paths using only local information [12]. This is termed "searchability." Kleinberg focuses on this characteristic in his lattice model and proves searchability for a precise set of input parameters, but his model lacks any heavy-tailed distributions [13], [3], [14].One promising model proposed recently is Kronecker graphs [10], [11], [15]. These graphs are simple to generate, are mathematically tractable, and have been shown to exhibit several important social network characteristics. In particular, these graphs can have heavy-tailed degree and eigenvalue distributions, a high-clustering coefficient, small diameter, and densify over time. Additionally, Leskovec developed an algorithm that could find an appropriate 2x2 or 3x3 initiator
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