Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.
As discussed in the introduction, a plethora of literature exists on theta graphs and their applications, though most of this work focuses on worst-case analysis. One notable exception is the work of Devroye al @cite_17 who study the maximum degree of theta graphs and show that, if @math is a set of @math points independently and uniformly distributed in a certain unit square, then @math has maximum degree concentrated around @math . Devroye al actually consider the closely-related Yao graphs @cite_5 @cite_11 , but their proofs apply, almost without modification, to theta graphs.
@cite_5 @cite_11 @cite_17
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We revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph @math with @math vertices, and define (for @math ) @math . Our aim is to estimate @math within a multiplicative error of @math (for a given approximation parameter @math ) in sublinear time. We consider the sparse graph model that allows access to: uniform random vertices, queries for the degree of any vertex, and queries for a neighbor of any vertex. For the case of @math (the average degree), @math queries suffice for any constant @math (Feige, SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral @math , by designing an algorithms that performs @math queries. We design a new, significantly simpler algorithm for this problem. In the worst-case, it exactly matches the bounds of Gonen-Ron-Shavitt, and has a much simpler proof. More importantly, the running time of this algorithm is connected to the degeneracy of @math . This is (essentially) the maximum density of an induced subgraph. For the family of graphs with degeneracy at most @math , it has a query complexity of @math . Thus, for the class of bounded degeneracy graphs (which includes all minor closed families and preferential attachment graphs), we can estimate the average degree in @math queries, and can estimate the variance of the degree distribution in @math queries. This is a major improvement over the previous worst-case bounds. Our key insight is in designing an estimator for @math that has low variance when @math does not have large dense subgraphs. Lov asz and Schrijver [SIAM J. Optim., 1 (1991), pp. 166--190] devised a lift-and-project method that produces a sequence of convex relaxations for the problem of finding in a graph an independent set (or a clique) of maximum size. Each relaxation in the sequence is tighter than the one before it, while the first relaxation is already at least as strong as the Lov asz theta function [IEEE Trans. Inform. Theory, 25 (1979), pp. 1--7]. We show that on a random graph Gn,1 2, the value of the rth relaxation in the sequence is roughly 0pt 7pt @math , almost surely. It follows that for those relaxations known to be efficiently computable, namely, for r=O(1), the value of the relaxation is comparable to the theta function. Furthermore, a perfectly tight relaxation is almost surely obtained only at the @math relaxation in the sequence. Motivated by applications of Gabriel graphs and Yao graphs in wireless ad-hoc networks, we show that the maximum degree of a random Gabriel graph or Yao graph defined on n points drawn uniformly at random from a unit square grows as ?(log n log log n) in probability.