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Star sampling (SS) is a random sampling procedure on a graph wherein each sample consists of a randomly selected vertex (the star center) and its (one-hop) neighbors (the star points). We consider the use of SS to find any member of a target set of vertices in a graph, where the figure of merit (cost) is either the expected number of samples (unit cost) or the expected number of star centers plus star points (linear cost) until a vertex in the target set is encountered, either as a star center or as a star point. We analyze these two performance measures on three related star sampling paradigms: SS with replacement (SSR), SS without center replacement (SSC), and SS without star replacement (SSS). Exact and approximate expressions are derived for the expected unit and linear costs of SSR, SSC, and SSS on Erd\H{o}s-R\'{e}nyi (ER) random graphs. The approximations are seen to be accurate. SSC/SSS are notably better than SSR under unit cost for low-density ER graphs, while SSS is notably better than SSR/SSC under linear cost for low- to moderate-density ER graphs. Simulations on twelve "real-world" graphs shows the cost approximations to be of variable quality: the SSR and SSC approximations are uniformly accurate, while the SSS approximation, derived for an ER graph, is of variable accuracy., Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1901.03393

Star sampling (SS) is a random sampling procedure on a graph wherein each sample consists of a randomly selected vertex (the star center) and its one-hop neighbors (the star endpoints). We consider the use of star sampling to find any member of an arbitrary target set of vertices in a graph, where the figure of merit (cost) is either the expected number of samples (unit cost) or the expected number of star centers plus star endpoints (linear cost) until a vertex in the target set is encountered, either as a star center or as a star point. We analyze this performance measure on three related star sampling paradigms: SS with replacement (SSR), SS without center replacement (SSC), and SS without star replacement (SSS). We derive exact and approximate expressions for the expected unit and linear costs of SSR, SSC, and SSS on Erdos-Renyi (ER) graphs. Our results show there is i) little difference in unit cost, but ii) significant difference in linear cost, across the three paradigms. Although our results are derived for ER graphs, experiments on "real-world" graphs suggest our performance expressions are reasonably accurate for non-ER graphs., Comment: Superseded by arXiv:1910.00431

We examine two greedy heuristics - wiring and rewiring - for constructing maximum assortative graphs over all simple connected graphs with a target degree sequence. Counterexamples show that natural greedy rewiring heuristics do not necessarily return a maximum assortative graph, even though it is known that the meta-graph of all simple connected graphs with given degree is connected under rewiring. Counterexamples show an elegant greedy graph wiring heuristic from the literature may fail to achieve the target degree sequence or may fail to wire a maximally assortative graph., Comment: 14 pages, 5 figures

We consider the problem of estimating the expected time to find a maximum degree node on a graph using a (parameterized) biased random walk. For assortative graphs the positive degree correlation serves as a local gradient for which a bias towards selecting higher degree neighbors will on average reduce the search time. Unfortunately, although the expected absorption time on the graph can be written down using the theory of absorbing Markov chains, computing this time is infeasible for large graphs. With this motivation, we construct an absorbing Markov chain with a state for each degree of the graph, and observe computing the expected absorption time is now computationally feasible. Our paper finds preliminary results along the following lines: i) there are graphs for which the proposed Markov model does and graphs for which the model does not capture the absorbtion time, ii) there are graphs where random sampling outperforms biased random walks, and graphs where biased random walks are superior, and iii) the optimal bias parameter for the random walk is graph dependent, and we study the dependence on the graph assortativity., Comment: 7 pages, 7 figures, submitted to CISS 2016