2,906 research outputs found

    The skew energy of random oriented graphs

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    Given a graph GG, let GσG^\sigma be an oriented graph of GG with the orientation σ\sigma and skew-adjacency matrix S(Gσ)S(G^\sigma). The skew energy of the oriented graph GσG^\sigma, denoted by ES(Gσ)\mathcal{E}_S(G^\sigma), is defined as the sum of the absolute values of all the eigenvalues of S(Gσ)S(G^\sigma). In this paper, we study the skew energy of random oriented graphs and formulate an exact estimate of the skew energy for almost all oriented graphs by generalizing Wigner's semicircle law. Moreover, we consider the skew energy of random regular oriented graphs Gn,dσG_{n,d}^\sigma, and get an exact estimate of the skew energy for almost all regular oriented graphs.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1011.6646 by other author

    Rainbow kk-connectivity of random bipartite graphs

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    A path in an edge-colored graph GG is called a rainbow path if no two edges of the path are colored the same. The minimum number of colors required to color the edges of GG such that every pair of vertices are connected by at least kk internally vertex-disjoint rainbow paths is called the rainbow kk-connectivity of the graph GG, denoted by rck(G)rc_k(G). For the random graph G(n,p)G(n,p), He and Liang got a sharp threshold function for the property rck(G(n,p))≀drc_k(G(n,p))\leq d. In this paper, we extend this result to the case of random bipartite graph G(m,n,p)G(m,n,p).Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.1942 by other author

    False Discovery Rate Controlled Heterogeneous Treatment Effect Detection for Online Controlled Experiments

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    Online controlled experiments (a.k.a. A/B testing) have been used as the mantra for data-driven decision making on feature changing and product shipping in many Internet companies. However, it is still a great challenge to systematically measure how every code or feature change impacts millions of users with great heterogeneity (e.g. countries, ages, devices). The most commonly used A/B testing framework in many companies is based on Average Treatment Effect (ATE), which cannot detect the heterogeneity of treatment effect on users with different characteristics. In this paper, we propose statistical methods that can systematically and accurately identify Heterogeneous Treatment Effect (HTE) of any user cohort of interest (e.g. mobile device type, country), and determine which factors (e.g. age, gender) of users contribute to the heterogeneity of the treatment effect in an A/B test. By applying these methods on both simulation data and real-world experimentation data, we show how they work robustly with controlled low False Discover Rate (FDR), and at the same time, provides us with useful insights about the heterogeneity of identified user groups. We have deployed a toolkit based on these methods, and have used it to measure the Heterogeneous Treatment Effect of many A/B tests at Snap
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