42,768 research outputs found

    On Large-Scale Graph Generation with Validation of Diverse Triangle Statistics at Edges and Vertices

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    Researchers developing implementations of distributed graph analytic algorithms require graph generators that yield graphs sharing the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree distribution) with efficiently calculable ground-truth solutions to the desired output. Reproducibility for current generators used in benchmarking are somewhat lacking in this respect due to their randomness: the output of a desired graph analytic can only be compared to expected values and not exact ground truth. Nonstochastic Kronecker product graphs meet these design criteria for several graph analytics. Here we show that many flavors of triangle participation can be cheaply calculated while generating a Kronecker product graph. Given two medium-sized scale-free graphs with adjacency matrices AA and BB, their Kronecker product graph has adjacency matrix C=ABC = A \otimes B. Such graphs are highly compressible: E|{\cal E}| edges are represented in O(E1/2){\cal O}(|{\cal E}|^{1/2}) memory and can be built in a distributed setting from small data structures, making them easy to share in compressed form. Many interesting graph calculations have worst-case complexity bounds O(Ep){\cal O}(|{\cal E}|^p) and often these are reduced to O(Ep/2){\cal O}(|{\cal E}|^{p/2}) for Kronecker product graphs, when a Kronecker formula can be derived yielding the sought calculation on CC in terms of related calculations on AA and BB. We focus on deriving formulas for triangle participation at vertices, tC{\bf t}_C, a vector storing the number of triangles that every vertex is involved in, and triangle participation at edges, ΔC\Delta_C, a sparse matrix storing the number of triangles at every edge.Comment: 10 pages, 7 figures, IEEE IPDPS Graph Algorithms Building Block

    Mathematics and the Internet: A Source of Enormous Confusion and Great Potential

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    Graph theory models the Internet mathematically, and a number of plausible mathematically intersecting network models for the Internet have been developed and studied. Simultaneously, Internet researchers have developed methodology to use real data to validate, or invalidate, proposed Internet models. The authors look at these parallel developments, particularly as they apply to scale-free network models of the preferential attachment type

    Systematic Topology Analysis and Generation Using Degree Correlations

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    We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS- and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.Comment: Final versio

    Gravity-Inspired Graph Autoencoders for Directed Link Prediction

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    Graph autoencoders (AE) and variational autoencoders (VAE) recently emerged as powerful node embedding methods. In particular, graph AE and VAE were successfully leveraged to tackle the challenging link prediction problem, aiming at figuring out whether some pairs of nodes from a graph are connected by unobserved edges. However, these models focus on undirected graphs and therefore ignore the potential direction of the link, which is limiting for numerous real-life applications. In this paper, we extend the graph AE and VAE frameworks to address link prediction in directed graphs. We present a new gravity-inspired decoder scheme that can effectively reconstruct directed graphs from a node embedding. We empirically evaluate our method on three different directed link prediction tasks, for which standard graph AE and VAE perform poorly. We achieve competitive results on three real-world graphs, outperforming several popular baselines.Comment: ACM International Conference on Information and Knowledge Management (CIKM 2019

    Spatial and performance optimality in power distribution networks

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    (c) 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.Complex network theory has been widely used in vulnerability analysis of power networks, especially for power transmission ones. With the development of the smart grid concept, power distribution networks are becoming increasingly relevant. In this paper, we model power distribution systems as spatial networks. Topological and spatial properties of 14 European power distribution networks are analyzed, together with the relationship between geographical constraints and performance optimization, taking into account economic and vulnerability issues. Supported by empirical reliability data, our results suggest that power distribution networks are influenced by spatial constraints which clearly affect their overall performance.Peer ReviewedPostprint (author's final draft
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