2,206 research outputs found

    Matched Filters for Noisy Induced Subgraph Detection

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    The problem of finding the vertex correspondence between two noisy graphs with different number of vertices where the smaller graph is still large has many applications in social networks, neuroscience, and computer vision. We propose a solution to this problem via a graph matching matched filter: centering and padding the smaller adjacency matrix and applying graph matching methods to align it to the larger network. The centering and padding schemes can be incorporated into any algorithm that matches using adjacency matrices. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks. However, there are currently no efficient algorithms for solving this problem. To illustrate the possibilities and challenges of such problems, we use an algorithm that can exploit a partially known correspondence and show via varied simulations and applications to {\it Drosophila} and human connectomes that this approach can achieve good performance.Comment: 41 pages, 7 figure

    Matched filters for noisy induced subgraph detection

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    First author draftWe consider the problem of finding the vertex correspondence between two graphs with different number of vertices where the smaller graph is still potentially large. We propose a solution to this problem via a graph matching matched filter: padding the smaller graph in different ways and then using graph matching methods to align it to the larger network. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks, though there are currently no efficient algorithms for solving this problem. We consider an approach that exploits a partially known correspondence and show via varied simulations and applications to the Drosophila connectome that in practice this approach can achieve good performance.https://arxiv.org/abs/1803.02423https://arxiv.org/abs/1803.0242

    Graph Isomorphism and Identification Matrices: Sequential Algorithms

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    AbstractA number of properties on identification matrices are presented here. For example, we prove that adjacency matrices are identification matrices for all bipartite graphs. We also study the application of the theory of identification matrices to solving the graph isomorphism problem, a famous open problem. We show that, given two graphs represented by two identification matrices with respect to a certain relation, isomorphism can be decided efficiently if at least one matrix satisfies the consecutive 1's property or a relaxed property thereof. Graphs which have identification matrices satisfying the consecutive 1's property include, among others, proper interval graphs and doubly convex bipartite graphs. This work leads to the first efficient isomorphism testing algorithms for certain classes of graphs and more efficient algorithms for some other classes of graphs. The algorithms for some classes of graphs including convex bipartite graphs run in linear time and are optimal

    An adaptive prefix-assignment technique for symmetry reduction

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    This paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system. The technique is based on McKay's canonical extension framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the technique are (i) adaptability---the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries; (ii) parallelizability---prefix-assignments can be processed in parallel independently of each other; (iii) versatility---the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability---the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the technique and carry out a set of experiments that demonstrate ability to reduce symmetry on hard instances. Furthermore, we demonstrate that the implementation effectively parallelizes to compute clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie

    Blind identification of an unknown interleaved convolutional code

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    We give here an efficient method to reconstruct the block interleaver and recover the convolutional code when several noisy interleaved codewords are given. We reconstruct the block interleaver without assumption on its structure. By running some experimental tests we show the efficiency of this method even with moderate noise

    Baby-Step Giant-Step Algorithms for the Symmetric Group

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    We study discrete logarithms in the setting of group actions. Suppose that GG is a group that acts on a set SS. When r,sSr,s \in S, a solution gGg \in G to rg=sr^g = s can be thought of as a kind of logarithm. In this paper, we study the case where G=SnG = S_n, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets A,BSnA, B \subseteq S_n such that every permutation of SnS_n can be written as a product abab of elements aAa \in A and bBb \in B. Our deterministic procedure is optimal up to constant factors, in the sense that AA and BB can be computed in optimal asymptotic complexity, and A|A| and B|B| are a small constant from n!\sqrt{n!} in size. We also analyze randomized "collision" algorithms for the same problem

    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
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