12,019 research outputs found

    Distance-Dependent Kronecker Graphs for Modeling Social Networks

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    This paper focuses on a generalization of stochastic Kronecker graphs, introducing a Kronecker-like operator and defining a family of generator matrices H dependent on distances between nodes in a specified graph embedding. We prove that any lattice-based network model with sufficiently small distance-dependent connection probability will have a Poisson degree distribution and provide a general framework to prove searchability for such a network. Using this framework, we focus on a specific example of an expanding hypercube and discuss the similarities and differences of such a model with recently proposed network models based on a hidden metric space. We also prove that a greedy forwarding algorithm can find very short paths of length O((log log n)^2) on the hypercube with n nodes, demonstrating that distance-dependent Kronecker graphs can generate searchable network models

    The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

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    Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

    On the Fiedler value of large planar graphs

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    The Fiedler value λ2\lambda_2, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs GG with nn vertices, denoted by λ2max\lambda_{2\max}, and we show the bounds 2+Θ(1n2)λ2max2+O(1n)2+\Theta(\frac{1}{n^2}) \leq \lambda_{2\max} \leq 2+O(\frac{1}{n}). We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on λ2max\lambda_{2\max} for two more classes of graphs, those of bounded genus and KhK_h-minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and Its Application

    Dynamic Programming for Graphs on Surfaces

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    We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure
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