Showing papers on "Vertex (graph theory) published in 2015"

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Abstract: Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.

Exploiting vertex relationships in speeding up subgraph isomorphism over large graphs

Abstract: Subgraph Isomorphism is a fundamental problem in graph data processing. Most existing subgraph isomorphism algorithms are based on a backtracking framework which computes the solutions by incrementally matching all query vertices to candidate data vertices. However, we observe that extensive duplicate computation exists in these algorithms, and such duplicate computation can be avoided by exploiting relationships between data vertices. Motivated by this, we propose a novel approach, BoostIso, to reduce duplicate computation. Our extensive experiments with real datasets show that, after integrating our approach, most existing subgraph isomorphism algorithms can be speeded up significantly, especially for some graphs with intensive vertex relationships, where the improvement can be up to several orders of magnitude.

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter.

Abstract: The radius and diameter are fundamental graph parameters They are defined as the minimum and maximum of the eccentricities in a graph, respectively, where the eccentricity of a vertex is the largest distance from the vertex to another node In directed graphs, there are several versions of these problems For instance, one may choose to define the eccentricity of a node in terms of the largest distance into the node, out of the node, the sum of the two directions (ie roundtrip) and so on All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step Solving APSP, however, on $n$-node graphs requires $\Omega(n^2)$ time even in sparse graphs, as one needs to output $n^2$ distances Motivated by known and new negative results on the impossibility of computing these measures exactly in general graphs in truly subquadratic time, under plausible assumptions, we search for \emph{approximation} and \emph{fixed parameter subquadratic} algorithms, and for reasons why they do not exist Our results include: - Truly subquadratic approximation algorithms for most of the versions of Diameter and Radius with \emph{optimal} approximation guarantees (given truly subquadratic time), under plausible assumptions In particular, there is a $2$-approximation algorithm for directed Radius with one-way distances that runs in $\tilde{O}(m\sqrt{n})$ time, while a $(2-\delta)$-approximation algorithm in $O(n^{2-\epsilon})$ time is unlikely - On graphs with treewidth $k$, we can solve the problems in $2^{O(k\log{k})}n^{1+o(1)}$ time We show that these algorithms are near optimal since even a $(3/2-\delta)$-approximation algorithm that runs in time $2^{o(k)}n^{2-\epsilon}$ would refute the plausible assumptions

Consistency Thresholds for the Planted Bisection Model

Abstract: The planted bisection model is a random graph model in which the nodes are divided into two equal-sized communities and then edges are added randomly in a way that depends on the community membership. We establish necessary and sufficient conditions for the asymptotic recoverability of the planted bisection in this model. When the bisection is asymptotically recoverable, we give an efficient algorithm that successfully recovers it. We also show that the planted bisection is recoverable asymptotically if and only if with high probability every node belongs to the same community as the majority of its neighbors.Our algorithm for finding the planted bisection runs in time almost linear in the number of edges. It has three stages: spectral clustering to compute an initial guess, a "replica" stage to get almost every vertex correct, and then some simple local moves to finish the job. An independent work by Abbe, Bandeira, and Hall establishes similar (slightly weaker) results but only in the sparse case where pn, qn = Θ(log n /n).

Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts

Abstract: Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4-vertex patterns is highly challenging, and there are few practical results known that can scale to massive sizes. Indeed, even a highly tuned enumeration code takes more than a day on a graph with millions of edges. Most previous work that runs for truly massive graphs employ clusters and massive parallelization. We provide a sampling algorithm that provably and accurately approximates the frequencies of all 4-vertex pattern subgraphs. Our algorithm is based on a novel technique of 3-path sampling and a special pruning scheme to decrease the variance in estimates. We provide theoretical proofs for the accuracy of our algorithm, and give formal bounds for the error and confidence of our estimates. We perform a detailed empirical study and show that our algorithm provides estimates within 1% relative error for all subpatterns (over a large class of test graphs), while being orders of magnitude faster than enumeration and other sampling based algorithms. Our algorithm takes less than a minute (on a single commodity machine) to process an Orkut social network with 300 million edges.

Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

Abstract: Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process that takes values in the vertex set of a graph G, which is more likely to cross edges it has visited before. We show that it can be interpreted as an annealed version of the Vertex-reinforced jump process (VRJP), conceived by Werner and first studied by Davis and Volkov (2002,2004), a continuous-time process favouring sites with more local time. We calculate, for any finite graph G, the limiting measure of the centred occupation time measure of VRJP, and interpret it as a supersymmetric hyperbolic sigma model in quantum field theory. This enables us to deduce that VRJP is recurrent in any dimension for large reinforcement, using a localisation result of Disertori and Spencer (2010).

Semidefinite Programs on Sparse Random Graphs and their Application to Community Detection

Abstract: Denote by $A$ the adjacency matrix of an Erdos-Renyi graph with bounded average degree. We consider the problem of maximizing $\langle A-E\{A\},X\rangle$ over the set of positive semidefinite matrices $X$ with diagonal entries $X_{ii}=1$. We prove that for large (bounded) average degree $d$, the value of this semidefinite program (SDP) is --with high probability-- $2n\sqrt{d} + n\, o(\sqrt{d})+o(n)$. For a random regular graph of degree $d$, we prove that the SDP value is $2n\sqrt{d-1}+o(n)$, matching a spectral upper bound. Informally, Erdos-Renyi graphs appear to behave similarly to random regular graphs for semidefinite programming. We next consider the sparse, two-groups, symmetric community detection problem (also known as planted partition). We establish that SDP achieves the information-theoretically optimal detection threshold for large (bounded) degree. Namely, under this model, the vertex set is partitioned into subsets of size $n/2$, with edge probability $a/n$ (within group) and $b/n$ (across). We prove that SDP detects the partition with high probability provided $(a-b)^2/(4d)> 1+o_{d}(1)$, with $d= (a+b)/2$. By comparison, the information theoretic threshold for detecting the hidden partition is $(a-b)^2/(4d)> 1$: SDP is nearly optimal for large bounded average degree. Our proof is based on tools from different research areas: $(i)$ A new `higher-rank' Grothendieck inequality for symmetric matrices; $(ii)$ An interpolation method inspired from statistical physics; $(iii)$ An analysis of the eigenvectors of deformed Gaussian random matrices.

$${C_2}$$ -Cofiniteness of Cyclic-Orbifold Models

Abstract: We prove an orbifold conjecture for conformal field theory with a solvable automorphism group. Namely, we show that if V is a \({C_2}\) -cofinite simple vertex operator algebra and G is a finite solvable automorphism group of V, then the fixed point vertex operator subalgebra \({V^G}\) is also \({C_2}\) -cofinite, where \({C_2}\) -cofiniteness is equivalent to the condition that V has only finitely many isomorphism classes of simple V-modules (including weak modules) and all finitely generated V-modules have composition series. This result offers a mathematically rigorous background to orbifold theories with solvable automorphism groups.

Graph invariant kernels

Abstract: We introduce a novel kernel that upgrades the Weisfeiler-Lehman and other graph kernels to effectively exploit high-dimensional and continuous vertex attributes. Graphs are first decomposed into subgraphs. Vertices of the subgraphs are then compared by a kernel that combines the similarity of their labels and the similarity of their structural role, using a suitable vertex invariant. By changing this invariant we obtain a family of graph kernels which includes generalizations of Weisfeiler-Lehman, NSPDK, and propagation kernels. We demonstrate empirically that these kernels obtain state-of-the-art results on relational data sets.

Balance between complexity and quality: local search for minimum vertex cover in massive graphs

Abstract: The problem of finding a minimum vertex cover (MinVC) in a graph is a well known NP-hard problem with important applications. There has been much interest in developing heuristic algorithms for finding a "good" vertex cover in graphs. In practice, most heuristic algorithms for MinVC are based on the local search method. Previously, local search algorithms for MinVC have focused on solving academic benchmarks where the graphs are of relatively small size, and they are not suitable for solving massive graphs as they usually have highcomplexity heuristics. In this paper, we propose a simple and fast local search algorithm called FastVC for solving MinVC in massive graphs, which is based on two low-complexity heuristics. Experimental results on a broad range of real world massive graphs show that FastVC finds much better vertex covers (and thus also independent sets) than state of the art local search algorithms for MinVC.

Accurate band gaps of extended systems via efficient vertex corrections in G W

Abstract: We propose the use of an approximate bootstrap exchange-correlation kernel to account for vertex corrections in self-consistent GW calculations. We show that the approximate kernel gives accurate band gaps for a variety of extended systems, including simple sp semiconductors, wide band-gap insulators, and transition-metal compounds with either closed or open d shells. The accuracy is comparable with that obtained via the solution of the Bethe-Salpeter equation but only at a fraction of the computational cost.

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal

Abstract: In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = (V, E) with a degree upper bound Bv on each vertex v ∈ V, and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this article we present a polynomial-time algorithm which returns a spanning tree T of cost at most OPT and dT(v) ≤ Bv + 1 for all v, where dT(v) denotes the degree of v in T. This generalizes a result of Furer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degree upper bound Bv, and returns a spanning tree with cost at most OPT and Av - 1 ≤ dT(v) ≤ Bv + 1 for all v ∈ V. This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms.

Connectivity is a poor indicator of fast quantum search.

Abstract: A randomly walking quantum particle evolving by Schrodinger's equation searches on d-dimensional cubic lattices in O(√N) time when d≥5, and with progressively slower runtime as d decreases. This suggests that graph connectivity (including vertex, edge, algebraic, and normalized algebraic connectivities) is an indicator of fast quantum search, a belief supported by fast quantum search on complete graphs, strongly regular graphs, and hypercubes, all of which are highly connected. In this Letter, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search. The second example is a novel two-stage quantum walk algorithm in which the walking rate must be adjusted to yield high search probability.

The quark-gluon vertex in Landau gauge bound-state studies

Abstract: We present a practical method for the solution of the quark-gluon vertex for use in Bethe-Salpeter and Dyson-Schwinger calculations. The efficient decomposition into the necessary covariants is detailed, with the numerical algorithm outlined for both real and complex Euclidean momenta. A truncation of the quark-gluon vertex, that neglects explicit back-coupling to enable the application to bound-state calculations, is given together with results for the quark propagator and quark-gluon vertex for different quark flavours. The relative impact of the various components of the quark-gluon vertex is highlighted with the flavour dependence of the effective quark-gluon interaction obtained, thus providing insight for the construction of phenomenological models within the rainbow ladder. Finally, we solve the corresponding Green's functions for complex Euclidean momenta as required in future bound-state calculations.

Scaling limits of random planar maps with a unique large face

Abstract: We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n(-1/2) is described by a Brownian excursion. The planar maps, with the graph metric resealed by n(-1/2), are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

Cheeger inequalities for unbounded graph laplacians

Abstract: We use the concept of intrinsic metrics to give a new denition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Construction and Classification of Holomorphic Vertex Operator Algebras

Abstract: We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

-Algebras, False Theta Functions and Quantum Modular Forms, I

Abstract: In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal eld theory. We prove a variety of results concerning the asymptotics of modied characters of irreducible modules of certain W-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity ( a la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many \suciently nice" irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier’s example coming from Kontsevich’s \strange series". In the last part, slightly independent of all this, we also discuss Nahm-type q-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the (1;p)-singlet algebra.

Parameterized streaming: maximal matching and vertex cover

Abstract: As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams.In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems. Here, we are given a parameter k and the objective is to decide whether there is a solution bounded by k. By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of Maximal Matching and Vertex Cover. We consider various models for a graph stream on n nodes: the insertion-only model where the edges can only be added, and the dynamic model where edges can be both inserted and deleted. More formally, we show the following results:• In the insertion only model, there is a one-pass deterministic algorithm for the parameterized Vertex Cover problem which computes a sketch using O(k2) space such that at each timestamp in time O(2k) it can either extract a solution of size at most k for the current instance, or report that no such solution exists. We also show a tight lower bound of Ω(k2) for the space complexity of any (randomized) streaming algorithms for the parameterized Vertex Cover, even in the insertion-only model.• In the dynamic model, and under the promise that at each timestamp there is a maximal matching of size at most k, there is a one-pass O(k2)-space (sketch-based) dynamic algorithm that maintains a maximal matching with worst-case update time O(k2). This algorithm partially solves Open Problem 64 from [1]. An application of this dynamic matching algorithm is a one-pass O(k2)-space streaming algorithm for the parameterized Vertex Cover problem that in time O(2k) extracts a solution for the final instance with probability 1 − δ/nO(1), where δ

Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations

Abstract: We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painleve equations, using expansions of irregular conformal blocks at an irregular singular point.

Distributed coloring algorithms for triangle-free graphs

Abstract: Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree-Δ graphs may require palettes of Δ + 1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find ( Δ / k ) -coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees. The parameter k can be at most ( 1 4 - o ( 1 ) ) ln ? Δ in triangle-free graphs and at most ( 1 - o ( 1 ) ) ln ? Δ in girth-5 graphs and trees, where o ( 1 ) is a function of Δ. Specifically, for Δ sufficiently large we can find such a coloring in O ( k + log * ? n ) time. Moreover, for any Δ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O ( log ? Δ + log Δ ? log ? n ) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most ( 4 + o ( 1 ) ) Δ ln ? Δ , which improves on Jamall's recent bound of ( 67 + o ( 1 ) ) Δ ln ? Δ . Finally, we show that ( Δ + 1 ) -coloring for triangle-free graphs can be obtained in sublogarithmic time for any Δ.