Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

Gupta, Bhaskar Das; Karpiński, Marek; Mobasheri, Nasim; Yahyanejad, Farzane

doi:10.1007/s00453-017-0291-7

Cited by 11 publications

(4 citation statements)

References 40 publications

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“…This is due to the fact that many of these graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) possess certain geometric and topological characteristics. Hence, for many applications, including the design of efficient algorithms (cf., e.g., [4,[9][10][11][12][13]17,20,33]), it is useful to know an accurate approximation of the hyperbolicity δ(G) of a graph G.…”

Section: Introductionmentioning

confidence: 99%

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Chalopin

Chepoi

Dragan

et al. 2019

Discrete Comput Geom

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In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs, via a new parameter which we call rooted insize. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomialtime, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm (with an additive constant 1) for computing the hyperbolicity of an n-vertex graph G = (V, E) in optimal time O(n 2 ) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space.We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space (X, d) has such a geodesic spanning tree. 1 The O(·) notation hides polyloglog factors. 2 Informally, (y|z)w can be viewed as half the detour you make, when going over w to get from y to z.

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Section: Introductionmentioning

confidence: 99%

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Chalopin

Chepoi

Dragan

et al. 2019

Discrete Comput Geom

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“…Moreover, hyperbolicity has been found to capture important properties of several large practical graphs such as the Internet [27] or database relations [32]. Due to its importance in discrete mathematics, algorithms, metric graph theory, researchers have studied various algorithmic aspects of hyperbolic graphs [8,13,9,14]. Note that graphs with diameter 2 are hyperbolic, which may contain any graph as an induced subgraph.…”

Section: Hyperbolic Graphsmentioning

confidence: 99%

Isometric path antichain covers: beyond hyperbolic graphs

Chakraborty¹,

Chalopin²,

Foucaud³

et al. 2023

Preprint

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The isometric path antichain cover number of a graph G, denoted by ipacc (G), is a graph parameter that was recently introduced to provide a constant factor approximation algorithm for Isometric Path Cover, whose objective is to cover all vertices of a graph with a minimum number of isometric paths (i.e. shortest paths between their end-vertices). This parameter was previously shown to be bounded for chordal graphs and, more generally, for graphs of bounded chordality and bounded treelength. In this paper, we show that the isometric path antichain cover number remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results imply a constant factor approximation algorithm for Isometric Path Cover on all the above graph classes. Our results also show that the distance functions of these (structurally) different graph classes are more similar than previously thought. * This research was partially financed by the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE initiative 16-IDEX-0001 .

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“…Our paper seeks to adapt the definition of curvature from the non-network domains in a suitable way for detecting network anomalies. For example, in networks with sufficiently small Gromov-hyperbolicity and sufficiently large diameter a suitably small subset of nodes or edges can be removed to stretch the geodesics between two distinct parts of the network by an exponential amount leading to extreme implications on the expansion properties of such networks [10,24], which is akin to the characterization of singularities (an extreme anomaly) by geodesic incompleteness (i.e., stretching all geodesics passing through the region infinitely) [39]. It is our hope that research works in this paper will stimulate further research concerning the exciting interplay between curvatures from network and nonnetwork domains, a much desired goal in our opinion.…”

Section: Why Use Network Curvature Measures?mentioning

confidence: 99%

Why Did the Shape of Your Network Change? (On Detecting Network Anomalies via Non-local Curvatures)

2020

Self Cite

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Anomaly detection problems (also called change-point detection problems) have been studied in data mining, statistics and computer science over the last several decades (mostly in non-network context) in applications such as medical condition monitoring, weather change detection and speech recognition. In recent days, however, anomaly detection problems have become increasing more relevant in the context of network science since useful insights for many complex systems in biology, finance and social science are often obtained by representing them via networks. Notions of local and non-local curvatures of higher-dimensional geometric shapes and topological spaces play a fundamental role in physics and mathematics in characterizing anomalous behaviours of these higher dimensional entities. However, using curvature measures to detect anomalies in networks is not yet very common. To this end, a main goal in this paper to formulate and analyze curvature analysis methods to provide the foundations of systematic approaches to find critical components and detect anomalies in networks. For this purpose, we use two measures of network curvatures which depend on non-trivial global properties, such as distributions of geodesics and higher-order correlations among nodes, of the given network. Based on these measures, we precisely formulate several computational problems related to anomaly detection in static or dynamic networks, and provide non-trivial computational complexity results for these problems. This paper must not be viewed as delivering the final word on appropriateness and suitability of specific curvature measures. Instead, it is our hope that this paper will stimulate and motivate further theoretical or empirical research concerning the exciting interplay between notions of curvatures from network and non-network domains, a much desired goal in our opinion.In this paper we seek to address research questions of the following generic nature:"Given a static or dynamic network, identify the critical components of the network that "encode" significant non-trivial global properties of the network".To identify critical components, one first needs to provide details for following four specific items:(i) network model selection,(ii) network evolution rule for dynamic networks, (iii) definition of elementary critical components, and (iv) network property selection (i.e., the global properties of the network to be investigated).The specific details for these items for this paper are as follows:(i) Network model selection: Our network model will be undirected graphs.(ii) Network evolution rule for dynamic networks: Our dynamic networks follow the time series model and are given as a sequence of networks over discrete time steps, where each network is obtained from the previous one in the sequence by adding and/or deleting some nodes and/or edges.(iii) Critical component definition: Individual edges are elementary members of critical components.(iv) Network property selection: The network measure for this paper will be appro...

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Effect of Gromov-Hyperbolicity Parameter on Cuts and Expansions in Graphs and Some Algorithmic Implications

Cited by 11 publications

References 40 publications

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs

Isometric path antichain covers: beyond hyperbolic graphs

Why Did the Shape of Your Network Change? (On Detecting Network Anomalies via Non-local Curvatures)

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