Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Jonckheere, Edmond; Lohsoonthorn, Poonsuk; Ariaei, F.

doi:10.1080/15427951.2011.601233

Cited by 12 publications

(17 citation statements)

References 10 publications

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“…We use the notation D u 1 ,u 2 ,u 3 ,u 4 = max i, j∈{1,2,3,4} d u i ,u j to indicate the diameter of the subset of four nodes u 1 , u 2 , u 3 and u 4 . By using theoretical or empirical calculations, the authors in [42] provide the bounds shown in Table III.…”

Section: Checking Hyperbolicity Via the Scaled Hyperbolicity Approachmentioning

confidence: 99%

Topological implications of negative curvature for biological and social networks

2014

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Network measures that reflect the most salient properties of complex large-scale networks are in high demand in the network research community. In this paper we adapt a combinatorial measure of negative curvature (also called hyperbolicity) to parametrized finite networks, and show that a variety of biological and social networks are hyperbolic. This hyperbolicity property has strong implications on the higher-order connectivity and other topological properties of these networks. Specifically, we derive and prove bounds on the distance among shortest or approximately shortest paths in hyperbolic networks. We describe two implications of these bounds to crosstalk in biological networks, and to the existence of central, influential neighborhoods in both biological and social networks.

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Section: Checking Hyperbolicity Via the Scaled Hyperbolicity Approachmentioning

confidence: 99%

Topological implications of negative curvature for biological and social networks

2014

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“…In this case, we return B G−C (p, (∆/3) − α ∆) = B G (p, (∆/3) − α ∆) as our solution. The size requirement follows since | B G (p, (∆/3) − α ∆) | < ζ n was shown in (22). Note that nodes in B G (p, (∆/3) − α ∆) can only be connected to nodes in C, and thus h G (B G (p, (∆/3) − α ∆) ) ≤ | C | / | B G (p, (∆/3) − α ∆) | ≤ (∆/3)d α∆ / n ε log d e/8 < n α−(ε log d e/8) log n < n 1/(7 ∆ 1/2 log(2d)) − (ε/(8 ln d) ) log n < ε where the penultimate inequality follows since ∆ = ω(1).…”

Section: Application To the Small Set Expansion Problemmentioning

confidence: 96%

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

et al. 2017

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δ-hyperbolic graphs, originally conceived by Gromov in 1987, occur often in many network applications; for fixed δ, such graphs are simply called hyperbolic graphs and include non-trivial interesting classes of "non-expander" graphs. The main motivation of this paper is to investigate the effect of the hyperbolicity measure δ on expansion and cut-size bounds on graphs (here δ need not be a constant), and the asymptotic ranges of δ for which these results may provide improved approximation algorithms for related combinatorial problems. To this effect, we provide constructive bounds on node expansions for δ-hyperbolic graphs as a function of δ, and show that many witnesses (subsets of nodes) for such expansions can be computed efficiently even if the witnesses are required to be nested or sufficiently distinct from each other. To the best of our knowledge, these are the first such constructive bounds proven. We also show how to find a large family of s-t cuts with relatively small number of cut-edges when s and t are sufficiently far apart. We then provide algorithmic consequences of these bounds and their related proof techniques for two problems for δ-hyperbolic graphs (where δ is a function f of the number of nodes, 2 Bhaskar DasGupta et al.the exact nature of growth of f being dependent on the particular problem considered). Mathematics Subject Classification1 IntroductionUseful insights for many complex systems such as the world-wide web, social networks, metabolic networks, and protein-protein interaction networks can often be obtained by representing them as parameterized networks and analyzing them using graph-theoretic tools. Some standard measures used for such investigations include degree based measures (e.g., maximum/minimum/average degree or degree distribution) connectivity based measures (e.g., clustering coefficient, claw-free property, largest cliques or densest sub-graphs), and geodesic based measures (e.g., diameter or betweenness centrality). It is a standard practice in theoretical computer science to investigate and categorize the computational complexities of combinatorial problems in terms of ranges of these parameters. For example:◮ Bounded-degree graphs are known to admit improved approximation as opposed to their arbitrary-degree counter-parts for many graph-theoretic problems. ◮ Claw-free graphs are known to admit improved approximation as opposed to general graphs for graph-theoretic problems such as the maximum independent set problem.In this paper we consider a topological measure called Gromov-hyperbolicity (or, simply hyperbolicity for short) for undirected unweighted graphs that has recently received significant attention from researchers in both the graph theory and the network science community. This hyperbolicity measure δ was originally conceived in a somewhat different group-theoretic context by Gromov [20]. The measure was first defined for infinite continuous metric space via properties of geodesics [10], but was later also adopted for finite graphs. Lately, there have been...

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“…There are many well-known measures of curvature of a continuous surface or other similar spaces (e.g., curvature of a manifold) that are widely used in many branches of physics and mathematics. It is possible to relate Gromov-hyperbolic curvature to such other curvature notions indirectly via its scaled version, e.g., see [45,46,56].…”

Section: Gromov-hyperbolic Curvaturementioning

confidence: 99%

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

2020

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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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Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Cited by 12 publications

References 10 publications

Topological implications of negative curvature for biological and social networks

Topological implications of negative curvature for biological and social networks

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

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

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