Node Expansions and Cuts in Gromov-hyperbolic Graphs ∗ Bhaskar DasGupta Department of Computer Science University of Illinois at Chicago Chicago, IL 60607, USA bdasgup@uic.edu March 28, 2016 Joint work with � Marek Karpinski (University of Bonn) � Nasim Mobasheri (UIC) � Farzaneh Yahyanejad ∗ Supported by NSF grant IIS-1160995 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 1 / 29
Outline of talk Introduction and Motivation 1 Basic definitions and notations 2 Effect of δ on Expansions and Cuts in δ -hyperbolic Graphs 3 Algorithmic Applications 4 Conclusion and Future Research 5 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 2 / 29
Introduction Various network measures Graph-theoretical analysis leads to useful insights for many complex systems, such as � World-Wide Web � social network of jazz musicians � metabolic networks � protein-protein interaction networks Examples of useful network measures for such analyses � degree based , e.g. ⊲ maximum/minimum/average degree, degree distribution, ...... � connectivity based , e.g. ⊲ clustering coefficient, largest cliques or densest sub-graphs, ...... � geodesic based , e.g. ⊲ diameter, betweenness centrality, ...... � other more complex measures Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 3 / 29
Introduction Gromov-hyperbolicity as a network measure network measure for this talk Gromov-hyperbolicity measure δ δ δ � originally proposed by Gromov in 1987 in the context of group theory ⊲ observed that many results concerning the fundamental group of a Riemann surface hold true in a more general context ⊲ defined for infinite continuous metric space via properties of geodesics ⊲ can be related to standard scalar curvature of Hyperbolic manifold � adopted to finite graphs using a 4-node condition or equivalently using thin triangles Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 4 / 29
Basic definitions and notations Hyperbolicity of real-world networks Are there real-world networks that are hyperbolic? Yes, for example: � Preferential attachment networks were shown to be scaled hyperbolic ⊲ [Jonckheere and Lohsoonthorn, 2004; Jonckheere, Lohsoonthorn and Bonahon, 2007] � Networks of high power transceivers in a wireless sensor network were empirically observed to have a tendency to be hyperbolic ⊲ [Ariaei, Lou, Jonckeere, Krishnamachari and Zuniga, 2008] � Communication networks at the IP layer and at other levels were empirically observed to be hyperbolic ⊲ [Narayan and Saniee, 2011] � Extreme congestion at a very limited number of nodes in a very large traffic network was shown to be caused due to hyperbolicity of the network together with minimum length routing ⊲ [Jonckheerea, Loua, Bonahona and Baryshnikova, 2011] � Topology of Internet can be effectively mapped to a hyperbolic space ⊲ [Bogun, Papadopoulos and Krioukov, 2010] Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 5 / 29
Motivation Effect of δ on expansion and cut-size Standard practice to investigate/categorize computational complexities of combinatorial problems in terms of ranges of topological measures: � 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. Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 6 / 29
Motivation Effect of δ on expansion and cut-size Standard practice to investigate/categorize computational complexities of combinatorial problems in terms of ranges of topological measures: � 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. Motivation for this paper: Effect of δ δ on expansion and cut-size δ � What is the effect of δ δ δ on expansion and cut-size bounds on graphs ? � For what asymptotic ranges of values of δ δ δ can these bounds be used to obtain improved approximation algorithms for related combinatorial problems ? Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 6 / 29
Outline of talk Introduction and Motivation 1 Basic definitions and notations 2 Effect of δ on Expansions and Cuts in δ -hyperbolic Graphs 3 Algorithmic Applications 4 Conclusion and Future Research 5 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 7 / 29
Basic definitions and notations Graphs, geodesics and related notations Graphs, geodesics and related notations G = ( V , E ) G = ( V , E ) G = ( V , E ) connected undirected graph of n ≥ 4 n ≥ 4 n ≥ 4 nodes � between nodes u u P u P u P � � � � � � v � v � v path P ≡ P ≡ P ≡ = u , u 1 ,..., u k − 1 , u k u 0 u 0 = u , u 1 ,..., u k − 1 , u k u 0 = u , u 1 ,..., u k − 1 , u k u u and v v v = v = v = v length (number of edges) of the path u P u P u P ℓ ( P ) ℓ ( P ) ℓ ( P ) � v � v � v � � � � � � P P P sub-path of P P between nodes u i u i and u j u i u i u i � u j � u j � u j u i , u i + 1 ,..., u j u i , u i + 1 ,..., u j u i , u i + 1 ,..., u j P u i u j u j u s u s u s � v � v � v a shortest path between nodes u u u and v v v d u , v d u , v d u , v length of a shortest path between nodes u u u and v v v u 4 u 4 u 4 P � � � � � � P P u 2 u 2 u 2 u 2 u 2 u 2 � u 6 � u 6 � u 6 is the path P ≡ P ≡ P ≡ u 2 , u 4 , u 5 , u 6 u 2 , u 4 , u 5 , u 6 u 2 , u 4 , u 5 , u 6 ℓ ( P ) ℓ ( P ) ℓ ( P ) = 3 = 3 = 3 u 5 u 5 u 5 u 6 u 6 u 6 u 1 u 1 u 1 d u 2 , u 6 d u 2 , u 6 d u 2 , u 6 = 2 = 2 = 2 u 3 u 3 u 3 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 8 / 29
Basic definitions and notations 4 node condition d u 2 , u 4 d u 2 , u 4 d u 2 , u 4 u 2 u 2 u 2 u 4 u 4 u 4 Consider four nodes u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 and d u 2 , u 3 d u 2 , u 3 d u 2 , u 3 d u 1 , u 4 d u 1 , u 4 d u 1 , u 4 the six shortest paths among pairs of these nodes u 1 u 3 u 1 u 1 u 3 u 3 d u 3 , u 4 d u 1 , u 2 d u 1 , u 2 d u 1 , u 2 d u 3 , u 4 d u 3 , u 4 d u 1 , u 3 d u 1 , u 3 d u 1 , u 3 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 9 / 29
Basic definitions and notations 4 node condition d u 2 , u 4 d u 2 , u 4 d u 2 , u 4 u 2 u 2 u 2 u 4 u 4 u 4 Consider four nodes u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 and d u 2 , u 3 d u 2 , u 3 d u 2 , u 3 d u 1 , u 4 d u 1 , u 4 d u 1 , u 4 the six shortest paths among pairs of these nodes u 1 u 3 u 1 u 1 u 3 u 3 d u 3 , u 4 d u 1 , u 2 d u 1 , u 2 d u 1 , u 2 d u 3 , u 4 d u 3 , u 4 d u 1 , u 3 d u 1 , u 3 d u 1 , u 3 Assume, without loss of generality, that + + + d u 1 , u 4 + d u 2 , u 3 d u 1 , u 4 + d u 2 , u 3 d u 1 , u 4 + d u 2 , u 3 ≥ d u 1 , u 3 + d u 2 , u 4 ≥ d u 1 , u 3 + d u 2 , u 4 ≥ d u 1 , u 3 + d u 2 , u 4 ≥ d u 1 , u 2 + d u 3 , u 4 ≥ d u 1 , u 2 + d u 3 , u 4 ≥ d u 1 , u 2 + d u 3 , u 4 + + + ≥ ≥ ≥ ≥ ≥ ≥ + + + � � � �� �� �� � � � � � � �� �� �� � � � � � � �� �� �� � � � = L = L = L = M = M = M = S = S = S Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 9 / 29
Basic definitions and notations 4 node condition d u 2 , u 4 d u 2 , u 4 d u 2 , u 4 u 2 u 2 u 2 u 4 u 4 u 4 Consider four nodes u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 and d u 2 , u 3 d u 2 , u 3 d u 2 , u 3 d u 1 , u 4 d u 1 , u 4 d u 1 , u 4 the six shortest paths among pairs of these nodes u 1 u 3 u 1 u 1 u 3 u 3 d u 3 , u 4 d u 1 , u 2 d u 1 , u 2 d u 1 , u 2 d u 3 , u 4 d u 3 , u 4 d u 1 , u 3 d u 1 , u 3 d u 1 , u 3 Assume, without loss of generality, that + + + d u 1 , u 4 + d u 2 , u 3 d u 1 , u 4 + d u 2 , u 3 d u 1 , u 4 + d u 2 , u 3 ≥ d u 1 , u 3 + d u 2 , u 4 ≥ d u 1 , u 3 + d u 2 , u 4 ≥ d u 1 , u 3 + d u 2 , u 4 ≥ d u 1 , u 2 + d u 3 , u 4 ≥ d u 1 , u 2 + d u 3 , u 4 ≥ d u 1 , u 2 + d u 3 , u 4 + + + ≥ ≥ ≥ ≥ ≥ ≥ + + + � � � �� �� �� � � � � � � �� �� �� � � � � � � �� �� �� � � � = L = L = L = M = M = M = S = S = S − ( ( ( ) ) ) − − + + + Let δ u 1 , u 2 , u 3 , u 4 = L − M δ u 1 , u 2 , u 3 , u 4 = L − M δ u 1 , u 2 , u 3 , u 4 = L − M + + + 2 2 2 2 2 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, 2016 9 / 29
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