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) Negative curvature for networks November 29, 2014 9 / 52
Basic definitions and notations Average hyperbolicity measure, computational issues Definition ( average hyperbolicity) δ ave ( G ) = 1 δ ave ( G ) = 1 δ ave ( G ) = 1 expected value of δ u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 if δ u 1 , u 2 , u 3 , u 4 � � � δ u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 � n � n � n � � � u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 are picked uniformly at random u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 4 4 4 Computation of δ ( G ) δ ( G ) δ ( G ) and δ ave ( G ) δ ave ( G ) δ ave ( G ) � � � n 4 � n 4 � n 4 � � Trivially in O O O time Floyd–Warshall algorithm ⊲ Compute all-pairs shortest paths � � � n 3 � n 3 � n 3 � O O O time � � � n 4 � n 4 � n 4 � ⊲ For each combination u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 u 1 , u 2 , u 3 , u 4 , compute δ u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 O O O time � n 4 − ε � � � n 4 − ε � n 4 − ε � � Open problem: can we compute in O O O time? Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 10 / 52
Outline of talk Introduction 1 Basic definitions and notations 2 Computing hyperbolicity for real networks 3 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 11 / 52
Computing hyperbolicity for real networks Direct calculation Real networks used for empirical validation 20 well-known biological and social networks � 11 biological networks that include 3 transcriptional regulatory, 5 signalling, 1 metabolic, 1 immune response and 1 oriented protein-protein interaction networks � 9 social networks range from interactions in dolphin communities to the social network of jazz musicians � hyperbolicity of the biological and directed social networks was computed by ignoring the direction of edges � hyperbolicity values were calculated by writing codes in C using standard algorithmic procedures Next slide: List of 20 networks � Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 12 / 52
Computing hyperbolicity for real networks Direct calculation 11 biological networks 9 social networks # nodes # edges # nodes # edges 1. E. coli transcriptional 1. Dolphin social network 62 160 311 451 2. American 2. Mammalian signaling 512 1047 115 612 3. E. coli transcriptional College Football 418 544 4. T-LGL signaling 3. Zachary Karate Club 34 78 58 135 5. S. cerevisiae 4. Books about 105 442 690 1082 transcriptional US politics 6. C. elegans metabolic 453 2040 5. Sawmill communication 36 62 7. Drosophila network segment polarity 78 132 6. Jazz Musician (6 cells) 198 2742 network 8. ABA signaling 55 88 7. Visiting ties 9. Immune response 75 144 18 42 in San Juan network 8. World Soccer 10. T cell receptor 35 118 94 138 Data, Paris 1998 signaling 9. Les Miserables 11. Oriented yeast PPI 786 2445 77 251 characters Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 13 / 52
Computing hyperbolicity for real networks Direct calculation Biological networks social networks Average Average δ ave δ ave δ ave δ δ δ δ ave δ ave δ ave δ δ δ degree degree 1. E. coli transcriptional 1. Dolphins social network 5.16 5.16 5.16 0.262 0.262 0.262 2 2 2 1.45 1.45 1.45 0.132 0.132 0.132 2 2 2 10.64 10.64 10.64 0.312 0.312 0.312 2 2 2 2. American College Football 2.04 0.013 3 2. Mammalian Signaling 2.04 2.04 0.013 0.013 3 3 3. Zachary Karate Club 4.58 4.58 4.58 0.170 0.170 0.170 1 1 1 3. E. Coli transcriptional 1.30 1.30 1.30 0.043 0.043 0.043 2 2 2 4. Books about US Politics 8.41 8.41 8.41 0.247 0.247 0.247 2 2 2 4. T LGL signaling 2.32 2.32 2.32 0.297 0.297 0.297 2 2 2 5. Sawmill communication 3.44 3.44 3.44 0.162 0.162 0.162 1 1 1 5. S. cerevisiae transcriptional 1.56 1.56 1.56 0.004 0.004 0.004 3 3 3 6. Jazz musician 27.69 27.69 27.69 0.140 0.140 1.5 0.140 1.5 1.5 6. C. elegans Metabolic 4.50 4.50 4.50 0.010 0.010 1.5 0.010 1.5 1.5 3.84 0.422 3 7. Visiting ties in San Juan 3.84 3.84 0.422 0.422 3 3 7. Drosophila segment polarity 1.69 1.69 1.69 0.676 0.676 0.676 4 4 4 8. World Soccer data, 1998 1998 1998 3.37 3.37 3.37 0.270 0.270 0.270 2.5 2.5 2.5 1.60 1.60 1.60 0.302 0.302 0.302 2 2 2 6.51 0.278 2 8. ABA signaling 9. Les Miserable 6.51 6.51 0.278 0.278 2 2 2.33 0.286 1.5 9. Immune Response Network 2.33 2.33 0.286 0.286 1.5 1.5 10. T Cell Receptor Signalling 1.46 1.46 1.46 0.323 0.323 0.323 3 3 3 11. Oriented yeast PPI 3.11 3.11 3.11 0.001 0.001 0.001 2 2 2 � Hyperbolicity values of almost all networks are small � For all networks δ ave δ ave δ ave is one or two orders of magnitude smaller than δ δ δ ⊲ Intuitively, this suggests that value of δ δ δ may be a rare deviation from typical values of δ u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 δ u 1 , u 2 , u 3 , u 4 for most combinations of 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 } � No systematic dependence of δ δ δ on number of nodes/edges or average degree Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 14 / 52
Computing hyperbolicity for real networks Direct calculation Definition ( Diameter of a graph) Fact � � � � � � δ ≤ D / 2 δ ≤ D / 2 δ ≤ D / 2 D = max u , v D = max u , v D = max u , v d u , v d u , v d u , v small diameter implies small hyperbolicity longest shortest path We found no systematic dependence of δ δ δ on D D D Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 15 / 52
Computing hyperbolicity for real networks Direct calculation Definition ( Diameter of a graph) Fact � � � � � � δ ≤ D / 2 δ ≤ D / 2 δ ≤ D / 2 D = max u , v D = max u , v D = max u , v d u , v d u , v d u , v small diameter implies small hyperbolicity longest shortest path We found no systematic dependence of δ δ δ on D D D For more rigorous checks of hyperbolicity of finite graphs and for evaluation of statistical significance of the hyperbolicity measure see our paper R. Albert, B. DasGupta and N. Mobasheri, Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 15 / 52
Outline of talk Introduction 1 Basic definitions and notations 2 Computing hyperbolicity for real networks 3 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 16 / 52
Implications of hyperbolicity We discuss topological implications of hyperbolicity somewhat informally Precise Theorems and their proofs are available in our paper R. Albert, B. DasGupta and N. Mobasheri, Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 17 / 52
Outline of talk Introduction 1 Basic definitions and notations 2 Computing hyperbolicity for real networks 3 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 18 / 52
Implications of hyperbolicity Hyperbolicity and crosstalk in regulatory networks Definition ( Path chord and chord) u 5 u 0 u 5 u 0 p a t h - c h o r d c h o r d u 4 u 1 u 4 u 1 v u 3 u 2 u 3 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 19 / 52
Implications of hyperbolicity Hyperbolicity and crosstalk in regulatory networks Definition ( Path chord and chord) u 5 u 0 u 5 u 0 p a t h - c h o r d c h o r d u 4 u 1 u 4 u 1 v u 3 u 2 u 3 u 2 Theorem ( large cycle without path-chord imply large hyperbolicity) G G G has a cycle of k k nodes which has no path-chord ⇒ k ⇒ δ ≥ ⌈ k / 4 ⌉ δ ≥ ⌈ k / 4 ⌉ δ ≥ ⌈ k / 4 ⌉ ⇒ Corollary Any cycle containing more than 4 δ 4 δ 4 δ nodes must have a path-chord Example δ < 1 δ < 1 δ < 1 ⇒ G ⇒ ⇒ G G is chordal graph Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 19 / 52 Next slide: implications for regulatory networks �
Implications of hyperbolicity Hyperbolicity and crosstalk in regulatory networks An example of a regulatory network Network associated to the Drosophila segment polarity G. von Dassow, E. Meir, E. M. Munro and G. M. Odell, Nature 406, 188-192 (2000) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 20 / 52
Implications of hyperbolicity Hyperbolicity and crosstalk in regulatory networks Hyperbolicity and crosstalk in regulatory networks short-cuts in long feedback loops node regulates itself through a long feedback loop ⇒ ⇒ ⇒ this loop must have a path-chord ⇒ ⇒ ⇒ a shorter feedback cycle through the same node a node interpreting chord or short path-chord as crosstalk “source” regulates “target” through two long paths ⇒ ⇒ ⇒ must exist a crosstalk path between these two paths c r o s s source target t a l k number of crosstalk paths increases at least linearly with total length of two paths R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 21 / 52
Outline of talk Introduction 1 Basic definitions and notations 2 Computing hyperbolicity for real networks 3 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 22 / 52
Implications of hyperbolicity Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths u 1 u 1 u 1 u 0 u 0 u 0 u 2 u 2 u 2 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths u 1 u 1 u 1 shortest path u 0 u 0 u 0 u 2 u 2 u 2 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths � d u 0 , u 1 + d u 0 , u 2 − d u 1 , u 2 � d u 0 , u 0,1 = u 1 u 1 u 1 2 shortest path � d u 1 , u 2 + d u 1 , u 0 − d u 2 , u 0 � d u 1 , u 0,1 = 2 d u 1 , u 0,1 = d u 1 , u 1,2 d u 0 , u 0,1 = d u 0 , u 0,2 u 0,1 u 0,1 u 0,1 d u 2 , u 0,2 = d u 2 , u 1,2 u 1,2 u 1,2 u 1,2 u 0 u 0 u 0 u 2 u 2 u 2 u 0,2 u 0,2 u 0,2 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths ✎ ☞ � � � � � � ∃ v ′ in the other path such that ∃ v ′ ∃ v ′ ∀ v d v , v ′ ≤ max d v , v ′ ≤ max d v , v ′ ≤ max 6 δ , 2 ∀ v ∀ v in one path 6 δ , 2 6 δ , 2 ✍ ✌ � d u 0 , u 1 + d u 0 , u 2 − d u 1 , u 2 � d u 0 , u 0,1 = u 1 u 1 u 1 2 shortest path � d u 1 , u 2 + d u 1 , u 0 − d u 2 , u 0 � d u 1 , u 0,1 = 2 d d u 1 , u 0,1 = d u 1 , u 1,2 d u 0 , u 0,1 = d u 0 , u 0,2 v , v ′ u 0,1 u 0,1 u 0,1 d u 2 , u 0,2 = d u 2 , u 1,2 u 1,2 u 1,2 u 1,2 d v , v ′ d v , v ′ u 0 u 0 u 0 u 2 u 2 u 2 u 0,2 u 0,2 u 0,2 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 23 / 52
Implications of hyperbolicity Implications of geodesic triangles and crosstalk paths for regulatory networks Implications of geodesic triangles for regulatory networks Consider feedback or feed-forward loop formed by the shortest paths among three nodes We can expect short cross-talk paths between these shortest paths ⇓ ⇓ ⇓ Feedback/feed-forward loop is nested with additional feedback/feed-forward loops R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 24 / 52
Implications of hyperbolicity Implications of geodesic triangles and crosstalk paths for regulatory networks Implications of geodesic triangles for regulatory networks Consider feedback or feed-forward loop formed by the shortest paths among three nodes We can expect short cross-talk paths between these shortest paths ⇓ ⇓ ⇓ Feedback/feed-forward loop is nested with additional feedback/feed-forward loops Empirical evidence [R. Albert, Journal of Cell Science 118, 4947-4957 (2005)] Network motifs a are often nested Two generations of nested assembly for a common E. coli motif [DeDeo and Krakauer, 2012] a e.g. , feed-forward or feedback loops of small number of nodes R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 24 / 52
Implications of hyperbolicity Hausdorff distance between shortest paths Definition ( Hausdorff distance between two paths P 1 P 1 P 1 and P 2 P 2 P 2 ) � � � � � � � � � � � � � � � � � � d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def = max max min d v 1 , v 2 , max min d v 1 , v 2 = max = max max max min min d v 1 , v 2 d v 1 , v 2 , , max max min min d v 1 , v 2 d v 1 , v 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 small Hausdorff distance implies every node of either path is close to some node of the other path R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity Hausdorff distance between shortest paths Definition ( Hausdorff distance between two paths P 1 P 1 P 1 and P 2 P 2 P 2 ) � � � � � � � � � � � � � � � � � � d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def = max max min d v 1 , v 2 , max min d v 1 , v 2 = max = max max max min min d v 1 , v 2 d v 1 , v 2 , , max max min min d v 1 , v 2 d v 1 , v 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 small Hausdorff distance implies every node of either path is close to some node of the other path d v , v ′ d v , v ′ u 0 u 0 u 0 u 2 u 2 u 2 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity Hausdorff distance between shortest paths Definition ( Hausdorff distance between two paths P 1 P 1 P 1 and P 2 P 2 P 2 ) � � � � � � � � � � � � � � � � � � d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def = max max min d v 1 , v 2 , max min d v 1 , v 2 = max = max max max min min d v 1 , v 2 d v 1 , v 2 , , max max min min d v 1 , v 2 d v 1 , v 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 small Hausdorff distance implies every node of either path is close to some node of the other path P 2 P 2 P 2 d v , v ′ d v , v ′ u 0 u 0 u 0 u 2 u 2 u 2 P 1 P 1 P 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity Hausdorff distance between shortest paths Definition ( Hausdorff distance between two paths P 1 P 1 P 1 and P 2 P 2 P 2 ) � � � � � � � � � � � � � � � � � � d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def d H ( P 1 , P 2 ) def = max max min d v 1 , v 2 , max min d v 1 , v 2 = max = max max max min min d v 1 , v 2 d v 1 , v 2 , , max max min min d v 1 , v 2 d v 1 , v 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 2 ∈ P 2 v 1 ∈ P 1 v 1 ∈ P 1 v 1 ∈ P 1 small Hausdorff distance implies every node of either path is close to some node of the other path P 2 P 2 P 2 d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d v , v ′ d v , v ′ u 0 u 0 u 0 u 2 u 2 u 2 P 1 P 1 P 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 25 / 52
Implications of hyperbolicity Hausdorff distance between shortest paths this result versus our previous path-chord result path-chord result this result P 2 P 2 P 2 d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } P 2 P 2 P 2 d v , v ′ d v , v ′ u 0 u 0 u 0 u 2 u 2 u 2 P 1 P 1 P 1 u 0 u 0 u 0 u 2 u 2 u 2 long cycle ⇒ ⇒ there is a path chord ⇒ P 1 P 1 P 1 Which result is more general in nature ? R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 26 / 52
Implications of hyperbolicity Hausdorff distance between shortest paths this result versus our previous path-chord result path-chord result this result P 2 P 2 P 2 d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } d H ( P 1 , P 2 ) ≤ max{ 6 δ , 2 } P 2 P 2 P 2 � � � d v , v ′ d v , v ′ u 0 u 0 u 0 u 2 u 2 u 2 P 1 P 1 P 1 u 0 u 0 u 0 u 2 u 2 u 2 long cycle ⇒ ⇒ there is a path chord ⇒ P 1 P 1 P 1 Which result is more general in nature ? R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 26 / 52
Implications of hyperbolicity A notational simplification A notational simplification unless G is a tree or a complete graph ( K n K n K n ), δ > 0 δ > 0 δ > 0 δ > 0 ≡ δ ≥ 1 / 2 δ > 0 ≡ δ ≥ 1 / 2 δ > 0 ≡ δ ≥ 1 / 2 δ ≥ 1 / 2 ⇒ max{ 6 δ , 2 } = 6 δ δ ≥ 1 / 2 ⇒ max{ 6 δ , 2 } = 6 δ δ ≥ 1 / 2 ⇒ max{ 6 δ , 2 } = 6 δ 6 δ max{ 6 δ , 2 } Hence, we will simply write 6 δ 6 δ instead of max{ 6 δ , 2 } max{ 6 δ , 2 } Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 27 / 52
Implications of hyperbolicity Distance between geodesic and arbitrary path s s s P P P � u 1 � u 1 � u 1 to another arbitrary path u 0 � u 1 � u 1 � u 1 Distance from a shortest path u 0 u 0 u 0 u 0 u 0 n ℓ ( P ) n is the number of nodes in the graph n ℓ ( P ) ℓ ( P ) is length of path P P P P P P � u 1 P ≡ P ≡ P ≡ u 0 u 0 u 0 � u 1 � u 1 arbitrary node u 0 u 1 u 0 u 0 v v v u 1 u 1 s s s shortest path u 0 u 0 u 0 � u 1 � u 1 � u 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 28 / 52
Implications of hyperbolicity Distance between geodesic and arbitrary path s s s P P P � u 1 � u 1 � u 1 to another arbitrary path u 0 � u 1 � u 1 � u 1 Distance from a shortest path u 0 u 0 u 0 u 0 u 0 n ℓ ( P ) n n is the number of nodes in the graph ℓ ( P ) ℓ ( P ) is length of path P P P ∃ v ′ d v , v ′ ≤ 6 δ log 2 ℓ ( P ) ∃ v ′ ∃ v ′ d v , v ′ ≤ 6 δ log 2 ℓ ( P ) d v , v ′ ≤ 6 δ log 2 ℓ ( P ) � � � �� �� �� � � � < 6 δ log 2 n < 6 δ log 2 n < 6 δ log 2 n (log n ) δ P P P O (log n ) (log n ) if δ δ is constant � u 1 P ≡ P ≡ u 0 P ≡ u 0 u 0 � u 1 � u 1 v ′ v ′ v ′ d v , v ′ d v , v ′ d v , v ′ arbitrary node u 0 u 1 u 0 u 0 v v v u 1 u 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 28 / 52
Implications of hyperbolicity Distance between geodesic and arbitrary path An interesting implication of this bound ∃ v ′ d v , v ′ ≤ 6 δ log 2 ℓ ( P ) ∃ v ′ d v , v ′ ≤ 6 δ log 2 ℓ ( P ) ∃ v ′ d v , v ′ ≤ 6 δ log 2 ℓ ( P ) P P P v ′ v ′ v ′ d v , v ′ d v , v ′ d v , v ′ u 0 u 0 u 0 u 1 u 1 u 1 v v v shortest path R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 29 / 52
Implications of hyperbolicity Distance between geodesic and arbitrary path An interesting implication of this bound assume ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ P P P = = = γ γ γ u 0 u 0 u 0 u 1 u 1 u 1 v v v R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 29 / 52
Implications of hyperbolicity Distance between geodesic and arbitrary path An interesting implication of this bound assume ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ γ γ γ � � � 2 Ω ( γ ) � 2 Ω ( γ ) � 2 Ω ( γ ) � 6 δ = 6 δ = 6 δ = ⇒ ⇒ ⇒ ℓ ( P ) ≥ 2 ℓ ( P ) ≥ 2 ℓ ( P ) ≥ 2 Ω Ω Ω P P P if δ if δ if δ δ δ is constant δ is constant δ δ is constant δ = = = γ γ γ u 0 u 0 u 0 u 1 u 1 u 1 v v v Next: better bounds for approximately short paths � R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 29 / 52
Implications of hyperbolicity Approximately short path Why consider approximately short paths ? Regulatory networks Up/down-regulation of a target node is mediated by two or more “close to shortest” paths starting from the same regulator node Additional “very long” paths between the same regulator and target node do not contribute significantly to the target node’s regulation target source ε Definition ε ε -additive-approximate short path P P P ℓ ( P ) ℓ ( P ) ℓ ( P ) P ≤ P ≤ length of shortest path + ε P ≤ + ε + ε P length of P length of P length of P P P Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 30 / 52
Implications of hyperbolicity Approximately short path Why consider approximately short paths ? Algorithmic efficiency reasons Routing and navigation problems (traffic networks) Approximate short path may be faster to compute as opposed to exact shortest path Routing via approximate short path P � � � � � � P P Definition µ µ µ -approximate short path u 0 u 0 u 0 � u k = � u k = � u k = u 0 , u 1 ,..., u k u 0 , u 1 ,..., u k u 0 , u 1 ,..., u k � � � � P P P � � ℓ u i � u j � u j � u j ≤ µ d u i , u j 0 ≤ i < j ≤ k ℓ ℓ u i u i ≤ µ ≤ µ d u i , u j d u i , u j for all 0 ≤ i < j ≤ k 0 ≤ i < j ≤ k length of sub-path distance from ui to uj between ui and uj P � � � � � � P P v 1 v 3 v 1 v 1 v 3 v 3 2 2 -approximate path u 0 2 u 0 u 0 � u 7 = � u 7 = � u 7 = u 0 , u 1 ,..., u 7 u 0 , u 1 ,..., u 7 u 0 , u 1 ,..., u 7 s s s shortest path u 0 u 0 u 0 � u 7 � u 7 � u 7 u 0 = v 0 u 0 = v 0 u 0 = v 0 u 1 u 1 u 1 u 2 u 2 u 2 u 4 u 4 u 4 u 5 u 5 u 5 u 6 u 6 u 6 u 7 = v 4 u 7 = v 4 u 7 = v 4 u 3 = v 2 u 3 = v 2 u 3 = v 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 31 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P P P � u 1 � u 1 � u 1 Distance from shortest path to an approximately short path u 0 u 0 u 0 � �� � ε ε ε -additive approximate or, µ µ µ -approximate P P P � u 1 P ≡ P ≡ u 0 P ≡ u 0 u 0 � u 1 � u 1 v ′ v ′ v ′ d v , v ′ d v , v ′ d v , v ′ arbitrary node u 0 u 0 u 0 v v v u 1 u 1 u 1 s s s � u 1 shortest path u 0 u 0 u 0 � u 1 � u 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 32 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P P P path u 0 u 0 u 0 � u 1 � u 1 � u 1 Distance from shortest path to an approximately short � �� � ε ε ε -additive approximate or, µ µ µ -approximate P P P P ≡ P ≡ P ≡ u 0 u 0 u 0 � u 1 � u 1 � u 1 v ′ v ′ v ′ d v , v ′ d v , v ′ d v , v ′ arbitrary node u 0 u 0 u 0 v u 1 u 1 u 1 v v s s s shortest path u 0 u 0 u 0 � u 1 � u 1 � u 1 P P P � u 1 u 0 u 0 u 0 � u 1 � u 1 is ε ε ε -additive approximate � � � � � � � � � � � � ∀ v ∃ v ′ d v , v ′ ≤ � � � � � � � � � � � � ∀ v ∃ v ′ d v , v ′ ≤ ∀ v ∃ v ′ d v , v ′ ≤ + 1 + ε + 1 + ε + 1 + ε 6 δ + 2 6 δ + 2 6 δ + 2 log 2 log 2 log 2 8 8 8 6 δ + 2 6 δ + 2 6 δ + 2 log 2 log 2 log 2 (6 δ + 2) (4 + 2 ε ) (6 δ + 2) (4 + 2 ε ) (6 δ + 2) (4 + 2 ε ) 2 2 2 � � � � � � �� �� �� O O O δ log δ log δ log ε + δ log ε ε + δ log ε ε + δ log ε depends only on δ δ δ and ε ε ε ε δ short crosstalk path for small ε ε and δ δ R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 32 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P P P Distance from shortest path to an approximately short path u 0 u 0 u 0 � u 1 � u 1 � u 1 � �� � ε ε ε -additive approximate or, µ µ µ -approximate P P P � u 1 � u 1 � u 1 P ≡ P ≡ P ≡ u 0 u 0 u 0 v ′ v ′ v ′ d v , v ′ d v , v ′ d v , v ′ arbitrary node u 0 u 0 u 0 v v v u 1 u 1 u 1 s s s � u 1 � u 1 � u 1 shortest path u 0 u 0 u 0 P P P u 0 � u 1 is µ � u 1 � u 1 µ u 0 u 0 µ -approximate � � � � � � �� �� �� � � � �� �� � � � � � � � � � � � � � � � � � ∀ v ∃ v ′ d v , v ′ ≤ ∀ v ∃ v ′ d v , v ′ ≤ ∀ v ∃ v ′ d v , v ′ ≤ 6 δ + 2 6 δ + 2 6 δ + 2 log 2 log 2 log 2 6 µ + 2 6 µ + 2 6 µ + 2 6 δ + 2 6 δ + 2 6 δ + 2 log 2 log 2 log 2 (6 δ + 2) (6 δ + 2) (6 δ + 2) 3 µ + 1 3 µ + 1 3 µ + 1 µ µ µ + µ + µ + µ � � � � � � �� �� �� O O O δ log δ log δ log µδ µδ µδ depends only on δ δ δ and µ µ µ µ δ short crosstalk path for small µ µ and δ δ R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 32 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path � � � � � � d v , v ′ = O δ log ℓ ( P ) Contrast the new bounds with the old bound of d v , v ′ = O d v , v ′ = O δ log ℓ ( P ) δ log ℓ ( P ) d u 0 , u 1 d u 0 , u 1 d u 0 , u 1 is the length of a shortest path between u 0 u 0 u 0 and u 1 u 1 u 1 P P P P P P u 0 u 0 u 0 � u 1 � u 1 is ε � u 1 ε ε -additive approximate u 0 u 0 u 0 � u 1 � u 1 � u 1 is µ µ -approximate µ ℓ ( P ) ≤ d u 0 , u 1 + ε ℓ ( P ) ≤ d u 0 , u 1 + ε ℓ ( P ) ≤ d u 0 , u 1 + ε ℓ ( P ) ≤ µ d u 0 , u 1 ℓ ( P ) ≤ µ d u 0 , u 1 ℓ ( P ) ≤ µ d u 0 , u 1 Old bound Old bound � � � � � � �� �� �� � � � � � � � � � ��� ��� ��� O O O δ log δ log δ log ε + d u 0 , u 1 ε + d u 0 , u 1 ε + d u 0 , u 1 O O O δ δ δ log log log µ d u 0 , u 1 µ d u 0 , u 1 µ d u 0 , u 1 New bound New bound �� no dependency �� no dependency � � � � � � �� �� � � � � � � �� �� O O O δ log δ log δ log ε + δ log ε ε + δ log ε ε + δ log ε O O O δ log δ log δ log µδ µδ µδ d u 0 , u 1 d u 0 , u 1 on d u 0 , u 1 d u 0 , u 1 on d u 0 , u 1 d u 0 , u 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 33 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P P P � u 1 Distance from an approximately short path u 0 u 0 u 0 � u 1 to a shortest path � u 1 � �� � ε -additive approximate ε ε or, µ µ µ -approximate for simplified exposition, we show bounds only in asymptotic O ( · ) ( · ) notation ( · ) please refer to our paper for more precise bounds P P P � u 1 P ≡ P ≡ P ≡ u 0 u 0 u 0 � u 1 � u 1 v ′ v ′ v ′ arbitrary node d v ′ , v d v ′ , v d v ′ , v u 0 u 0 u 0 v u 1 u 1 u 1 v v s s s � u 1 shortest path u 0 u 0 u 0 � u 1 � u 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 34 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P P P Distance from an approximately short path u 0 u 0 u 0 � u 1 � u 1 � u 1 to a shortest path � �� � ε ε ε -additive approximate or, µ µ µ -approximate for simplified exposition, we show bounds only in asymptotic O ( · ) ( · ) ( · ) notation please refer to our paper for more precise bounds P P P P ≡ P ≡ P ≡ u 0 u 0 u 0 � u 1 � u 1 � u 1 v ′ v ′ v ′ arbitrary node d v ′ , v d v ′ , v d v ′ , v u 0 u 0 u 0 v u 1 u 1 u 1 v v s s s shortest path u 0 u 0 u 0 � u 1 � u 1 � u 1 P P P u 0 u 0 u 0 � u 1 � u 1 � u 1 is ε ε ε -additive approximate � � � � � � �� �� �� ∀ v ′ ∃ v d v ′ , v ≤ O ∀ v ′ ∃ v d v ′ , v ≤ ∀ v ′ ∃ v d v ′ , v ≤ ε + δ log ε + δ log ε depends only on δ δ and ε δ ε ε O O ε + δ log ε + δ log ε + δ log ε ε + δ log ε R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 34 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P P P � u 1 Distance from an approximately short path u 0 u 0 u 0 � u 1 to a shortest path � u 1 � �� � ε -additive approximate ε ε µ or, µ µ -approximate for simplified exposition, we show bounds only in asymptotic O ( · ) ( · ) notation ( · ) please refer to our paper for more precise bounds P P P � u 1 P ≡ P ≡ P ≡ u 0 u 0 u 0 � u 1 � u 1 v ′ v ′ v ′ arbitrary node d v ′ , v d v ′ , v d v ′ , v u 0 u 0 u 0 v u 1 u 1 u 1 v v s s s � u 1 shortest path u 0 u 0 u 0 � u 1 � u 1 P P P u 0 u 0 u 0 � u 1 is µ � u 1 � u 1 µ µ -approximate � � �� ∀ v ′ ∃ v d v ′ , v ≤ ∀ v ′ ∃ v d v ′ , v ≤ ∀ v ′ ∃ v d v ′ , v ≤ O � � � � �� �� depends only on δ δ and µ δ µ µ O O µδ log µδ log µδ log µδ µδ µδ R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 34 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Distance from approximate short path P 1 P 1 P 1 to approximate short path P 2 P 2 P 2 � �� � � �� � arbitrary node v v v nearest node v ′ v ′ v ′ P 2 P 2 P 2 u 0 u 0 u 0 u 1 u 1 u 1 arbitrary node v v v P 1 P 1 P 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 35 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Distance from approximate short path P 1 P 1 P 1 to approximate short path P 2 P 2 P 2 � �� � � �� � arbitrary node v v v nearest node v ′ v ′ v ′ P 2 P 2 P 2 go to any shortest path v ′′ v ′′ v ′′ u 0 u 0 u 0 u 1 u 1 u 1 shortest path arbitrary node v v v P 1 P 1 P 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 35 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Distance from approximate short path P 1 P 1 P 1 to approximate short path P 2 P 2 P 2 � �� � � �� � arbitrary node v v v nearest node v ′ v ′ v ′ P 2 P 2 P 2 v ′ v ′ v ′ continue to the other path v ′′ v ′′ v ′′ u 0 u 0 u 0 u 1 u 1 u 1 shortest path arbitrary node v v v P 1 P 1 P 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 35 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path P 1 P 2 Distance from approximate short path P 1 P 1 to approximate short path P 2 P 2 � �� � � �� � nearest node v ′ v ′ v ′ arbitrary node v v v we sometimes overestimate quantities to simplify expression P 1 P 1 P 1 is ε 1 ε 1 ε 1 -additive approximate P 1 is ε P 1 P 1 ε -additive approximate ε P 2 P 2 is ε 2 ε 2 ε 2 -additive approximate P 2 is µ P 2 µ -approximate µ P 2 P 2 � � � � � � � � � � � � � � � + δ 2 loglog ε + δ 2 loglog ε + δ 2 loglog ε � � � O ε 1 + δ log( ε 1 ε 2 ) + δ log δ O O ε 1 + δ log( ε 1 ε 2 ) + δ log δ ε 1 + δ log( ε 1 ε 2 ) + δ log δ O O O ε + δ log ε + δ log ε + δ log εµ εµ εµ µ µ 1 P 1 is µ P 1 P 1 µ -approximate P 1 P 1 is µ 1 P 1 µ 1 -approximate ε µ 2 P 2 P 2 P 2 is ε ε -additive approximate P 2 is µ 2 P 2 P 2 µ 2 -approximate � � � � � � � � � � � � � � � � � � � � � � � � O O O µδ log µδ log µδ log µδ µδ µδ + ε + δ log ε + ε + δ log ε + ε + δ log ε O O O µ 1 δ log µ 1 δ log µ 1 δ log µ 1 δ µ 1 δ µ 1 δ + δ log µ 2 + δ log µ 2 + δ log µ 2 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 36 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Interesting implications of these improved bounds approximately short path P P P v ′ v ′ v ′ d v , v ′ d v , v ′ d v , v ′ u 0 u 0 u 0 v u 1 u 1 u 1 v v shortest path R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Interesting implications of these improved bounds assume ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ approximately short path P P P = = = γ γ γ u 0 u 0 u 0 v u 1 u 1 u 1 v v R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Interesting implications of these improved bounds if P P is ε P ε -additive-approximate short then ε assume ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ � � � � � � ⇒ ⇒ ⇒ 2 γ / δ 2 γ / δ 2 γ / δ ε = Ω ε = Ω ε = Ω − log δ − log δ − log δ δ δ δ approximately short path P P P = = = γ γ γ u 0 u 0 u 0 v u 1 u 1 u 1 v v R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path Interesting implications of these improved bounds if P P is µ P µ -approximate short then µ assume ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ ∀ v ′ ∈ P d v , v ′ ≥ γ � � � � � � ⇒ ⇒ ⇒ 2 γ / δ 2 γ / δ 2 γ / δ µ = Ω µ = Ω µ = Ω γ γ γ approximately short path P P P = = = γ γ γ u 0 u 0 u 0 v u 1 u 1 u 1 v v R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 37 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path To wrap it up, approximate shortest paths look like the following cartoon R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 38 / 52
Implications of hyperbolicity Distance between geodesic and approximately short path To wrap it up, approximate shortest paths look like the following cartoon Interpretation for regulatory networks � It is reasonable to assume that, when up- or down-regulation of a target node is mediated by two or more approximate short paths starting from the same regulator node, additional very long paths between the same regulator and target node do not contribute significantly to the target node’s regulation � We refer to the short paths as relevant, and to the long paths as irrelevant � Then, our finding can be summarized by saying that almost all relevant paths between two nodes have crosstalk paths between each other R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 38 / 52
Outline of talk Introduction 1 Basic definitions and notations 2 Computing hyperbolicity for real networks 3 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 39 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3( κ − 2) δ r > 3( κ − 2) δ r > 3( κ − 2) δ Example: 5 1 9 δ + 1 9 δ + 1 9 δ + 1 u 0 u 0 u 0 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3( κ − 2) δ r > 3( κ − 2) δ r > 3( κ − 2) δ Example: 5 1 9 δ + 1 9 δ + 1 9 δ + 1 u 2 u 2 u 2 u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ u 1 u 1 u 1 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3( κ − 2) δ r > 3( κ − 2) δ r > 3( κ − 2) δ Example: 5 1 9 δ + 1 9 δ + 1 9 δ + 1 u 3 u 3 u 3 α α α α α α + + + r r r u 2 u 2 u 2 u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ u 1 u 1 u 1 α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3( κ − 2) δ r > 3( κ − 2) δ r > 3( κ − 2) δ Example: 5 1 9 δ + 1 9 δ + 1 9 δ + 1 consider any shortest path P P between u 3 P u 3 and u 4 u 3 u 4 u 4 u 3 u 3 u 3 P P must look like this P α α α u 2 u 2 u 2 u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ u 1 u 1 u 1 α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3( κ − 2) δ r > 3( κ − 2) δ r > 3( κ − 2) δ Example: 5 1 9 δ + 1 9 δ + 1 9 δ + 1 consider any shortest path P P between u 3 P u 3 and u 4 u 3 u 4 u 4 u 3 u 3 u 3 P P must look like this P α α α u 2 u 2 u 2 � 3 � 3 � 3 � � � P P P γ = d u 0 , v ≤ γ = d u 0 , v ≤ γ = d u 0 , v ≤ r − r − r − 2 κ − 1 2 κ − 1 2 κ − 1 δ δ δ r − Θ ( κδ ) r − Θ ( κδ ) r − Θ ( κδ ) u 0 u 0 u 0 r v v v u 1 u 1 u 1 γ γ γ α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ ≥ 4 κ ≥ 4 κ ≥ 4 α > 0 α > 0 α > 0 r > 3( κ − 2) δ r > 3( κ − 2) δ r > 3( κ − 2) δ Example: 5 1 9 δ + 1 9 δ + 1 9 δ + 1 consider any shortest path P P P between u 3 u 3 u 3 and u 4 u 4 u 4 u 3 u 3 u 3 P P must look like this P α α α u 2 u 2 u 2 � 3 � 3 � 3 � � � P P P γ = d u 0 , v ≤ γ = d u 0 , v ≤ γ = d u 0 , v ≤ r − r − r − 2 κ − 1 2 κ − 1 2 κ − 1 δ δ δ r − Θ ( κδ ) r − Θ ( κδ ) r − Θ ( κδ ) u 0 u 0 u 0 r v v v ℓ ( P ) ≥ (3 κ − 2) δ + 2 α ℓ ( P ) ≥ ℓ ( P ) ≥ (3 κ − 2) δ + 2 α (3 κ − 2) δ + 2 α Ω ( κδ + α ) Ω ( κδ + α ) Ω ( κδ + α ) u 1 u 1 u 1 γ γ γ α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 40 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary ( of previous results) consider any path P P P between u 3 u 3 u 3 and u 4 u 4 u 4 suppose that P P P does not intersect the shaded region u 3 u 3 u 3 α α α u 2 u 2 u 2 P P P u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ u 1 u 1 u 1 α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary ( of previous results) consider any path P P between u 3 P u 3 and u 4 u 3 u 4 u 4 suppose that P P does not intersect the shaded region P 6 δ + κ 6 δ + κ 6 δ + κ α α α 4 4 4 ℓ ( P ) ≥ ℓ ( P ) ≥ ℓ ( P ) ≥ 2 2 2 u 3 u 3 u 3 � α � α � α � � � α α α 2 Ω 2 Ω 2 Ω δ + κ δ + κ δ + κ u 2 u 2 u 2 P P P u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ very long u 1 u 1 u 1 path α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary ( of previous results) consider any path P P between u 3 P u 3 u 3 and u 4 u 4 u 4 suppose that P P does not intersect the shaded region P 6 δ + κ 6 δ + κ 6 δ + κ α α α 4 4 4 ℓ ( P ) ≥ ℓ ( P ) ≥ ℓ ( P ) ≥ 2 2 2 u 3 u 3 u 3 � α � α � α � � � α α α 2 Ω 2 Ω 2 Ω δ + κ δ + κ δ + κ u 2 u 2 u 2 P ε P P ε -additive-approximate ⇒ ε ⇒ ⇒ 6 δ + κ α P P P ε > 2 4 48 δ − log 2 (48 δ ) u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ � 2 Θ ( α + κ ) � Ω if δ δ δ is constant very long u 1 u 1 u 1 path α α α u 4 u 4 u 4 R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary ( of previous results) consider any path P P between u 3 P u 3 and u 4 u 3 u 4 u 4 suppose that P P does not intersect the shaded region P 6 δ + κ 6 δ + κ 6 δ + κ α α α 4 4 4 ℓ ( P ) ≥ ℓ ( P ) ≥ ℓ ( P ) ≥ 2 2 2 u 3 u 3 u 3 � α � α � α � � � α α α 2 Ω 2 Ω 2 Ω δ + κ δ + κ δ + κ u 2 u 2 u 2 P P P ε ε ε -additive-approximate ⇒ ⇒ ⇒ 6 δ + κ α P P P ε > 2 4 48 δ − log 2 (48 δ ) u 0 u 0 u 0 r ≥ 3 κδ ≥ 3 κδ ≥ 3 κδ � 2 Θ ( α + κ ) � Ω if δ δ is constant δ very P P µ µ -approximate ⇒ µ ⇒ P ⇒ long u 1 u 1 u 1 path 6 δ + κ α 4 2 µ ≥ � � α α α 12 α + 6 δ 3 κ − 26 � � u 4 u 4 u 4 2 Θ ( α δ + κ ) Ω α + κδ R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 41 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks u target u target u target u source u source u source shortest path R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks u middle u middle u middle u target u target u target u source u source u source R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks u middle u middle u middle u target u target u target u source u source u source ξ = O ( δ ) ξ = O ( δ ) ξ = O ( δ ) R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks All shortest paths between u source u source and u target u source u target u target must intersect the ξ ξ ξ -neighborhood Therefore, “knocking out” nodes in ξ ξ ξ -neighborhood cuts off all shortest regulatory paths between u source u source and u target u source u target u target o r h h t e a t s s p t s h o t p a h r t e s t u middle u middle u middle u target u target u target u source u source u source a t h t e s p o r t h s t s s p h e a o r t ξ = O ( δ ) ξ = O ( δ ) ξ = O ( δ ) t h R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks But, it gets even more interesting ! o r h h t e a t s s p t s h o t p a h r t e s t u middle u middle u middle u target u target u target u source u source u source a t h t e s p o r t h s t s s p h e a o r t ξ = O ( δ ) ξ = O ( δ ) ξ = O ( δ ) t h R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks But, it gets even more interesting ! shifting the ξ ξ ξ -neighborhood does not change claim t h a p s t h t e a t s p r h o o h r t e s t s u target u target u target u source u source u source s h e s o o r t t h r s p t a e s t h t p a t h R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks how about enlarging the ξ ξ ξ -neighborhood ? o r t h h t e a s s p t s h s t p a h o r t e t u middle u middle u middle u target u target u target u source u source u source a t h o r t e p h s s t t s s p h e o r t ξ = O ( δ ) ξ = O ( δ ) ξ = O ( δ ) a t h R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks how about enlarging the ξ ξ ξ -neighborhood ? approximately short paths start intersecting the neighborhood t e m a l i a x m y t o e s i r p l x h y p o o a r s r t h p p o a r t r t a t h h o e p s s t s s t p e h o r t a t h u middle u middle u middle u target u target u target u source u source u source h a t o r t p h e s s t s t 2 ξ 2 ξ 2 ξ h s p o r t e h a t R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks Consider a ball (neighborhood) of radius ξ log n ξ log n ξ log n ( n n n is the number of nodes) u middle u middle u middle u target u target u target u source u source u source ξ log n ξ log n ξ log n R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks Consider a ball (neighborhood) of radius ξ log n ξ log n ξ log n ( n n is the number of nodes) n All paths intersect the neighborhood × × × u middle u middle u middle u target u target u target u source u source u source ξ log n ξ log n ξ log n R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 42 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks Empirical estimation of neighborhoods and number of essential nodes We empirically investigated these claims on relevant paths passing through a neighborhood of a central node for the following two biological networks: � E. coli transcriptional � T-LGL signaling by selecting a few biologically relevant source-target pairs Our results show much better bounds for real networks compared to the worst-case pessimistic bounds in the mathematical theorems see our paper for further details R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 43 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions a p p r o x i m a t e g e o d e s i c Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions a p p i x o r m a t e g e o d e s i c Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions p a m i x o r p the further we move from the central node a t e g e o d e s the more a shortest path bends inward towards the central node i c Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks � � � � � � � � � � � � � � � eavesdropper eavesdropper may succeed with limited sensor range ☛ � � � eavesdropper need not be a hub � � � � � � � � � � � � � � Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks � � � � � � � Traffic network � � � � � � need not be a hub � � � � � � � � � Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 44 / 52
Outline of talk Introduction 1 Basic definitions and notations 2 Computing hyperbolicity for real networks 3 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 45 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks Visual illustration of a well-known social network Zachary’s Karate Club ( http://networkdata.ics.uci.edu/data.php?id=105 ) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 46 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks Structural hole in a social network [Burt, 1995; Borgatti, 1997] Definition ( Adjacency matrix of an undirected unweighted graph) u u u � 1, ... ... ... ... ... . . . . . if { u , v } is an edge . . . . . . . . . . a u , v = v v v ... ... a u , v a u , v a u , v ... ... 0, otherwise . . . . . . . . . . . . . . . ... ... ... ... ... Definition ( measure of structural hole at node u u u [Burt, 1995; Borgatti, 1997]) (assume u u u has degree at least 2) � [ 1 − a u , y + a y , u a v , y + a y , v a u , v + a v , u too complicated � def � � M u = = � � � � � � a u , x + a x , u a v , z + a z , v max max a u , x + a x , u v ∈ V y ∈ V z �= y x �= u y �= u , v x �= u Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 47 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks Structural hole in a social network [Burt, 1995; Borgatti, 1997] Definition ( Adjacency matrix of an undirected unweighted graph) u u u � 1, ... ... ... ... ... . . . . . if { u , v } is an edge . . . . . . . . . . a u , v = v v v ... ... a u , v a u , v a u , v ... ... 0, otherwise . . . . . . . . . . . . . . . ... ... ... ... ... Definition ( measure of structural hole at node u u [Burt, 1995; Borgatti, 1997]) u (assume u u u has degree at least 2) Nbr ( u ) u Let Nbr ( u ) Nbr ( u ) be set of nodes adjacent to u u � � � a v , y a v , y a v , y v , y ∈ Nbr ( u ) v , y ∈ Nbr ( u ) v , y ∈ Nbr ( u ) � � � � � � � − � − � − M u = M u = M u = � Nbr ( u ) � Nbr ( u ) � Nbr ( u ) � � � � � � � Nbr ( u ) � Nbr ( u ) � Nbr ( u ) � � � Next: An intuitive interpretation of M u M u M u � Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 47 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks An intuitive interpretation of M u M u M u Definition ( weak dominance ≺ ρ , λ ≺ ρ , λ ≺ ρ , λ Definition ( strong dominance ≺ ρ , λ ≺ ρ , λ ≺ ρ , λ weak ) strong ) strong strong weak weak ( ρ , λ ) Nodes v , y v , y v , y are weakly ( ρ , λ ) ( ρ , λ ) -dominated by v , y ( ρ , λ ) Nodes v , y v , y are strongly ( ρ , λ ) ( ρ , λ ) -dominated by node u u u provided u node u u provided � ρ < d u , v , d u , y ≤ ρ + λ ρ < d u , v , d u , y ≤ ρ + λ � ρ < d u , v , d u , y ≤ ρ + λ ρ < d u , v , d u , y ≤ ρ + λ , and ρ < d u , v , d u , y ≤ ρ + λ , and ρ < d u , v , d u , y ≤ ρ + λ � for at least one shortest path P � for every shortest path P P P P P between v v v v y between v v and y y , P P P contains a node and y y , P y P contains a node z P z z such that z z such that d u , z ≤ ρ d u , z ≤ ρ d u , z ≤ ρ z d u , z ≤ ρ d u , z ≤ ρ d u , z ≤ ρ λ = 2 λ = 2 y y u u v v ρ = 1 ρ = 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 48 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks An intuitive interpretation of M u M u M u Notation (boundary of the ξ ξ ξ -neighborhood of node u u u ) � � � � � � B ξ ( u ) = B ξ ( u ) = B ξ ( u ) = v | d u , v = ξ v | d u , v = ξ v | d u , v = ξ the set of all nodes at a distance of precisely ξ ξ ξ from u u u Observation v is selected uniformly ran- v v � v , y number of pairs of nodes v , y v , y such that � � � � � domly from B j ( u ) B j ( u ) B j ( u ) M u M u M u = = = E v , y (0 ρ ,1 ) u v , y v , y is weakly (0 (0 ρ ,1 ρ ,1 ) ) -dominated by u u � � ρ < j ≤ 1 0 ρ < j ≤ 1 0 0 ρ < j ≤ 1 λ λ λ λ λ λ v is selected uniformly ran- v v � number of pairs of nodes v , y v , y v , y such that � � � � ≥ ≥ ≥ E � domly from B j ( u ) B j ( u ) B j ( u ) v , y v , y is strongly (0,1)-dominated by u v , y u u � 0 < j ≤ 1 0 < j ≤ 1 0 < j ≤ 1 y y y always true u u u equality does not hold in general v v v R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 49 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks Generalize M u M u M u to M u , ρ , λ M u , ρ , λ M u , ρ , λ for larger ball of influence of a node replace (0,1) (0,1) (0,1) by ( ρ , λ ) ( ρ , λ ) ( ρ , λ ) v is selected uniformly ran- v v � v , y number of pairs of nodes v , y v , y such that � � � � � domly from B j ( u ) B j ( u ) B j ( u ) M u M u = M u = = E v , y (0 ρ ,1 ) u v , y is weakly (0 v , y (0 ρ ,1 ρ ,1 ) ) -dominated by u u � � ρ < j ≤ 1 ρ < j ≤ 1 0 0 ρ < j ≤ 1 0 λ λ λ λ λ λ v is selected uniformly ran- v v � number of pairs of nodes v , y v , y v , y such that � � � � M u , ρ , λ = M u , ρ , λ M u , ρ , λ = = E � domly from B j ( u ) B j ( u ) B j ( u ) v , y v , y is weakly ( ρ , λ ) v , y ( ρ , λ ) -dominated by u ( ρ , λ ) u u � ρ < j ≤ λ ρ < j ≤ λ ρ < j ≤ λ R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 50 / 52
Implications of hyperbolicity Effect of hyperbolicity on structural holes in social networks Generalize M u M u M u to M u , ρ , λ M u , ρ , λ M u , ρ , λ for larger ball of influence of a node (0,1) ( ρ , λ ) replace (0,1) (0,1) by ( ρ , λ ) ( ρ , λ ) Lemma ( equivalence of strong and weak domination) If λ ≥ 6 δ log 2 n λ ≥ 6 δ log 2 n λ ≥ 6 δ log 2 n then v � v v is selected uniformly ran- number of pairs of nodes v , y v , y v , y such that def def def � � � � M u , ρ , λ M u , ρ , λ M u , ρ , λ = = = = = = E � B j ( u ) domly from B j ( u ) B j ( u ) ( ρ , λ ) v , y is weakly ( ρ , λ ) v , y v , y ( ρ , λ ) -dominated by u u u � ρ < j ≤ λ ρ < j ≤ λ ρ < j ≤ λ v v is selected uniformly ran- v � number of pairs of nodes v , y v , y v , y such that � � � � = = E = � domly from B j ( u ) B j ( u ) B j ( u ) v , y v , y v , y is strongly ( ρ , λ ) ( ρ , λ ) ( ρ , λ ) -dominated by u u u � ρ < j ≤ λ ρ < j ≤ λ ρ < j ≤ λ equality holds now R. Albert, B. DasGupta and N. Mobasheri, Physical Review E 89(3), 032811 (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, 2014 50 / 52
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