Hyperbolic metric spaces and their applications to large complex real world networks II

Transcription

1 Hyperbolic metric spaces Hyperbolic metric spaces and their applications to large complex real world networks II Gabriel H. Tucci Bell Laboratories Alcatel-Lucent May 17, 2010 Tucci Hyperbolic metric spaces II

2 Review Last week we discussed: Euclidean, spherical and hyperbolic geometry Euclidean fifth postulate and its consequences Curvature Regular tessellations Definition of Gromov s hyperbolicity Linear isoperimetric inequality and Cheeger constant Exponential growth Random walks and electric resistance

3 Review of Gromov s hyperbolicity definition Let X be a geodesic metric space. Definition A geodesic triangle consists of three points x, y, z and three geodesic segments [x, y],[y, z],[z, x]. A geodesic triangle is δ slim if each side is contained in a δ neighborhood of the other two sides. The space is δ hyperbolic if all the geodesic triangles are δ slim. Figure: δ slim triangle

4 Remarks Trees are 0 hyperbolic

5 Remarks Trees are 0 hyperbolic The classical hyperbolic space is δ hyperbolic and therefore all its (p, q)-regular tessellations

6 Remarks Trees are 0 hyperbolic The classical hyperbolic space is δ hyperbolic and therefore all its (p, q)-regular tessellations Hyperbolic spaces satisfy a linear isoperimetric inequality, i.e. the perimeter of a closed curve and the area are of the same order!

7 Remarks Trees are 0 hyperbolic The classical hyperbolic space is δ hyperbolic and therefore all its (p, q)-regular tessellations Hyperbolic spaces satisfy a linear isoperimetric inequality, i.e. the perimeter of a closed curve and the area are of the same order! They have exponential growth

8 Remarks Trees are 0 hyperbolic The classical hyperbolic space is δ hyperbolic and therefore all its (p, q)-regular tessellations Hyperbolic spaces satisfy a linear isoperimetric inequality, i.e. the perimeter of a closed curve and the area are of the same order! They have exponential growth Random walks in hyperbolic spaces are transient

9 Remarks Trees are 0 hyperbolic The classical hyperbolic space is δ hyperbolic and therefore all its (p, q)-regular tessellations Hyperbolic spaces satisfy a linear isoperimetric inequality, i.e. the perimeter of a closed curve and the area are of the same order! They have exponential growth Random walks in hyperbolic spaces are transient Hyperbolic graphs as electric networks have finite resistance

10 Remarks Trees are 0 hyperbolic The classical hyperbolic space is δ hyperbolic and therefore all its (p, q)-regular tessellations Hyperbolic spaces satisfy a linear isoperimetric inequality, i.e. the perimeter of a closed curve and the area are of the same order! They have exponential growth Random walks in hyperbolic spaces are transient Hyperbolic graphs as electric networks have finite resistance Hyerbolicity is a large scale property! It is not a local property

11 Higuchi s curvature Determine if a given graph is hyperbolic it is not easy. However if the graph is planar one can prove hyperbolicity by studying the local structure!

12 Higuchi s curvature Determine if a given graph is hyperbolic it is not easy. However if the graph is planar one can prove hyperbolicity by studying the local structure! Given G a graph we can define the curvature of G at a vertex v as K(v) := 1 deg(v) 2 deg(v) 1 + d(f i ) where F 1, F 2,..., F deg(v) are the incident faces at v and d(f i ) is the number of sides. i=1

13 Higuchi s curvature Determine if a given graph is hyperbolic it is not easy. However if the graph is planar one can prove hyperbolicity by studying the local structure! Given G a graph we can define the curvature of G at a vertex v as K(v) := 1 deg(v) 2 deg(v) 1 + d(f i ) where F 1, F 2,..., F deg(v) are the incident faces at v and d(f i ) is the number of sides. Example : In the following graph i=1 K(v) = = 1 6

14 Higuchi s curvature Example : Consider the following graph K(v) = = 7 60 The points in the exterior circle have curvature K(v) = = 1 12

15 Higuchi s curvature It is not difficult to see that for every H p,q tessellation the curvature is ( ) 1 K := p p + 1 q 1 2 for every vertex. Therefore it is positive for spherical tessellations, zero for planar and negative for hyperbolic.

16 Higuchi s curvature It is not difficult to see that for every H p,q tessellation the curvature is ( ) 1 K := p p + 1 q 1 2 for every vertex. Therefore it is positive for spherical tessellations, zero for planar and negative for hyperbolic. Theorem (Higuchi) Let G be a planar graph if K(v) < 0 for every vertex v G then G is hyperbolic.

17 Higuchi s curvature It is not difficult to see that for every H p,q tessellation the curvature is ( ) 1 K := p p + 1 q 1 2 for every vertex. Therefore it is positive for spherical tessellations, zero for planar and negative for hyperbolic. Theorem (Higuchi) Let G be a planar graph if K(v) < 0 for every vertex v G then G is hyperbolic. This theorem allow us to go from the local structure to the global. Planarity plays a fundamental role!

18 Large complex networks For more than 40 years the scientific community treated large complex networks as being completely random objects. Figure: Realization of ER random graph

19 Large complex networks For more than 40 years the scientific community treated large complex networks as being completely random objects. Figure: Realization of ER random graph This paradigm has its roots in the work of the mathematicians Paul Erdös and Alfred Rényi. In the 60 s they study many of their properties.

20 G(n, p) and other classical models Their results included that: If p > (1+ɛ) log n n then G(n, p) will almost surely be connected. The diameter of G(n, p) is small with respect to n, this is called the small world property, diam(g(n, p)) = (1 + o(1)) log(n) log(pn).

21 G(n, p) and other classical models Their results included that: If p > (1+ɛ) log n n then G(n, p) will almost surely be connected. The diameter of G(n, p) is small with respect to n, this is called the small world property, diam(g(n, p)) = (1 + o(1)) log(n) log(pn). Other random graphs model have been consider : Barabási Albert preferential attachment model Watts-Strogatz small-world model others

22 New idea : networks as geometric objects Recent advances in the theory of complex networks show that even though many characteristic of the Erdös and Rényi, Barabási Albert, and others models appear natural in large networks as the Internet, the World Wide Web and many social networks: small diameter (ER, WS) power law degree distribution (BA) etc these models alone are not completely appropriate for their study. The great majority of complex real world networks have a more complex architecture than classical random graphs. Abstracting the network details away allows one to concentrate on the phenomena intrinsically connected with the underlying geometry, and discover connections between the metric properties and the network characteristics.

23 New idea : networks as geometric objects We believe that it is important to look at the large scale geometry of these large complex networks.

24 New idea : networks as geometric objects We believe that it is important to look at the large scale geometry of these large complex networks. Randomness alone is not sufficient. These networks should be modeled as random graphs but with certain underlying geometry!

25 New idea : networks as geometric objects We believe that it is important to look at the large scale geometry of these large complex networks. Randomness alone is not sufficient. These networks should be modeled as random graphs but with certain underlying geometry! We believe that many of the complex real world networks have characteristics of negatively curved or more generally Gromov s hyperbolic spaces.

26 Hyperbolic spaces in real life

27 Hyperbolic spaces in real life

28 Hyperbolic spaces in real life

29 Hyperbolic spaces in real life

30 Hyperbolic spaces in real life

31 Hyperbolic spaces in real life

32 New picture of the Internet We believe that many complex real networks have characteristics of HS

33 New picture of the Internet We believe that many complex real networks have characteristics of HS Figure: [Carmi, Havlin, Kirkpatrick, Shavitt and Shir] published in PNAS, 2007

34 Core of the Internet They showed that at the center of the Internet are about 80 core nodes through which most traffic flows. Figure: [Carmi, Havlin, Kirkpatrick, Shavitt and Shir] published in PNAS, 2007

35 Hyperbolic metric spaces Periphery of the Internet At the very edge of the Internet are 5,000 or so isolated nodes that are the most dependent upon the core and become cut off if the core is removed or shut down. Yet those nodes within this periphery are able to stay connected because of their peer-to-peer connections. Figure: [Carmi, Havlin, Kirkpatrick, Shavitt and Shir] published in PNAS, 2007 Tucci Hyperbolic metric spaces II

36 Large scale phenomena We are interested in studying large complex networks. Is it a good idea to study the asymptotic behavior? After all real networks are finite.

37 Large scale phenomena We are interested in studying large complex networks. Is it a good idea to study the asymptotic behavior? After all real networks are finite. In many situations to model systems from the physical world we need many independent variables.

38 Large scale phenomena We are interested in studying large complex networks. Is it a good idea to study the asymptotic behavior? After all real networks are finite. In many situations to model systems from the physical world we need many independent variables. It is a curious phenomenon that the cumulative effect of many independent variables in a system becomes more predictable, and easy to analyze, as the number of variables increases, rather than less!

39 Large scale phenomena We are interested in studying large complex networks. Is it a good idea to study the asymptotic behavior? After all real networks are finite. In many situations to model systems from the physical world we need many independent variables. It is a curious phenomenon that the cumulative effect of many independent variables in a system becomes more predictable, and easy to analyze, as the number of variables increases, rather than less! In many cases, the exact behaviour of each variable becomes irrelevant and one gets the same observed behaviour for the system regardless of what the individual components are doing.

40 Large scale phenomena We are interested in studying large complex networks. Is it a good idea to study the asymptotic behavior? After all real networks are finite. In many situations to model systems from the physical world we need many independent variables. It is a curious phenomenon that the cumulative effect of many independent variables in a system becomes more predictable, and easy to analyze, as the number of variables increases, rather than less! In many cases, the exact behaviour of each variable becomes irrelevant and one gets the same observed behaviour for the system regardless of what the individual components are doing. Typical examples are : Statistical Mechanics Law of large numbers and central limit theorem Asymptotic eigenvalue distribution for large random matrices

41 Large scale phenomena We are interested in studying large complex networks. Is it a good idea to study the asymptotic behavior? After all real networks are finite. In many situations to model systems from the physical world we need many independent variables. It is a curious phenomenon that the cumulative effect of many independent variables in a system becomes more predictable, and easy to analyze, as the number of variables increases, rather than less! In many cases, the exact behaviour of each variable becomes irrelevant and one gets the same observed behaviour for the system regardless of what the individual components are doing. Typical examples are : Statistical Mechanics Law of large numbers and central limit theorem Asymptotic eigenvalue distribution for large random matrices It is for this reason that we will study the characteristic of traffic as the size of the graph increases.

42 Asymptotic Traffic in a Tree Let {k l } l=1 be a sequence of positive integers, this sequence defines a tree T. Let T n be the finite tree generated by the first n generations of T.

43 Asymptotic Traffic in a Tree Let {k l } l=1 be a sequence of positive integers, this sequence defines a tree T. Let T n be the finite tree generated by the first n generations of T. N(n) := T n. Every node communicates with every node a unit load Total traffic in T n is N(N 1) 2

44 Asymptotic Traffic in a Tree Let {k l } l=1 be a sequence of positive integers, this sequence defines a tree T. Let T n be the finite tree generated by the first n generations of T. Question N(n) := T n. Every node communicates with every node a unit load Total traffic in T n is N(N 1) 2 What is the proportion of the traffic passing through the root as n? What about other nodes in T n?

45 Notation Hyperbolic metric spaces For A T n we denote by L n (A) the total traffic passing through A. We define the asymptotic proportion of the traffic passing through A as Note that in this case ap(a) := lim inf n L n (A) L n (T n ) where N = T n. L n (T n ) = N(N 1) 2 Tucci Hyperbolic metric spaces II

46 Hyperbolic metric spaces Theorem (Baryshnikov, T.) Let v be a node in T n and let l be its depth. The asymptotic proportion of the traffic through v is ( ) ap(v) = β(l) β(l) 1, β(l + 1) where β(l) := k 1 k 2... k l Tucci Hyperbolic metric spaces II

47 Hyperbolic metric spaces Theorem (Baryshnikov, T.) Let v be a node in T n and let l be its depth. The asymptotic proportion of the traffic through v is ( ) ap(v) = β(l) β(l) 1, β(l + 1) where β(l) := k 1 k 2... k l Corollary If the tree is k regular. Then ( ) ap(v) = 1 k l 2 1 k l 1 k l+1. Then the asymptotic proportion of the traffic through the root is : ap(root) = 1 1 k Tucci Hyperbolic metric spaces II

48 Asymptotic Traffic in Z d If we do the same in the grid Z d then for every finite subset A Z d the asymptotic traffic is zero! ap(a) := lim inf n as a matter of fact it can be probed that L n (A) L n (Z d ) = 0 L n (A) N 2 1 d (Saniee, Narayan) They also consider the traffic in 10 Rocketfuel networks.

49 General Case Let X be an infinite, simple and locally finite graph, and let x 0 be a node in X that will be considered the root or base point. Let such that : {x 0 } = X 0 X 1 X 2... X n... X X n is finite and star like n 1 X n = X

50 General Case Let X be an infinite, simple and locally finite graph, and let x 0 be a node in X that will be considered the root or base point. Let such that : {x 0 } = X 0 X 1 X 2... X n... X X n is finite and star like n 1 X n = X µ n a measure in X n, this measure defines a load demand function: given u and v nodes in X n the load between nodes is µ n (v)µ n (u)

51 General Case Let X be an infinite, simple and locally finite graph, and let x 0 be a node in X that will be considered the root or base point. Let such that : {x 0 } = X 0 X 1 X 2... X n... X X n is finite and star like n 1 X n = X µ n a measure in X n, this measure defines a load demand function: given u and v nodes in X n the load between nodes is µ n (v)µ n (u) for A X n we denote by L n (A) the total traffic passing through A

52 General Case Let X be an infinite, simple and locally finite graph, and let x 0 be a node in X that will be considered the root or base point. Let such that : {x 0 } = X 0 X 1 X 2... X n... X X n is finite and star like n 1 X n = X µ n a measure in X n, this measure defines a load demand function: given u and v nodes in X n the load between nodes is µ n (v)µ n (u) for A X n we denote by L n (A) the total traffic passing through A Definition A point y X is in the asymptotic α core if lim inf n for some r α independent on n. L n (B(y, r α )) L n (X n ) α

53 Main Results on the core for HN Theorem (Baryshnikov, T.) Assume that X is δ hyperbolic and non-elementary. Then x 0 belongs to the α core for all α < 1.

54 Main Results on the core for HN Theorem (Baryshnikov, T.) Assume that X is δ hyperbolic and non-elementary. Then x 0 belongs to the α core for all α < 1. The core is well localized. Theorem (Baryshnikov, T.) For every r > 0 and every α [0, 1), there exists a positive constant R = R(α, r) such that if x X and d(x 0, x) > R then lim sup n L n (B X (x, r)) L n (X n ) < α.

55 Hyperbolic metric spaces Distance dependent traffic load For each n 1, we have traffic going from X n to X n. We will also assume that there is a non increasing function f : [0, ) [0, ), such that the traffic rate between x and y in X n is equal to R(x, y) = f (d(x, y)). We will pay special attention to the case where f (t) = β t for β > 1. Tucci Hyperbolic metric spaces II

56 Hyperbolic metric spaces Distance dependent traffic load For each n 1, we have traffic going from X n to X n. We will also assume that there is a non increasing function f : [0, ) [0, ), such that the traffic rate between x and y in X n is equal to R(x, y) = f (d(x, y)). We will pay special attention to the case where f (t) = β t for β > 1. Question What can we say about the traffic pattern? Is there an α core? Tucci Hyperbolic metric spaces II

57 Hyperbolic metric spaces Assume that X is a δ hyperbolic graph and X n := {x X : d(x 0, x) n} and the traffic load decays exponentially with the distance. Meaning, the traffic rate between x and y in X n is equal to R(x, y) = β d(x,y). Tucci Hyperbolic metric spaces II

58 Hyperbolic metric spaces Assume that X is a δ hyperbolic graph and X n := {x X : d(x 0, x) n} and the traffic load decays exponentially with the distance. Meaning, the traffic rate between x and y in X n is equal to R(x, y) = β d(x,y). Theorem (Baryshnikov, T.) There exists a constant D > 1 such that if 1 < β < D the traffic is global and x 0 is in the α core for all α β D the traffic is local and all the cores are empty. Tucci Hyperbolic metric spaces II

59 Hyperbolic metric spaces Assume that X is a δ hyperbolic graph and X n := {x X : d(x 0, x) n} and the traffic load decays exponentially with the distance. Meaning, the traffic rate between x and y in X n is equal to R(x, y) = β d(x,y). Theorem (Baryshnikov, T.) There exists a constant D > 1 such that if 1 < β < D the traffic is global and x 0 is in the α core for all α β D the traffic is local and all the cores are empty. Example For the k regular tree the value of D = k. Tucci Hyperbolic metric spaces II

60 Conclusions Hyperbolic metric spaces Hyperbolic metric spaces seem to be a good model to study the traffic behavior in large complex networks. Tucci Hyperbolic metric spaces II

61 Conclusions Hyperbolic metric spaces Hyperbolic metric spaces seem to be a good model to study the traffic behavior in large complex networks. There is no universally accepted notion of hyperbolicity for finite networks. Tucci Hyperbolic metric spaces II

62 Conclusions Hyperbolic metric spaces Hyperbolic metric spaces seem to be a good model to study the traffic behavior in large complex networks. There is no universally accepted notion of hyperbolicity for finite networks. What happens if only a fraction of the triangles are δ slim? Tucci Hyperbolic metric spaces II

63 Conclusions Hyperbolic metric spaces Hyperbolic metric spaces seem to be a good model to study the traffic behavior in large complex networks. There is no universally accepted notion of hyperbolicity for finite networks. What happens if only a fraction of the triangles are δ slim? Thanks! Tucci Hyperbolic metric spaces II