Shortest Paths & Link Weight Structure in Networks Piet Van - PowerPoint PPT Presentation

Shortest Paths & Link Weight Structure in Networks Piet Van Mieghem CAIDA WIT (May 2006) Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 1

Outline Introduction The Art of Modeling Conclusions Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 2

Telecommunication: e2e B A NETWORK G(N,L) Main purpose : Transfer information from A B • Nearly always : Transport of packets along shortest (or optimal) paths • Optimality criterion : in terms of Quality of Service (QoS) parameters • (delay, loss, jitter, distance, monetary cost,...) • Broad focus: What is the role of the graph/network on e2e QoS ? What is the collective behavior of flows, the network dynamics ? Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 3

Fractal Nature of the Internet Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 4

Large Graphs protein Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 5

Internet observed by RIPE boxes

Ad-hoc network Despite node movements, at any instant in time the network can be considered as a graph with a certain topology Physics of the evolution of Ad-hoc graphs Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 7

Outline Introduction The Art of Modeling: The Basic Model Conclusions Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 8

Network Modeling • Any network can be representated as a graph G with N nodes and L links: – adjaceny matrix, degree (=number of direct neighbors), connectivity, etc... – link weight structure: importance of a link • Link weights: also known as quality of service (QoS) parameters – delay, available capacity, loss, monetary cost, physical distance, etc... • Assumption that transport follows shortest paths – correct in more than 70% cases in the Internet Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 9

Network Modeling Confinement to: • Hop Count (or number of hops) of the shortest path: – Apart from end systems, QoS degradation occurs in routers (= node). – QoS measures (packet delay, jitter, packet loss) depend on the number of traversed routers and not on the ‘length’ of shortest path. – Relatively easy to measure (trace-route utility) and to compute (initial assumption) • Weight (End-to-end delay) of the shortest path: – perhaps the most interesting QoS measure for applications – difficult to measure accurately – capacity of a link: • measurement is related to a delay measurement (dispersion of two IP packets back-to-back) Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 10

Simple Topology Model • Link weights w(i → j )>0: 2 – unknown, but very likely not constant, w(i → j ) ≠ 1 1 2 5/2 – assumption : i.i.d. uniformly/exponentially distributed 3 2/3 5 – bi-directional links: w(i → j )= w(j → i ) 3 1 4 1 • Complete graph K N • One level of complexity higher: ER Random graphs of class G p (N) – N: number of nodes – p: link density or probability of being edge i → j equals p – only connected graphs: p > p c ~ logN/N Is this the right structure? Are exponential weights reasonable? Not a good model for the Internet graph, but reasonably good for Peer to Peer networks (Gnutella, KaZaa) and Ad Hoc networks Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 11

Markov Discovery Process in the complete graph K N 1) property of i.i.d. exponential r.v.’s ⎛ ⎞ ( ) n ∑ ⎜ ⎟ = Exp a Exp a min ( ) ⎜ ⎟ j j ≤ ≤ j n ⎝ ⎠ 1 = j 1 2) memoryless property of exponential distribution λ = − n N n ( ) 3) transition rates for K N : node A node A + n n , 1 t t 0 0 t t B 1 1 t 2 t 3 time time T Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 12

Markov Discovery Process and Uniform Recursive Tree v 8 Markov v 7 corresponding URT Discovery v 6 Process H = 0 0 v 5 v 4 v 3 H = 1 2 6 1 5 v 2 6 H = 2 v 1 3 5 4 1 0 time H = 3 7 8 2 Growth rule URT: 4 Every not yet discovered node 3 has equal probability to be attached to any node of the 8 τ 6 k ∑ = τ URT. v Discovery times: k j where τ j is 7 = j 1 Hence, the number of URTs with N nodes equals (N-1)! exponentially distributed [ ] 1 τ = with mean E Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 13 j − j N j ( )

Uniform Recursive Tree = X ( 0 ) Root 1 N 1 ( = X 1 ) 5 N 6 2 3 5 26 [ ] k E X ( ) [ ] = = y k N Pr = X ( 2 ) 9 N N N 12 18 4 9 7 8 10 11 15 = ≥ k X ( ) 0 if k N N = X ( 3 ) 7 N 22 14 21 13 16 20 17 = X ( 4 ) 4 N 24 19 23 25 [ ] [ ] − − k N E X ( 1 ) 1 ∑ Recursion from Growth rule URT : = k m E X ( ) N m = m k Recursion for generating function: ( ) ( ) + ϕ = + ϕ N z N z z 1 ( ) ( ) + N N 1 [ ] Γ + N z ( ) ϕ = = Solution: y z E z ( ) N N Γ + Γ + N z ( 1 ) ( 1 ) Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 14

Hopcount of the shortest path in K N with exp. link weights • Shortest path (SP) described via Markov discovery process. • Shortest path tree (SPT) = Uniform recursive tree (URT) • The generating function of the hopcount H N of shortest path between different nodes is [ ] Γ + N ⎛ ⎞ N z 1 ( ) = 1 ϕ − ϕ = H N E z ⎜ z ⎟ with z ( ) ( ) N N − Γ + Γ + N ⎝ N ⎠ N z ( 1 ) ( 1 ) from which (via Taylor expansion around z = 0 and Stirling’s approximation) follows that − k m k ] ∑ N [ log = ≈ P H k c N m − N k m ( )! = m 0 [ ] and ≈ + γ − E H N log 1 N close to Poisson π 2 [ ] ≈ + γ − H N N var log 6 Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 15

The Weight of SP The k -th discovered node is attached at time v k = v 8 Markov τ 1 + τ 2 +...+ τ k where τ n is exponentially distributed with v 7 Discovery rate n(N-n) and the τ ’s are independent (Markov v 6 Process property): [ ] ∏ − k n N n ( ) − zv = E e v 5 k + − z n N n ( ) v 4 = n 1 v 3 The weight W N of the SP in the complete graph with 5 v 2 exponential link weights is 6 [ ] [ ] [ v 1 N ] ∑ − − 1 zW = zv E e E e Pr endnode is k - th attached node in URT N k 0 = k 1 time or [ ] − 2 k N n N n 1 ( ) ∑∏ − ϕ = zW = z E e ( ) N W − + − N z n N n N 1 ( ) 4 = = k n 1 1 3 From this pgf, the mean weight (length) is derived as 8 τ 6 − N 1 [ ] 1 1 ∑ = − ϕ = E W ' ( 0 ) W N N − N n 1 7 = n 1 Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 16

Additional Results • The asymptotic distribution of the weight of the SP is [ ] − x − − log ≤ = e NW N x e lim Pr N → ∞ N • The flooding time (= minimum time to inform all ∑ − N 1 − = τ v nodes in network) is N j 1 = j 1 • The weight W U of the shortest path tree can be computed explicitly. • Other instances of trees (see Multicast). • The degree distribution (= number of direct neighbors) in the URT can be computed explicitly, ⎛ ⎞ − k 1 N [ ] log − ⎜ ⎟ = = k + D k O Pr 2 ⎜ ⎟ N 2 ⎝ ⎠ Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 17

Outline Introduction The Art of Modeling: Extension & Measurements Conclusions Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 18

Extending the Basic Model • From the complete graph K N to the Erdös-Rényi random graph G p (N): from p = 1 to p < 1 . • Result (which can be proved rigorously): for large N and fixed p > p c ~ logN/N holds that the SPT in the class of G p (N) with exponential link weights is a URT. Intuitively: G p (N) with p > p c is sufficiently dense (there are enough links) and the link weights can be arbitrarily small such that the thinning effect of the link weights is precisely the same in all connected random graphs. • Implications: – basic model (SPT = URT) seems more widely applicable! – influence of the link weight structure is important Faculty of Electrical Engineering, Mathematics and Computer Science P. Van Mieghem 19