Adventures in random graphs: Models, structures and algorithms

Transcription

1 BCAM January Adventures in random graphs: Models, structures and algorithms Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu

2 BCAM January LECTURE 5 Small Worlds

3 BCAM January Letter-relaying experiment by S. Milgram Objective: A source individual must forward a letter to a destination individual Rules: Limited information about destination individual: Address, name and profession Letter cannot be addressed directly to the destination individual Can only forward letter to someone known on a first name basis

4 BCAM January Outcome (1967): A significant fraction of the letters reached their destination in at most six hops! Six degrees of separation Relevance: Structure of social networks and importance of social ties Routing with limited information in communication networks Browsing behavior in WWW

5 BCAM January Modeling, modeling, modeling! Model social networks as (random) graphs Erdős-Renyi graphs G(n; p)? NO! Because

6 BCAM January In Erdős-Renyi graphs G(n; p): Small diameter: Diam(G(n; p)) = O(log n) when p λ n No structure such as geography, professional occupation, etc Clustering: Clustering coefficient (G(n; p)) = p and this will be small when p λ n Small World: Random graph with small diameter but high clustering!

7 BCAM January Random graph models of small worlds

8 BCAM January Q: How do we construct a random graph model of small worlds? Regular networks + Random short cuts Typically, Regular networks: High average path length but high clustering Random shortcuts: Shortens average path length but may decrease clustering

9 BCAM January According to Strogratz and Watts One-dimensional lattice organized on a ring L nodes Each bound is rewired with probability φ Shortcuts Difficulties: Lack of uniformity, Rewiring can make the network disconnected

10 BCAM January According to Newman and Watts One-dimensional lattice organized on a ring L nodes Add shortcuts at random

11 BCAM January The random graph SW(n; p) Two-dimensional lattice or grid with n = L 2 nodes G L = {1,..., L} {1,..., L} Local edges to grid neighbors Deterministic x y if and only if d(x, y) = x y 1 = 1 so between 2 and 4 neighbors Long haul edges With probability p (0, 1), each node x G L creates a random connection with one node selected uniformly from G L {x}.

12 BCAM January Theorem 1 For each p (0, 1), there exists A = A(p) > 0 such that lim P [Diam (SW(n; p)) A log n] n In the case p = 0: Diam (SW(n; 0)) = 2L = 2 n and Clustering coefficient (SW(n; 0)) = 0

13 BCAM January According to Kleinberg Two-dimensional lattice or grid G L = {1,..., L} {1,..., L} Local edges to grid neighbors Deterministic x y if and only if d(x, y) = x y 1 = 1 Long haul edges Each node x has q random connections with destinations y 1,..., y q (in G L {x}) selected with probability for some α > 0. d(x, y) α z G, z x d(x, z) α, y = y 1,..., y q

14 BCAM January The impact of α > 0 With α 0, increasingly uniform selection With α = 0 and q = 1, we recover SW(L; p) (with p = 1) With α, only very close neighbors are selected with very high probability

15 BCAM January Navigation on small worlds

16 BCAM January Distributed routing A routing algorithm is distributed if the decisions made t step t depends only on the knowledge of the nodes x 0,..., x t visited so far the coordinates of the destinations of shortcuts generated at these nodes

17 BCAM January Figure of merit: sup x y (E [T Algo(x, y)]) where T Algo (x, y) = Number of steps used by Algo to reach y from x Three cases concerning navigability α < 2 α = 2 α > 2

18 BCAM January Efficient routing when α = 2 Greedy (decentralized) algorithm: Always forward the message to a grid node as close to the target node as possible This greedy decentralized algorithm makes the small world navigable! Theorem 2 We have sup x y (E [T Greedy(x, y)]) O((log n) 2 )

19 BCAM January Algo generates iterates {x t, t = 0, 1,...} with x 0 = x In phase j at iterate t if 2 j < d (x t, y) 2 j+1 How many phases? No more than log 2 2L phases How long in a phase? On the average, no more than 144 (1 + log 2L) steps

20 BCAM January Thus E [T Greedy (x, y)] 144 (1 + log 2L) log 2 2L C (log n) 2 (1) for some C > 0 If x t is in phase j, then the probability that a shortcut to a node w leads to a phase k < j is bounded below by min x t : 2 j <d(x t,y) 2 j+1 w: d(y,w) 2 d(x j t, w) 2 z x t d(x t, z) 2

21 BCAM January Impossibility of efficient routing when α 2 When 0 < α < 2, short paths exists but individuals cannot determine them in a decentralized manner, i.e., for some δ > 0 E [T Algo (x, y)] δl 2 α 3 When 2 < α, short path no longer exist!