Routing in Poisson small-world networks

1. Routing in Poisson small-world networks A. J. Ganesh Microsoft Research, Cambridge Joint work with Moez Draief

2. What is a small world network? • Milgram (1967) • Sent letters to various people in the US addressed to targets in Boston • Demographic information about target provided: name, address, occupation • Letters had to be forwarded to target via contacts known on first-name basis • Average length of chain for successful delivery was found to be six.

3. Modelling social networks • Random graphs have small diameter • Diameter of Erdos-Renyi random graph on n nodes is O(log n) at connectivity threshold • Similar results for power-law random graphs • But not a good model of social networks • Social networks are more clustered

4. Small-world network models • Watts & Strogatz • Lattice plus random shortcuts (uniform) • Benjamini & Berger • d-dimensional lattice, edge (x,y) present with probability |x-y|-r • Coppersmith et al. • Diameter is O(log n) if r=d • Polynomial in n if r>2d

5. Outline of talk • Background: can short paths be found using local information? • Kleinberg, Franceschetti & Meester • Model: Poisson point process with local links and shortcuts • Results • Open problems

6. Kleinberg: can short paths be found? • 2-dimensional lattice on N2 points • Each node has q shortcuts • Each shortcut from x is incident on y with probability proportional to d(x,y)-r, independent of others • Need to route a message from source s to destination t using a decentralised algorithm

7. What is a decentralised algorithm? • Each node knows co-ordinates of its shortcut contacts & of destination t • Suppose algorithm has currently visited nodes x0,…,xk • Next node to visit must be chosen from among contacts (lattice or shortcut) of these nodes.

8. r=2: Greedy routing • At each node visited by algorithm • Choose shortcut contact if it is closer to destination • Else choose lattice contact which is closer • Theorem: Number of hops to destination is O(log2N) • Key idea: Find good shortcut that halves distance every O(log N) steps

9. r ≠ 2: Impossibility result • r≠2: No decentralised algorithm can find path shorter than a polynomial in N • Reason: • Short-cuts lack range if r>2 • Short-cuts spread too uniformly if r<2, can’t close in on target (using only local information)

10. Continuum model • Franceschetti and Meester • Each point in plane has shortcut to other points according to an inhomogeneous Poisson process • Intensity at distance x proportional to x-r (not integrable) • Objective is to deliver message from s to  neighbourhood of t

11. Results • If r=2, greedy algorithm has expected  delivery time O(log d(s,t) + log(1/)) • If r>2, any decentralised algorithm needs at least d(s,t)β steps • If r<2, any decentralised algorithm needs at least (1/)β steps

12. Poisson small-world model • Nodes: located at points of unit rate Poisson point process on square of area n. • Local links: to all other nodes within distance √c log(n) • Shortcuts:q per node, in expectation • Probability of shortcut to node at distance d: c(q,n) d-r

13. Remarks • For sufficiently large c>0, graph formed by local links alone is connected. • Hence, message can always be routed in O(√n/log(n)) hops. • Do shortcuts help us do better? • Can we route in polylog(n) hops?

14. Results: r=2 • r=2: there is a decentralised algorithm that can route a message between any pair of nodes in O(log2n) hops, with high probability, for sufficiently large c>0 and any q>0.

15. Results: r≠2 • r<2: Any decentralised routing algorithm needs more than n hops on average, for any  < (2-r)/6 • r>2: Any decentralised routing algorithm needs more than n hops on average, for any  < (r-2)/2(r-1)

16. Algorithm for r=2 • At each hop, algorithm maintains a `radius’ , initialised to d(s,t). • If current node x has shortcut to a node in circle of radius /2 centred at t, message is delivered to this node. • Else, it is delivered to one of the local contacts of x which is closer to t • If d(x,t)</2,  is updated to /2

17. In pictures + modification

18. Sketch proof: r=2 • If c large enough, every node x has a local contact which is closer to t • P(good shortcut) depends on number of nodes in annulus • It is O(1/log n) if number of nodes is large enough • Probability that number of nodes is small is negligible • Hence, good shortcut found after geometric O(log n) steps

19. Sketch proof: r<2 • Suppose algorithm finds a path with fewer than n hops • There has to be at least one shortcut which takes path into circle of radius n+ centred at t • P(shortcut between u and v) is small for any u,v. • Very unlikely to find shortcut into this circle, by union bound.

20. Sketch proof: r>2 • Long-range contacts penalised • P(shortcut has length > n0.5--) is too small • With high probability, there is no such shortcut within first n nodes seen by routing algorithm

21. Remarks • Results hold for inhomogeneous Poisson processes with intensity bounded away from zero and infinity • Impossibility results continue to hold with • 1-step look-ahead: each node knows locations not only of its local and long-range contacts but also their contacts • or k-step look-ahead, fixed k • Mean number of shortcuts per node being polylog(n) instead of constant

22. Open problems • Other density functions for shortcuts • r=2: variants of proposed algorithm should also work, but hard to prove • r=2: what if there are no local contacts but c log(n) shortcuts? Is graph connected? Is efficient routing possible? • Still doesn’t explain Milgram’s experiments!