Eclectism Shrinks Even Small Worlds

Pierre Fraigniaud (CNRS, Univ. Paris Sud)

joint work with

Cyril Gavoille (Univ. Bordeaux)Christophe Paul (Univ. Montpellier)

Milgram’s Experiment

• Source person s (e.g., in Wichita)• Target person t (e.g., in Cambridge)

• Name, occupation, etc.

• Letter transmitted via a chain of individuals related on a personal basis

• Result: The “six degrees of separation”

Formal support to the 6 degrees

Watts and Strogatz: augmented graphs H=(G,D)

• Individuals as nodes of a graph G• Edges of G model relations between individuals

deducible from their societal positions• D = probabilistic distribution • “Long links” = links added to G at random,

according to D• Long links model relations between individuals

that cannot be deduced from their societal positions

Kleinberg’s model d-dimensional meshes augmented with d-harmonic links

uv

prob(uv) ≈ 1/dist(u,v)d

Exactly 1 long link per node

Greedy Routing

• Source s = (s1,s2,…,sd)

• Target t = (t1,t2,…,td)

• Current node x selects, among its 2d+1 neighbors, the closest to t in the mesh, y.

Action: Node x sends to y.

Performances of Greedy Routing

t

x

distG(x,t)=m

B=ball radius m/2

“jump”

“jump”

O(log n) expect. #stepsto enter BO(log2n) expect. #stepsto reach t from s

Limit of Kleinberg’s model

• d = #dimensions of the mesh ≈ #criterions for the search of t • Performances of greedy routing in d-dimensional meshes: O(log2n)

expected #steps independent of #criterions

Intermediate destination

OccupationGeography André

Marc

AliceRobertMary

Anne

Awareness

x

Nx = {(x,v1),(x,v2),…,(x,v2d)}Ax = {e1,e2,…,ek}

ex

Indirect-Greedy Routing

Two phases: Phase 1: Among all edges in Ax U Nx

current node x picks e such that head(e) is closest to t in the mesh.

Phase 2: Current node x selects, among its 2d+1 neighbors, the closest to tail(e) in the mesh, y.

Action: Node x sends to y.

Example

x

ttail(e)

e

y

Convergence of Indirect Greedy Routing

Definition: A system of awareness {Au/uV} is monotone if for every u, for every eAu-{eu}, the first node v on the greedy path from u to tail(e) satisfies eAv.

Theorem: IGR converges if and only if the system of awareness is monotone.

Example: Au = long links of the k closest neighbors of u in the mesh

Performances of IGR

mm/r

u

tBall of k nodesRadius ≈ k1/d

Tradeoff

• Large awareness large expected #steps to reach ID small expected #phases “m m/r”• Small awareness small expected #steps to reach ID large expected #ID before

“mm/2”

Case |Au|=O(log n)

• Theorem: If every node is aware of the long links of its O(log n) closest neighbhors, then IGR performs in O(log1+1/dn) expected #steps.

• Proof: O(log1/dn) exp. #steps to reach ID O(log n) exp. #steps mm/2

Consequences

• GR does not take #criterions into account O(log2n) exp. #steps

• IGR takes #criterions into account O(log1+1/dn) exp. #steps

Eclecticism shrinks even small worlds

|Au|=O(log n) is optimal

Size awareness

Exp. #steps

log2n

log n logdn

log1+1/dn

#phase too large

ID too far

KGR is betterKGR

ConclusionE(#steps) |

awareness|

Greedy (harm.)

Θ(log2n / c) c log n

Greedy (any) Ω(log2n / (c loglog n))

c log n

Decentralized O(log2n / log2c) c log n

NoN-greedy O(log2n / (c log c)) c2 log n

Indirect-gdy O(log1+1/dn / c1/d) log2nc = #long-range links per node