Small world effect

P. Fraigniaud, E. Lebhar, Z. Lotker

Recovering the long range links in augmented graphs

Large interaction networks Spontaneous & local construction. Very large number of nodes. Small diameter. High clustering. Power-law degree distr

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

Name, professional occupation, city of living, etc.

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

Result: “six degrees of separation”

The small world effect

M. SmithFarmerWyoming

M. SMOOTHPhysicianBosto

Very short paths exists

Milgram 1967

People are able to discover them locally. (navigability)

Navigability Jon Kleinberg (2000)

Why should there exist short chains of acquaintances linking together arbitrary pairs of strangers?

Why should arbitrary pairs of strangers be able to find short chains of acquaintances that link them together?

In other words: how to navigate in a small worlds?

Augmented graphs H=(G,D) Individuals as nodes of a graph G

Edges of G model relations between individuals deducible from their societal positions

A number k of “long links” are added to Gat random, according to some probability distribution D: Du(v)=prob(u→v) Long links model relations between individuals

that cannot be deduced from their societal positions

Example

Greedy Routing in augmented graphs

Source s ∈ V(G) Target t ∈ V(G) Current node x selects among its degG(x)+k

neighbors the closest to t in G, that is according to the distance function distG().

Greedy routing in augmented graphs aims at modeling the routing process performed by social entities in Milgram’s experiment.

Example

4 3 2

Augmented meshesKleinberg (2000)

d-dimensional n-node meshes augmented with d-harmonic links

uv

Du(v)=prob(u→v) ≈ 1/(log(n)*dist(u,v)d)

Kleinberg’s theorems Greedy routing performs in O(log2n / k)

expected #steps in d-dimensional meshes augmented with k links per node, chosen according to the d-harmonic distribution. Note: k = log n ⇒ O(log n) expect. #steps

Greedy routing in d-dimensional meshes augmented with a h-harmonic distribution, h≠d, performs in Ω(nε) expected #steps.

Navigable graphs Let f : N → R be a function

Definition: An n-node graph G is f-navigable if there exist an augmentation D for G such that greedy routing in (G,D) performs in at most f(n) expected #steps.

O(polylog(n))-Navigable graphs

Bounded growth graphs Definition: |B(x,2r)| ≤ k |B(x,r)| Duchon, Hanusse, Lebhar, Schabanel (2005)

Bounded doubling dimension Definition: every B(x,2r) can be covered by at

most 2d balls of radius r, B(xi,r) Slivkins (2005)

Graphs of bounded treewidth Fraigniaud (2005)

Question Does it exist a function f∈O(polylog(n))

such that all graphs are f-navigable?

Doubling dimensionBounded doubling dimension

Definition: every B(x,2r) can be covered by at most 2d balls of radius r, B(xi,r)

Doubling dimension

Our resultTheorem (Fraigniaud, Lebhar, Lotker)

Let d such that limn→+∞ loglog n / d(n) = 0

Let F be the family of n-node graphs with doubling dimension at most d(n).

F is not O(polylog(n))-navigable.

Examples

d(n) = (loglog n)1+ε for ε>0 d(n) = (log n)1/2

Remark: 2d(n) is not in O(polylog(n))

Proof of non-navigability The family F contains the graph Hd:

x = x1 x2 ... xdis connected to all nodes

y = y1 y2 ... ydsuch that yi = xi + ai where

ai ∈ -1,0,+1

Hd has doubling dimension d

Intuitive approach Large doubling dimension d implies that

every nodes x ∈ Hd has choices over exponentially many directions

The underlying metric of Hd is L∞:

Directions

+1,+1

+1,0

+1,-1

-1,+1

-1,0

-1,-1

0,+1

0,-1

Case of symmetric distribution

Source s

Target t

Disadvantageddirection

At every step, probability ≤ 1/2d to go in the right direction

Diagonals+1,+1

+1,0

+1,-1

-1,+1

-1,0

-1,-1

0,+1

0,-1

Lines

p lines in each direction

p

p

Intervals

J

Certificates

J

v

v is a certificate for J

Counting argument 2d directions Lines are split in intervals of length L n/L × 2d intervals in total Every node belongs to many intervals,

but can be the certificate of at most one interval

If L<2d there is one interval J0 without certificate

L-1 steps from s to t

J0

source s

target t

QED

Conclusion Remark: The proof still holds if the long

links are not set independently. Theorem (Peleg)

Every n-node graph is O(n1/2)-navigable, by using a uniform setting of the long links

Open problem: look for the smallest function f such that every n-node graph is f(n)-navigable

Navigability in Kleinbergmodel

A routing algorithm is claimed decentralized if:

1. It knows all links of the mesh,2. It discovers locally the extra random

links. Greedy routing computes paths of

expected length O(log2n) between any pair in this model.

Augmented graph models Augmented graph= (H,φ)

H= base graph, globally known φ= augmented links distribution

φu(v)= probability that v is the long range contact of u.

Question Can we detect, in a “small world”

graph, which links are the long-range links?

Recovering the longrange links

1. Validation of the model: Are there long range links? What kind of real links are they?

2. Efficient routing with light encoding: The set of grid coordinates is enough to

route. Distances easy to compute with small

labels.

Goal Design an algorithm

with input G, small world graph that outputs H and a set E of long links such that if G is H’ augmented with E’,

(H,E) is a good approximation of (H’,E’). Good approximation: H close to H’ and

greedy routing efficient knowing only H.

Too big H may destroythe routing

Remaining shortcuts can be problematic!

Greedy diameter Ω(n1/√log n)

Hypothesis on the input G= small world graph We assume that G comes from a

density based augmentation. G= H + E, H of bounded growth and E

produced by φ, density based. We assume that the base metric H has

a high clustering.

high clustering. An edge u-v has clustering C/n iff u and

v share at least C neighbors.

Extracting the long range linksIntuition

The base metric has high clustering. Extraction algorithm: tests if links are

highly clustered.

Highly clustered:chances to be close.

Few triangles:chances to be long range.

Extracting the long range links Input: a bounded growth graph G,

clustered augmented by a density based distribution of shortcuts (unknown set E).

Extraction algorithm: for each edge, test # of triangles if it does not correspond to the clustering,

label it as shortcut.

Result Theorem:

if the clustering is high enough (≥logn/loglogn), the algorithm detects all shortcuts of large stretch (≥polylog(n)) except polylog(n) of them w.h.p..

Greedy routing with the output map computes routes of length at most poylylog(n) longer than it should with the original map.

Clustering hypothesis The clustering hypothesis is essential to

recover the long links with a local maximum likelihood algorithm. Theorem: On the ring of n nodes

harmonically augmented, for any 0<ε<1/5, any local maximum

likelihood algorithm misses Ω(n5ε/logn) links of stretch Ω(n1/(5-ε)).

The End