In this presentation we show that there exist graphs which greedy routing thak long time. This show that Small world is a Social phenomena and not mathematical one.

1. Small world effect
P. Fraigniaud, E. Lebhar, Z. Lotker

2. 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

3. 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”

4. The small world effect
M. Smith
Farmer
Wyoming
M. SMOOTH
Physician
Bosto
Very short paths exists
Milgram 1967
People are able to discover them
locally.
(navigability)

5. 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?

6. 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 G
at 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

7. Example

8. 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.

9. Example
4 3 2

10. Augmented meshes
Kleinberg (2000)
d-dimensional n-node meshes
augmented with d-harmonic links
u
v
Du(v)=prob(u→v) ≈ 1/(log(n)*dist(u,v)d)

11. 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.

12. 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.

13. 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)

14. Question
 Does it exist a function f∈O(polylog(n))
such that all graphs are f-navigable?

15. Doubling dimension
Bounded doubling dimension
 Definition: every B(x,2r) can be covered by
at most 2d balls of radius r, B(xi,r)

16. Doubling dimension

17. Our result
Theorem (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.

18. 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))

19. Proof of non-navigability
 The family F contains the graph Hd:
x = x1 x2 ... xd
is connected to all nodes
y = y1 y2 ... yd
such that yi = xi + ai where
ai ∈ {-1,0,+1}
Hd has doubling dimension d

20. 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∞:

21. Directions
+1,+1
+1,0
+1,-1
-1,+1
-1,0
-1,-1
0,+1
0,-1

22. Case of symmetric distribution
Source s
Target t
Disadvantaged
direction
At every step, probability ≤ 1/2d to go in the right direction

23. Diagonals
+1,+1
+1,0
+1,-1
-1,+1
-1,0
-1,-1
0,+1
0,-1

24. Lines
p lines in each direction
p
p

25. Intervals
J

26. Certificates
J
v
v is a certificate for J

27. 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

28. L-1 steps from s to t
J0
source s
target t
QED

29. 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

30. Navigability in Kleinberg
model
 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.

31. 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.

32. Question
 Can we detect, in a “small world”
graph, which links are the long-range
links?

33. Recovering the long
range 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.

34. 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.

35. Too big H may destroy
the routing
 Remaining shortcuts can be
problematic!
Greedy diameter Ω(n1/√log n)

36. 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.

37. high clustering.
 An edge u-v has clustering C/n iff u and
v share at least C neighbors.

38. Extracting the long range links
Intuition
 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.

39. 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.

40. 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.

41. 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)
).

42. The End