Eclectism Shrinks Even Small Worlds

1
Eclectism Shrinks Even Small Worlds

• Pierre Fraigniaud (CNRS, Univ. Paris Sud)
• joint work with
• Cyril Gavoille (Univ. Bordeaux)
• Christophe Paul (Univ. Montpellier)

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Milgrams 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

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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

4
Kleinbergs model

• d-dimensional meshes augmented
• with d-harmonic links

u
prob(u?v) 1/dist(u,v)d
Exactly 1 long link per node
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Greedy Routing

• Source s (s1,s2,,sd)
• Target t (t1,t2,,td)
• Current node x selects, among its 2d1 neighbors,
  the closest to t in the mesh, y.
• Action Node x sends to y.

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Performances of Greedy Routing
Bball radius m/2
t
O(log n) expect. steps to enter B
x
O(log2n) expect. steps to reach t from s
distG(x,t)m
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Limit of Kleinbergs 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

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Intermediate destination
André
Occupation
Geography
Mary
Robert
Alice
Marc
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Awareness
x
Nx (x,v1),(x,v2),,(x,v2d)
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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 2d1
  neighbors, the closest to tail(e) in the mesh, y.
• Action Node x sends to y.

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Example
x
y
tail(e)
t
e
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Convergence of Indirect Greedy Routing

• Definition A system of awareness Au/u?V is
  monotone if for every u, for every e?Au-eu, the
  first node v on the greedy path from u to tail(e)
  satisfies e?Av.
• 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

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Performances of IGR
Ball of k nodes Radius k1/d
t
m/r
m
u
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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 m?m/2

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Case AuO(log n)

• Theorem If every node is aware of the long links
  of its O(log n) closest neighbhors, then IGR
  performs in O(log11/dn) expected steps.
• Proof
• O(log1/dn) exp. steps to reach ID
• O(log n) exp. steps m?m/2

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Consequences

• GR does not take criterions into account ?
  O(log2n) exp. steps
• IGR takes criterions into account
• ? O(log11/dn) exp. steps
• Eclecticism shrinks even small worlds

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AuO(log n) is optimal
Exp. steps
log2n
log11/dn
Size awareness
log n
logdn
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Conclusion
c long-range links per node