Augmented graph models • Density based augmentations: [Kleinberg 00, Kleinberg 01, Duchon et al. 05, Slivkins 05] If H is of bounded growth i.e. B u (2R) ≤ c.B u (R) for all node u and radius R 20
Augmented graph models • Density based augmentations: [Kleinberg 00, Kleinberg 01, Duchon et al. 05, Slivkins 05] If H is of bounded growth i.e. B u (2R) ≤ c.B u (R) for all node u and radius R and if φ is density based i.e. φ u (v) proportional to 1/B u ( dist(u,v) ) 20
Augmented graph models • Density based augmentations: [Kleinberg 00, Kleinberg 01, Duchon et al. 05, Slivkins 05] If H is of bounded growth i.e. B u (2R) ≤ c.B u (R) for all node u and radius R and if φ is density based i.e. φ u (v) proportional to 1/B u ( dist(u,v) ) Then H is navigable i.e. greedy routing computes polylog(n) paths knowing only H. 20
Augmented graph models • Density based augmentations 21
Augmented graph models • Density based augmentations small probability v u B u ( d(u,v) ) 21
Augmented graph models • Density based augmentations small probability big probability v v u u B u ( d(u,v) ) B u ( d(u,v) ) 21
Augmented graph models • Density based augmentations small probability big probability v v u u B u ( d(u,v) ) B u ( d(u,v) ) Liben-Nowell et al 05 : it fits observations on electronic social networks. 21
- Navigability: Dynamic properties 22
Question • Harmonic augmentation 1/x is crucial. 23
Question • Harmonic augmentation 1/x is crucial. • But : natural spontaneous networks. 23
Question • Harmonic augmentation 1/x is crucial. • But : natural spontaneous networks. • How does it emerge? 23
Question • Harmonic augmentation 1/x is crucial. • But : natural spontaneous networks. • How does it emerge? We look for a natural dynamic process arising the shortcuts that produces navigability. (e.g. 1 -harmonic distribution on the ring.) 23
Previous attempts [Clauset & Moore 2003] • Hypothesis: shortcuts comes from successive searches. 24
Previous attempts [Clauset & Moore 2003] • Hypothesis: shortcuts comes from successive searches. new hyperlink: bookmark - u stops after t steps (random) and rewire its link. 24
Previous attempts [Clauset & Moore 2003] • Hypothesis: shortcuts comes from successive searches. new hyperlink: bookmark hyperlink Threshold of frustration - u tries to route greedily towards v (random). - u stops after t steps (random) and rewire its link. 24
Previous attempts 25
Previous attempts • Simulation results: 1 0.95 convergence to the 0.9 rewired exponent, � rewired 0.85 harmonic distribution 0.8 0.75 and short greedy 0.7 � =75 � =150 routes. 0.65 � =300 � =500 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 initial exponent, � 0 25
Previous attempts • Simulation results: 1 0.95 convergence to the 0.9 rewired exponent, � rewired 0.85 harmonic distribution 0.8 0.75 and short greedy 0.7 � =75 � =150 routes. 0.65 � =300 � =500 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 initial exponent, � 0 • No analytical result: complex dependencies of the shortcuts. 25
Previous attempts [Clarke & Sandberg 2006] • Same hypothesis: shortcuts comes from successive searches. 26
Previous attempts [Clarke & Sandberg 2006] • Same hypothesis: shortcuts comes from successive searches. - u computes a greedy path P to v (random). 26
Previous attempts [Clarke & Sandberg 2006] • Same hypothesis: shortcuts comes from successive searches. - u computes a greedy path P to v (random). - each node of P rewires its link to v with probability p. 26
Previous attempts [Clarke & Sandberg 2006] • Same hypothesis: shortcuts comes from successive searches. - u computes a greedy path P to v (random). - each node of P rewires its link to v with probability p. • Analytical results: convergence to the harmonic distribution. • Simulation results: short greedy routes. 26
Our approach [Chaintreau, Fraigniaud, L. 08] • A different hypothesis : 1. Links comes from people who met once. 2. Links tend to be forgotten with time. 27
Our approach [Chaintreau, Fraigniaud, L. 08] • A different hypothesis : 1. Links comes from people who met once. 2. Links tend to be forgotten with time. • Our process can be fully analyzed. 27
Move and Forget process Z k 28
Move and Forget process Z k Individuals: token moving on the grid. 28
Move and Forget process Z k Individuals: token moving on the grid. 28
Move and Forget process Z k Each token moves independently according to a random walk. 29
Move and Forget process Z k Each token moves independently according to a random walk. 29
Move and Forget process Z k Each token moves independently according to a random walk. 29
Move and Forget process Z k Each token moves independently according to a random walk. 29
Move and Forget process Z k Each token moves independently according to a random walk. 29
Move and Forget process Z k Each token drags the head of a shortcut rooted at its departure position. 30
Move and Forget process But after some time, we forget people. 31
Move and Forget process But after some time, we forget people. How sad. 31
Move and Forget process But after some time, we forget people. Forgetting mechanism: • In each step, if our link is of age a, we forget it with probability ∝ 1 /a. 31
Move and Forget process But after some time, we forget people. Forgetting mechanism: • In each step, if our link is of age a, we forget it with probability ∝ 1 /a. I.e. : there is less chance to forget your very old friend than the one met at the bar yesterday. 31
Move and Forget process Z k When a link is forgotten, it is rewired to its departure and a new token is launched. 32
Move and Forget process Z k When a link is forgotten, it is rewired to its departure and a new token is launched. 32
Move and Forget process Z k When a link is forgotten, it is rewired to its departure and a new token is launched. 33
Move and Forget process Z k When a link is forgotten, it is rewired to its departure and a new token is launched. 33
More precisely • At time t, the long range contact of u is (x 1 (t),x 2 (t),...,x k (t)). 34
More precisely • At time t, the long range contact of u is (x 1 (t),x 2 (t),...,x k (t)). • It is forgotten with probability ∝ 1/a (a=the time since it has been launched). 34
More precisely • At time t, the long range contact of u is (x 1 (t),x 2 (t),...,x k (t)). • It is forgotten with probability ∝ 1/a (a=the time since it has been launched). • If it survives, for all i: x i (t+1)= x i (t)+1 with probability 1/2 x i (t+1)= x i (t)-1 with probability 1/2 34
More precisely • At time t, the long range contact of u is (x 1 (t),x 2 (t),...,x k (t)). • It is forgotten with probability ∝ 1/a (a=the time since it has been launched). • If it survives, for all i: x i (t+1)= x i (t)+1 with probability 1/2 x i (t+1)= x i (t)-1 with probability 1/2 • Otherwise, (x 1 (t+1),..., x k (t+1))=u. 34
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