Existence & Emergence of Navigability in Social Networks - - PowerPoint PPT Presentation

Existence & Emergence

• f Navigability

in Social Networks

Emmanuelle Lebhar (CNRS & CMM-U. de Chile)

1

1- Social networks

2

Social networks

Networks of human beings interactions.

High school dating Co-authorship in Physics

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Social networks

Networks of human beings interactions.

Disease spreading through human contacts Blogs network

4

Social networks

• Billions of nodes.
• Network built “spontaneously”, no map.
• Recently: huge increase of available

data! (emails, Facebook, blogs)

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Social networks

• Billions of nodes.
• Network built “spontaneously”, no map.
• Recently: huge increase of available

data! (emails, Facebook, blogs)

5

Specific properties common to

• ther “complex networks”:

(Internet, electricity, proteins..)

Social networks

• Billions of nodes.
• Network built “spontaneously”, no map.
• Recently: huge increase of available

data! (emails, Facebook, blogs)

5

Specific properties common to

• ther “complex networks”:

(Internet, electricity, proteins..)

➡High clustering

Social networks

• Billions of nodes.
• Network built “spontaneously”, no map.
• Recently: huge increase of available

data! (emails, Facebook, blogs)

5

Specific properties common to

• ther “complex networks”:

(Internet, electricity, proteins..)

➡Power law degree

distribution

➡High clustering

Social networks

• Billions of nodes.
• Network built “spontaneously”, no map.
• Recently: huge increase of available

data! (emails, Facebook, blogs)

5

Specific properties common to

• ther “complex networks”:

(Internet, electricity, proteins..)

➡Small diameter ➡Power law degree

distribution

➡High clustering

The small world effect

• Very short paths exists.
• People are able to discover them locally.

=Navigability

• M. Smith

Farmer Wyoming

• M. SMOOTH

Physician Boston

Milgram 1967

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“6 degrees of separation”

2- Random models

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Static random models

• Why random? To reflect human

behaviors...

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Static random models

• Why random? To reflect human

behaviors...

8

• First model: Erdös-Rényi random

graph.

Static random models

• Why random? To reflect human

behaviors...

8

• First model: Erdös-Rényi random

graph. Probability p that two people know each other.

Static random models

• Why random? To reflect human

behaviors...

8

• First model: Erdös-Rényi random

graph. Probability p that two people know each other.

n=#nodes p

Static random models

• Why random? To reflect human

behaviors...

8

• First model: Erdös-Rényi random

graph. Probability p that two people know each other.

If p> (log n)/n, diameter ≤ log n. n=#nodes p

Why a small diameter?

• Expected degree of a random node u: np.

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p

u

Why a small diameter?

• Expected degree of a random node u: np.

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p

• (very roughly) # of k-neighbors: (np)k

u

Why a small diameter?

• Expected degree of a random node u: np.

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p

• (very roughly) # of k-neighbors: (np)k

u

• Distance k to reach all nodes: ln(np).

Why a small diameter?

• Expected degree of a random node u: np.

9

• True result: log(n)/log(<d>) as soon as

<d>=np>1.

p

• (very roughly) # of k-neighbors: (np)k

u

• Distance k to reach all nodes: ln(np).

Static random models

Main features of Erdös-Rényi random graph:

• Poisson degree distribution.
• Exponential neighborhood growth.

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Static random models

Main features of Erdös-Rényi random graph:

• Poisson degree distribution.
• Exponential neighborhood growth.

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• Consequences on virus spreading models.

Static random models

Main features of Erdös-Rényi random graph:

• Poisson degree distribution.
• Exponential neighborhood growth.

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• Consequences on virus spreading models.
• But uniform random graph turned out

to be very far from real observations!

Importance of random models accuracy

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• Q°: when do we reach the epidemic state?

Importance of random models accuracy

Virus spread: if there is a threshold of

infection to reach before the epidemy, we can try to slow down the spread not to reach it.

11

• Q°: when do we reach the epidemic state?

Importance of random models accuracy

Virus spread: if there is a threshold of

infection to reach before the epidemy, we can try to slow down the spread not to reach it.

11

• Q°: when do we reach the epidemic state?
• Network with Poisson degree

distribution: threshold exists.

Importance of random models accuracy

Virus spread: if there is a threshold of

infection to reach before the epidemy, we can try to slow down the spread not to reach it.

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• Q°: when do we reach the epidemic state?
• Network with Poisson degree

distribution: threshold exists.

• Network with power-law degree distrib°

with exponent in [2,3] (i.e. like real): threshold=0.

Uniform random graphs

• No clustering: friends of friends not

well connected.

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Uniform random graphs

• No clustering: friends of friends not

well connected.

• It makes sense! No reason why since the

distribution is uniform.

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Uniform random graphs

• No clustering: friends of friends not

well connected.

• It makes sense! No reason why since the

distribution is uniform.

• Small world navigation does not work:

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Uniform random graphs

• No clustering: friends of friends not

well connected.

• It makes sense! No reason why since the

distribution is uniform.

• Small world navigation does not work:

Pr( reach √n - ball around target) ≤ 1/√n

➡ at least √n expected path length.

12

Static random models

13

• Mixing randomization and regularity:

Watts & Strogatz 1999.

Static random models

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• Mixing randomization and regularity:
• Ok for clustering.

Watts & Strogatz 1999.

Static random models

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• Mixing randomization and regularity:
• Ok for clustering.
• But still not navigable (unifrom distribution
• ver distances).

Watts & Strogatz 1999.

v u

Pr(u→v)∝1/|u-v|2

Modeling navigability

Kleinberg model of augmented graphs [2000]:

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v u

• 2-dim grid: distances globally known.

Pr(u→v)∝1/|u-v|2

Modeling navigability

Kleinberg model of augmented graphs [2000]:

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v u

• 2-dim grid: distances globally known.
• Extra random shortcuts: private links

(e.g. friends).

Pr(u→v)∝1/|u-v|2

Modeling navigability

Kleinberg model of augmented graphs [2000]:

14

v u

Pr(u→v)∝1/|u-v|s

Modeling navigability

Kleinberg model of augmented graphs [2000]:

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v u

Pr(u→v)∝1/|u-v|s

Theorem [K00]

Modeling navigability

Kleinberg model of augmented graphs [2000]:

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v u

• s=2: greedy routing reaches any node

in O(log2n) steps.

Pr(u→v)∝1/|u-v|s

Theorem [K00]

Modeling navigability

Kleinberg model of augmented graphs [2000]:

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v u

• s=2: greedy routing reaches any node

in O(log2n) steps.

• s≠2: greedy takes at least Ω(nc) steps.

Pr(u→v)∝1/|u-v|s

Theorem [K00]

Modeling navigability

Kleinberg model of augmented graphs [2000]:

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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.

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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.

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Navigability : Structural properties

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Augmented graph (H,φ) :

Augmented graph models

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Augmented graph (H,φ) : ➡ H= base graph , globally known

Augmented graph models

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Augmented graph (H,φ) : ➡ H= base graph , globally known

➡ φ = augmented links distribution

Augmented graph models

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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.

Augmented graph models

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Augmented graph models

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Augmented graph models

• Bounded growth graphs [Duchon et al. 05]

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Augmented graph models

• Bounded treewidth graphs [Fraigniaud 05]
• Bounded growth graphs [Duchon et al. 05]

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Augmented graph models

• Bounded treewidth graphs [Fraigniaud 05]
• Bounded growth graphs [Duchon et al. 05]
• Bounded doubling dimension metrics [Slivkins

05]

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Augmented graph models

• Graphs excluding a fixed minor

[Abraham&Gavoille 06]

• Bounded treewidth graphs [Fraigniaud 05]
• Bounded growth graphs [Duchon et al. 05]
• Bounded doubling dimension metrics [Slivkins

05]

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Augmented graph models

• Graphs excluding a fixed minor

[Abraham&Gavoille 06]

• Bounded treewidth graphs [Fraigniaud 05]
• Bounded growth graphs [Duchon et al. 05]
• Bounded doubling dimension metrics [Slivkins

05] But for some graphs, greedy paths are of length at least Ω(n1/√log n) for any augmentation. [Fraigniaud,L.,Lotker 06]

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Augmented graph models

• Density based augmentations: [Kleinberg 00,

Kleinberg 01, Duchon et al. 05, Slivkins 05]

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Augmented graph models

• Density based augmentations: [Kleinberg 00,

Kleinberg 01, Duchon et al. 05, Slivkins 05]

If H is of bounded growth

i.e. Bu(2R)≤c.Bu(R) for all node u and radius R

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Augmented graph models

• Density based augmentations: [Kleinberg 00,

Kleinberg 01, Duchon et al. 05, Slivkins 05]

If H is of bounded growth

i.e. Bu(2R)≤c.Bu(R) for all node u and radius R

and if φ is density based

i.e. φu(v) proportional to 1/Bu( dist(u,v) )

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Augmented graph models

• Density based augmentations: [Kleinberg 00,

Kleinberg 01, Duchon et al. 05, Slivkins 05]

If H is of bounded growth

i.e. Bu(2R)≤c.Bu(R) for all node u and radius R

and if φ is density based

i.e. φu(v) proportional to 1/Bu( dist(u,v) )

Then H is navigable

i.e. greedy routing computes polylog(n) paths knowing only H.

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Augmented graph models

• Density based augmentations

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Augmented graph models

• Density based augmentations

u v

Bu(d(u,v))

small probability

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Augmented graph models

• Density based augmentations

u v

Bu(d(u,v))

small probability u v

Bu(d(u,v))

big probability

21

Augmented graph models

• Density based augmentations

u v

Bu(d(u,v))

small probability u v

Bu(d(u,v))

big probability Liben-Nowell et al 05 : it fits observations on electronic social networks.

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• Navigability:

Dynamic properties

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Question

• Harmonic augmentation 1/x is crucial.

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Question

• Harmonic augmentation 1/x is crucial.
• But : natural spontaneous networks.

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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.

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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.

Threshold of frustration hyperlink

• u tries to route greedily towards v (random).

new hyperlink: bookmark

• u stops after t steps (random) and rewire its

link.

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Previous attempts

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Previous attempts

• Simulation results:

convergence to the harmonic distribution and short greedy routes.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 initial exponent, 0 rewired exponent, rewired =75 =150 =300 =500

25

Previous attempts

• No analytical result: complex

dependencies of the shortcuts.

• Simulation results:

convergence to the harmonic distribution and short greedy routes.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 initial exponent, 0 rewired exponent, rewired =75 =150 =300 =500

25

Previous attempts

[Clarke & Sandberg 2006]

• Same hypothesis: shortcuts comes

from successive searches.

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Previous attempts

[Clarke & Sandberg 2006]

• Same hypothesis: shortcuts comes

from successive searches.

• u computes a greedy path P to v (random).

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Previous attempts

[Clarke & Sandberg 2006]

• Same hypothesis: shortcuts comes

from successive searches.

• each node of P rewires its link to v with

probability p.

• u computes a greedy path P to v (random).

26

Previous attempts

[Clarke & Sandberg 2006]

• Same hypothesis: shortcuts comes

from successive searches.

• each node of P rewires its link to v with

probability p.

• u computes a greedy path P to v (random).
• Simulation results: short greedy routes.
• Analytical results: convergence to the

harmonic distribution.

26

Our approach

• A different hypothesis :
• 1. Links comes from people who met once.
• 2. Links tend to be forgotten with time.

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[Chaintreau, Fraigniaud, L. 08]

Our approach

• 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

[Chaintreau, Fraigniaud, L. 08]

Move and Forget process

Zk

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Move and Forget process

Zk Individuals: token moving

• n the grid.

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Move and Forget process

Zk Individuals: token moving

• n the grid.

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Move and Forget process

Zk Each token moves independently according to a random walk.

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Move and Forget process

Zk Each token moves independently according to a random walk.

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Move and Forget process

Zk Each token moves independently according to a random walk.

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Move and Forget process

Zk Each token moves independently according to a random walk.

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Move and Forget process

Zk Each token moves independently according to a random walk.

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Move and Forget process

Zk Each token drags the head of a shortcut rooted at its departure position.

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Move and Forget process

But after some time, we forget people.

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Move and Forget process

But after some time, we forget people. How sad.

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Move and Forget process

Forgetting mechanism:

• In each step, if our link is of age a,

we forget it with probability ∝ 1/a.

But after some time, we forget people.

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Move and Forget process

Forgetting mechanism:

• In each step, if our link is of age a,

we forget it with probability ∝ 1/a.

But after some time, we forget people. I.e. : there is less chance to forget your very

• ld friend than the one met at the bar

yesterday.

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Move and Forget process

Zk When a link is forgotten, it is rewired to its departure and a new token is launched.

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Move and Forget process

Zk When a link is forgotten, it is rewired to its departure and a new token is launched.

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Move and Forget process

Zk When a link is forgotten, it is rewired to its departure and a new token is launched.

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Move and Forget process

Zk When a link is forgotten, it is rewired to its departure and a new token is launched.

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More precisely

• At time t, the long range contact of

u is (x1(t),x2 (t),...,xk(t)).

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More precisely

• At time t, the long range contact of

u is (x1(t),x2 (t),...,xk(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 (x1(t),x2 (t),...,xk(t)).

• It is forgotten with probability ∝ 1/a

(a=the time since it has been launched).

• If it survives, for all i:

xi(t+1)= xi(t)+1 with probability 1/2 xi(t+1)= xi(t)-1 with probability 1/2

34

More precisely

• At time t, the long range contact of

u is (x1(t),x2 (t),...,xk(t)).

• It is forgotten with probability ∝ 1/a

(a=the time since it has been launched).

• If it survives, for all i:

xi(t+1)= xi(t)+1 with probability 1/2 xi(t+1)= xi(t)-1 with probability 1/2

• Otherwise, (x1(t+1),..., xk(t+1))=u.

34

M&F convergence

• If Φ(i) is the forget probability

distribution, the process converges iff:

Σa Πa

i=1(1-Φ(i)) is finite.

35

M&F convergence

• If Φ(i) is the forget probability

distribution, the process converges iff:

Σa Πa

i=1(1-Φ(i)) is finite.

Pr(to be of age a)

35

M&F convergence

• If Φ(i) is the forget probability

distribution, the process converges iff:

Σa Πa

i=1(1-Φ(i)) is finite.

Pr(to be of age a)

• f(d)= probability for a shortcut to be of

35

M&F convergence

• If Φ(i) is the forget probability

distribution, the process converges iff:

Σa Πa

i=1(1-Φ(i)) is finite.

Pr(to be of age a)

• f(d)= probability for a shortcut to be of

f(d)= Σa≥0 π(a) . Pr{|(x1(a), x2(a), ..., xk

35

M&F convergence

• If Φ(i) is the forget probability

distribution, the process converges iff:

Σa Πa

i=1(1-Φ(i)) is finite.

Pr(to be of age a)

• f(d)= probability for a shortcut to be of

f(d)= Σa≥0 π(a) . Pr{|(x1(a), x2(a), ..., xk Pr(to be of age a in stationary state)

35

M&F convergence

Theorem [CFL08]: If Φ is1-harmonic, then in stationary state:

c ||d||k ·ln1+ε||d|| ≤ f(d) ≤ c′lnk/2||d|| ||d||k ·ln1+ε||d||

36

M&F convergence

Theorem [CFL08]: If Φ is1-harmonic, then in stationary state:

c ||d||k ·ln1+ε||d|| ≤ f(d) ≤ c′lnk/2||d|| ||d||k ·ln1+ε||d||

The process converges to the k-harmonic distribution in Zk.

36

Intuition

Expected translation of a random walk after t steps=√t.

37

Intuition

⇒At age << d2, exponentially small probability to be of length d. Expected translation of a random walk after t steps=√t.

37

Intuition

⇒At age << d2, exponentially small probability to be of length d. Expected translation of a random walk after t steps=√t. For large age a, probability to be of length d ∝ (1/√a)k

37

Intuition

⇒At age << d2, exponentially small probability to be of length d. Expected translation of a random walk after t steps=√t. For large age a, probability to be of length d ∝ (1/√a)k ⇒f(d)= Σa≤d2 0.(1/a) + Σa≥d2 (1/√a)k.(1/a) π(a) π(a)

37

Intuition

⇒At age << d2, exponentially small probability to be of length d. Expected translation of a random walk after t steps=√t. For large age a, probability to be of length d ∝ (1/√a)k =∫d2 1/tk/2+1 dt = 1/dk. ⇒f(d)= Σa≤d2 0.(1/a) + Σa≥d2 (1/√a)k.(1/a) π(a) π(a)

37

Navigability

Theorem [CFL08]:

In Zk augmented by the shortcuts of M&F at stationary state, greedy routing computes paths of O(log2+εD) hops on expectation between any pair at distance D.

38

Navigability

Theorem [CFL08]:

In Zk augmented by the shortcuts of M&F at stationary state, greedy routing computes paths of O(log2+εD) hops on expectation between any pair at distance D.

u v

x/2 x/2

38

Navigability

Theorem [CFL08]:

In Zk augmented by the shortcuts of M&F at stationary state, greedy routing computes paths of O(log2+εD) hops on expectation between any pair at distance D.

u v

x/2 x/2

Ω((x/2)k) nodes

38

Navigability

Theorem [CFL08]:

In Zk augmented by the shortcuts of M&F at stationary state, greedy routing computes paths of O(log2+εD) hops on expectation between any pair at distance D.

u v

x/2 x/2

Ω((x/2)k) nodes

Pr ≥ 1/(xk log1+εx)

38

Navigability

Theorem [CFL08]:

In Zk augmented by the shortcuts of M&F at stationary state, greedy routing computes paths of O(log2+εD) hops on expectation between any pair at distance D.

u v

x/2 x/2

Ω((x/2)k) nodes

Pr ≥ 1/(xk log1+εx)

⇒Pr ≥ 1/(log1+εx)

to get twice closer.

38

Perspectives

39

Random models for large networks

• Random mechanisms:
• 1. To reflect irrational human behaviors.
• 2. To reflect unexpected failures.
• 3. To deal with untractability of billions

nodes networks.

40

Random models for large networks

• But:
• 1. Individuals are not uniformly random
• 2. Try to match random mechanism with

social behaviors.

41

Random models for large networks

• But:
• 1. Individuals are not uniformly random
• 2. Try to match random mechanism with

social behaviors.

41

• Another way of looking at it: game

theory.

Game theory & social networks

• Most of the results come from research

in economics.

42

Game theory & social networks

• Most of the results come from research

in economics.

42

• Ex: co-authorship models based on

amount of time players are willing to pay.

Game theory & social networks

• Most of the results come from research

in economics.

42

• Ex: co-authorship models based on

amount of time players are willing to pay.

• Most of the results in economics are

based on very small networks (because

• f experiments or model objective)

Conclusion

• Underlying social network dynamics

produce specific efficient structures.

43

• Unable to produce efficient networks

from scratch, locally.

• Question still widely open: no structural

characterization.