The Contact Process on Evolving Scale-free Networks Motivation and - - PowerPoint PPT Presentation

The Contact Process on Evolving Scale-free Networks

Motivation and results

Amitai Linker (Universidad de Chile) Joint work with Peter M¨

• rters (University of Bath)

Emmanuel Jacob (ENS Lyon)

The contact process

The contact process

Given a finite or countably infinite connected graph G = (V , E), the contact process is a c` adl` ag Markov process {Xt}t≥0 with state space {0, 1}V , where − → healthy 1 − → infected

1

The contact process

Given a finite or countably infinite connected graph G = (V , E), the contact process is a c` adl` ag Markov process {Xt}t≥0 with state space {0, 1}V , where − → healthy 1 − → infected

1

The contact process

Given a finite or countably infinite connected graph G = (V , E), the contact process is a c` adl` ag Markov process {Xt}t≥0 with state space {0, 1}V , where − → healthy 1 − → infected If a vertex y is infected, it recovers at rate 1, while if it is healthy, it gets infected at rate λ

• z∼y

Xt(z) where λ > 0 is called the infection rate.

1

The contact process

Given a finite or countably infinite connected graph G = (V , E), the contact process is a c` adl` ag Markov process {Xt}t≥0 with state space {0, 1}V , where − → healthy 1 − → infected If a vertex y is infected, it recovers at rate 1, while if it is healthy, it gets infected at rate λ

• z∼y

Xt(z) where λ > 0 is called the infection rate. The configuration {0}V is the only absorbing state, and the event of reaching is called extinction.

1

Phase transition

Calling Pt

1 the distribution of Xt given X0 = {1}V , it can easily be seen that Pt 1

converges weakly to the upper invariant measure ¯ ν satisfying δ0 ≤ µ ≤ ¯ ν for all invariant measure µ.

2

Phase transition

Calling Pt

1 the distribution of Xt given X0 = {1}V , it can easily be seen that Pt 1

converges weakly to the upper invariant measure ¯ ν satisfying δ0 ≤ µ ≤ ¯ ν for all invariant measure µ. One of the main features of the contact process on infinite graphs G is the existence of some 0 ≤ λ0(G) < ∞ such that

• λ < λ0 =

⇒ δ0 = ¯ ν

• λ > λ0 =

⇒ δ0 ⊥ ¯ ν

2

Phase transition

Calling Pt

1 the distribution of Xt given X0 = {1}V , it can easily be seen that Pt 1

converges weakly to the upper invariant measure ¯ ν satisfying δ0 ≤ µ ≤ ¯ ν for all invariant measure µ. One of the main features of the contact process on infinite graphs G is the existence of some 0 ≤ λ0(G) < ∞ such that

• λ < λ0 =

⇒ δ0 = ¯ ν

• λ > λ0 =

⇒ δ0 ⊥ ¯ ν Which is equivalent to λ > λ0 ⇐ ⇒ for all x ∈ V , Px(Text = ∞) > 0

2

Phase transition

Our starting point is the following theorem, by Pemantle: Theorem: Pemantle (1992) Let G be supercritical Galton-Watson tree with descendant distribution having heavy tails, then λ0(G) = 0.

3

Phase transition

Our starting point is the following theorem, by Pemantle: Theorem: Pemantle (1992) Let G be supercritical Galton-Watson tree with descendant distribution having heavy tails, then λ0(G) = 0.

3

Phase transition

The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough?

4

Phase transition

The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? Lemma (Berger et al. 2005) There exists C > 0 such that if G is a star graph with center x and degree k, then Px

• Text > eCkλ2

= 1 − o(1/k).

4

Phase transition

The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? Lemma (Berger et al. 2005) There exists C > 0 such that if G is a star graph with center x and degree k, then Px

• Text > eCkλ2

= 1 − o(1/k).

4

Phase transition

The process survives locally around powerful vertices, but how can we know if a vertex is “powerful” enough? Lemma (Berger et al. 2005) There exists C > 0 such that if G is a star graph with center x and degree k, then Px

• Text > eCkλ2

= 1 − o(1/k). In this case, the central vertex is “powerful” if kλ2 >> 1. In particular we notice that this definition depends on λ.

4

Finite networks

• For finite networks, {0, 1}V is finite, so the process gets extinct a.s. in

finite time.

• If G is finite but sufficiently large, it seems “infinite” for a single infected

vertex.

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• For finite networks, {0, 1}V is finite, so the process gets extinct a.s. in

finite time.

• If G is finite but sufficiently large, it seems “infinite” for a single infected

vertex.

• If {Gn}n∈N is a sequence of finite graphs converging locally to some

infinite graph G, then for n large we expect Xt to behave differently on Gn depending on whether λ < λ0(G) − ε

• r

λ0(G) + ε < λ

5

New phase transition

Even though Gn is finite and Pt

1 → δ0 weakly, if λ > λ0 the existence of ¯

ν for G is reflected into the existence of some ¯ νn ⊥ δ0 which acts as a virtual equilibrium for a long time.

6

New phase transition

Even though Gn is finite and Pt

1 → δ0 weakly, if λ > λ0 the existence of ¯

ν for G is reflected into the existence of some ¯ νn ⊥ δ0 which acts as a virtual equilibrium for a long time.

• 1. We have Fast extinction if E1(Text) → ∞ at most polynomially with |Vn|.
• 2. We have Slow extinction if E1(Text) → ∞ exponentially with |Vn|.

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New phase transition

Even though Gn is finite and Pt

1 → δ0 weakly, if λ > λ0 the existence of ¯

ν for G is reflected into the existence of some ¯ νn ⊥ δ0 which acts as a virtual equilibrium for a long time.

• 1. We have Fast extinction if E1(Text) → ∞ at most polynomially with |Vn|.
• 2. We have Slow extinction if E1(Text) → ∞ exponentially with |Vn|.

With slow extinction we can have metastability, where the density of infected individuals, ρ(Xt) stabilizes around a fixed value (called the metastable density) until an excursion takes the process to extinction.

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Our main motivation

Consider the configuration model G = (V , E) whose degree distribution follows a power law of parameter τ, that is, for any vertex x, P

• deg(x) = k
• ≍ k−τ

7

Our main motivation

Consider the configuration model G = (V , E) whose degree distribution follows a power law of parameter τ, that is, for any vertex x, P

• deg(x) = k
• ≍ k−τ

In 2002, Pastor-Satorras and Vespignani used mean-field approximations to predict:

• If τ < 3, there is Slow extinction for all λ > 0
• If τ > 3, there is Fast extinction for λ small.

7

Our main motivation

Consider the configuration model G = (V , E) whose degree distribution follows a power law of parameter τ, that is, for any vertex x, P

• deg(x) = k
• ≍ k−τ

In 2002, Pastor-Satorras and Vespignani used mean-field approximations to predict:

• If τ < 3, there is Slow extinction for all λ > 0
• If τ > 3, there is Fast extinction for λ small.

What is more, if 2 < τ < 3 their method predicts a metastable density of the form λ

1 3−τ for λ small.

7

The main motivation

The prediction was wrong! Chaterjee and Durret (2009) proved that there is slow extinction for all values of τ and λ. Later, Mountford et al. (2013) proved that in this case the metastable density has the form λe+o(1) where e =

• 1

3−τ

if 2 < τ < 2.5 2τ − 3 if 2.5 < τ

8

• This correction shows that the instant loss of “connection memory” of the

mean-field approximations can hurt the infection, destroying the effect studied by Pemantle.

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• This correction shows that the instant loss of “connection memory” of the

mean-field approximations can hurt the infection, destroying the effect studied by Pemantle.

• On the other hand, if τ is small, there is some survival mechanism for the

process which does not rely on local survival.

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• This correction shows that the instant loss of “connection memory” of the

mean-field approximations can hurt the infection, destroying the effect studied by Pemantle.

• On the other hand, if τ is small, there is some survival mechanism for the

process which does not rely on local survival.

• We want to understand mild versions of this effect by running the C.P. on

graphs that change over time, giving the “connection memory” some finite lifespan.

9

Our evolving scale-free network model

Our construction:

• The set of vertices is {1, 2, . . . , N}
• Each pair of vertices i and j is initially connected independently with

probability 1 N p i N , j N

• for some continuous, symmetric, decreasing function p : (0, 1]2 → R+ such

that 1 p(a, s)ds ≍ a−γ for some γ ∈ (0, 1).

10

Our construction:

• The set of vertices is {1, 2, . . . , N}
• Each pair of vertices i and j is initially connected independently with

probability 1 N p i N , j N

• for some continuous, symmetric, decreasing function p : (0, 1]2 → R+ such

that 1 p(a, s)ds ≍ a−γ for some γ ∈ (0, 1). For such function p, E

• deg(i)
• ≍

i

N

−γ, so τ = 1 + 1

γ . 10

Our construction:

• The set of vertices is {1, 2, . . . , N}
• Each pair of vertices i and j is initially connected independently with

probability 1 N p i N , j N

• For such function p, E
• deg(i)
• ≍

i

N

−γ, so τ = 1 + 1

γ . 10

Our construction:

• The set of vertices is {1, 2, . . . , N}
• Each pair of vertices i and j is initially connected independently with

probability 1 N p i N , j N

• For such function p, E
• deg(i)
• ≍

i

N

−γ, so τ = 1 + 1

γ .

• Every vertex i has an exponential clock with parameter

κi = κ i N −γη with η ≥ 0, such that when it strikes, all adjacent edges are removed and new edges {i, j} are formed independently with probability

1 N p( i N , j N ). 10

Our construction:

• The set of vertices is {1, 2, . . . , N}
• Each pair of vertices i and j is initially connected independently with

probability 1 N p i N , j N

• For such function p, E
• deg(i)
• ≍

i

N

−γ, so τ = 1 + 1

γ .

• Every vertex i has an exponential clock with parameter

κi = κ i N −γη This way, κi ≍ E

• deg(i)

η.

10

Theorem: M¨

• rters, Jacob, L. (In preparation)

For the model with factor kernel p(x, y) = (xy)−γ and 0 ≤ η < 1/2 there is

• fast extinction for small λ when τ > 4 − 2η.
• slow extinction for all λ when τ < 4 − 2η,

In the latter case the metastable density is given by λe(τ,η)+o(1) where o(1) tends to zero as λ → 0 and e(τ, η) has the form e(τ, η) =     

2τ−2−2η 4−2η−τ

if

5 2 + η < τ 1 3−τ

if 2 < τ <

5 2 + η

In the case 1/2 < η there is

• fast extinction for small λ when τ > 3.
• slow extinction for all λ when τ < 3,

In the latter case the metastable density is given by λ

1 3−τ +o(1).

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We can partition the parameter space of η and τ into two regions.

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Proof idea: Lower bounds

For any fixed λ we divide V into powerful vertices (stars) and weak vertices (connectors), with the aid of some parameter 0 < a(λ) ≤ 1

• 1, . . . , ⌊aN⌋
• stars

, ⌊aN⌋ + 1, . . . , N

• connectors
• .

13

Proof idea: Lower bounds

For any fixed λ we divide V into powerful vertices (stars) and weak vertices (connectors), with the aid of some parameter 0 < a(λ) ≤ 1

• 1, . . . , ⌊aN⌋
• stars

, ⌊aN⌋ + 1, . . . , N

• connectors
• .

Defining ρa

t = 1 aN

⌊aN⌋

n=1 Xt(n) the density of infected stars, our proof relies on

finding some t and c1, c2 > 0 independent of λ, a and N such that P(ρa

t > c1 | X0) ≥ 1 − e−c2N

for any initial condition X0 such that ρa

0 > c1 (and any initial configuration of

the graph). This inequality suffices to show slow extinction for that given λ.

13

• Our aim is to show λ0 = 0, which translates into finding a function a that

can satisfy the inequality for all λ small.

• If the inequality holds, E(ρ(Xt)) ≥ λa1−γ which gives a lower bound for

the metastable density.

14

• Our aim is to show λ0 = 0, which translates into finding a function a that

can satisfy the inequality for all λ small.

• If the inequality holds, E(ρ(Xt)) ≥ λa1−γ which gives a lower bound for

the metastable density.

• We recognize four different survival strategies, from which we can
• btain with a large probability

ρa

0 > c1 =

⇒ ρa

t > c1.

Each strategy imposes a condition for a(λ), giving a range of admissible

• values. We will always choose the largest one, since it gives the largest

bound for the density.

14

Quick Survival Strategies

• Quick direct spreading: Each infected star infects a sufficient amount of

neighbouring stars before recovery.

15

Quick Survival Strategies

• Quick direct spreading: Each infected star infects a sufficient amount of

neighbouring stars before recovery. In this case t = 1, and naming aqd the threshold parameter of this strategy, then:

• Each star has approximately p(aqd, aqd)aqd neighbours that are stars

15

Quick Survival Strategies

• Quick direct spreading: Each infected star infects a sufficient amount of

neighbouring stars before recovery. In this case t = 1, and naming aqd the threshold parameter of this strategy, then:

• Each star has approximately p(aqd, aqd)aqd neighbours that are stars
• The amount of stars infected by other stars before recovery will be then

approximately c1aqdN · λ · p(aqd, aqd)aqd

15

Quick Survival Strategies

• Quick direct spreading: Each infected star infects a sufficient amount of

neighbouring stars before recovery. In this case t = 1, and naming aqd the threshold parameter of this strategy, then:

• Each star has approximately p(aqd, aqd)aqd neighbours that are stars
• The amount of stars infected by other stars before recovery will be then

approximately c1aqdN · λ · p(aqd, aqd)aqd

In order for ρa

1 > c1, aqd must solve

c1aN ≤ c1aN · λp(a, a)a. This way, we have slow extinction for all λ if we can find a function aqd such that 1 ≤ λp(aqd, aqd)aqd

15

Quick Survival Strategies

• Quick indirect spreading: Each infected star infects a sufficient amount
• f neighbouring connectors before recovery. Those connectors in turn

infect a sufficient amount of neighbouring stars before recovery.

16

Quick Survival Strategies

• Quick indirect spreading: Each infected star infects a sufficient amount
• f neighbouring connectors before recovery. Those connectors in turn

infect a sufficient amount of neighbouring stars before recovery. If aqi is the threshold parameter of this strategy, then:

• Each star has approximately a−γ

qi

neighbours that are connectors

16

Quick Survival Strategies

• Quick indirect spreading: Each infected star infects a sufficient amount
• f neighbouring connectors before recovery. Those connectors in turn

infect a sufficient amount of neighbouring stars before recovery. If aqi is the threshold parameter of this strategy, then:

• Each star has approximately a−γ

qi

neighbours that are connectors

• Each connector has approximately p(aqi, 1)aqi neighbours that are stars

16

Quick Survival Strategies

• Quick indirect spreading: Each infected star infects a sufficient amount
• f neighbouring connectors before recovery. Those connectors in turn

infect a sufficient amount of neighbouring stars before recovery. If aqi is the threshold parameter of this strategy, then:

• Each star has approximately a−γ

qi

neighbours that are connectors

• Each connector has approximately p(aqi, 1)aqi neighbours that are stars
• The amount of stars infected indirectly by other stars will be then

approximately c1aqiN · λ · a−γ

qi

· λ · p(aqi, 1)aqi

16

Quick Survival Strategies

• Quick indirect spreading: Each infected star infects a sufficient amount
• f neighbouring connectors before recovery. Those connectors in turn

infect a sufficient amount of neighbouring stars before recovery. If aqi is the threshold parameter of this strategy, then:

• Each star has approximately a−γ

qi

neighbours that are connectors

• Each connector has approximately p(aqi, 1)aqi neighbours that are stars
• The amount of stars infected indirectly by other stars will be then

approximately c1aqiN · λ · a−γ

qi

· λ · p(aqi, 1)aqi

As before, we have slow extinction for all λ if we can find a function aqi such that 1 ≤ λ2p(aqi, 1)a1−γ

qi 16

Delayed survival strategies

The delayed survival strategies, delayed direct spreading and delayed indirect spreading are versions of the previous strategies where now stars infect other vertices in a much larger time window, T(a, λ), which represents the amount of time powerful vertices can be sustained through local survival.

17

Delayed survival strategies

The delayed survival strategies, delayed direct spreading and delayed indirect spreading are versions of the previous strategies where now stars infect other vertices in a much larger time window, T(a, λ), which represents the amount of time powerful vertices can be sustained through local survival. This way, for delayed direct spreading, the threshold parameter add must satisfy 1 ≤ λp(a, a)aT(a, λ) and for delayed indirect spreading, the threshold parameter adi must satisfy 1 ≤ λ2p(a, 1)a1−γT(a, λ)

17

Delayed survival strategies

In our vertex updating scheme, T(a, λ) ≈ λ2a−γ(1−2η)

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Delayed survival strategies

In our vertex updating scheme, T(a, λ) ≈ λ2a−γ(1−2η) T R

18

Delayed survival strategies

In our vertex updating scheme, T(a, λ) ≈ λ2a−γ(1−2η) T R U U κ−1

a

κ−1

a 18

Delayed survival strategies

In our vertex updating scheme, T(a, λ) ≈ λ2a−γ(1−2η) T κ−1

a

κ−1

a

a−γ

18

Delayed survival strategies

In our vertex updating scheme, T(a, λ) ≈ λ2a−γ(1−2η) T κ−1

a

κ−1

a

a−γ

λκ−1

a

λκ−1

a 18

Delayed survival strategies

In our vertex updating scheme, T(a, λ) ≈ λ2a−γ(1−2η) T κ−1

a

κ−1

a

a−γ

λκ−1

a

λκ−1

a

It is clear that when η > 1/2, λ2a−γ(1−2η) tends to zero so the delayed strategies are worse than the quick ones.

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We can now recognize that the slow extinction region is divided depending on which mechanism is the predominant one.

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Upper Bounds

In order to deduce extinction or upper bounds for the metastable density, we use a new process Yt ≥ Xt called the Wait-and-see process where now think of present edges as revealed and missing ones as unrevealed.

• Every infected vertex recovers at rate 1
• The infection passes from an infected vertex x to another vertex y at rate:
• λ if the edge between them is revealed
• λpx,y if the edge between them is unrevealed

after this scenario the edge is revealed

• At rate κx all the edges {x, z} turn unrevealed

20

Fast Extinction

Using a function S : (0, 1] → R+ such that:

• 1

0 S(x)dx < ∞

• For some 0 < c and 0 < δ < 1

max{T(λ, ·), 1} ≤ cS(·)δ

• There is 0 < D small such that

max{T(λ, ·), 1} 1 S(y)λp(·, y)dy ≤ DS(·)

21

Fast Extinction

Using a function S : (0, 1] → R+ such that:

• 1

0 S(x)dx < ∞

• For some 0 < c and 0 < δ < 1

max{T(λ, ·), 1} ≤ cS(·)δ

• There is 0 < D small such that

max{T(λ, ·), 1} 1 S(y)λp(·, y)dy ≤ DS(·)

We construct a scoring function as mt(x) =    s(x) + (2κ−1

x λNt(x) ∧ 1 2)2t(x)

if Yt(x) = 1 (2κ−1

x λNt(x) ∧ 1)(s(x) + t(x))

if Yt(x) = 0

21

Fast Extinction

Lemma The process Mt = mt(x) is a supermartingale, and so it is Zt = Mδ

t + δξt

for some fixed ξ > 0

22

Fast Extinction

Lemma The process Mt = mt(x) is a supermartingale, and so it is Zt = Mδ

t + δξt

for some fixed ξ > 0 Using this result, from the optional stopping theorem we have δξE1[Text] = E1[ZText ] ≤ E1[Mδ

0 ] ≤ Nδ

1 S(x)dx δ giving fast extinction.

22

Upper Bounds for ρ

When 1

0 S(x)dx = ∞ we can’t deduce fast extinction from our method, but

we can still deduce upper bounds from our methods. From the duality of the contact process we know that E1(ρ(Xt)) = 1 N

N

• x=1

Px(Text > t)

23

Upper Bounds for ρ

When 1

0 S(x)dx = ∞ we can’t deduce fast extinction from our method, but

we can still deduce upper bounds from our methods. From the duality of the contact process we know that E1(ρ(Xt)) = 1 N

N

• x=1

Px(Text > t) Defining T = Thit ∧ Text where Ta is the first time a vertex y ≤ aN gets infected, we obtain Lemma The processes MT

t and Z T t

are supermartingales

23

Upper Bounds for ρ

When 1

0 S(x)dx = ∞ we can’t deduce fast extinction from our method, but

we can still deduce upper bounds from our methods. From the duality of the contact process we know that E1(ρ(Xt)) = 1 N

N

• x=1

Px(Text > t) Defining T = Thit ∧ Text where Ta is the first time a vertex y ≤ aN gets infected, we obtain Lemma The processes MT

t and Z T t

are supermartingales and using a similar technique we obtain E1(ρ(Xt)) ≤ a + 1 S(a) 1

a

S(x)dx + 1 δξt 1

a

S(x)δdx

23

The preferential attachment kernel

Theorem: M¨

• rters, Jacob, L. (In preparation)

For the network with preferential attachment kernel p(x, y) = β(x ∧ y)−γ(x ∨ y)γ−1 and 0 ≤ η < 1/2 there is only slow extinction and the metastable density is given by λe(τ,η)+o(1) where in this case e(τ, η) has the form e(τ, η) =             

3τ−5−2η 1−2η

if

8 3 + 2 3 η < τ 3τ−4−2η 4−2η−τ

if 2 + 2η < τ < 8

3 + 2 3 η τ−1 3−τ

if 2 < τ < 2 + 2η In the case η ≥ 1/2 there is

• slow extinction when τ < 3,
• fast extinction when τ > 3.

In the latter case the metastable density is given by λ

τ−1 3−τ +o(1).

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25

• In the case of the P.A. kernel the quick direct strategy is not viable, which

reflects the fact that in this graph, stars are directly connected to other stars very rarely.

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• In the case of the P.A. kernel the quick direct strategy is not viable, which

reflects the fact that in this graph, stars are directly connected to other stars very rarely.

• Again, when η > 1/2, only quick strategies are viable, and the critical

value for τ reflects the one obtained in the mean-field model.

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Thank you very much!

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