Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Metastable Densities for the Contact
Process on Power Law Random
Graphs
Qiang YAO
(Joint with Thomas MOUNTFORD and Daniel VALESIN)
School of Finance and Statistics,
East China Normal University
November 16th, 2013
Qiang YAO
Contents
1
Introduction
NSW random graphs
Contact processes
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Contents
2
Problem and Main Results
Assumptions and notation
Problem
Previous works
Our main results
3
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
4
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Introduction
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Newman-Strogatz-Watts random graphs
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
The graph: G n := (V n , E n ).
Vertex set: V n := {1, 2, · · · , n}.
Degree of vertex i is denoted by di (i = 1, 2, · · · , n),
which follows
(1) d1 , d2 , · · · , dn i.i.d.
(2) pk (:= P(d1 = k)) ∼ Ck −α (α > 1, C > 0) if k is
large. i.e.
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
α
lim k · pk = C ∈ (0, ∞).
k→∞
Newman-Strogatz-Watts random graphs
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
How can we construct the graph once given
a suitable realization of the degree sequence
(d1 (ω), d2 (ω), · · · , dn (ω))?
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Newman-Strogatz-Watts random graphs
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
M. E. J. Newman, S. H. Strogatz and D. J. Watts
(2001):
each vertex i was issued with di half edges and these
half edges were matched up in a uniformly chosen
manner.
If α > 3, then P(no loops or multiple edges) → 1 as
n → ∞. If 2 < α ≤ 3, not so.
We treat both cases in our work.
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Basic definitions of contact processes
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
First introduced by T. E. Harris (1974).
A model to describe the spread of diseases.
Two classical books:
T. M. Liggett (1985). Interacting Particle Systems.
T. M. Liggett (1999). Stochastic Interacting
Systems: Contact, Voter and Exclusion Processes.
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Basic definitions of contact processes
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
The process (ξt : t ≥ 0): a continuous-time Markov
process.
State space: {A : A ⊆ V n }.
At each t, each vertex is either healthy or infected.
ξt is the collection of infected vertices at time t.
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Transition rates:
Notation of measures
Two useful estimations
Sketch of proof
ξt → ξt \ {x} for x ∈ ξt at rate 1,
References
ξt → ξt ∪ {x} for x ∈
/ ξt at rate λ · |{y ∈ ξt : x ∼ y }|.
(ξtA : t ≥ 0): the process with initial state A.
Absorbing state: ∅.
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Self-duality
Qiang YAO
Introduction
For any A, B ∈ V n and t > 0, we have
P(ξtA
∩ B 6= ∅) =
P(ξtB
∩ A 6= ∅).
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Especially, for any x ∈ V n and t > 0, we have
n
P(ξtx 6= ∅) = P(x ∈ ξtV ).
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Previous
Results
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Assumptions and notation
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
Consider the contact process on the
Newman-Strogatz-Watts random graph.
Assumptions:
(1) p0 = p1 = p2 = 0;
(2) Conditioned on En := {d1 + · · · + dn is even}.
1
lim P(En ) =
n→∞
2
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Assumptions and notation
Given δ ∈ (0, 1), λ > 0, and x ∈ V n randomly
chosen. Define
x
ρn (λ, δ) := P ξexp(n
1−δ ) 6= ∅ .
ρn (λ, δ) = P x ∈
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
By self-duality,
Qiang YAO
Vn
ξexp(n
1−δ )

 n
V
ξexp(n
1−δ ) .
= E
n
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Define
ρ(λ, δ) := lim inf ρn (λ, δ),
n→∞
ρ(λ, δ) := lim sup ρn (λ, δ).
n→∞
Problem
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Problem: What is the asymptotic behavior
of ρ(λ, δ) and ρ(λ, δ) if λ > 0 is small?
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Previous works
Qiang YAO
N. Berger, C. Borges, J. T. Chayes and A. Saberi
(2005):
If α > 3, then ∀δ ∈ (0, 1),
C
bλ ≤ ρ(λ, δ) ≤ ρ(λ, δ) ≤ Bλ
c
when λ is small enough.
S. Chatterjee and R. Durrett (2009):
If α > 3, then ∀δ ∈ (0, 1), ε > 0,
cλ1+2(α−2)+ε ≤ ρ(λ, δ) ≤ ρ(λ, δ) ≤ C λ1+(α−2)+ε
when λ is small enough.
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
What are we interested in?
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Improve the bounds (find the actual behavior of
ρ(λ, δ) and ρ(λ, δ))
Extend the result to α > 2.
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Reference:
S. Chatterjee and R. Durrett: Contact processes on random graphs
with power law degree distributions have critical value 0, Ann.
Probab. 37 2332-2356 (2009).
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Our main results
If α > 3, there exist m1 , M1 > 0 so that, for any
δ ∈ (0, 1) and small enough λ > 0,
m1
λ1+2(α−2)
log2(α−2)
1
λ
≤ ρ(λ, δ) ≤ ρ(λ, δ) ≤ M1
Qiang YAO
Introduction
λ1+2(α−2)
log2(α−2)
λ1+2(α−2)
logα−2
logα−2
≤ ρ(λ, δ) ≤ ρ(λ, δ) ≤ M2
1
λ
If 2 < α ≤ 2 12 , there exist m3 , M3 > 0 so that, for
any δ ∈ (0, 1) and small enough λ > 0,
α−2
α−2
m3 λ1+ 3−α ≤ ρ(λ, δ) ≤ ρ(λ, δ) ≤ M3 λ1+ 3−α .
Results
Idea of Proof
λ1+2(α−2)
1
λ
.Problem and Main
Assumptions and notation
Problem
Previous works
Our main results
If 2 12 < α ≤ 3, there exist m2 , M2 > 0 so that, for
any δ ∈ (0, 1) and small enough λ > 0,
m2
1
λ
NSW random graphs
Contact processes
.
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Idea of Proof
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Some basic definitions
Qiang YAO
Introduction
Denote µ :=
P
npn , then µ ∈ (0, ∞) since α > 2.
n
e by q
en =
Denote the law q
npn
.
µ
en+1 .
Denote the law q by qn = q
(q is called the size-biased law.)
Given a graph G containing vertex x, denote by
BG (x, K ) the set of vertices in G at distance less
than or equal to K from x, including x itself.
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Connection with Galton-Watson trees
Proposition: For each n ∈ N, let G n be a
Newman-Strogatz-Watts random graph with degree
law p with associated exponent α > 2. Assume that
x is uniformly chosen in {1, . . . , n}, independently of
the graph. Then for any K ∈ N fixed, as n → ∞, the
law of BG n (x, K ) converges to the law of BT (o, K ),
where T is a Galton-Watson tree such that
(i) the degree of the root o is chosen ∼ p.
e (in
(ii) the degrees of subsequent vertices are i.i.d. ∼ q
other words, the offspring distribution for subsequent
generations is equal to q).
References:
[1] M. E. J. Newman, S. H. Strogatz and D. J. Watts: Random
graphs with arbitrary degree distributions and their applications,
Phys. Rev. E. 64 paper 026118 (2001).
[2] R. Durrett: Random Graph Dynamics. Cambridge University
Press, 2007.
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Notation of measures
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
e(p,q) the law under which a random
Denote by P
rooted tree T is obtained by choosing the degrees of
the vertices independently: that of the root o
according to probability p and those of “subsequent”
vertices according to q.
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
eq for the law under which every vertex has
Use P
degree i.i.d. according to q.
Idea of Proof
e(p,q),λ denotes the joint law of the
Notation P
contact process with parameter λ on a tree
e(p,q) .
independently generated according to law P
References
eq,λ denotes the joint law of the contact
Notation P
process with parameter λ on a tree independently
eq .
generated according to law P
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
Two useful estimations
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
Proposition 1: For any ε, δ > 0, there exists λ1 > 0
NSW random graphs
Contact processes
such that, for any λ < λ1 and R > 1,
Problem and Main

 Results
∃y ∈ BT (o, R), t ∈ R+ :
Assumptions and notation
Problem
1
o,


deg(y
)
>
,
y
∈
ξ
works
t
e(p,q),λ 
 . Previous
λ2+ε
ρ(λ, δ) ≥ P
 and the infection path from  Our main results
Idea of Proof
o to y lies entirely in BT (o, R)
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
Proposition 2: For any δ > 0,
e(p,q),λ ( ξ o 6= ∅ ∀t ) .
ρ(λ, δ) ≤ P
t
References
The case α > 3
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Lower bound: When α > 3, sites of “supercritical”
degree, that is, degree larger than 1/λ to some
power strictly larger than 2, are typically very far
from each other and far from the root of the tree.
However, if the infection starts at a site
whose
2 1
1
degree is a large multiple of λ2 log λ , then the
infection is maintained for a long time and can
therefore reach distant sites. We argue that at this
distance, supercritical sites can be found.
Upper bound: Argue the opposite direction. That
is: sites ofdegree smaller than a small multiple of
1
log2 λ1 do not maintain the infection for a time
λ2
that is sufficient to reach “supercritical” sites.
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
The case 2 12 < α ≤ 3
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
The expectation associated to the size-biased
distribution q is infinite.
Lower bound: Similar to the lower bound of the
case α > 3.
Upper bound: In contrast to the case α > 3, there
is a non-negligible probability that “supercritical”
sites are close to the root. Then, if the infection is
sustained close to the root for some time of order
1/λε , where ε is not very small, these supercritical
sites will be reached. The proof of the upper bound
depends on controlling the probability of this event.
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
The case 2 < α ≤ 2 12
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Here the bulk of the density no longer comes from
sites which are neighbors of sites with degree of the
order (neglecting log terms) λ12 .
Idea: Compare with branching processes.
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
References
N. Berger, C. Borgs, J. T. Chayes, and A. Saberi: On the
spread of viruses on the internet. Proceedings of the 16th
Symposium on Discrete Algorithms, 301-310 (2005).
S. Chatterjee and R. Durrett: Contact processes on random
graphs with power law degree distributions have critical value
0, Ann. Probab. 37 2332-2356 (2009).
R. Durrett: Random Graph Dynamics. Cambridge University
Press, 2007.
T. E. Harris: Contact interactions on a lattice, Ann. Probab. 2
969-988 (1974).
T. M. Liggett: Interacting Particle Systems. New York:
Springer-Verlag, 1985.
T. M. Liggett: Stochastic Interacting Systems: Contact, Voter
and Exclusion Processes. Springer, Berlin, Heidelberg, 1999.
M. E. J. Newman, S. H. Strogatz and D. J. Watts: Random
graphs with arbitrary degree distributions and their
applications, Phys. Rev. E. 64 paper 026118 (2001).
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References
Metastable Densities for
the Contact Process on
Power Law Random
Graphs
Qiang YAO
Introduction
Thank you!
NSW random graphs
Contact processes
Problem and Main
Results
Assumptions and notation
Problem
Previous works
Our main results
Idea of Proof
E-mail: [email protected]
Connection with GW trees
Notation of measures
Two useful estimations
Sketch of proof
References