From survival to extinction of the contact process by the removal of a single edge

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by  Réka Szabó

FROM SURVIVAL TO EXTINCTION OF THE CONTACT PROCESS
BY THE REMOVAL OF A SINGLE EDGE
RE´KA SZABO´, DANIEL VALESIN
Abstract. We give a construction of a tree in which the contact process with any
∗
positiveinfectionratesurvivesbut,ifacertainprivilegededgee isremoved,oneobtains
two subtrees in which the contact process with infection rate smaller than 1/4 dies out.
6
1
0 1. Introduction
2
In this paper, we present an example of interest to the discussion of how the behaviour
n
a of interacting particle systems can be affected by local changes in the graph on which
J
they are defined.

5
The contact process on a locally finite and connected graph G = (V,E) with rate λ ≥ 0
2
is a continuous-time Markov process (ξ ) with state space {0,1}V and generator
t t≥0
]
R
P (1) Lf(ξ) = f(ξ0→x)−f(ξ)+λ· f(ξ1→y)−f(ξ) ,
.  
h
t x∈VX:ξ(x)=1 y∈V:X{x,y}∈E(cid:0) (cid:1)
a  
m where f is a local function, ξ ∈ {0,1}V and, for i ∈ {0,1} and z ∈ V,
[
i, if w = z;
ξi→z(w) = w ∈ V.
1
ξ(w), otherwise,
v (
4
This process is usually seen as a model of epidemics: vertices are individuals, which can
6
5 be healthy (state 0) or infected (state 1); infected individuals recover with rate 1 and
6
transmit the infection to each neighbor with rate λ. A comprehensive exposition of the
0
. contact process can be found in [6].
1
0 Given A ⊂ V, we denote by (ξtA)t≥0 the contact process with initial configuration
6 ξA = 1 , the indicator function of A; if A = {x}, we write ξx instead of ξ{x}. We abuse
1 0 A t t
: notation and associate a configuration ξ ∈ {0,1}V with the set {x : ξ(x) = 1}.
v
The contact process admits a well-known graphical construction, which we now briefly
i
X
describe. WeletPλ beaprobabilitymeasureunder whichafamilyofindependent Poisson
G
r
a point processes on [0,∞) are defined:
D for x ∈ V, each with rate 1,
x
D for x,y ∈ V with {x,y} ∈ E, each with rate λ;
(x,y)
we regard each D and D as a random discrete subset of [0,∞). Given a realization of
x (x,y)
all these processes, aninfection path is a function γ : [t ,t ] → V which is right continuous
1 2
with left limits and satisfies, for all t ∈ [t ,t ],
1 2
t ∈/ D and γ(t) 6= γ(t−) implies t ∈ D .
γ(t) (γ(t−),γ(t))
We say that two points (x,s),(y,t) ∈ V ×[0,∞) with s ≤ t are connected by an infection
path if there exists an infection path γ : [s,t] → V with γ(s) = x and γ(t) = y. This
event is denoted by {(x,s) ↔ (y,t)}. Then, given A ⊆ V, the process
ξA(x) = 1{∃y ∈ A : (y,0) ↔ (x,t)}
t
1
2 RE´KA SZABO´,DANIELVALESIN
1
has the distribution of the contact process with initial configuration .
A
To motivate our result, we will now state some facts which follow immediately either
from the generator expression (1) or the graphical construction. First, the “all healthy”
configuration, represented as the empty set ∅, is a trap state for the dynamics. Second,
(2) PλG ∃t : ξtA = ∅ ≥ PλG′′ ∃t : ξtB = ∅ if λ ≤ λ′, A ⊆ B and G ⊆ G′
(G ⊆ G′ mea(cid:0)ns that the(cid:1)vertex s(cid:0)et and edge(cid:1)set of G are respectively contained in the
vertex set and edge set of G′). Third (using the fact that G is connected),
(3) Pλ ∃t : B ⊆ ξA > 0 for all finite A,B ⊆ V.
G t
Combining (2) and (3), i(cid:0)t is seen that(cid:1)the probability
(4) Pλ ∃t ≥ 0 : ξA = ∅
G t
is either equal to 1 for any finite A ⊆(cid:0)V or strictly less(cid:1)than 1 for any finite A ⊆ V. The
process is said to die out (or go extinct) in the first case and to survive in the latter.
Whether one has survival or extinction may depend on both G and λ, so one defines
the critical rate
λ (G) = sup{λ : Pλ ∃t : ξA = ∅ = 1 ∀A ⊆ V, A finite}.
c G t
It follows from this definition and (2(cid:0)) that the pr(cid:1)ocess dies out when λ < λ and survives
c
when λ > λ .
c
It is natural to expect that the critical rate λ of the contact process is not affected
c
by local changes on G, such as the addition or removal of edges (as long as G remains
connected). More precisely,
Conjecture 1 (Pemantle and Stacey, [10]). Assume G = (V,E) and G′ = (V,E′) are
two connected graphs with the same vertex set V and E′ = E ∪{{x,y}}, with x,y ∈ V.
Then, λ (G) = λ (G′).
c c
Jung [4] proved this conjecture for vertex-transitive graphs (a graph G is vertex transi-
tive if, for any two vertices x and y, there exists an automorphism in G that maps x into
y). Proving the conjecture in full generality is still an open problem.
Rather than making progress on this problem, we consider a slightly different line
of inquiry. Let G = (V ,E ) and G = (V ,E ) be two graphs with disjoint vertex
1 1 1 2 2 2
sets V and V . Let x ∈ V , y ∈ V and define G = (V,E) with V = V ∪ V and
1 2 1 2 1 2
E = E ∪ E ∪ {{x,y}} (that is, we connect the two graphs using the edge {x,y}).
1 2
It follows from (2) that λ (G) ≤ min(λ (G ),λ (G )), and it is natural to ask whether
c c 1 c 2
or not the inequality can be strict. Strict inequality would mean that, for some λ <
min(λ (G ),λ (G )), the contact process with rate λ on G survives. This leads to a
c 1 c 2
curious situation: since the process is subcritical on both G and G , under Pλ there are
1 2 G
almost surely no infinite infection paths entirely contained either in G or in G , so any
1 2
infinite infection path in G needs to traverse the edge {x,y} infinitely many times.
We present an example of a graph in which this situation indeed occurs:
Theorem 1. There exists a tree G = (V,E) with a privileged edge e∗ so that
(5) λ (G) = 0
c
and, letting G , G be the two subgraphs of G obtained by removing e∗, we have
1 2
1
(6) λ (G ),λ (G ) ≥ .
c 1 c 2
4
REMOVAL OF AN EDGE IN THE CONTACT PROCESS 3
We end this Introduction discussing some related works in the interacting particle sys-
tems literature (apart from the already mentioned [4]). In [7], Madras, Schinazi and
Schonmann considered the contact process on Z in deterministic inhomogeneous envi-
ronments – for them, this means that the recovery rates (that is, the rates of transition
from state 1 to state 0) are vertex-dependent and deterministic, while the infection rate
is the same everywhere. Among other results, they showed that if the recovery rate is
equal to 1 everywhere except for a sufficiently sparse set S ⊂ Z, where it is equal to
some other value b ∈ (0,1), then the critical infection rate λ is the same as that of the
c
original process on Z. In [9], Newman and Volchan studied a contact process on Z in an
environment in which the recovery rates are chosen randomly, independently among the
vertices (the infection rate is again constant). They give a condition for the recovery rate
distribution under which the process survives for any value of the infection rate (similarly
to what happens to our graph G of Theorem 1). In [3], Handjani exhibited a modified
version of the voter model (which is another class of interacting particle system) in which
modifications of the flip mechanism in a single site can change the probability of survival
of the set of 1’s from zero to positive.
2. Notation and preliminary results
Given a set A, the indicator function of A is denoted 1 and the cardinality of A is
A
denoted |A|.
Given a graph G = (V,E), the degree of x ∈ V is denoted deg (x), the graph distance
G
between x,y ∈ V is dist (x,y) and the ball of radius R with center x is B (x,R). We
G G
omit G from the notation when it is clear from the context. We sometimes abuse notation
and associate G with its set of vertices (so that, for example, |G| denotes the number of
vertices of G). A star graph S with hub o on n vertices is a tree with one internal node
(o) and n−1 leaves.
We always assume that the contact process is constructed from the graphical construc-
tion. Given A,B ⊆ V, J ,J ⊆ [0,∞), we write A×J ↔ B×J if (x,t ) ↔ (y,t ) holds
1 2 1 2 1 2
for some x ∈ A, y ∈ B, t ∈ J and t ∈ J .
1 1 2 2
In the remaining part of this section we will describe five preliminary results that will
be needed in the proof of Theorem 1. We start with the following.
Lemma 2. For any λ ≤ 1, letting I = {1,...,n}, we have
4 n
1
(7) Pλ ξIn ⊆ I ∀t ≥ .
Z t n 2
(cid:0) (cid:1)
Proof. Define
L = inf{x : ξIn(x) = 1}, R = sup{x : ξIn(x) = 1}, t ≥ 0,
t t t t
with inf∅ = ∞ and sup∅ = −∞. It is readily seen that R is stochastically smaller than
t
the continuous-time Markov chain (X ) on Z with X = n which jumps one unit to the
t 0
right with rate λ and jumps one unit to the left with rate 1. Hence,
3
Pλ(R < n+1 ∀t) ≥ P(X < n+1 ∀t) ≥ ,
Z t t 4
byanelementarycomputationforbiasedrandomwalkonZ. WesimilarlyhavePλ(L > 0 ∀t) ≥
Z t
3, so
4
1
Pλ ξIn ⊆ I ∀t = Pλ(R < n+1 and L > 0 ∀t) ≥ .
Z t n Z t t 2
(cid:3)
(cid:0) (cid:1)
4 RE´KA SZABO´,DANIELVALESIN
Our remaining four preliminary results are taken from [8]. The following shows that the
contact process survives on a large star graph S for a time that is exponential in λ2|S|.
It is a refinement of the first result to this effect that appeared in [1] in Lemma 5.3.
Lemma 3. ([8], Lemma 3.1) There exists c > 0 such that, if λ < 1, S is a star with hub
o so that deg(o) > 64e2 · 1 and |ξ | > 1 ·λdeg(o), then
λ2 0 16e
(8) Pλ(ξ 6= ∅) ≥ 1−e−cλ2deg(o).
S ecλ2deg(o)
In [8], this lemma was applied to guarantee that in a connected graph G an infection
around a vertex with sufficiently high degree is maintained long enough to produce an
infection path that reaches another vertex at a certain distance:
Lemma 4. ([8], Lemma 3.2) There exists λ > 0 such that, if 0 < λ < λ , the following
0 0
holds. If G is a connected graph and x,y are distinct vertices of G with
7 1 1 |ξ ∩B(x,1)| 1
0
deg(x) > log ·dist (x,y) and > ,
cλ2 λ G λ|B(x,1)| 16e
(cid:18) (cid:19)
then
|ξ ∩B(y,1)| 1
(9) Pλ ∃t : t > > 1−2e−cλ2deg(x).
G λ|B(y,1)| 16e
(cid:18) (cid:19)
(Note that c is the same constant that appeared Lemma 3).
In the opposite direction as Lemma 3, the following result bounds from below the
probability that the infection disappears from a star graph within time 3log(1/λ):
Lemma 5. ([8], Lemma 5.2) If λ < 1 and S is a star, then
4
1
(10) Pλ ξS = ∅ ≥ e−16λ2|S|.
S 3log(1) 4
λ
(cid:16) (cid:17)
It is interesting to note that it follows from this result that, if |S| is large, the infection
will with high probability disappear before time exp{Cλ2|S|} for any C > 16. This
estimate on the time until the infection disappears thus matches (except for the value of
the constant in the exponential) the one that follows from Lemma 3.
It follows from Lemma 3 that vertices of degree much larger than 1 will sustain the
λ2
infection for a long time. The last preliminary lemma in our list deals with tree graphs in
which such big vertices are absent; in this case, it is unlikely that the infection spreads:
Lemma 6. ([8], Lemma 5.1) Let λ < 1 and T be a finite tree with maximum degree
2
bounded by 1 . Then, for any x,y ∈ T and t > 0,
8λ2
(11) Pλ(ξT 6= ∅) ≤ |T|2 ·e−t/4 and
T t
(12) Pλ({x}×[0,t] ↔ {y}×R ) ≤ (t+1)·(2λ)distT(x,y).
T +
3. Proof of Theorem 1
3.1. Construction of G. Our graph G will be equal to the one-dimensional lattice Z
with the modification that a few vertices, denoted o ,o ,..., are given extra neighbors, so
1 2
that their degrees become increasingly large. The extra neighbors are vertices which we
add to the graph as leaves.
We start defining sequences of integers (o ) , (d ) satisfying
i i≥1 i i≥1
0 > o > o > ··· , 0 < o < o < ··· , 0 < d < d < ··· .
1 3 2 4 1 2
REMOVAL OF AN EDGE IN THE CONTACT PROCESS 5
The definition will be inductive. We set d = 1, o = −1 and o = 2. Assume we have
1 1 2
already defined o ,...,o and d ,...,d . Then, set
1 i 1 i−1
o +i· (i−1)/2d if i is odd,
i−1 j=1 2j
(13) o = d = i·|o −o |.
i+1 i i i+1
(o −i·P(i−2)/2d if i is even,
i−1 j=0 2j+1
We clearly have
P
(14) d > i!
i
Now, for each i ≥ 1, let {xi,...,xi } be a set with d distinct elements (for distinct
1 di i
values of i, these sets are assumed to be disjoint). Then let
∞ ∞ di
G = (V,E), with V = Z∪ {xi,...,xi }, E = E(Z)∪ {{o ,xi}},
1 di i j
i=1 i=1j=1
[ [[
where E(Z) is the set of edges of Z. The construction is illustrated on Figure 1. We let e∗
be the edge {0,1}. When e∗ is removed, G is split into two subgraphs: we let G denote
−
the one associated to the negative half-line, and G the one associated to the positive
+
half-line.
e∗
o o o 0 1 o o
5 3 1 2 4
Figure 1. The graph G.
The strategy in the construction of G, and in particular in the choice of (d ) and (o ),
i i
is as follows.
• As we will prove in the next subsection, for any λ > 0, if i is sufficiently large, the
star B(o ,1) is large enough to sustain the infection long enough that it reaches
i
B(o ,1) with high probability. The infection is then sustained there long enough
i+1
to reach B(o ,1) and so on. As the probability of the intersection of all these
i+2
events is close to 1, there is survival. Note that, as observed in the Introduction,
indeed the infection necessarily relies on infinitely many traversals of e∗ in order
to survive.
• In Subsection 3.3, we will show that, if λ = 1 and e∗ is absent, then the star
4
B(o ,1) is not quite large enough to hold the infection long enough to overcome
i
the distance to o . Hence, it becomes increasingly difficult for the infection to
i+2
travel from one star to the next in the same half-line, and consequently there is
extinction.
3.2. Proof of λ (G) = 0. We need to show that, for any λ > 0, the contact process with
c
rate λ on G survives. By (2), it is sufficient to show this for λ ∈ (0,λ ), where λ is as in
0 0
Lemma 4.
Fix λ ∈ (0,λ ). As explained in the Introduction, it is enough to show that there exists
0
a finite set A ⊂ V such that (4) is strictly less than 1. Assume i ∈ N is large enough that
7 1 1
deg(o ) = d +2 > i·|o −o | = i·dist (o ,o ) > log ·dist (o ,o ).
i i i i+1 G i i+1 cλ2 λ G i i+1
(cid:18) (cid:19)
Then, by Lemma 4,
|A∩B(o ,1)| 1 |ξA ∩B(o ,1)| 1
if A ⊆ V, i > , then Pλ ∃t : t i+1 > > 1−2e−cλ2di.
λ|B(o ,1)| 16e G λ·|B(o ,1)| 16e
i (cid:18) i+1 (cid:19)
6 RE´KA SZABO´,DANIELVALESIN
The desired result now follows from(14), the Strong Markov Property anda union bound.
3.3. Proof of λ (G ),λ (G ) ≥ 1. We will only carry out the proof of λ (G ) ≥ 1; the
c − c + 4 c + 4
proof for G is similar. As explained in the Introduction, it is sufficient to show that, in
−
the contact process on G with rate
+
1
(15) λ =
4
and started with a single infection located at vertex 1, the infection almost surely disap-
pears:
(16) Pλ ξ1 6= ∅ ∀t = 0.
G+ t
Define the sets of vertices (cid:0) (cid:1)
S = {o ,xj,...,xj }, j ∈ {2,4,...}
j j 1 dj
H = {1},
0
H = (o ,o )∩Z, j ∈ {2,4,...},
j j j+2
i/2
G = H ∪ (S ∪H ) , i ∈ {2,4,...}.
i 0 2j 2j
 
j=1
[
 
We will abuse notation and refer to the above sets as subgraphs of G ; for instance, H
+ 2
willbethesubgraphwithvertexsetdefinedaboveandsetofedgeshavingbothextremities
in this vertex set.
We now fix an arbitrary i ∈ 2N. Define
i
(17) τ = exp (d +d +···+d ) .
2 4 i
2
(cid:26) (cid:27)
We have
Pλ ξ1 6= ∅ ∀t ≤ Pλ ξ1 6= ∅
G+ t G+ τ
≤ P(cid:0)λG+ ξτ1 6= ∅(cid:1), ξt1 ⊆ G(cid:0)i ∀t ≤ τ(cid:1)+PλG+ ξτ1 6= ∅, ξt1 * Gi for some t ≤ τ
(18) ≤ PλGi (cid:0)ξτGi 6= ∅ +PλG+({oi}×(cid:1)[0,τ] ↔(cid:0){oi+2}×[0,τ]). (cid:1)
We bound the(cid:0)two term(cid:1)s on (18) separately, starting with the second:
Pλ ({o }×[0,τ] ↔ {o }×[0,τ])
G+ i i+2
= Pλ ({o }×[0,τ] ↔ {o }×[0,τ])
Hi∪{oi,oi+2} i i+2
(12)
≤ (τ +1)·(2λ)dist(oi,oi+2)
(13),(15),(17) 1
(19) ≤ 2exp i(d +d +···+d ) −log(2) < exp{−d }
2 4 i i
2
(cid:26) (cid:18) (cid:19)(cid:27)
if i is large enough.
We now turn to the first term in (18). First define
1
(20) t = 3log , L = ilog(dist(o ,o )) = ilog(i(d +d +···+d )).
1 i i+2 2 4 i
λ
(cid:18) (cid:19)
We will assume that i is large enough (depending on λ) so that L > t . Using the Markov
1
property and (2), we have
(21) Pλ ξGi 6= ∅ ≤ Pλ ξGi 6= ∅ ⌊τ/L⌋.
Gi τ Gi L
(cid:0) (cid:1) (cid:0) (cid:1)
REMOVAL OF AN EDGE IN THE CONTACT PROCESS 7
We then bound
Pλ ξGi = ∅
Gi L
(22) (cid:0) i/2 (cid:1) i/2
≥ Pλ ξS2j ⊆ S ∀t, ξS2j = ∅ · Pλ ξH2j ⊆ H ∀t, ξH2j = ∅ .
Gi t 2j t1 Gi t 2j L
Yj=1 (cid:16) (cid:17) Yj=0 (cid:16) (cid:17)
Now, for all j ≤ i/2,
Pλ ξH2j ⊆ H ∀t, ξH2j = ∅ ≥ Pλ ξH2j ⊆ H ∀t −Pλ ξH2j 6= ∅
Gi t 2j L Gi t 2j H2j L
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(7),(11) 1
≥ −|H |2 ·e−L/4
2j
2
1 1
(23) (=20) −(dist(o ,o ))2−i/4 ≥
i i+2
2 4
if i is large enough. Again for all j ≤ i/2, we have
Pλ ξS2j ⊆ S ∀t, ξS2j = ∅
Gi t 2j t1
(cid:16) (cid:17)
≥ Pλ ξS2j = ∅ ·Pλ D ∪D ∩[0,t ] = ∅
S2j t1 Gi {o2j,o2j−1} {o2j,o2j+1} 1
(cid:16) (cid:17)
(10) 1 (cid:0)(cid:0) (cid:1) (cid:1)
≥ exp −16λ2|S | ·exp{−2λt }
2j 1
4
(20) λ6λ (cid:8) (cid:9)
(24) ≥ exp −17λ2d .
2j
4
Using (23) and (24) in (22), we(cid:8)get (cid:9)
λ6λ 2i+1
Pλ ξGi = ∅ ≥ ·exp −17λ2(d +d +···+d )
Gi L 16 2 4 i
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:8) (cid:9)
(14)
≥ exp −18λ2(d +d +···+d )
2 4 i
if i is large enough; using this in (21(cid:8)), we get (cid:9)
exp i(d +d +···+d )
Pλ ξGi 6= ∅ ≤ exp −exp −18λ2(d +d +···+d ) · 2 2 4 i
Gi τ 2 4 i ilog(i(d +d +···+d ))
( (cid:8) 2 4 i (cid:9))
(cid:0) (cid:1) (cid:8) (cid:9)
(25) < exp{−d }
i
if i is large enough.
In conclusion, using (19) and (25) in (18), we see that Pλ (ξ1 6= ∅ ∀t) < 2exp{−d }
G+ t i
for all i, so (16) follows.
Acknowledgments
TheauthorswouldliketothankOriGurel-Gurevich, F´abioMachadoandPabloRodriguez
for helpful discussions and Aernout van Enter for suggestions on the writing of this work.
References
[1] N.Berger,C.Borgs,J.T.ChayesandA.SaberiOn thespread of viruses on theinternet,Proceedings
of the 16th Symposium on Discrete Algorithms (2005), 301–310MR-2298278
[2] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1 Wiley, 2. edition (1971)
[3] S. J. Handjani Inhomogeneous voter models in one dimension, Journal of Theoretical Probability 16
(2003), no. 2, 325–338
[4] P.JungThe critical value of the contact process with added and removed edges,JournalofTheoretical
Probability 18 (2005), no. 4, 949–955
8 RE´KA SZABO´,DANIELVALESIN
[5] T. Liggett Interacting Particle Systems, Grundelheren der matematischen Wissenschaften 276,
Springer (1985), MR-0776231
[6] T.LiggettStochastic InteractingSystems: Contact, Voter and Exclusion Processes,Grundelherender
matematischen Wissenschaften 324, Springer (1999), MR-1717346
[7] N. Madras, R. Schinazi and R. H. Schonmann On the critical behavior of the contact process in
deterministic inhomogeneous environments, The Annals of Probability 22 (1994), no. 3, 1140–1159
[8] T.Mountford,D.ValesinandQ.YaoMetastabledensitiesforthecontactprocessonpowerlawrandom
graphs, Electronic Journal of Probability 18 (2013), no. 103, 1–36
[9] C. Newman andS. VolchanPersistent survival of one-dimensional contact processes in random envi-
ronments, The Annals of Probability 24 (1996), no. 1, 411–421
[10] R.PemantleandA.M.StaceyThe branching random walk and contact process on non-homogeneous
and Galton–Watson trees, The Annals of Probability 29 (2001), no. 4, 1563–1590
University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands
E-mail address: [email protected], [email protected]