Aldous' Gossip process

32 Pages·2012·0.29 MB·English

Asymptotic Behavior of Aldous’ Gossip Process
Shirshendu Chatterjee
Joint work with Rick Durrett
Probability and Mathematical Physics Seminar, NYU
March 24, 2012
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Questions:
How does ξ grow?
t
When ξ = Λ(N)?
t
First passage percolation on a torus
Space is Λ(N) = (Z mod N)2.
Suppose one agent is present at each vertex of a Λ(N).
At time 0 the center receives an information.
Each neighbor of the center gets the information independently at
rate 1/4.
In general, whenever a vertex is informed, each of its uninformed
neighbor gets the information independently at rate 1/4.
ξ is the set of vertices informed by time t. ξ = {(0,0)}.
t 0
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First passage percolation on a torus
Space is Λ(N) = (Z mod N)2.
Suppose one agent is present at each vertex of a Λ(N).
At time 0 the center receives an information.
Each neighbor of the center gets the information independently at
rate 1/4.
In general, whenever a vertex is informed, each of its uninformed
neighbor gets the information independently at rate 1/4.
ξ is the set of vertices informed by time t. ξ = {(0,0)}.
t 0
Questions:
How does ξ grow?
t
When ξ = Λ(N)?
t
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Results for nearest neighbor case
(HW(’65) and Kesten(’86)) In any direction the information
percolates with a linear speed.
(Cox and Durrett(’81)) Diameter of ξ grows linearly and it has an
t
asymptotic shape..
Fluctuation results for the minimum passage time (on lattices):
(cid:73) lower bound due to Pemantle and Peres(’94), Newman and Piza(’95)
and Zhang(’08).
(cid:73) upper bound due to of Benjamini, Kalai and Schramm(’03).
(cid:73) conjectured behavior differs from both bounds.
If T is the time when every agent on the torus is informed, then T /N
N N
converges to a number.
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Question: How does ξ grow?
t
How does T scale?
N
Short-long FPP on Λ(N) (Aldous ’07)
State of the process is ξ ⊂ Λ(N), the set of informed vertices at time
t
t. ξ = {(0,0)}.
0
Information spreads from vertex i to j at rate ν , where
ij
(cid:40)
1/4 if j is a (nearest) neighbor of i
ν =
ij λ /(N2−5) if not.
N
If a vertex gets the information from a non-neighbor, we call it a
‘center’.
So each informed vertex tries to give birth to new centers at rate λ .
N
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Short-long FPP on Λ(N) (Aldous ’07)
State of the process is ξ ⊂ Λ(N), the set of informed vertices at time
t
t. ξ = {(0,0)}.
0
Information spreads from vertex i to j at rate ν , where
ij
(cid:40)
1/4 if j is a (nearest) neighbor of i
ν =
ij λ /(N2−5) if not.
N
If a vertex gets the information from a non-neighbor, we call it a
‘center’.
So each informed vertex tries to give birth to new centers at rate λ .
N
Question: How does ξ grow?
t
How does T scale?
N
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The state of our process at time t is C ⊂ Γ(N), the subset informed
t
by time t.
C starts with one center chosen uniformly from Γ(N) at time 0.
t
Each center corresponds to a disk, whose radius grows as
√
r(s) = s/ 2π.
At time t, birth rate of new centers is λ |C | = λ C .
N t N t
The location of each new center is chosen uniformly from the torus.
If the new center lands at x ∈ C , it has no effect. But we caount all
t
centers in X˜ .
t
Our model (‘balloon process C ’)
t
To simplify:
we remove randomness from nearest neighbor growth
we formulate on the (real) torus Γ(N) = (R mod N)2.
S.Chatterjee () Aldous’GossipProcess March24,2012 5/26
Our model (‘balloon process C ’)
t
To simplify:
we remove randomness from nearest neighbor growth
we formulate on the (real) torus Γ(N) = (R mod N)2.
The state of our process at time t is C ⊂ Γ(N), the subset informed
t
by time t.
C starts with one center chosen uniformly from Γ(N) at time 0.
t
Each center corresponds to a disk, whose radius grows as
√
r(s) = s/ 2π.
At time t, birth rate of new centers is λ |C | = λ C .
N t N t
The location of each new center is chosen uniformly from the torus.
If the new center lands at x ∈ C , it has no effect. But we caount all
t
centers in X˜ .
t
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Case 2: α = 3.
(cid:73) with probabilities bounded away from 0, (i) no long range jump and
√
T ≈ πN, and (ii) there is one that lands close enough to
N √
(N/2,N/2) to make T ≤(1−δ) πN.
N √
(cid:73) T /N converges weekly to a random variable with support [0, π] and
N √
an atom at π.
Case 3: α < 3.
(cid:73) Many long range jumps.
(cid:73) The cover time is significantly accelerated.
Phase transition
Consider λ = N−α.
N
Case 1: α > 3.
(cid:73) If the diameter of C grows linearly, then (cid:82)c0NC dt =O(N3).
t 0 t
(cid:73) So with high probability there is no long jump before the initial disk
covers the entire torus, and
(cid:73) the cover time satisfies
T √
N →P π.
N
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Case 3: α < 3.
(cid:73) Many long range jumps.
(cid:73) The cover time is significantly accelerated.
Phase transition
Consider λ = N−α.
N
Case 1: α > 3.
(cid:73) If the diameter of C grows linearly, then (cid:82)c0NC dt =O(N3).
t 0 t
(cid:73) So with high probability there is no long jump before the initial disk
covers the entire torus, and
(cid:73) the cover time satisfies
T √
N →P π.
N
Case 2: α = 3.
(cid:73) with probabilities bounded away from 0, (i) no long range jump and
√
T ≈ πN, and (ii) there is one that lands close enough to
N √
(N/2,N/2) to make T ≤(1−δ) πN.
N √
(cid:73) T /N converges weekly to a random variable with support [0, π] and
N √
an atom at π.
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