Connectivity of sparsified random geometric graphs
Nicolas Broutin, Inria
joint work with
Luc Devroye, McGill
Gábor Lugosi, ICREA and Pompeu Fabra
Random geometric graphs
G(n, r)
N = Poisson(n) uniform points in D ⊆ Rd
i ∼ j iif kXi − Xj k < r
Random geometric graphs
G(n, r)
N = Poisson(n) uniform points in D ⊆ Rd
i ∼ j iif kXi − Xj k < r
Random geometric graphs
G(n, r)
N = Poisson(n) uniform points in D ⊆ Rd
i ∼ j iif kXi − Xj k < r
Random geometric graphs
G(n, r)
N = Poisson(n) uniform points in D ⊆ Rd
i ∼ j iif kXi − Xj k < r
Random geometric graphs
G(n, r)
N = Poisson(n) uniform points in D ⊆ Rd
i ∼ j iif kXi − Xj k < r
Random geometric graphs
G(n, r)
N = Poisson(n) uniform points in D ⊆ Rd
i ∼ j iif kXi − Xj k < r
Connectivity of random geometric graphs
Theorem.
ϑd the volume of the unit ball in Rd
rn? such that
ϑd n(rn? )d = log n
For any ∈ (0, 1)
lim P (G(n, rn ) is connected) =
n→∞
0
1
if rn ≤ (1 − )rn?
if rn ≥ (1 + )rn?
Random irrigation graphs
G(n, r) gets connected when average degree is Θ(log n)
idea:
sparsify the graph
in a distributed way
while ensuring connected
irrigation graphs
Sn (r, c)
every point ”sees” his neighbours in G(n, r)
every point keeps c neighbours chosen at random
Random irrigation graphs
G(n, r) gets connected when average degree is Θ(log n)
idea:
sparsify the graph
in a distributed way
while ensuring connected
irrigation graphs
Sn (r, c)
every point ”sees” his neighbours in G(n, r)
every point keeps c neighbours chosen at random
Random irrigation graphs
G(n, r) gets connected when average degree is Θ(log n)
idea:
sparsify the graph
in a distributed way
while ensuring connected
irrigation graphs
Sn (r, c)
every point ”sees” his neighbours in G(n, r)
every point keeps c neighbours chosen at random
Random irrigation graphs
Random irrigation graphs
History and results
Previous results:
Dubhashi, Johansson, Häggström, Panconesi, and Sozio
r = Θ(1)
⇒
Sn (r, 2) is connected whp
Crescenzi, Nocentini, Pietracaprina, Pucci
p
d = 2 r > log n/n
⇒ Sn (r, c) is connected whp
c > γ2 log(1/r)
History and results
Previous results:
Dubhashi, Johansson, Häggström, Panconesi, and Sozio
r = Θ(1)
⇒
Sn (r, 2) is connected whp
but G(n, r) is then an expander !
Crescenzi, Nocentini, Pietracaprina, Pucci
p
d = 2 r > log n/n
⇒ Sn (r, c) is connected whp
c > γ2 log(1/r)
but for r < n− , c = Θ(log n)!
Connectivity for small visibility radius
D = [0, 1]d
Theorem. (B-Devroye-Lugosi-Fraiman)
Suppose γ large enough
Set
r∼γ
log n
n
s
1/d
and
c?n
=
2 log n
log log n
Then, for any ∈ (0, 1)
lim P (Sn (rn , cn )is connected) =
n→∞
0
1
if cn ≤ (1 − )c?n
if cn ≥ (1 + )c?n
Connectivity for large visibility radius
Take
rn ∼ γn−(1−δ)/d
with
δ ∈ (0, 1)
Theorem. There exists a constant c = c(δ) such that
lim P (Sn (rn , c) is connected) = 1
n→∞
Theorem. Define:
n
o
c? (δ) := sup c : lim P (Sn (rn , c) is connected) = 0
n→∞
n
o
c? (δ) := inf c : lim P (Sn (rn , c) is connected) = 1
n→∞
Then, for any ∈ (0, 1) and δ small enough,
(1 − )δ −1/2 ≤ c? (δ) ≤ (1 + )δ −1/2
Giant component
Refined model: (d = 2)
(ξi )i≥1 i.i.d. copies of a random variable ξ ≥ 1
Every node i chooses ξi neighbours
C1 (·) the size of the largest connected component
Theorem. Suppose E [ξ] > 1. For every δp∈ (0, 1),
there exists a γ > 0 such that, for rn ≥ γ n−1 log n
lim P (C1 (Sn (rn , ξ)) ≥ (1 − δ)n) = 1
n→∞
−1/3
Theorem. Suppose rn = o (n log n)
lim P (C1 (Sn (rn , 1)) ≥ δn) = 0
n→∞
. Then ∀δ > 0
“Explosive” phase transition
Usually: phase transition is continuous
1
C1 /n
G(n, p) random graphs
E[degree]
Here: as discontinuous as it can be...
1
Intuition from the mean-field model
Assume rn = ∞
Theorem. (Fenner–Frieze ’82)
lim P (Sn (∞, 2) is connected) = 1
n→∞
Theorem. There exists a constant δ > 0 such that
lim P (C1 (Sn (∞, 1)) ≥ δn) > 0
n→∞
Intuition from the mean-field model
Assume rn = ∞
Theorem. (Fenner–Frieze ’82)
lim P (Sn (∞, 2) is connected) = 1
n→∞
Theorem. There exists a constant δ > 0 such that
lim P (C1 (Sn (∞, 1)) ≥ δn) > 0
n→∞
Idea:
Sn (∞, 1) is a random mapping
Sn (∞, 2) is the union of 2 random mappings
Discretization I
Discretization I
Cells
krn
k large but fixed
1
Discretization I
Cells
krn
k large but fixed
1
Discretization I
krn
Cells
k large but fixed
1
Associate : events to nodes/cells
events to bonds/faces
o
percolation
Uniformity of the point set
Definition. A cell is δ-good if for all boxes B
nrn2
nrn2
(1 − δ) 2 ≤ |B ∩ X| ≤ (1 + δ) 2
4d
4d
p
Lemma. Suppose rn ≥ γ log n/n. For any δ > 0,
there exists γ0 > 0 such that if γ > γ0
lim P (every cell is δ-good) = 1
n→∞
Lemma. ∃α, β > 0 such that
lim P ∀x :
n→∞
αnrn2
≤ |B(x, rn ) ∩ X| ≤
βnrn2
=1
Node/Cell events
central box
Second level partition:
boxes of side length
of side length r/(2d)
d large but fixed
seed boxes
Node/Cell events
central box
Second level partition:
boxes of side length
of side length r/(2d)
d large but fixed
seed boxes
Node events II
Exploration: in a given cell
Start from a point x ∈ X in the central box
Explore the neighborhoods for k 2 generations
Kill the paths that leave the cell
Define:
∆x (i) the collection of points at generation i
2
Gx = {∀ box B, |∆x (k ) ∩ B| ≥ E [ξ]
k2 /2
}
Lemma. ∀η > 0, can choose all the constants such that
for all n large
P( Gx | X, δ-good) ≥ 1 − η
Link events
node event occurs
Link events II
Q Q0
kd/2 boxes
Jx (Q, Q0 ) the sequence of first neighbors of x reaches
the central box in a ladder fashion
JR = ∩x∈R Jx , with R the population of the seed box in Q
Lemma.
0
P( JR (Q, Q ) | X, R) ≥ 1 − exp
|R|
(10βd2 )kd
Link events II
Q
Q Q0
Q0
k2 /2
Recall |R| ≥ E [ξ]
if node event occurs in Q
kd/2 boxes
Jx (Q, Q0 ) the sequence of first neighbors of x reaches
the central box in a ladder fashion
JR = ∩x∈R Jx , with R the population of the seed box in Q
Lemma.
0
P( JR (Q, Q ) | X, R) ≥ 1 − exp
|R|
(10βd2 )kd
Coupling with a (site/bond) percolation process
Coupling with a (site/bond) percolation process
1
Coupling with a (site/bond) percolation process
1
2
Coupling with a (site/bond) percolation process
1
2
3
Coupling with a (site/bond) percolation process
1
2
3
4
Coupling with a (site/bond) percolation process
1
2
3
4
Coupling with a (site/bond) percolation process
1
2
3
4
Coupling with a (site/bond) percolation process
percolation process = random configuration, where
o
each edge open with probability pe
independently !
each node open with probability pn
Definition of the coupling:
1. Exploration of clusters =⇒ partial percolation process
on torus [m] × [m]
2. For unassigned values: complete with i.i.d. Bernoulli
Observation. If H is a cluster of cells in the percolation
process, then, there is a connected component of Sn (rn , ξ)
which “sees” every single point of S
Cluster built is ubiquitous but too sparse!
Lemma. For any > 0
2
lim P ∃H : |H| ≥ (1 − )m = 1
n→∞
Problems:
1) in Sn (rn , ξ) only constant number of points per cell
so only O(m2 ) = O(n/ log n) nodes in total !
2) close points do not connect locally !
`
• after ` generations about E [ξ] points
• need ` = log log n to have a proportion in a cell
√
• but these Ω(log n) are spread at distance Ω( log log n)
Finale: gathering the remaining points
Only ”used up” a vanishing proportion of nodes per cell
⇒ can play the same game again !
Finale: gathering the remaining points
Only ”used up” a vanishing proportion of nodes per cell
⇒ can play the same game again !
Consider x ∈ X: make its box central in the discretization
x
x
Cells may not be aligned, but boxes are aligned
Hooking up left over points
Observations.
1) P (x in second giant) ≥ 1 − 2) the two giants overlap on (1 − 2) proportion of the boxes
⇒ Ω(n/ log n) such boxes
The too giants are bound to hook up !
k2 /2
in each of the overlapping boxes: about E [ξ]
whose neighbors have not yet been chosen !
points
P (one node fails to hook) ≤ (1 − O(1/ log n))
P (all node fail to hook) = (1 − Ω(1/ log n))n/ log n = e−Ω(n)