Júlia Komjáthy
joint with Joost Jorritsma, Dieter Mitsche
Random Discrete Structures
March 2023
Papers: tiny.cc/cluster-size-ksrg
[Grimmett & Marstrand '90, Kesten & Zhang '90]
Hyperbolic random graph
Scale-free percolation
Long-range percolation
Age-dependent RCM
Random geom. graph
Nearest-neighbor percolation
Vertex set V∞
Edge set E∞
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
Vertex set V∞
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
P(u↔v∣V∞)
Vertex set V∞
Edge set E∞
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
Vertex set V∞
Edge set E∞
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
A parameterized kernel: σ≥0
κσ(wu,wv):=max{wu,wv}min{wu,wv}σ
SFP/GIRG
Hyperbolic RG
Age-dep. RCM
Scale-free Gilbert
Long-range percolation
Random geom. graph
Nearest-neighbor percolation
Theorem. When τ>2:
Theorem. When τ<2:
P(deg(0)≥k)∼k−(τdeg−1).
[Deijfen, v.d. Hofstad, Hooghiemstra '13]
[Gracar, Grauer, Lüchtrath, Mörters '18]
[Heydenreich, Hulshof, Jorritsma '17]
[Hirsch '17]
[v.d. Hofstad, v.d. Hoorn, Maitra, '22]
[Jorritsma, K., Mitsche, '23]
[Lüchtrath '22]
Theorem. When τ<2:
Theorem. When τ>2:
P(deg(0)≥k)∼k−(τdeg−1).
Questions
[Alexander & Chayes & Chayes '90, Grimmett (& Marstrand),
Kesten & Zhang '90,
...,
Lichev, Lodewijks, Mitsche, Schapira '22,
Penrose '05]
[Sly & Crawford'12]
[Kiwi & Mitsche '17]
When τ>2, there exists ζ∈(0,1) s.t.
When τ>2, there exists ζ∈(0,1) s.t.
Only four possible values
When τ>2, there exists ζ∈(0,1) s.t.
Only four possible values
d=1: ζnn>max{ζll,ζlh,ζhh}⟹subcritical
[Gracar, Mönch, Lüchtrath '22]
Consider supercritical interpolating model, σ≤τ−1.
If ζnn is not the unique maximum of Z:={ζll,ζlh,ζhh,ζnn}:
If ζnn is the unique maximum of Z, d>1:
d=2α=1.6
(2)
(4)
(1)
(3)
Aim: Find minimal ζ s.t.
Isolation events
Aim: Find minimal ζ s.t.
Isolation costs (vertices)
Aim: Find minimal ζ s.t.
Isolation costs (edges)
Possible optimizers
Edges, vertices
Aim: Find minimal ζ s.t.
Conclusion:
Aim: Find minimal ζ s.t.
nn
ll
lh
hh
Consider supercritical interpolating model, σ≤τ−1.
If ζnn is not the unique maximum of Z:={ζll,ζlh,ζhh,ζnn}:
Reveal graph in stages.
Three techniques: ζll,ζlh, and ζhh:
No control short edges/local geometry:
no upper bound (nn)
Prevent too large components
Consider supercritical interpolating model, σ≤τ−1.
If ζnn is not the unique maximum of Z:={ζll,ζlh,ζhh,ζnn}
If ζnn is the unique maximum of Z, d>1:
ζll=2−α,ζlh=ατ−1−(τ−2),ζhh=2−(τ−1)/α3−τ,ζnn=dd−1
Aim: Maximize ζ s.t.
When n≤exp(−21kζ)
When k=kn=(2log(n))1/ζ
Reveal graph in stages.
Three techniques: ζll,ζlh, and ζhh:
No control short edges/local geometry:
no upper bound (nn)
Aim: Find minimal ζ s.t.
Isolation costs (edges)
Possible optimizers