Cluster-size decay in kernel-based spatial random graphs
Joost Jorritsma
joint with Júlia Komjáthy, Dieter Mitsche
Recent trends in Spatial Stochastic Processes
October 2022
Hyperbolic random graph
Scale-free percolation
Long-range percolation
Age-dependent RCM
Random geom. graph
Nearest-neighbor percolation
Kernel-based spatial random graphs
Vertex set V∞
- Spatial locations xv∈Rd: PPP(1) or Zd,
- i.i.d. weights wv≥1: P(wv≥w)=w−(τ−1),
Edge set E∞
- Symmetric kernel κ(wu,wv),
- Long-range parameter α>1,
- Edge-density β>0,
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
Vertex set V∞
- Spatial locations xv∈Rd: PPP(1) or Zd,
- i.i.d. weights wv≥1: P(wv≥w)=w−(τ−1),
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∞)
Kernel-based spatial random graphs
Vertex set V∞
- Spatial locations xv∈Rd: PPP(1),
- i.i.d. weights wv≥1: Pareto(τ),
Edge set E∞
- Symmetric kernel κ(w1,w2),
- Edge-density β>0,
- Long-range parameter α>1,
Connection probability
P(u↔v∣V∞)=(β∥xu−xv∥dκ(wu,wv))α∧1
Vertex set V∞
- Spatial locations xv∈Rd: PPP(1), or Zd,
- i.i.d. weights wv≥1: P(wv≥w)=w−(τ−1),
Edge set E∞
- Symmetric kernel κ(w1,w2),
- Edge-density β>0,
- Long-range parameter α>1,
The interpolating kernel
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
- σ: assortativity
- σ: interpolation
τ−1σ
\frac{\sigma}{\tau-1}
τ−11
\frac1{\tau-1}
1
1
1
1
σ=τ−1
\sigma=\tau-1
τ=2
\tau=2
σ=1
\sigma=1
σ=1
\sigma=1
σ=τ−2
\sigma=\tau-2
σ=τ−2
\sigma=\tau-2
σ=0
\sigma=0
σ=0
\sigma=0
SFP/GIRG
Hyperbolic RG
Age-dep. RCM
Scale-free Gilbert
Long-range percolation
Random geom. graph
Nearest-neighbor percolation
Degree distribution
τ−1σ
\frac{\sigma}{\tau-1}
τ−11
\frac1{\tau-1}
1
1
1
1
σ=τ−1
\sigma=\tau-1
τ=2
\tau=2
σ=1
\sigma=1
σ=τ−2
\sigma=\tau-2
σ=0
\sigma=0
τdeg=τ
\tau_\mathrm{deg}=\tau
τdeg=σ−(τ−2)σ+1
\tau_\mathrm{deg}=\tfrac{\sigma+1}{\sigma-(\tau-2)}
deg(0)=∞
\mathrm{deg}(0)=\infty
Theorem. When τ>2:
Theorem. When τ<2:
- Infinite degrees
- Bounded diameter
P(deg(0)≥k)∼k−(τdeg−1).
[Deijfen, v.d. Hofstad, Hooghiemstra '13]
[Gracar, Grauer, Lüchtrath, Mörters '18]
[Heydenreich, Hulshof, J. '17]
[Hirsch '17]
[v.d. Hofstad, v.d. Hoorn, Maitra, '22]
[J., Komjáthy, Mitsche, '22+]
[Lüchtrath '22]
Theorem. When τ<2:
Theorem. When τ>2:
P(deg(0)≥k)∼k−(τdeg−1).
Small components
- C∞ : unique infinite component
(exists by assumption). - C(0): component containing 0
- Cn(2) : second-largest component in Gn
(graph in volume-n box)
Definition
- Cluster-size decay: P(k≤∣C(0)∣<∞)=
- Second-largest component: ∣Cn(2)∣=Θ(what)
Questions
Supercritical scaling exponent ζ
- Cluster-size decay: P(k≤∣C(0)∣<∞)=exp(−Θ(kζ));
- Second-largest component: ∣Cn(2)∣=Θ((log(n))1/ζ).
ζ=dd−1
\zeta=\frac{d-1}d
∣Cn(2)∣=O((log(n))??)
|\mathcal{C}_n^{(2)}|=\mathcal{O}\big((\log(n))^{??}\big)
∣Cn(2)∣=Θ((log(n))2/(3−τ))
|\mathcal{C}_n^{(2)}|=\Theta\big((\log(n))^{2/(3-\tau)}\big)
[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.
Supercritical scaling exponent ζ
- Cluster-size decay: P(k≤∣C(0)∣<∞)=exp(−Θ(kζ));
- Second-largest component: ∣Cn(2)∣=Θ((log(n))1/ζ).
When τ>2, there exists ζ∈(0,1) s.t.
τ−1σ
\frac{\sigma}{\tau-1}
τ−11
\frac1{\tau-1}
1
1
1
1
σ=τ−1
\sigma=\tau-1
τ=2
\tau=2
ζnn:=dd−1
\zeta_\mathrm{nn}:=\frac{d-1}{d}
ζhh:=2−(τ−1)/ασ+2−τ
\zeta_\mathrm{hh}:=\frac{\sigma+2-\tau}{2-(\tau-1)/\alpha}
ζlh:=ατ−1−(τ−2)
\zeta_\mathrm{lh}:=\frac{\tau-1}{\alpha}-(\tau-2)
ζll:=2−α
\zeta_\mathrm{ll}:=2-\alpha
Only four possible values
Phase diagram of ζ
- Cluster-size decay: P(k≤∣C(0)∣<∞)=exp(−Θ(kζ));
- Second-largest component: ∣Cn(2)∣=Θ((log(n))1/ζ).
When τ>2, there exists ζ∈(0,1) s.t.
ζnn=dd−1
\zeta_\mathrm{nn}=\frac{d-1}{d}
ζhh=2−(τ−1)/ασ+2−τ
\zeta_\mathrm{hh}=\frac{\sigma+2-\tau}{2-(\tau-1)/\alpha}
ζlh=ατ−1−(τ−2)
\zeta_\mathrm{lh}=\frac{\tau-1}{\alpha}-(\tau-2)
ζll=2−α
\zeta_\mathrm{ll}=2-\alpha
Only four possible values
ζ=max{ζll,ζlh,ζhh,ζnn}
\zeta=\max\{\zeta_\mathrm{ll}, \zeta_\mathrm{lh}, \zeta_\mathrm{hh}, \zeta_\mathrm{nn}\}
Conjecture "almost" proven [JKM'22+]
d=1: ζnn>max{ζll,ζlh,ζhh}⟹subcritical
If ζnn is the unique maximum of Z, d>1:
[Gracar, Mönch, Lüchtrath '22]
Consider supercritical interpolating model, σ≤τ−1.
- Cluster-size decay: P(k≤∣C(0)∣<∞)∼exp(−Θ(kmax(Z)));
- Second-largest component: ∣Cn(2)∣∼Θ((log(n))1/max(Z)).
If ζnn is not the unique maximum of Z:={ζll,ζlh,ζhh,ζnn}:
- Always lower bound;
- Upper bound: no weights, high edge-density β≫βc.
Gap: Monotonicity of ζ
d=2α=1.6
Proof strategy: 4 steps
P(k≤∣C(0)∣<∞)≤exp(−A′kζ)
\mathbb{P}\big(k\le |\mathcal{C}(0)|<\infty\big) \le \exp\big(-A'k^\zeta\big)
⟹
\Longrightarrow
∣Cn(2)∣≥A(log(n))1/ζ
|\mathcal{C}_n^{(2)}|\ge A(\log(n))^{1/\zeta}
⟹
\Longrightarrow
(2)
(4)
P(∣Cn(2)∣≥k)≤nexp(−Akζ)
\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge k\big)\le n\exp\big(-Ak^\zeta\big)
P(k≤∣C(0)∣<∞)≥exp(−A′kζ)
\mathbb{P}\big(k\le |\mathcal{C}(0)|<\infty\big) \ge \exp\big(-A'k^\zeta\big)
(1)
(3)
P(∣Cn(2)∣≥k)≤nexp(−Akζ)
\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge k\big)\le n\exp\big(-Ak^\zeta\big)
P(k≤∣C(0)∣<∞)≥exp(−A′kζ)
\mathbb{P}\big(k\le |\mathcal{C}(0)|<\infty\big) \ge \exp\big(-A'k^\zeta\big)
(1)
(3)
Lower bounds
Upper bounds
Lower bounds: an isolated component
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≥exp(−Akζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\ge\exp\big(-Ak^\zeta\big);
Isolation events
- C(0)≥k in Θ(k)-volume box,
-
Θ(k)-volume box disconnected:
- Empty boundary
- High-weight vertices are dangerous
- Especially close to boundary
- No edges
- Trade-off γ
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
Lower bounds: an isolated component
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≥exp(−Akζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\ge\exp\big(-Ak^\zeta\big);
Isolation costs (vertices)
P(empty yellow)=exp(−k(d−1)/d)
\mathbb{P}(\text{empty yellow})=\phantom{\exp\big(-k^{(d-1)/d}\big)}
k(d−1)/d
k^{(d-1)/d}
E[∣boundary∣]∼k(d−1)/d
\mathbb{E}[|\text{boundary}|]\sim k^{(d-1)/d}
E[∣other yellow∣]∼
\mathbb{E}[|\text{other yellow}|]\sim
k1−γ(τ−1)
k^{1-\gamma(\tau-1)}
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
P(empty yellow)=exp(−E[∣yellow∣])
\mathbb{P}(\text{empty yellow})=\exp\big(-\mathbb{E}[|\text{yellow}|]\big)
E[∣boundary∣]∼
\mathbb{E}[|\text{boundary}|]\sim
⟹ζ≥max{dd−1,1−γ(τ−1)}
\Longrightarrow \quad \zeta\ge\max\big\{\tfrac{d-1}{d}, 1-\gamma(\tau-1)\big\}
P(W≥w)=w−(τ−1)
\mathbb{P}(W\ge w)=w^{-(\tau-1)}
Lower bounds: an isolated component
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≥exp(−Akζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\ge\exp\big(-Ak^\zeta\big);
Isolation costs (edges)
k(d−1)/d
k^{(d-1)/d}
P(no edges)∼exp(−E[∣red↔blue∣])
\mathbb{P}(\text{no edges})\sim\exp\big(-\mathbb{E}[|\text{red}\leftrightarrow\text{blue}|])
ζ≥max{dd−1,1−γ(τ−1)}.
\zeta\ge\max\big\{\tfrac{d-1}{d}, 1-\gamma(\tau-1)\big\}.
E[∣red↔blue∣]∼0≤γ1≤γ2≤γmaxE[∣kγ1-vert.∣]⋅E[∣kγ2-vert.∣]⋅q((0,kγ1),(k,kγ2))
\begin{aligned}
\mathbb{E}[|\text{red}\leftrightarrow\text{blue}|]&\sim \max_{0\le\gamma_1\le\gamma_2\le\gamma}\mathbb{E}[|k^{\gamma_1}\text{-vert.}|] \cdot\mathbb{E}[|k^{\gamma_2}\text{-vert.}|]\\
&\hspace{50pt}\cdot q\big((0, k^{\gamma_1}), (k, k^{\gamma_2})\big)
\end{aligned}
E[∣red↔blue∣]∼
\begin{aligned}
\mathbb{E}[|\text{red}\leftrightarrow\text{blue}|]&\sim
\end{aligned}
⟹ζ≥0≤γ1≤γ2≤γmax{2−α+γ1(σα−(τ−1))+γ2(α−(τ−1))}
\begin{aligned}\Longrightarrow \quad\zeta\ge
\max_{0\le\gamma_1\le\gamma_2\le\gamma}
\{2-\alpha&+\gamma_1(\sigma\alpha-(\tau-1))\\&+ \gamma_2(\alpha-(\tau-1))\}\end{aligned}
⟹ζ≥0≤γ1≤γ2≤γmax{2−α+γ1(σα−(τ−1))+γ2(α−(τ−1))}
\begin{aligned}\Longrightarrow \quad\zeta\ge
\max_{\phantom{0\le\gamma_1\le\gamma_2\le\gamma}}
\{\phantom{2-\alpha}&\phantom{+\gamma_1(\sigma\alpha-(\tau-1))}\\&\phantom{+ \gamma_2(\alpha-(\tau-1))}\}\end{aligned}
Isolation costs (edges)
ζ≥0≤γ1≤γ2≤γmax{2−α+γ1(σα−(τ−1))+γ2(α−(τ−1))}
\begin{aligned}\zeta\ge
\max_{0\le\gamma_1\le\gamma_2\le\gamma}
\{2-\alpha&+\gamma_1(\sigma\alpha-(\tau-1))\\&+ \gamma_2(\alpha-(\tau-1))\}\end{aligned}
Possible optimizers
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
(0, 0),
\phantom{(0, \gamma),
(\gamma, \gamma)}
\big\}
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
(0, 0),
(0, \gamma),
\phantom{(\gamma, \gamma)}
\big\}
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
(0, 0),
(0, \gamma),
(\gamma, \gamma)
\big\}
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
\phantom{(0, 0),
(0, \gamma),
(\gamma, \gamma)}
\big\}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&\phantom{2-\alpha,}\\
&\phantom{2-\alpha + \gamma(\alpha-(\tau-1)),}\\
&\phantom{2-\alpha + \gamma((\sigma+1)-2(\tau-1))}\}
\end{aligned}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&2-\alpha,\\
&\phantom{2-\alpha + \gamma(\alpha-(\tau-1)),}\\
&\phantom{2-\alpha + \gamma((\sigma+1)-2(\tau-1))}\}
\end{aligned}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&2-\alpha,\\
&2-\alpha + \gamma(\alpha-(\tau-1)),\\
&\phantom{2-\alpha + \gamma((\sigma+1)-2(\tau-1))}\}
\end{aligned}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&2-\alpha,\\
&2-\alpha + \gamma(\alpha-(\tau-1)),\\
&2-\alpha + \gamma((\sigma+1)-2(\tau-1))\}
\end{aligned}
Minimizing ζ
ζ=γminmax{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1)),1−γ(τ−1),dd−1}
\begin{aligned}\zeta=\min_\gamma
\max
\{&2-\alpha,\\
&2-\alpha + \gamma(\alpha-(\tau-1)),\\
&2-\alpha + \gamma((\sigma+1)-2(\tau-1)),\\
&1-\gamma(\tau-1),
\\
&\tfrac{d-1}d
\}
\end{aligned}
Edges, vertices
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≥exp(−Akζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\ge\exp\big(-Ak^\zeta\big);
Lower bound wrap-up
Conclusion:
ζ=max{ζll,ζlh,ζhh,ζnn}
\begin{aligned}\zeta=
\max
\{&\zeta_\mathrm{ll}, \zeta_\mathrm{lh}, \zeta_\mathrm{hh}, \zeta_\mathrm{nn}
\}
\end{aligned}
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≥exp(−Akζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\ge\exp\big(-Ak^\zeta\big);
nn
ll
lh
hh
ζ=max{ζll,ζlh,ζhh,ζnn}
\begin{aligned}\zeta=
\max
\{&\phantom{\zeta_\mathrm{ll}, \zeta_\mathrm{lh}, \zeta_\mathrm{hh}, \zeta_\mathrm{nn}}
\}
\end{aligned}
ζ=max{ζll,ζlh,ζhh,ζnn}
\begin{aligned}\zeta=
\max
\{&\zeta_\mathrm{ll}, \phantom{\zeta_\mathrm{lh}, \zeta_\mathrm{hh}, \zeta_\mathrm{nn}}
\}
\end{aligned}
ζ=max{ζll,ζlh,ζhh,ζnn}
\begin{aligned}\zeta=
\max
\{&\zeta_\mathrm{ll}, \zeta_\mathrm{lh}, \phantom{\zeta_\mathrm{hh}, \zeta_\mathrm{nn}}
\}
\end{aligned}
ζ=max{ζll,ζlh,ζhh,ζnn}
\begin{aligned}\zeta=
\max
\{&\zeta_\mathrm{ll}, \zeta_\mathrm{lh}, \zeta_\mathrm{hh}, \phantom{\zeta_\mathrm{nn}}
\}
\end{aligned}
Thank you!
Consider supercritical interpolating model, σ≤τ−1.
If ζnn is not the unique maximum of Z:={ζll,ζlh,ζhh,ζnn}
- Cluster-size decay: P(k≤∣C(0)∣<∞)∼exp(−Θ(kmax(Z)));
- Second-largest component: ∣Cn(2)∣∼Θ((log(n))1/max(Z)).
If ζnn is the unique maximum of Z, d>1:
- Always lower bound;
- Upper bound: no weights, high edge-density β≫βc.
Upper bound: second-largest component
Reveal graph in stages.
-
Form "backbone"
with "(kγ1,kγ2)"-edges of length k1/d. - Merge all large components with backbone with edges of length k1/d.
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
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}
- Cluster-size decay: P(k≤∣C(0)∣<∞)∼exp(−Θ(kmax(Z)));
- Second-largest component: ∣Cn(2)∣∼Θ((log(n))1/max(Z)).
If ζnn is the unique maximum of Z, d>1:
- Always lower bound;
- Upper bound: no weights, high edge-density β≫βc.
ζll=2−α,ζlh=ατ−1−(τ−2),ζhh=2−(τ−1)/α3−τ,ζnn=dd−1
Upper bounds
Aim: Maximize ζ s.t.
P(∣Cn(2)∣≥k)≲nexp(−kζ).
\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big).
P(∣Cn(0)≥k,0∈/Cn(1))≤P(∣Cn(2)∣≥k)≲exp(−21kζ)
\mathbb{P}\big(|\mathcal{C}_n(0)\ge k, 0\notin\mathcal{C}_n^{(1)}\big)\le\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge k\big)\lesssim\exp(-\tfrac12k^\zeta)
When n≤exp(−21kζ)
When k=kn=(2log(n))1/ζ
P(∣Cn(2)∣≥(2log(n))1/ζ)≲nexp(−2log(n)).
\mathbb{P}\big(|\mathcal{C}_n^{(2)}|\ge (2\log(n))^{1/\zeta}\big)\lesssim n\exp\big(-2\log(n)\big).
Upper bound: second-largest component
Reveal graph in stages.
-
Form "backbone"
with "(kγ1,kγ2)"-edges of length k1/d. - Merge all large components with backbone.
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
k1/d
k^{1/d}
2k1/d
2k^{1/d}
0
0
Three techniques: ζll,ζlh, and ζhh:
No control short edges/local geometry:
no upper bound (nn)
Lower bounds: an isolated component
Aim: Find minimal ζ s.t.
P(k≤∣C(0)∣<∞)≥exp(−Akζ);
\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\ge\exp\big(-Ak^\zeta\big);
Isolation costs (edges)
k(d−1)/d
k^{(d-1)/d}
P(no edges)∼exp(−E[∣red↔blue∣])
\mathbb{P}(\text{no edges})\sim\exp\big(-\mathbb{E}[|\text{red}\leftrightarrow\text{blue}|])
ζ≥max{dd−1,1−γ(τ−1)}.
\zeta\ge\max\big\{\tfrac{d-1}{d}, 1-\gamma(\tau-1)\big\}.
E[∣red↔blue∣]∼0≤γ1≤γ2≤γmaxE[∣kγ1-vert.∣]⋅E[∣kγ2-vert.∣]⋅q((0,kγ1),(k,kγ2))
\begin{aligned}
\mathbb{E}[|\text{red}\leftrightarrow\text{blue}|]&\sim \max_{0\le\gamma_1\le\gamma_2\le\gamma}\mathbb{E}[|k^{\gamma_1}\text{-vert.}|] \cdot\mathbb{E}[|k^{\gamma_2}\text{-vert.}|]\\
&\hspace{50pt}\cdot q\big((0, k^{\gamma_1}), (k, k^{\gamma_2})\big)
\end{aligned}
E[∣red↔blue∣]∼
\begin{aligned}
\mathbb{E}[|\text{red}\leftrightarrow\text{blue}|]&\sim
\end{aligned}
⟹ζ≥0≤γ1≤γ2≤γmax{2−α+γ1(σα−(τ−1))+γ2(α−(τ−1))}
\begin{aligned}\Longrightarrow \quad\zeta\ge
\max_{0\le\gamma_1\le\gamma_2\le\gamma}
\{2-\alpha&+\gamma_1(\sigma\alpha-(\tau-1))\\&+ \gamma_2(\alpha-(\tau-1))\}\end{aligned}
⟹ζ≥0≤γ1≤γ2≤γmax{2−α+γ1(σα−(τ−1))+γ2(α−(τ−1))}
\begin{aligned}\Longrightarrow \quad\zeta\ge
\max_{\phantom{0\le\gamma_1\le\gamma_2\le\gamma}}
\{\phantom{2-\alpha}&\phantom{+\gamma_1(\sigma\alpha-(\tau-1))}\\&\phantom{+ \gamma_2(\alpha-(\tau-1))}\}\end{aligned}
Isolation costs (edges)
ζ≥0≤γ1≤γ2≤γmax{2−α+γ1(σα−(τ−1))+γ2(α−(τ−1))}
\begin{aligned}\zeta\ge
\max_{0\le\gamma_1\le\gamma_2\le\gamma}
\{2-\alpha&+\gamma_1(\sigma\alpha-(\tau-1))\\&+ \gamma_2(\alpha-(\tau-1))\}\end{aligned}
Possible optimizers
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
(0, 0),
\phantom{(0, \gamma),
(\gamma, \gamma)}
\big\}
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
(0, 0),
(0, \gamma),
\phantom{(\gamma, \gamma)}
\big\}
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
(0, 0),
(0, \gamma),
(\gamma, \gamma)
\big\}
(γ1⋆,γ2⋆)∈{(0,0),(0,γ),(γ,γ)}
(\gamma_1^\star, \gamma_2^\star)\in\big\{
\phantom{(0, 0),
(0, \gamma),
(\gamma, \gamma)}
\big\}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&\phantom{2-\alpha,}\\
&\phantom{2-\alpha + \gamma(\alpha-(\tau-1)),}\\
&\phantom{2-\alpha + \gamma((\sigma+1)-2(\tau-1))}\}
\end{aligned}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&2-\alpha,\\
&\phantom{2-\alpha + \gamma(\alpha-(\tau-1)),}\\
&\phantom{2-\alpha + \gamma((\sigma+1)-2(\tau-1))}\}
\end{aligned}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&2-\alpha,\\
&2-\alpha + \gamma(\alpha-(\tau-1)),\\
&\phantom{2-\alpha + \gamma((\sigma+1)-2(\tau-1))}\}
\end{aligned}
ζ≥max{2−α,2−α+γ(α−(τ−1)),2−α+γ((σ+1)−2(τ−1))}
\begin{aligned}\zeta\ge
\max
\{&2-\alpha,\\
&2-\alpha + \gamma(\alpha-(\tau-1)),\\
&2-\alpha + \gamma((\sigma+1)-2(\tau-1))\}
\end{aligned}
Cluster-size decay in kernel-based spatial random graphs Joost Jorritsma joint with Júlia Komjáthy, Dieter Mitsche Recent trends in Spatial Stochastic Processes October 2022