Spatial random graphs with assortativity

Spatial random graphs with tunable assortativity

Joost Jorritsma

joost.jorritsma@stats.ox.ac.uk

Intriguing maths: classical network models

Erdős–Rényi random graph $G(n, \lambda /n)$

Bernoulli percolation on $\mathbb{Z}^2$ w.p. $p$

Intriguing maths: classical network models

Motivation

Do there exist graphs with property X?
Is there a percolation phase transition?
What happens at the critical point?
What is the component-size distribution?

Intriguing maths: classical network models

Motivation

Universality: do all network (models) behave like ...?
What are the implications for real-world networks?

Aim for this talk

Introduce model for type-1 question
Give modelling interpretation: degree assortativity
Is this model useful for applications?
What should we investigate?

Universal properties demand null models

Heavy-tailed degrees (power law: $p_k\sim k^{-\tau}$)
Small world: small distances compared to network size
Clustering, local communities
Geometry: (natural) embedding of nodes into space
Hierarchy

Adapted from Network Science (2015), A.L. Barabási

Geometric inhomogeneous random graphs (GIRG)

Vertex set $\mathcal{V}_\infty$

Spatial locations $x_v\in\mathbb{R}^d$
i.i.d. weights $w_v\ge 1$: $\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

Edge set $\mathcal{E}_\infty$

High weight: high degree
Local clustering
Long-range parameter $\alpha>1$,
Edge-density $\beta>0$,

Connection probability

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{w_uw_v}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1$$

Vertex set $\mathcal{V}_\infty$

Spatial locations $x_v\in\mathbb{R}^d$,
i.i.d. weights $w_v\ge 1$: $\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{w_uw_v}{\|x_u-x_v\|^d}\bigg)^\alpha$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{\bigg(\beta}\frac{w_uw_v}{\phantom{\|x_u-x_v\|^d}}\phantom{\bigg)^\alpha\wedge 1}$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{\bigg(\beta}\frac{w_uw_v}{\|x_u-x_v\|^d}\phantom{\bigg)^\alpha\wedge 1}$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\phantom{\beta}\frac{w_uw_v}{\|x_u-x_v\|^d}\bigg)^\alpha\phantom{\wedge 1}$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)$$

Geometric inhomogeneous random graphs (GIRG)

GIRG generalizes random hyperbolic graph (HRG)

Figure by Tobias Müller

Hyperbolic random graph models Internet

Figure by Tobias Müller

Boguñá, Papadopoulos & Krioukov (2010, Nature comm.)

GIRGs for understanding epidemics

[Odor, Czifra, Komjáthy, Lovasz, Karsai '21; Komjáthy, Lapinskas, Lengler '21]

Back to math: epidemics via percolation

Vertex set $\mathcal{V}_\infty$

Spatial locations $x_v\in\mathbb{R}^d$
i.i.d. weights : $\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}$,

Edge set $\mathcal{E}_\infty$

Long-range parameter $\alpha>1$,
Edge-density $\beta>0$,
Percolation probability $p\in[0,1]$

Connection probability

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\phantom{p\bigg(}\bigg(\beta\frac{w_uw_v}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1\phantom{\bigg)}$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=p\bigg(\bigg(\beta\frac{w_uw_v}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1\bigg)$$

Question: Is there a non-trivial phase transition?

Bernoulli percolation on $\mathbb{Z}^2$ w.p. $p$

Geometric inhomogeneous random graph

Q1: Is there a non-trivial phase transition?

GIRGs contain an infinite component when

Weights have infinite variance ($\tau\in(2,3)$).
Weights have finite variance, but
enough short and long edges
($\alpha\in(1,2), p, \beta$ large).
Weights have finite variance, few long edges ($\alpha>2$), enough short edges, and dimension at least 2.

Deijfen, vdHofstad, Hooghiemstra '13; Fountoulakis, Müller '18; Bringmann, Keusch, Lengler '19; Gracar, Lüchtrath, Mönch '25]

Q2: What do small components look like?

[J., Komjáthy, Mitsche '24, '24, '25]

Second-largest component,

(Finite) Component of the origin

Asymptotic size-distribution, and shape
determined by variational problem

Trade-off between

long edges - high weights
long edges - low weights
short edges - dimension

\zeta=\max\bigg(\frac{3-\tau}{2-(\tau-1)/\alpha}, \,2-\alpha, \,\frac{d-1}d\bigg)

Are these the only three possible shapes?

Hyperbolic random graph

GIRG

Long-range percolation

Spatial preferential attachment

Random geom. graph

Nearest-neighbor percolation

Kernel-based spatial random graphs (KSRG)

Connection probability

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{w_uw_v}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1$$

$$\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta\frac{\kappa(w_u,w_v)}{\|x_u-x_v\|^d}\bigg)^\alpha\wedge 1$$

Kernel $\kappa$:

Komjáthy, Lodewijks '18; Gracar, Heydenreich, Mönch, Mörters '19; Jorritsma, Komjáthy, Mitsche '24; Gracar, Lüchtrath, Mönch '25]

The interpolating kernel generalizes many models

Connection probability

$${\color{grey}\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta}\frac{\kappa(w_u, w_v)}{\color{grey}\|x_u-x_v\|^d}{\color{grey}\bigg)^\alpha\wedge 1}$$

A parameterized kernel: $\sigma\ge 0$

$$\kappa_{\sigma}(w_u, w_v):=\max\{w_u, w_v\}\min\{w_u, w_v\}^\sigma$$

$\tau: \mathbb{P}(w_v\ge w)=w^{-(\tau-1)}.$
$\sigma$: interpolation

\frac{\sigma}{\tau-1}

\frac1{\tau-1}

1

\sigma=\tau-1

\tau=2

\sigma=1

\sigma=\tau-2

\sigma=0

GIRG

Hyperbolic RG

Spatial pref. attachment

Scale-free Gilbert

Long-range percolation

Random geom. graph

Nearest-neighbor percolation

Interpolation kernel tunes degree assortativity

Connection probability

$${\color{grey}\mathbb{P}\big(u\leftrightarrow v\mid \mathcal{V}_\infty\big)=\bigg(\beta}\frac{\kappa(w_u, w_v)}{\color{grey}\|x_u-x_v\|^d}{\color{grey}\bigg)^\alpha\wedge 1}$$

A parameterized kernel, $\sigma\in\mathbb{R}:$

$$\kappa_{\sigma}(w_u, w_v):=\max\{w_u, w_v\}\min\{w_u, w_v\}^\sigma$$

Assortativity parameter