Distances and component sizes

in scale-free random graphs

Joost Jorritsma, PhD

Florence Nightingale Bicentennial Fellowship

Research talk

Erdős–Rényi Random Graph

Nearest-Neighbour Percolation on \(\mathbb{Z}^d\)

Random Graphs: Probability, Combinatorics, Physics, Epidemiology

Motivation

  • Real-world networks:
    hubs and long edges
  • New phenomena

Inhomogeneous Percolation

Preferential attachment

3x Information diffusion in random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE

Internet: a growing network of routers and servers

~1969: 2 connected sites

Time

~1989: 0.5 million users

~2023: billions of devices

  • [Faloutsos, Faloutsos & Faloutsos, '99]:
    • Short average distance:
      Quick spread of information

~1999: 248 million users

Distance evolution in a growing network

1999

\(\mathrm{dist}_{\color{red}{'99}}(u_{'99}, v_{'99}) = 4\)

2005

\(\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3\)

2023

\(\mathrm{dist}_{{\color{red}'23}}(u_{'99}, v_{'99}) = 2\)

21 possible networks

Attachment rule:

Favour connecting to high-degree vertices, \(\tau\): tail of power-law degree distribution

2005

\(\phantom{\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3}\)

Distance evolution

Distance evolution: hydrodynamic limit

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume \(\tau<3\).
Let \(t'=T_t(a):=t\exp\big(\log^a(t)\big)\) for \(a\in[0,1]\), then

$$ \mathbb{P}\bigg(\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|>\varepsilon\bigg)\longrightarrow 0.$$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume \(\tau<3\).
Let \(t'=T_t(a):=t\exp\big(\log^a(t)\big)\) for \(a\in[0,1]\), then

$$ \phantom{\mathbb{P}\bigg(}\phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\phantom{>\varepsilon\bigg)\phantom{\longrightarrow} 0.}$$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume \(\tau<3\).
Let \(\phantom{t'=T_t(a):=t\exp\big(\log^a(t)\big)}\) for \(\phantom{a\in[0,1]}\), then

$$ \phantom{\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}{\longrightarrow} 0.}$$

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume \(\tau<3\).
Let \(t'=T_t(a):=t\exp\big(\log^a(t)\big)\) for \(a\in[0,1]\), then

$$ \phantom{\mathbb{P}\bigg(}\phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - \phantom{(1-a)\frac{4}{|\log(\tau-2)|}}\right|\phantom{>\varepsilon\bigg)\longrightarrow 0.}$$

  • Dynamics in PAMs.
  • Generalization with edge weights: 
random transmission times
Novelties

  • Fast spreading among influentials;

Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume \(\tau<3\).
Let \(t'=T_t(a):=t\exp\big(\log^a(t)\big)\) for \(a\in[0,1]\), then

$$ \mathbb{P}\bigg(\sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|>\varepsilon\bigg)\phantom{\longrightarrow 0.}$$

Information spreading on random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE

Intervention strategies

Real networks contain many triangles!

Do real networks look like this?

Large deviations (rare events) of cluster sizes:

  • Much larger/smaller (final) size of epidemic than expected.

What is the influence of long edges and high-degree vertices?

Hyperbolic random graph

Scale-free percolation

Long-range percolation

Soft Poisson-Boolean model

Random geom. graph

Nearest-neighbour percolation

Kernel-based spatial random graphs

Only four parameters

Vertex set

  • Spatial locations, \(\mathbb{Z}^d\) or Poisson point process
  • Vertex weights
    (iid Pareto or constant)

Edge more likely if

  • Spatially nearby,
  • High weight

Clusters in supercritical graphs

  • Largest component \({\color{blue}\mathcal{C}_n^{(1)}}\):
    • Linear in box size: giant component
    • Law of large numbers
    • Much smaller giant
    • Much larger giant

Questions

\exp\big(-\Theta(n^\zeta)\big).

Theorem [J., Komjáthy, Mitsche, '23+]
We find explicit \(\zeta\in[1/2,1)\), \(\theta\in(0,1)\) s.t.

  • Lower tail: small final size
    If enough long edges or no hubs
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=
Novelties

  • Reversed discrepancy: slower upper tail.
  • Long edges can beat surface tension: any         ;
  • First (upper) LDP for giant in spatial graph;
\zeta\!\in\![1/2, 1)
  •   drives also other cluster-size distributions
\zeta

What is the influence of long edges and high-degree vertices?

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)= \phantom{\Theta\big(n^{-I(\varepsilon)}\big)}
  • Upper tail: large final size
    If hubs are present, we find rate funtion \(I(\varepsilon)\): 

What is the influence of long edges and high-degree vertices?

\Theta\big(n^{-I(\varepsilon)}\big).

Information spreading on random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE

Distances and component sizes

in scale-free random graphs

Joost Jorritsma, PhD

Florence Nightingale Bicentennial Fellowship

Research talk

Surface tension in supercritical nearest-neighbour percolation

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n<\theta - \varepsilon\big)=
\exp\big(-\Theta(\sqrt{n})\big)

[Grimmett & Marstrand '90, Kesten & Zhang '90, Pisztora '96]

 

\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta + \varepsilon\big)=
\exp\big(-\Theta(n)\big)

[Lebowitz & Schonmann '88]

 

  • Discrepancy: faster upper tail
  • No precise asymptotics 

What is the influence of long edges and high-degree vertices?

Lower tail: small final size 

Upper tail: large final size 

Kernel-based spatial random graphs

Edge set \(\mathcal{E}_\infty\)

  • Symmetric kernel \(\kappa(w_u, w_v)\),
  • Long-range parameter \(\alpha\in(1,\infty]\),
  • Edge-density \(\beta>0\),

Connection probability

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

Vertex set \(\mathcal{V}_\infty\)

  • Spatial locations, either
    • Lattice \(\mathbb{Z}^d\)
    • Poisson point process (unit intensity)
  • Power-law i.i.d. weights \(w_v\ge 1\):
    \(\mathbb{P}(w_v\ge w)=w^{-(\tau-1)}\),

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

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

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

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

Components in supercritical graphs

  • Largest component \({\color{blue}\mathcal{C}_n^{(1)}}\):
    • Linear in box size
    • Law of large numbers
    • Lower tail large deviations
    • Upper tail large deviations

Questions


    Component of the origin \({\color{green}\mathcal{C}(0)}\): $$\mathbb{P}\big(k\le |{\color{green}\mathcal{C}(0)}|<\infty\big)= $$

    Second-largest component: $$|{\color{red}\mathcal{C}_n^{(2)}}|=\Theta\big(\phantom{what}\big)$$

Lower bounds: cluster-size decay

Aim: Find minimal \(\zeta\) s.t.

\mathbb{P}\big(k \le |\mathcal{C}(0)|<\infty\big)\gtrsim\exp\big(-k^\zeta\big);
|\mathcal{C}(0)|\ge k
\Bigg\}
\Theta(k)
k^\zeta\sim\mathbb{E}\big[|\{v\in\Lambda_k: v\leftrightarrow \Lambda_k^c\}|\big]
\mathbb{E}[|\phantom{\square\square}|]\sim\mathbb{E}[|\phantom{\square\square}\leftrightarrow \phantom{\square\square}|]

Aim: Find \(\gamma\) s.t.

\sim k^\zeta
{\color{green}\text{Cluster-size decay}}

Upper bounds

\mathbb{P}\big(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\big)\lesssim n\exp\big(-k^\zeta\big)
\mathbb{P}\big(|{\color{blue}\text{largest}}|/n\le \varepsilon \big)\lesssim \phantom{n} \exp\big(-n^\zeta\big)
\text{Law of large numbers}
{\text{\color{blue}Lower tail large deviations}}

Challenge: Delocalized components

\binom{n}k \sim n^k \gg n\exp\big(k^\zeta\big)

# possibilities for \(|{\color{red}2^{\mathrm{nd}}\text{-largest}}|\ge k\)

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