SC9 Probability on Graphs & Lattices
Joost Jorritsma
joost.jorritsma@stats.ox.ac.uk
Math,
Music,
Biking
Understanding network models
Probability, Combinatorics
& Physics, Epidemiology, ...
Information diffusion in random graphs
Distances
Component sizes
Intervention strategies
FINAL SIZE
Randomized optimization algorithms
Optimization under chance constraints
Sampling algorithms for explainable AI
Inhomogeneous degrees, triangles, and small world
Inhomogeneous degrees, triangles, and small world
Inhomogeneous degrees, triangles, and small world
Inhomogeneous degrees, triangles, and small world
Vertex set
- Spatial locations
- Vertex weights
Edge more likely if
- Spatially nearby,
- High weight
Spatial random graphs
Percolation: Reed-Frost epidemic
Four parameters: new and classical models
New phenomena
- Delocalized connected components
- Double-exponential neighbourhood growth
- Large outbreaks more likely than small outbreaks
Challenges
- Geometry:
- Short/Long edges
- Presence of hubs
New direction: Scaling limits of trees and random graphs
- Only vertex weights, no location
- Edge lengths scale with graph size
Figure by Igor Kortchemski
Setup
- Include asymmetry in edge probabilities
- Uncountable and unbounded weights
Goal
References
- The Continuum Random Tree; David Aldous; Annals of Probability (1991).
-
Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems;
Nicolas Broutin, Thomas Duquesne, Minmin Wang; Probability Theory & Related Fields (2021). -
Cluster-size decay in supercritical long-range percolation;
With Júlia Komjáthy, Dieter Mitsche; Electronic Journal of Probability (2024). -
Cluster-size decay in supercritical kernel-based spatial random graphs;
With Júlia Komjáthy, Dieter Mitsche; Min. revision in Annals of Probability, arXiv: 2303.00712. -
Large deviations of the giant in supercritical kernel-based spatial random graphs;
With Júlia Komjáthy, Dieter Mitsche; Preprint arXiv: 2404.02984. -
Surface order large deviations for Ising, Potts and percolation models;
Agoston Pisztora; Probability Theory and Related Fields (2024).
Department Away Day `24
Joost Jorritsma
Random graph models for real-world networks
Department Away Day `24
Joost Jorritsma
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
-
Short average distance:
~1999: 248 million users
Distance evolution in a growing network
1999
\(\mathrm{dist}_{\color{red}{'99}}(u_{'99}, v_{'99}) = 4\)
2005
\(\mathrm{dist}_{{\color{red}'05}}(u_{'99}, v_{'99}) = 3\)
2024
\(\mathrm{dist}_{{\color{red}'24}}(u_{'99}, v_{'99}) = 2\)
21 possible networks
Attachment rule:
Prefer 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
$$ \sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\overset{\mathbb{P}}\longrightarrow 0.$$
Theorem [J., Komjáthy, Annals of Applied Probability '22]. Assume \(\tau<3\).
Let \(\phantom{t'=T_t(a):=t\exp\big(\log^a(t)\big)}\) 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
$$ \phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} \phantom{- (1-a)\frac{4}{|\log(\tau-2)|}}\right|\phantom{\overset{\mathbb{P}}\longrightarrow 0.}$$
-
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
$$ \sup_{a\in[0,1]} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\phantom{\overset{\mathbb{P}}\longrightarrow 0.}$$
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
$$ \phantom{\sup_{a\in[0,1]}} \left| \frac{\mathrm{dist}_{T_t(a)}(U_t, V_t)}{\log\log(t)} - (1-a)\frac{4}{|\log(\tau-2)|}\right|\phantom{\overset{\mathbb{P}}\longrightarrow 0.}$$
Information diffusion in random graphs
Distances
Component sizes
Intervention strategies
FINAL SIZE
Real networks contain many triangles!
Do real networks look like this?
Large deviations (rare events) of cluster sizes
Kernel-based spatial random graphs
Only four parameters
Vertex set
- Spatial locations
- Vertex weights
Edge more likely if
- Spatially nearby,
- High weight
Hyperbolic random graph
Scale-free percolation
Long-range percolation
Bond percolation on \(\mathbb{Z}^d\)
\exp\big(-\Theta(\sqrt{n})\big).
Theorem (\(\mathbb{Z}^d\)-like graphs)
[Lebowitz & Schonmann '88; Gandolfi '89; Grimmett, Marstrand '90;
Kesten, Zhang '90; Alexander, Chayes, Chayes, Newman '90; Pisztora '96;
Cerf '97; Contreras, Martineau, Tassion '2024; ...]
Lower tail:
Surface tension drives too small cluster
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)=
\phantom{\Theta\big(n^{-I(\varepsilon)}\big)}
Upper tail:
Large clusters are very unlikely
\exp(-\Theta(n)\big).
Figure by Tobias Muller
Hyperbolic random graph
Scale-free percolation
Long-range percolation
Bond percolation on \(\mathbb{Z}^d\)
\exp\big(-\Theta(\sqrt{n})\big).
Theorem (\(\mathbb{Z}^d\)-like graphs)
[Lebowitz & Schonmann '88; Gandolfi '89; Grimmett, Marstrand '90;
Kesten, Zhang '90; Alexander, Chayes, Chayes, Newman '90; Pisztora '96;
Cerf '97; Contreras, Martineau, Tassion '2024; ...]
Lower tail:
Surface tension drives too small cluster
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n< \theta-\varepsilon\big)=
\mathbb{P}\big(|{\color{blue}\mathcal{C}_n^{(1)}}|/n>\theta +\varepsilon\big)=
\phantom{\Theta\big(n^{-I(\varepsilon)}\big)}
Upper tail:
Large clusters are very unlikely
\exp(-\Theta(n)\big).
Question:
Long edges and high-degree vertices,
do they matter?
\exp\big(-\Theta(n^\zeta)\big).
Theorem [J., Komjáthy, Mitsche, '24+]
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: large outbreak likelier than small.
-
Long edges can beat surface tension: any ;
\zeta\!\in\
- Random friend of a friend tree
Sofiya Burova (PhD student Barcelona),
Dieter Mitsche (Santiago de Chile) - Cover time of long-range percolation Carlos (MSc student Lima), D. Mitsche
- (Evolution of) the mixing time of preferential attachment models (PAM)
- Generating PAMs via random walks
Rajat Hazra & PhD student
Building a network: Attend and organize workshops, seminars
Near future (invitations)
- One-world probability seminar
- Probability meets Combinatorics (IST, Vienna)
- Long-range phenomena in Percolation (Köln)
- Mathematical Foundations of Network Models and their Applications (Chennai)
Opportunities Leiden: Lorentz Center
Organisational experience:
RandNET Workshop (with Serte Donderwinkel)
- 10 days, 80 participants
- Mini-courses, Open problem sessions (4+ papers)
- Budget: €55k
- Attending prof:
"You set the gold standard for running such an event"
Introduction
By joostjor
