Research Talk

Applied Probability

Tenure Track Interview

Joost Jorritsma

slides.com/joostjor/leiden-interview

Academic positions 

Understanding network models 

Probability, Combinatorics

& Physics, Epidemiology, ...

2 papers

6 papers

Information diffusion in random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE
Randomized optimization algorithms

 

Optimization under chance constraints

 

Sampling algorithms for explainable AI

 

Annals of Applied Probability (2x),
Random Structures & Algorithms

Electronic Journal of Prob.,

Annals of Probability (revision),

Prob. Theory & Rel. Fields (submitted)

PLOS One; Chaos, Solitons & Fractals
Complex Networks '20,
Coronavirus Modeling PIMS '20

GECCO'23,
Algorithmica (revision)

IDA'23
Runner up Frontier Prize

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

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

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\![1/2, 1)
  •   governs second-largest cluster, and cluster of 0
\zeta
  • Techniques: probability, combinatorics, optimization.

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 diffusion in random graphs

Distances

Component sizes

Intervention strategies

FINAL SIZE

Research plan: Large deviations in percolation and random graphs

Ongoing

Near future

Opportunities Leiden

  • Large deviations  of the giant in inhomogeneous random graphs
    Bert Zwart (CWI)
  • Tall or small trees
    Serte Donderwinkel (Groningen)
  • Applying for CIRM Fellowship
    Luisa Andreis (Milan), 
  • MSc Students (Oxford)
  • Dalia Terhesu

Research plan: Random walks on random graphs

Ongoing

Opportunities Leiden

  • 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

Research plan: Scaling limits of trees and random graphs

Near future

Opportunities Leiden

  • Scaling limit of infinite-type branching processes
    Christina Goldschmidt (Oxford),
    Louigi Addario-Berry (McGill)
  • Component sizes in critical
    inhomogeneous random graphs (IRG)
  • Metric-space scaling limit of IRGs
  • Semester at SLMath (Berkeley, California)
  • VENI Grant ('25 or '26)
  • PhD student

Figure by Igor Kortchemski

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"

References

  1. Weighted distances in scale-free preferential attachment models;
    With Júlia Komjáthy; Random Structures & Algorithms (2020).
  2. Distance evolutions in growing preferential attachment graphs;
    With Júlia Komjáthy; Annals of Applied Probability (2022).
  3. Cluster-size decay in supercritical long-range percolation;
    With Júlia Komjáthy, Dieter Mitsche; Electronic Journal of Probability (2024).
  4. Cluster-size decay in supercritical kernel-based spatial random graphs;
    With Júlia Komjáthy, Dieter Mitsche;
    Revision at Annals of Probability, arXiv: 2303.00712.
  5. Large deviations of the giant in supercritical kernel-based spatial random graphs;
    With Júlia Komjáthy, Dieter Mitsche;
    Submitted to Probability Theory & Related Fields,     arXiv: 2404.02984.

Research talk

Applied Probability

Tenure Track Interview

Joost Jorritsma

Teaching Vision

Applied Probability

Tenure Track Interview

Joost Jorritsma

Teaching Experience

Courses

Thesis supervision

  • Teaching assistant (BSc and MSc). Examples:
    • Random graphs: design, supervise research projects
    • Statistical Learning Theory: Certificate of Excellence
  • Lecturer (service teaching):
    • Course material; automated examination
  • Lecturing in Oxford: Probability on Graphs & Lattices
  • Co-supervision 3 MSc students
    • Probability (2x),
    • Data Science (with KLM & TU Delft, 1x)
  • Oxford: 3 MSc in Statistics, 3 MSc in Probability

Teaching Vision

Philosophy

Potential teaching in Leiden

  • "Good researcher \(\neq\) Good teacher"   \(\Rightarrow\)   BKO
  • Goal: Exciting to understand and solve problems
    • Collaboration: \(1+1>2\)
    • Research projects/seminars: communication skills
    • Work atmosphere: mistakes are essential
  • Everybody is different:
    academia vs. industry; introvert vs. extravert
  • Inleiding kansrekening; Complex Networks;
    Ruin Theory & Extremes; Stochastic Simulation
  • Potential MasterMath courses:
    Percolation; Random graphs; Scaling limits in probability
  • Supervision: Probability (Rajat Hazra), OR (Michel Mandjes), LIACS (Frank Takes, Akrati Saxena)

Teaching Vision

Applied Probability

Tenure Track Interview

Joost Jorritsma

Leiden research presentation

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