IS02: Heavy-tailed phenomena in networks

date: 7/15/2025, time: 16:00-17:30, room: ICS 141

Organizer: Bert Zwart (CWI Amsterdam)

Chair: Agnieszka Janicka (Eindhoven University of Technology)

Condensation in geometric inhomogenous random graphs with excess edges

Céline Kerriou (University of Cologne)

We consider geometric inhomogeneous random graphs (GIRGs), where vertices are distributed according to a Poisson point process in \(\mathbb{R}^d\), and each vertex carries an independent heavy-tailed weight representing its "influence." The probability of an edge between two vertices depends on their Euclidean distance and their weights. In this talk, we identify the upper large deviation probability for the number of edges in such graphs and show that the mechanism behind the large deviation is based on a condensation effect. Loosely speaking, the mechanism randomly selects a finite number of vertices and increases their power, so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. The focus in this talk is on GIRGs but our result covers a range of models such as the Boolean model with heavy-tailed radius distribution and the age-dependent random connection model.
This is joint work with Remco van der Hofstad, Pim van der Hoorn, Neeladri Maitra and Peter Mörters.

Emergence of heavy-tailed cascades in flow networks through a unified stochastic overload framework.

Agnieszka Janicka (Eindhoven University of Technology)

From power blackouts to traffic jams and financial crises, cascading failures pose a persistent threat to networks. These events rarely remain isolated — an initial failure can propagate through the system, leading to widespread disruption. Notably, across domains, the associated costs often follow heavy-tailed distributions. Our work provides a mathematically rigorous modeling framework that explains the emergence of this phenomenon. In this talk, I will present our stochastic, multi-commodity model for overload cascading failures — events triggered when flows exceed component capacities. Flows and resources are allocated optimally, following the principle of minimum energy dissipation. This formulation offers a flexible, yet analytically tractable framework applicable across diverse systems. Our results suggest that, despite domain-specific complexities, a common underlying driver — exogenous heavy-tailed inputs — may be responsible for the heavy-tailed nature of cascade costs. We derive probabilistic results that characterize when and why these costs inherit the input’s tail behavior. In particular, we show that large costs arise with high probability when a single location exhibits disproportionately high resource requirements — a phenomenon reminiscent of the catastrophe principle. The analysis also relies on continuity and scale-invariance properties of the cascade cost function, which we establish for several broad classes of modeling choices. This work deepens our understanding of cascading failures in complex systems and provides a unifying framework for analyzing large-scale disruptions, laying the groundwork for resilience analysis and policy design.

Ornstein-Uhlenbeck process with heavy tailed noise

Janusz Gajda (University of Warsaw)

We discuss some results related to Ornstein-Uhlenbeck (OU) processes defined as the unique solution of the stochastic differential equation (SDE) \[\tag{1} dU(t) = -\lambda U(t)dt + \sigma dR(t),\; U(0) = u_0.\] In Eq. \((1)\) \(\lambda,\sigma>0, u_0\in\mathbb{R}\) and \(\{R(t)\}_{t\geq 0}\) is either a Gaussian or symmetric \(\alpha\)-stable noise. Moreover we discuss subordination of this processes by independent process \(\{G_\eta(t)\}_{t\geq 0}\) being a Gamma distributed Lévy subordinator such that the increments \(G_\eta(t) - G_\eta(s) \sim\) Gamma\((\eta, (t-s))\). The PDF of \(G_\eta(t)\) is given by \[\begin{aligned} g(y,t) = \frac{\eta^{ t}}{\Gamma{( t)}} y^{ t-1} e^{-\eta y}, \;y>0, \;\eta>0.\end{aligned}\]

Thus the subordinated OU process is then defined via \[%\label{def:SGOU} X(t) = U\left(G_\eta(t)\right).\]

Bibliography

\([1]\) J. Gajda, A. Grzesiek, A. Wyłomańska, Ornstein-Uhlenbeck process driven by \(\alpha\)-stable process and its Gamma subordination, Methodology and Computing in Applied Probability, 25: 9 (2023) 1–17.