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We investigate the lifetime of dynamical regimes under the impact of noise. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state space faster. However, for relevant systems with complexities ranging from phenomenological toy models to reduced models of atmospheric dynamics, this intuition has proven misleading. As long as the noise is sufficiently small, the noisy system stays in regimes of interest on average longer than its deterministic counterpart, an effect we call "stochastic inertia". We derive a formula to compute whether or not stochastic inertia occurs. Additionally, we propose a numerical technique for testing the occurrence of stochastic inertia, using a Markov chain on the set of points given by a sufficiently long trajectory of the system without noise. The method is shown to correctly predict the presence of stochastic inertia in simple systems, and its utility is demonstrated on a paradigm model of atmospheric dynamics. This is joint work with R. Chemnitz, P. Koltai, S. Pfahl and H. Schoeller.

We aim to provide the foundation for extending normal form theory from stochastic to rough differential equations. We address the existence of normal forms for rough ordinary differential equations, assuming suitable smoothness and the hyperbolicity of an equilibrium point. In this context, we establish local formal equivalence of the two solution flows generated by a random nonlinear RDE and its linearized version.

Given a simple graph G = (V, E) and a map l0 : V → {+1, −1}, the majority dynamics on G with initial assignment of states l0 is a process that begins on day 0, and for each t ≥ 0 produces a new assignment of states lt+1 where each vertex takes the state of the majority of its neighbours, and remains at its previous state in the case of a tie. Specifically, for each v ∈ V , lt+1(v) = ( +1 if P u∈N (v) lt(u) > 0, or P u∈N (v) lt(u) = 0 and lt(v) = +1, −1 otherwise. (1) This process is a model for opinion exchange dynamics, with applications in many areas, such as politics, sociology, biophysics. While there exist results about the process even- tually reaching a 2-periodic stable state on all graphs, a natural question to study would be under which initial conditions is unanimity reached, and how quickly. When considering the question specifically in the case of Binomial Random Graphs, a longstanding conjecture, due to Benjamini, Chan, O’Donnel, Tamuz and Tan (2016) is the following: Conjecture. Let G ∼ G(n, p) be the binomial random graph with p = ω(1/n) and l0(v) be sampled uniformly at random from {+1, −1} for each v ∈ V . Then w.h.p. the majority dynamics process reaches unanimity after sufficiently many steps t. Steps towards proving the conjecture have been taken taken by gradually improving the range densities d = np for which the conjecture is known to be true, with the current best bound being d ≫ n1/3 log2/3 n due to Kim and Tran. Our work aims to prove the conjecture for the range n1/4 ≪ d ≤ O(n1/3), and lays the groundwork for proving the conjecture in the general case n1/(k+1) ≪ d ≤ O(n1/k). This is based on joint work with Nikolaos Fountoulakis, University of Birmingham.

In network science, as in other applied fields, answering key questions often requires distinguishing causal mechanisms from context-dependent associations. One formalization of causality uses invariance: a relationship between a target variable and a set of covariates is causal if it remains stable across diverse environments or interventions. Originally developed for linear models, this approach has been extended to more flexible frameworks, including generalized linear models, where causality is expressed via Pearson risk invariance. In this talk, we extend causal invariance to dynamic networks, focusing on Relational Event Models (REMs), which describe instantaneous interactions over time between social actors. Because the target —the dynamic network— and potential causal drivers are stochastic processes, dependences among them are characterized using conditional local independence. By exploiting a connection between REMs and logistic regression, we extend Pearson risk invariance to this dynamic setting, producing a causal discovery algorithm that only requires data from a single observational environment. We show how this approach is able to identify, and distinguish, important causal mechanisms in network science, such as social influence and homophily. This is joint work with Melania Lembo and Ernst Wit (USI).