Funded by the EU under REA
Grant Agreement No. 101120085

Welcome to, the home of the MSCA Doctoral Network BeyondTheEdge: Higher-Order Networks and Dynamics funded by the European Union under REA Grant Agreement No. 101120085.


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Many systems that govern our everyday lives—from communication networks to the human brain—can be seen as networks of interconnected units. Traditionally, networks are equated with graphs where edges give pairwise relations between two units. However, in network dynamical systems, nonlinear higher-order interactions between more than two units often play a critical role in shaping the collective dynamical behaviour of all units: For example, the spread of a disease depends not only on our behaviour as pairs of individuals but also how we behave in groups of more than two. Thus, elucidating the role of these higher-order interactions is critical to understand and control the dynamics of complex systems that determine our lives and livelihoods, whether it is the spreading of a disease or the proper functioning of the human brain as a network of billions of neurons.

The doctoral network BeyondTheEdge will identify the role of nonpairwise higher-order interactions in the emergence of complex dynamical behaviour of networks of interacting units. BeyondTheEdge brings together key researchers in an international network that is interdisciplinary (from mathematics to neuroscience) and intersectorial (including academia, private research institutes, and industry) to develop new mathematical insights relevant for real-world problems. BeyondTheEdge will train a cohort of 10 PhD students through research, education, and complementary skills training. This will enable the PhD students to innovate, collaborate, and become leading professionals in academia, industry, or the public sector: Innovative training activities will ensure that all PhD students can apply their skills beyond the academic context and put them in perspective of the wider world. Supervisor training activities ensure that the more junior project partners can shape the PhD education of the future. Thus, BeyondTheEdge will make a lasting contribution that will far outlive the duration of the project.

What are Higher-Order Networks?

C. Bick, E. Gross, H. A. Harrington, M. T. Schaub. What Are Higher-Order Networks? SIAM Rev. 65, 686–731 (2023). doi:10.1137/21M1414024 [Open Access]

G. Bianconi. Higher-Order Networks: An introduction to simplicial complexes. Cambridge University Press.

F. Battiston et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021). doi:10.1038/s41567-021-01371-4

F. Battiston et al. Networks beyond pairwise interactions: Structure and dynamics Phys. Rep. 874, 1–92 (2020). doi:10.1016/j.physrep.2020.05.004

We are recruiting!

BeyondTheEdge is recruiting 10 PhD students across the consortium members; these positions are subject to eligibility requirements.

BeyondTheEdge PhD Position at HUN-REN Alfréd Rényi Institute of Mathematics (Budapest, HU)
The theory of graph limits turned out to be a powerful tool for solving problems in extremal combinatorics and random graphs. The limit object is usually not a graph anymore (e.g., it can be a probability measure or an operator). The goal of the project is to discover extensions of this theory to hypergraphs, describe limit objects of random hypergraphs and higher-order networks, and find applications for mean-field limit theories for dynamics on heterogeneous networks. [Prospective research secondments at TU Munich and VU Amsterdam]
Contact: Ágnes Backhausz
Prospective starting date: September 2, 2024
Applications at now closed.
BeyondTheEdge PhD Position at the University of Porto (Porto, PT)
We will develop a coupled cell hypernetwork formalism, which captures higher-order interactions in networks given by differential equations, to determine the existence and stability of network synchronization patterns. Applying this framework to research concrete model equations (e.g., ecological models) will allow to understand and control synchrony, synchrony breaking, as well as the emergence of periodic (e.g., cyclic competition between species) and heteroclinic behavior. [Prospective research secondments at VU Amsterdam and Universita degli Studi di Napoli Federico II]
Contact: Ana Dias
Prospective starting date: October 1, 2024
Applications at now closed.
BeyondTheEdge PhD Position at the Technical University of Munich (Munich, DE)
We are going to study mean-field limit dynamics on hypergraphs using a combination of tools from dynamical systems and graph limit theory. A focus will be the case of adaptive (or co-evolutionary) hypergraph dynamics for several concrete models arising in applications and their analysis using bifurcation theory. [Prospective research secondments at Aix-Marseille University (AMU) and CWTS]
Contact: Christian Kuehn
Prospective starting date: On or before October 1, 2024
Apply now at:; application deadline March 31, 2024
BeyondTheEdge PhD position at Vrije Universiteit (Amsterdam, NL)
Coupled oscillator networks give rise to higher-order phase interactions. This project will consider the structure and properties of such derived higher-order interaction networks and how these relate to collective phenomena—synchronization and beyond—of the oscillator network. [Prospective research secondments at Technical University of Munich (TUM) and University of Sao Paulo (USP)]
Contact: Christian Bick
Prospective starting date: On or before October 1, 2024
Applications at now closed.
BeyondTheEdge PhD position at University of Naples Federico II (Napoli, IT)
Analysis, Control and Application of Network with Higher-Order Interactions: Control of complex networks typically involves devising strategies for networks characterized by pairwise interactions. The project will extend these approaches to many-body interactions in real world applications offering new opportunities for manipulating the collective dynamics arising in complex systems. [Prospective research secondments at University of California, Davis and BT Wireless]
Contact: Mario di Bernardo
Prospective starting date: November 1, 2024
Apply at: To be announced in March/April 2024
BeyondTheEdge PhD position at Aix-Marseille University (Marseille, FR)
The PhD will consist in a data-driven development of new approaches and tools to deal with higher order networks (HONs). We will define tools to characterize relevant and central structures in HONs, develop comparison methods between HONs, tools to extract timescales and states in temporal HONs. We will also consider models of dynamical processes involving higher-order structures and study the role of the various structures we have defined in these processes. [Prospective research secondments at CENTAI and Universidad de Zaragoza]
Contact: Alain Barrat
Prospective starting date: October 1, 2024
Apply now at:; application deadline March 22, 2024
BeyondTheEdge PhD position at CENTAI Institute (Torino, IT)
The PhD project aims to develop methods for inferring higher-order dynamical mechanisms in systems from biology and neuroscience, where data is often incomplete or aggregated. It focuses on distinguishing between network (pairwise) and higher-order interactions using limited data. The project will create new analysis and modeling techniques, using both empirical and synthetic models, to study the impact of data aggregation on identifying different process types. [Prospective research secondments at Rheinisch-Westfaelische Technische Hochschule (RWTH) Aachen and Amsterdam UMC - VUmc]
Contact: Giovanni Petri
Prospective starting date: October 31, 2024
Apply now at:; application deadline April 7, 2024
BeyondTheEdge PhD position at Vrije Universiteit (Amsterdam, NL)
Development, evaluation, and (clinical) application of statistical methods for assessing higher-order interactions in functional magnetic resonance (fMRI) data. The project focuses on the formulation, statistical inference, and goodness-of-fit analysis of parametric models for multivariate non-Gaussian observations. In these models, higher-order interactions are quantified by higher-order multivariate cumulants. In a second stage, the models are complemented by hemodynamic observation models. [Prospective research secondments at Amsterdam UMC - VUmc and Kadir Has Universitesi]
Contact: Rikkert Hindriks
Prospective starting date: On or before October 1, 2024
Apply now at:; application deadline June 15, 2024
BeyondTheEdge PhD position at University of Zaragoza (Zaragoza, ES)
The PhD student will work on the general topic of developing a mathematical and computational framework to characterize the structure and dynamics of higher-order networked systems. More specifically, we aim at elucidating when graph metrics are suited to characterize higher-order interactions (HOIs), both in terms of the structure and the dynamics, and when and which new metrics need to be introduced. Critical questions include what metrics reveal relevant hypergraph structural and dynamical features and how to properly capture their interdependency. [Prospective research secondments at CENTAI and CWTS]
Contact: Yamir Moreno
Prospective starting date: Between September 1 and October 15, 2024
Apply now at:; application deadline May 12, 2024
BeyondTheEdge PhD position at Kadir Has University (Istanbul, TR)
The PhD student will address the problem of reconstructing complex systems' network structure from data to understand and predict sudden changes in network dynamics. Using the framework of complex systems, we will pioneer methods for characterizing the network dynamics from data that can feature in diverse fields such as geology, climate science, chemistry, and neuroscience by blending tools from the fields of Dynamical Systems and Machine Learning. This project will investigate mathematical brain hypernetworks. [Prospective research secondments at Universidade de Sao Paulo (USP) and Universiteit van Amsterdam]
Contact: Deniz Eroglu
Prospective starting date: October 1, 2024
Apply now at:; application deadline March 31, 2024



Vrije Universiteit Amsterdam
Netherlands (NL)


HUN-REN Alfréd Rényi Institute of Mathematics
Hungary (HU)
Universidade do Porto
Portugal (PT)
Technische Universität München
Germany (DE)
Universita degli Studi di Napoli Federico II
Italy (IT)
Aix Marseille Université
France (FR)
CENTAI Institute SPA
Italy (IT)
Universidad de Zaragoza
Spain (ES)
Kadir Has Universitesi
Turkey (TR)

Associated Partners

Amsterdam UMC - VUmc
Netherlands (NL)
Universiteit van Amsterdam
Netherlands (NL)
Universidade de Sao Paulo
Brazil (BR)
Rheinisch–Westfälische Technische Hochschule Aachen
Germany (DE)
University of California, Davis
United States of America (US)
Netherlands (NL)
BT Wireless
United Kingdom (UK)
Politecnico di Torino
Italy (IT)
Eötvös Loránd University
Hungary (HU)

Scientific Advisory Board

Peter Ashwin (University of Exeter, UK)
Federico Battiston (Central European University, AT)
Ginestra Bianconi (Queen Mary, University of London, UK)
Marjeta Kramar Fijavž (University of Ljubljana, SI)
Heather Harrington (University of Oxford, UK)
Peter Mucha (Dartmouth College, US)
Arkady Pikovsky (University of Potsdam, DE)
Lai-Sang Young (New York University, US)