Structural balance is a classical notion in network science, originally introduced in social psychology to describe globally consistent patterns of positive and negative relations. Here we present an overview of a series of works that progressively generalise this concept from signed graphs to richer classes of weighted networks. Starting from weighted signed networks, where each edge carries a real value with a sign, we revisit the classification into balanced, antibalanced, and strictly unbalanced regimes, and show how each is reflected in spectral properties and in the behaviour of spreading processes and other dynamics. We then extend the framework to networks whose edges are weighted by complex numbers, replacing the binary sign with a continuous phase, and further to matrix-weighted networks, where interactions act on multidimensional states through matrix couplings. In all these settings, the same fundamental principle applies: balance, or more generally coherence, describes whether signals propagating along different paths combine coherently or destructively interfere. This unifying viewpoint connects structural consistency to the spectrum of generalised adjacency and Laplacian matrices, and thereby to the long-term behaviour of both linear and nonlinear dynamics. We illustrate the reach of this perspective through consensus dynamics, random walks, spectral clustering, and synchronisation of higher-dimensional Kuramoto oscillators on networks.

Bio: Renaud Lambiotte has a PhD in Physics from the Université Libre de Bruxelles. Following postdocs at ENS Lyon, Université de Liège, UCLouvain and Imperial College London, and a Professorship in Mathematics at the University of Namur, he is currently Professor of Networks and Nonlinear Systems at the Mathematical Institute of Oxford University. His main research interests are the modelling and analysis of large networks, with a particular focus on clustering and temporal networks, and applications in social and neuronal systems. He is Associate Editor for Science Advances, an INET Fellow, External Faculty at the Complexity Hub in Vienna and Teaching Fellow at Somerville College.

Web page: https://www.maths.ox.ac.uk/people/renaud.lambiotte