Opinion dynamics, the study of how individual beliefs and collective public opinion evolve, is a fertile domain for applying statistical physics to complex social phenomena. Like physical systems, societies exhibit macroscopic regularities from localized interactions, leading to outcomes such as consensus or fragmentation. This field has grown significantly, attracting interdisciplinary methods and driven by a surge in large-scale behavioral data. This review covers its rapid progress, bridging the literature dispersion. We begin with essential concepts and definitions, encompassing the nature of opinions, microscopic and macroscopic dynamics. This foundation leads to an overview of empirical research, from lab experiments to large-scale data analysis, which informs and validates models of opinion dynamics. We then present individual-based models, categorized by their macroscopic phenomena (e.g., consensus, polarization, echo chambers) and microscopic mechanisms (e.g., homophily, assimilation). We also review social contagion phenomena, highlighting their connection to opinion dynamics. Furthermore, the review covers common analytical and computational tools, including stochastic processes, treatments, simulations, and optimization. Finally, we explore emerging frontiers, such as connecting empirical data to models and using AI agents as testbeds for novel social phenomena. By systematizing terminology and emphasizing analogies with traditional physics, this review aims to consolidate knowledge, provide a robust theoretical foundation, and shape future research in opinion dynamics.
This article engages with two prominent defenses of unequal power from the standpoint of social equality, by Daniel Viehoff (2019) and Ryan Cox (2022), which aim to debunk the Constitution Claim, or the idea that equal power is constitutively necessary for social equality. This paper shows that the Constitution Claim is not defeated by the proposals such as the ones mentioned to tease apart power equality and social equality. Adequate social justification of power inequality, understood by Viehoff (2019) as moral-equality-respectful social justification, is not sufficient to ground social equality. If understood as the “best interpretation available,” adequate social justification either collapses into the actual justification of society lacking a normative framework for determining the best interpretation or comes closer to an objective interpretation, which loses the social character of social inequality. Pace Cox (2022), power inequality is not objectionable only because it gives rise to consideration inequality (when it does) or only when it is known about. Relational egalitarians should worry about power inequality even when it is secret because those with lesser power are vulnerable to those with greater power, which can, in certain conditions, raise the worry of domination. I conclude that the Constitution Claim is not proven wrong.
We further investigate the metalogical properties of the ordered fragment. First, we provide a simplified proof of the satisfiability invariance under A. Herzig’s translation of the ordered fragment into modal logic KD . Second, based on the notion of bisimulation developed by B. Bednarczyk and R. Jaakkola, we show that each ordered formula is equivalent to a disjunction of ‘ordered types’. Third, we show that the fragment enjoys uniform interpolation, and that uniform interpolants can be effectively constructed from ‘ordered types’. Finally, we establish the Łoś-Tarski Preservation Theorem for the fragment, and therefore conclude that the ordered fragment is nice.
Multilayer network science has emerged as a central framework for analysing interconnected and interdependent complex systems. Its relevance has grown substantially with the increasing availability of rich, heterogeneous data, which makes it possible to uncover and exploit the inherently multilayered organisation of many real-world networks. In this review, we summarise recent developments in the field. On the theoretical and methodological front, we outline core concepts and survey advances in community detection, dynamical processes, temporal networks, higher-order interactions, and machine-learning-based approaches. On the application side, we discuss progress across diverse domains, including interdependent infrastructures, spreading dynamics, computational social science, economic and financial systems, ecological and climate networks, science-of-science studies, network medicine, and network neuroscience. We conclude with a forward-looking perspective, emphasizing the need for standardised datasets and software, deeper integration of temporal and higher-order structures, and a transition toward genuinely predictive models of complex systems.
Higher-order interactions that nonlinearly couple more than two nodes are important in many networked systems, and their effects on collective dynamics are increasingly being studied. Here, we provide an overview of this rapidly growing field and of the techniques that can be used to describe and analyse them. We focus in particular on new phenomena and challenges that emerge when non-pairwise interactions are considered. We conclude by discussing open questions and promising future directions on the collective dynamics of higher-order networks. This Review surveys how higher-order interactions, which link more than two units at a time, reshape collective dynamics in complex systems. New synchronization phenomena, analytical frameworks and emerging methods to reduce or infer higher-order structure from data, are highlighted.
