Lorenzo Zino

Martina Alutto, Lorenzo Zino, Karl H. Johansson et al.

Leadership in social groups emerges dynamically through interaction and opinion exchange. Empirical evidence indicates that individuals expressing strong opinions tend to gain influence, while sustained leadership critically depends on maintaining alignment with the surrounding social context. Motivated by these observations, we introduce a coupled dynamical model describing the simultaneous evolution of opinions and leadership in a networked population. Extending the Friedkin-Johnsen framework, we represent leadership as a time-varying susceptibility to social influence, which evolves according to a game-theoretic mechanism, consistent with social psychology evidence. Within this setting, agents strengthen their leadership by expressing decisive yet socially coherent opinions, whereas misalignment with the collective state results in a loss of influence. We analyze the coupled dynamics and establish sufficient conditions to identify which agents necessarily emerge as leaders and which act as followers in the social network.

Lorenzo Zino, Giacomo Como, Fabio Fagnani

We consider a broad class of stochastic imitation dynamics over networks, encompassing several well known learning models such as the replicator dynamics. In the considered models, players have no global information about the game structure: they only know their own current utility and the one of neighbor players contacted through pairwise interactions in a network. In response to this information, players update their state according to some stochastic rules. For potential population games and complete interaction networks, we prove convergence and long-lasting permanence close to the evolutionary stable strategies of the game. These results refine and extend the ones known for deterministic imitation dynamics as they account for new emerging behaviors including meta-stability of the equilibria. Finally, we discuss extensions of our results beyond the fully mixed case, studying imitation dynamics where agents interact on complex communication networks.

Roberta Raineri, Mengbin Ye, Lorenzo Zino

In this paper, we consider a population of individuals who have actions and opinions, which coevolve, mutually influencing one another on a complex network structure. In particular, we formulate a control problem for this social network, in which we assume that we can inject into the network a committed minority -- a set of stubborn nodes -- with the objective of steering the population, initially at a consensus, to a different consensus state. Our study focuses on two main objectives: i) determining the conditions under which the committed minority succeeds in its goal, and ii) identifying the optimal placement for such a committed minority. After deriving general monotone convergence result for the controlled dynamics, we leverage these results to build a computationally-efficient algorithm to solve the first problem and an effective heuristics for the second problem, which we prove to be NP-complete. For both algorithms, we establish theoretical guarantees. The proposed methodology is illustrated though academic examples, and demonstrated on a real-world case study.

Lorenzo Zino, Mengbin Ye, Brian D. O. Anderson

We study a continuous-time dynamical system of nodes diffusively coupled over a hierarchical network to examine the efficiency and performance tradeoffs that organizations, teams, and command and control units face while achieving coordination and sharing information across layers. Specifically, after defining a network structure that captures real-world features of hierarchical organizations, we use linear systems theory and perturbation theory to characterize the rate of convergence to a consensus state, and how effectively information can propagate through the network, depending on the breadth of the organization and the strength of inter-layer communication. Interestingly, our analytical insights highlight a fundamental performance tradeoff. Namely, networks that favor fast coordination will have decreased ability to share information that is generated in the lower layers of the organization and is to be passed up the hierarchy. Numerical results validate and extend our theoretical results.

Ella C. Davidson, Lorenzo Zino, Ming Cao et al.

This paper proposes a mathematical model for the coevolution of actions and opinions for a population facing a social dilemma. In particular, we assume each person participates in a Public Goods Game (PGG), with their action being to cooperate or defect, and holds an opinion about which action they prefer. We propose a payoff function that combines the PGG with the Friedkin--Johnsen model from opinion dynamics to form a coevolutionary game. According to a discrete-time process, players asynchronously update their actions and opinions, aiming to maximise their individual payoff for the coevolutionary game using myopic best-response. We study the equilibria and provide conditions for the existence of the all-defection and all-cooperation consensus equilibria. We also establish conditions for global convergence to the all-defection equilibrium.

Martina Alutto, Lorenzo Zino, Karl H. Johansson et al.

Leadership in social groups is often a dynamic characteristic that emerges from interactions and opinion exchange. Empirical evidence suggests that individuals with strong opinions tend to gain influence, at the same time maintaining alignment with the social context is crucial for sustained leadership. Motivated by the social psychology literature that supports these empirical observations, we propose a novel dynamical system in which opinions and leadership co-evolve within a social network. Our model extends the Friedkin-Johnsen framework by making susceptibility to peer influence time-dependent, turning it into the leadership variable. Leadership strengthens when an agent holds strong yet socially aligned opinions, and declines when such alignment is lost, capturing the trade-off between conviction and social acceptance. After illustrating the emergent behavior of this complex system, we formally analyze the coupled dynamics, establishing sufficient conditions for convergence to a non-trivial equilibrium, and examining two time-scale separation regimes reflecting scenarios where opinion and leadership evolve at different speeds.

Mengbin Ye, Lorenzo Zino, Ming Cao

For a society to remain healthy and prosperous, people must collectively behave and act to contribute to the common good, even if there is often a tradeoff against their individual benefit. Paradigmatic examples include the adoption of sustainable behaviors and technologies to combat the climate crisis, and the mobilization for collective action to promote the rights and freedoms of repressed minorities. In this tutorial, we illustrate how game theory and network systems theory can be powerful tools to model and study this collective decision-making problem. We provide examples of how awareness of this tradeoff can impact collective change toward the societal good, exploring different problem contexts such as sustainable behavior and collective action. Finally, we review recent developments using systems and control-theoretic approaches to generate awareness and guide the emergent population dynamics towards a desired outcome, and conclude by highlighting new research and application frontiers.

Alessandro Licciardi, Roberta Raineri, Anton Proskurnikov et al.

Federated Learning (FL) enables privacy-preserving collaborative model training, but its effectiveness is often limited by client data heterogeneity. We introduce a client-selection algorithm that (i) dynamically forms nonoverlapping coalitions of clients based on asymptotic agreement and (ii) selects one representative from each coalition to minimize the variance of model updates. Our approach is inspired by social-network modeling, leveraging homophily-based proximity matrices for spectral clustering and techniques for identifying the most informative individuals to estimate a group's aggregate opinion. We provide theoretical convergence guarantees for the algorithm under mild, standard FL assumptions. Finally, we validate our approach by benchmarking it against three strong heterogeneity-aware baselines; the results show higher accuracy and faster convergence, indicating that the framework is both theoretically grounded and effective in practice.

Lorenzo Zino, Giacomo Como, Fabio Fagnani

Imitation dynamics for population games are studied and their asymptotic properties analyzed. In the considered class of imitation dynamics - that encompass the replicator equation as well as other models previously considered in evolutionary biology - players have no global information about the game structure, and all they know is their own current utility and the one of fellow players contacted through pairwise interactions. For potential population games, global asymptotic stability of the set of Nash equilibria of the sub-game restricted to the support of the initial population configuration is proved. These results strengthen (from local to global asymptotic stability) existing ones and generalize them to a broader class of dynamics. The developed techniques highlight a certain structure of the problem and suggest possible generalizations from the fully mixed population case to imitation dynamics whereby agents interact on complex communication networks.