The COVID-19 epidemic hit Italy particularly hard, yielding the implementation of strict national lockdown rules. Previous modelling studies at the national level overlooked the fact that Italy is divided into administrative regions which can independently oversee their own share of the Italian National Health Service. Here, we show that heterogeneity between regions is essential to understand the spread of the epidemic and to design effective strategies to control the disease. We model Italy as a network of regions and parameterize the model of each region on real data spanning over two months from the initial outbreak. We confirm the effectiveness at the regional level of the national lockdown strategy and propose coordinated regional interventions to prevent future national lockdowns, while avoiding saturation of the regional health systems and mitigating impact on costs. Our study and methodology can be easily extended to other levels of granularity to support policy- and decision-makers.

Coordinating ensembles of dynamical systems in a decentralized manner is of central importance when controlling complex networks. Pinning control is a much used technique where only a small fraction of the network nodes is directly controlled, with control gains being typically selected uniformly across the control layer. In this paper, we tackle the problem of optimally selecting the control gains of a pinning control strategy. Indeed, in networks of nonlinear dynamical systems fulfilling the so-called QUAD assumption, pinning controllability improves as the smallest eigenvalue λ1 of an extended Laplacian matrix increases. Based on this observation, we pose a constrained optimization problem on the network. Rather than solving it in a centralized fashion, we propose a fully decentralized multilayer approach. Specifically, one layer is used to evaluate the sensitivity of λ1 to the variation of the gains, while the second layer uses such an estimate to dynamically tune the control gains. The effectiveness of the approach is demonstrated via a representative example.

In this paper, we model the problem of influencing the opinions of groups of individuals as a containment control problem, as in many practical scenarios, the control goal is not full consensus among all the individual opinions, but rather their containment in a certain range, determined by a set of leaders. As in classical bounded confidence models, we consider individuals affected by the confirmation bias, thus tending to influence and to be influenced only if their opinions are sufficiently close. However, here we assume that the confidence level, modeled as a proximity threshold, is not constant and uniform across the individuals, as it depends on their opinions. Specifically, in an extremist society, the most radical agents (i.e., those with the most extreme opinions) have a higher appeal and are capable of influencing nodes with very diverse opinions. The opposite happens in a moderate society, where the more connected (i.e., influential) nodes are those with an average opinion. In three artificial societies, characterized by different levels of extremism, we test through extensive simulations the effectiveness of three alternative containment strategies, where leaders have to select the set of followers they try to directly influence. We found that, when the network size is small, a stochastic time-varying pinning strategy that does not rely on information on the network topology proves to be more effective than static strategies where this information is leveraged, while the opposite happens for large networks where the relevance of the topological information is prevalent.

Network reciprocity has been successfully put forward (since M. A. Nowak and R. May’s, 1992, influential paper) as the simplest mechanism—requiring no strategical complexity—supporting the evolution of cooperation in biological and socioeconomic systems. The mechanism is actually the network, which makes agents’ interactions localized, while network reciprocity is the property of the underlying evolutionary process to favor cooperation in sparse rather than dense networks. In theoretical models, the property holds under imitative evolutionary processes, whereas cooperation disappears in any network if imitation is replaced by the more rational best-response rule of strategy update. In social experiments, network reciprocity has been observed, although the imitative behavior did not emerge. What did emerge is a form of conditional cooperation based on direct reciprocity—the propensity to cooperate with neighbors who previously cooperated. To resolve this inconsistency, network reciprocity has been recently shown in a model that rationally confronts the two main behaviors emerging in experiments—reciprocal cooperation and unconditional defection—with rationality introduced by extending the best-response rule to a multi-step predictive horizon. However, direct reciprocity was implemented in a non-standard way, by allowing cooperative agents to temporarily cut the interaction with defecting neighbors. Here, we make this result robust to the way cooperators reciprocate, by implementing direct reciprocity with the standard tit-for-tat strategy and deriving similar results.

In this paper, we deal with the containment control problem in presence of antagonistic interactions. In particular, we focus on the cases in which it is not possible to contain the entire network due to a constrained number of control signals. In this scenario, we study the problem of selecting the nodes where control signals have to be injected to maximize the number of contained nodes. Leveraging graph condensations, we find a suboptimal and computationally efficient solution to this problem, which can be implemented by solving an integer linear problem. The effectiveness of the selection strategy is illustrated through representative simulations.