When agent-based models are developed to capture opinion formation in large-scale populations, the opinion update equations often need to embed several complex psychological traits. The resulting models are more realistic, but also challenging to assess analytically, and hence numerical analysis techniques have an increasing importance in their study. Here, we propose the Qualitative Outcome Likelihood (QOL) analysis, a novel probabilistic analysis technique aimed to unravel behavioural patterns and properties of agent-based opinion formation models, and to characterise possible outcomes when only limited information is available. The QOL analysis reveals which qualitative categories of opinion distributions a model can produce, brings to light their relation to model features such as initial conditions, agent parameters and underlying digraph, and allows us to compare the behaviour of different opinion formation models. We exemplify the proposed technique by applying it to four opinion formation models: the classical Friedkin-Johnsen model and Bounded Confidence model, as well as the recently proposed Backfire Effect and Biased Assimilation model and Classification-based model.

We proposed network-decentralized control strategies, in which each actuator can exclusively rely on local information, without knowing the network topology and the external input, ensuring that the flow asymptotically converges to the optimal one with respect to the p -norm. For 1 < p < ∞ , the flow converges to a unique constant optimal up∗. We show that the state converges to the optimal Lagrange multiplier of the optimization problem. Then, we consider networks where the flows are affected by unknown spontaneous dynamics and the buffers need to be driven exactly to a desired set-point. We propose a network-decentralized proportional-integral controller that achieves this goal along with asymptotic flow optimality; now it is the integral variable that converges to the optimal Lagrange multiplier. The extreme cases p=1 and p=∞ are of some interest since the former encourages sparsity of the solution while the latter promotes fairness. Unfortunately, for p=1 or p=∞ these strategies become discontinuous and lead to chattering of the flow, hence no optimality is achieved. We then show how to approximately achieve the goal as the limit for p 1 or p ∞.

Comparing model predictions with real data is crucial to improve and validate a model. For opinion formation models, validation based on real data is uncommon and difficult to obtain, also due to the lack of systematic approaches for a meaningful comparison. We introduce a framework to assess opinion formation models, which can be used to determine the qualitative outcomes that an opinion formation model can produce, and compare model predictions with real data. The proposed approach relies on a histogram-based classification algorithm, and on transition tables. The algorithm classifies an opinion distribution as perfect consensus, consensus, polarization, clustering, or dissensus; these qualitative categories were identified from World Values Survey data. The transition tables capture the qualitative evolution of the opinion distribution between an initial and a final time. We compute the real transition tables based on World Values Survey data from different years, as well as the predicted transition tables produced by the French-DeGroot, Weighted-Median, Bounded Confidence, and Quantum Game models, and we compare them. Our results provide insight into the evolution of real-life opinions and highlight key directions to improve opinion formation models.

We give a sufficient and a necessary condition for the topology-independent robust stability of networked systems formed by uncertain MIMO systems. Both conditions involve constants associated with the nominal node dynamics and arc interconnection matrices, the uncertainty bounds, and the maximum connectivity degree of the network; they are scalable (they can be checked locally), independent of the network topology and even of the number of nodes and arcs, and hold for networks of heterogeneous MIMO systems and interconnection matrices, with heterogeneous uncertainties. The dual cases of 1-norm and ∞-norm bounds are considered. In both cases, if the systems at the nodes are diagonal, we get a necessary and sufficient condition. We apply our results to the topology-independent robust stability analysis of a case-study from cancer biology.

We study dynamic networks described by a directed graph where the nodes are associated with MIMO systems with transfer-function matrix F(s), representing individual dynamic units, and the arcs are associated with MIMO systems with transfer-function matrix G(s), accounting for the dynamic interactions among the units. In the nominal case, we provide a topology-independent condition for the stability of all possible dynamic networks with a maximum connectivity degree, regardless of their size and interconnection structure. When node and arc transfer-function matrices are affected by norm-bounded homogeneous uncertainties, the robust condition for size- and topology-independent stability depends on the uncertainty magnitude. Both conditions, expressed as constraints for the Nyquist diagram of the poles of the transfer-function matrix H(s) = F(s)G(s), are scalable and can be checked locally to guarantee stability-preserving “plug-and-play” addition of new nodes and arcs.