Agent-based models of opinion formation are becoming increasingly complex, because of their size and of the embedding of several individual psychological traits of the agents, aimed at realistically capturing the multifaceted aspects of social interaction. Therefore, the characterisation of the model properties mostly relies on simulation-based numerical approaches: more techniques are needed to analyse, contrast, and compare the properties of different models. We propose a novel graphical technique, which relies on the Agreement Plot to visualise the evolution of opinion distributions over time, that allows us to unveil behavioural patterns and capabilities of agent-based opinion formation models. Our proposed approach can be used to characterise the relation between global properties of the model evolution and the model features (initial opinion distributions, agent parameters, underlying digraphs), and is here showcased through its application to both seminal and recently proposed opinion formation models.@en

An interconnected system is composed of multiple well-defined self-contained subsystems that interact among them and that together create collective behaviours. We can find many examples of interconnected systems in real life. Ranging from biological systems, such as the growth and interaction of populations in diverse and spatially distributed environments, to electric grids connecting power-generating sources, buildings and infrastructures in a country. When studying interconnected systems, a fundamental and natural question is how the properties and characteristics of the individual subsystems and the way they are connected relate to the collective behaviour of the complete system. That is the driving question of the present dissertation. Given that interconnected systems can be found in a wide variety of contexts, their representation and specific research interests can be equally varied. Because of this, it is impossible to answer the aforementioned question uniquely for all interconnected systems, and specific cases must be considered. In this dissertation, we consider two types of interconnected systems: a general class of uncertain multiple-input-multipleoutput (MIMO) systems, and agent-based opinion formation models. The investigation of uncertain MIMO interconnected systems is focused on providing topology-independent conditions for robust stability. The primary motivation for this approach is that, in real systems, it is costly or even impossible to have complete and accurate information on the network topology and subsystem parameters and dynamics. However, it is of critical interest to guarantee the system’s stability. Therefore we need stability conditions that require only partial information about the network and the subsystems to ensure the system’s stability. By studying these systems both in the time and frequency domain, we are able to provide conditions thatmeet these requirements. As for agent-based opinion formation models, we assume that each individual (or agent) in a population has an opinion about a statement. By exchanging opinions among themselves, the agents update their own internal opinion, resulting in a collective dynamic of opinion evolution. When studying these systems, the interests shifts from stability conditions, to a characterisation of the relation between the agents’ individual traits and qualitative properties of the opinion distribution in the population. Several techniques and approaches to analyse opinion formation models are proposed and applied to multiple models, one of which is new to this dissertation. The collective study of the previously mentioned interconnected systems requires the use of multiple and diverse analysis techniques and approaches, from analytical methods based on the Nyquist criterion, Bauer-Fike theorem, and Lyapunov functions to qualitative and numerical analysis techniques like histograms and binomial proportion confidence intervals. It is our hope that some of the presented results, methods, or ideas may advance the knowledge frontier in this scientific field, sparkle new research directions, and either directly or indirectly prove some value to society. @en

We propose an agent-based opinion formation model characterised by a two-fold novelty. First, we realistically assume that each agent cannot measure the opinion of its neighbours about a given statement with infinite resolution and accuracy, and hence it can only perceive the opinion of others as agreeing much more, or more, or comparably, or less, or much less (than itself) with that given statement. This leads to a classification-based rule for opinion update. Second, we consider three complementary agent traits suggested by significant sociological and psychological research: conformism, radicalism and stubbornness. We rely on World Values Survey data to show that the proposed model has the potential to predict the evolution of opinions in real life: the classification-based approach and complementary agent traits produce rich collective behaviours, such as polarisation, consensus, and clustering, which can yield predicted opinions similar to survey results. @en

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.

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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 ∞.

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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.

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We consider network systems where the node dynamics are described by identical MIMO LTI subsystems with transfer-function matrix F(s), while the dynamic interactions associated with the bidirectional arcs are described by identical MIMO LTI subsystems with transfer-function matrix G(s); the dynamics of the individual nodes and arcs are affected by heterogeneous, norm-bounded uncertainties. We provide a topology-independent condition for the robust stability of all possible network systems with a maximum connectivity degree, regardless of their size and interconnection structure. We also give a topology-independent condition that robustly guarantees not only stability, but also α-convergence (i.e. all poles having real part less than a negative – α). The proposed frequency-domain conditions are scalable and can be evaluated locally, also for large-scale networks where nodes and arcs can be added or removed in real time. The conditions are applied to assess the robust α-convergence of a suspension bridge system of arbitrary size.@en

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.

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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.@en

We consider a multi-compartment evolutionary model representing growth, mutation and migration of cancer cells, as well as the effect of drugs, and we design optimal switching targeted cancer therapies where a single drug, or suitable drug combination, is given at each time so as to minimise not only the overall tumor size over a finite horizon, but also drug-provoked side effects. The strong diagonally- dominant structure of the model allows to solve the problem via convex optimisation. We provide an algorithm that yields optimality throughout the whole treatment duration by solving the convex optimisation problem with different horizons, and show how dwell time can be enforced via heuristics. Also the optimal treatment duration can be computed via convex optimisation. The proposed approaches are applied to a model of ALK-rearranged lung carcinoma.@en