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In this paper we propose an extended version of the Friedkin–Johnsen (FJ) model where we analyze the effects that a homophily-based influence matrix has on the opinion formation process. In particular, we assume that the influence matrix varies over time, and its entries (that represent the appraisals that individuals have of the other individuals) update based on the comparison between their opinions and can take both positive and negative values. This leads to a system of two difference equations, where the first one corresponds to the standard FJ model, except that the influence matrix is no longer constant, and the second one models the dynamics of this matrix. We show that a necessary and sufficient condition for the convergence of this modified version of the classical FJ model is that the influence matrix becomes constant in a finite number of steps. Moreover, we provide the explicit expression for the agents' asymptotic opinions in some special cases: the purely cooperative setting, the case of a structurally balanced network and the case of a single discussion topic. Finally, the case when the topics are correlated and the influence matrix depends on the average opinion vectors is investigated. [ABSTRACT FROM AUTHOR]

In this paper we investigate the consensus problem under arbitrary switching for homogeneous multi-agent systems with switching communication topology, by assuming that each agent is described by a single-input stabilizable state–space model and that the communication graph is connected at every time instant. Under these assumptions, we construct a common quadratic positive definite Lyapunov function for the switched system describing the evolution of the disagreement vector, thus showing that the agents always reach consensus. In addition, the proof leads to the explicit construction of a constant state-feedback matrix that allows the multi-agent system to achieve consensus. [ABSTRACT FROM AUTHOR]

In this paper we investigate the herdability property, namely the capability of a system to be driven towards the (interior of the) positive orthant, for linear time-invariant state space models. Herdability of certain matrix pairs (A , B) , where A is the adjacency matrix of a multi-agent network, and B a selection matrix that singles out a subset of the agents (the "network leaders"), is explored. The cases when the graph associated with A , G (A) , is directed and clustering balanced (in particular, structurally balanced), or it has a tree topology and there is a single leader, are investigated. [ABSTRACT FROM AUTHOR]

Recently a model for the interplay between homophily-based appraisal dynamics and influence-based opinion dynamics has been proposed. The model explores for the first time how the opinions of a group of agents on a certain number of issues/topics is influenced by the agents' mutual appraisal and, conversely, the agents' mutual appraisal is updated based on the agents' opinions on the various issues, according to a homophily model. In this paper we show that a simplified (and, in some situations, more feasible) version of the model, that accounts only for the signs of the agents' appraisals rather than for their numerical values, provides an equally accurate and effective model of the opinion dynamics in small networks. The equilibria reached by this model correspond, almost surely, to situations in which the agents' network is complete and structurally balanced. On the other hand, we ensure that such equlibria can always be reached in a finite number of steps, and, differently from the original model, we rule out other types of equilibria that correspond to disconnected social networks. [ABSTRACT FROM AUTHOR]

In this paper we address the consensus problem in the context of networked agents whose communication graph splits into clusters: interactions between agents in the same cluster are cooperative, while interactions between agents belonging to different clusters are antagonistic. This problem set-up arises in the context of social networks and opinion dynamics, where reaching a consensus means that the opinions of the agents in the same cluster converge to the same decision, that is typically different for the different clusters. Under the assumption that agents belonging to the same cluster have the same amount of trust (/distrust) to be distributed among their cooperators (/adversaries), we propose a modified version of DeGroot's law. By simply constraining how much agents in each group should be conservative about their own opinions, it is possible to achieve a nontrivial solution by means of a distributed algorithm. The result is then particularized to unweighted complete communication graphs, and subsequently extended to a class of nonlinear multi-agent systems. [ABSTRACT FROM AUTHOR]

Abstract: In this note we first characterize the periodic trajectories (or, equivalently, the limit cycles) of a Boolean network, and their global attractiveness. We then investigate under which conditions all the trajectories of a Boolean control network may be forced to converge to the same periodic trajectory. If every trajectory can be driven to such a periodic trajectory, this is possible by means of a feedback control law. [Copyright &y& Elsevier]

Abstract: In this paper we consider discrete-time positive switched systems, switching among autonomous subsystems, characterized either by monomial matrices or by circulant matrices. Necessary and sufficient conditions are provided guaranteeing either (global uniform) asymptotic stability or stabilizability (i.e. the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence). Such conditions lead to simple algorithms that allow to easily detect, under suitable conditions, whether a given positive switched system is not stabilizable. [Copyright &y& Elsevier]

Abstract: The aim of this paper is that of investigating the asymptotic exponential cone of a single Metzler matrix, introduced in [23], and of defining and analysing the new concept of asymptotic exponential cone of a family of Metzler matrices (along a certain direction). These results will provide necessary and/or sufficient conditions for the solvability of an interesting algebraic problem that arises in the context of continuous-time positive switched systems and, specifically, in the investigation of the reachability property [21,22,25]. [Copyright &y& Elsevier]

Abstract: In this paper, the zero pattern properties and the asymptotic evolution of the trajectories of a continuous-time positive system are investigated. To this end, we need to introduce some new tools and to derive some new results, within the broad research area of nonnegative matrix theory, which enable use to explore the zero pattern and the elementary modes of the exponential of a Metzler matrix. Specifically, a normal form for the exponential of a Metzler matrix is provided, and the concept of echelon basis (consisting of eigenvectors and generalized eigenvectors of the Metzler matrix) is introduced. By making use of these two ingredients, a detailed result about the dominant mode of each single block appearing in the normal form of the exponential matrix is provided. This allows to obtain a “modal decomposition” of the exponential matrix, emphasizing the column dominant modes. As a result, the zero pattern as well as the asymptotic behavior of every free state evolution, depending on the zero pattern of the initial state, can be easily determined. [Copyright &y& Elsevier]

Abstract: Reachability and observability of two-dimensional (2D) discrete state-space models are introduced in two different forms: a local form, which refers to single local states, and a global form, which pertains to the infinite set of local states lying on a separation set [M. Bisiacco, State and output feedback stabilizability of 2D systems, IEEE Trans. Circ. Syst., CAS-32 (1985) 1246–1249; E. Fornasini, G. Marchesini, Global properties and duality in 2-D systems, Syst. Control Lett. 2 (1) (1982) 30–38]. While local reachability and observability can be naturally characterized by resorting to classical state space techniques, their global counterparts are better addressed by means of polynomial techniques. In this paper, reachability and observability are introduced in the context of 2D positive systems and their global versions investigated via a polynomial approach. Necessary and sufficient conditions for the existence of these properties are provided and, in particular, polynomial canonical forms for globally reachable/observable positive systems with scalar inputs/scalar outputs are provided. [Copyright &y& Elsevier]

In this paper, discrete time systems defined on Z+ are analyzed within the behavioral approach. Once the main internal properties of such behaviors have been defined and characterized, our attention is focused on those aspects that make behaviors defined on Z+ dramatically different from behaviors defined on Z. Specifically, the general lack of permanence and the role of finite support trajectories. Several consequences of these two facts are analyzed. In particular, it is shown that while controllable behaviors with trajectories on Z can always be identified by means of a finite set of finite support trajectories, for controllable behaviors with trajectories on Z+ this is no more possible. [Copyright &y& Elsevier]

In the paper output feedback control of Boolean control networks (BCNs) is investigated. First, necessary and sufficient conditions for the existence of a time-invariant output feedback (TIOF) law, stabilizing the BCN to some equilibrium point, are given, and constructive algorithms to test the existence of such a feedback law are proposed. Two sufficient conditions for the existence of a stabilizing time-varying output feedback (TVOF) are then given. Finally, an example concerning the lac Operon in the bacterium Escherichia Coli is presented, to illustrate the effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

Abstract: We consider Boolean control networks (BCNs), and in particular Boolean networks (BNs), in the framework of symbolic dynamics (SD). We show that the set of state-space trajectories of a BCN is a shift space of finite type (SFT). This observation allows us to extend two important analysis tools from SD, namely, the Artin–Mazur zeta function and the topological entropy, to BNs and BCNs. Some of the theoretical results are illustrated using a BCN model of the core network regulating the mammalian cell cycle. [Copyright &y& Elsevier]