Este estudo visa, por meio de simulações computacionais, compreender de que modo a centralidade dos agentes, os diferentes tipos de rede e o não- compartilhamento da informação afetam os processos de adoção e de difusão de diferentes tipos de inovações. Para esta tarefa, foram feitas simulações com os modelos de rede descritos por Watts e Strogatz (WATTS & STROGATZ, 1998), com um número fixo de 100 nós ou agentes (n=100), em que foram variados os parâmetros de mi (centralidade de grau inicial) e p (probabilidade de reconexão desses nós). Foram utilizadas redes regulares (p=0), de pequeno-mundo (p=0.5) e aleatórias (p=1.0). Escolhemos os adotantes iniciais por 5 diferentes métodos: por maiores centralidades de grau, de proximidade, de intermediação e de Bonacich, além da escolha aleatória. Também consideramos dois tipos de agentes com diferentes características: o primeiro, chamado de social, compartilhava e recebia informação, o segundo, chamado de egoísta, recebia e não compartilhava informação. Para parte das simulações, utilizamos uma inovação contínua e, para outra parte, utilizamos uma inovação disruptiva. Como resultado, constatamos, para as simulações realizadas neste modelo, o maior crescimento do número de adotantes não dependeu somente das características da rede, mas também dos agentes que a compõem, além da própria inovação. Por esse motivo, temos que o modelo disruptivo favorece a inovação e a presença de agentes egoístas funciona como um obstáculo ou barreira / This study aims, through computational simulations, to understand how the influence of the different types of network, the centrality of agents and the non-sharing of information affect the processes of adoption and diffusion in different types of innovations. For this task, simulations were made with the network models described by Watts and Strogatz (WATTS & STROGATZ, 1998), with a fixed number of 100 nodes or agents (n = 100), in which the parameters mi (initial degree centrality) and p (probability of reconnection of these nodes) were varied. We used regular (p=0), small- world (p=0.5) and random networks (p=1.0). We choose the initial adopters by 5 different methods: by greater degree, closeness, of betweeness and eingevector centralities, besides the random choice. We also considered two types of agents with different characteristics: the first, called social, shared and received information, the second, called selfish, received and did not share information. For part of the simulations, we use a continuous innovation and, for another part, we use a disruptive innovation. As a result, for the simulations carried out in this model, the greatest growth in the number of adopters was not only dependent on the characteristics of the network, but also on the agents that compose it, besides the innovation itself. For this reason, we think that the disruptive model favors innovation and the presence of selfish agents as an obstacle or barrier

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Die vorliegende Arbeit behandelt die kollektive Dynamik identischer Klasse-I-anregbarer Elemente. Diese können im Rahmen der nichtlinearen Dynamik als Systeme nahe einer Sattel-Knoten-Bifurkation auf einem invarianten Kreis beschrieben werden. Der Fokus der Arbeit liegt auf dem Studium aktiver Rotatoren als Prototypen solcher Elemente. In Teil eins der Arbeit besprechen wir das klassische Modell abstoßend gekoppelter aktiver Rotatoren von Shinomoto und Kuramoto und generalisieren es indem wir höhere Fourier-Moden in der internen Dynamik der Rotatoren berücksichtigen. Wir besprechen außerdem die mathematischen Methoden die wir zur Untersuchung des Aktive-Rotatoren-Modells verwenden. In Teil zwei untersuchen wir Existenz und Stabilität periodischer Zwei-Cluster-Lösungen für generalisierte aktive Rotatoren und beweisen anschließend die Existenz eines Kontinuums periodischer Lösungen für eine Klasse Watanabe-Strogatz-integrabler Systeme zu denen insbesondere das klassische Aktive-Rotatoren-Modell gehört und zeigen dass (i) das Kontinuum eine normal-anziehende invariante Mannigfaltigkeit bildet und (ii) eine der auftretenden periodischen Lösungen Splay-State-Dynamik besitzt. Danach entwickeln wir mit Hilfe der Averaging-Methode eine Störungstheorie für solche Systeme. Mit dieser können wir Rückschlüsse auf die asymptotische Dynamik des generalisierten Aktive-Rotatoren-Modells ziehen. Als Hauptergebnis stellen wir fest dass sowohl periodische Zwei-Cluster-Lösungen als auch Splay States robuste Lösungen für das Aktive-Rotatoren-Modell darstellen. Wir untersuchen außerdem einen "Stabilitätstransfer" zwischen diesen Lösungen durch sogenannte Broken-Symmetry States. In Teil drei untersuchen wir Ensembles gekoppelter Morris-Lecar-Neuronen und stellen fest, dass deren asymptotische Dynamik der der aktiven Rotatoren vergleichbar ist was nahelegt dass die Ergebnisse aus Teil zwei ein qualitatives Bild für solch kompliziertere und realistischere Neuronenmodelle liefern. / We study the collective dynamics of class I excitable elements, which can be described within the theory of nonlinear dynamics as systems close to a saddle-node bifurcation on an invariant circle. The focus of the thesis lies on the study of active rotators as a prototype for such elements. In part one of the thesis, we motivate the classic model of repulsively coupled active rotators by Shinomoto and Kuramoto and generalize it by considering higher-order Fourier modes in the on-site dynamics of the rotators. We also discuss the mathematical methods which our work relies on, in particular the concept of Watanabe-Strogatz (WS) integrability which allows to describe systems of identical angular variables in terms of Möbius transformations. In part two, we investigate the existence and stability of periodic two-cluster states for generalized active rotators and prove the existence of a continuum of periodic orbits for a class of WS-integrable systems which includes, in particular, the classic active rotator model. We show that (i) this continuum constitutes a normally attracting invariant manifold and that (ii) one of the solutions yields splay state dynamics. We then develop a perturbation theory for such systems, based on the averaging method. By this approach, we can deduce the asymptotic dynamics of the generalized active rotator model. As a main result, we find that periodic two-cluster states and splay states are robust periodic solutions for systems of identical active rotators. We also investigate a 'transfer of stability' between these solutions by means of so-called broken-symmetry states. In part three, we study ensembles of higher-dimensional class I excitable elements in the form of Morris-Lecar neurons and find the asymptotic dynamics of such systems to be similar to those of active rotators, which suggests that our results from part two yield a suitable qualitative description for more complicated and realistic neural models.