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Abstract

Standard two-state (Boolean) threshold networks have been used to model a broad range of social and biological systems. In this paper, we generalize this class of systems to arbitrary finite sets with nonsymmetric thresholds. For this new class of systems, we derive sufficient conditions on the threshold parameters to ensure that the limit sets are fixed points. In contrast to the standard Boolean threshold networks, this broader class can have long periodic orbits and here we identify bifurcation points of these systems. Our focus is mainly on asynchronous systems, but we also discuss synchronous systems. The extension we introduce is directly motivated by applications in the social sciences. However, we also expect that our results will be useful for modeling biological phenomena where a finer level of expression than 0/1 or on/off is needed.

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... The GDSC application can be used for both research and education. As evidence for the former point, we note that three works [8][9][10] used GDSC to identify experimentally dynamical system behaviors that were then rigorously proved as general characterizations of GDSs. Thus, GDSC is a useful tool for experimental mathematics and computational mathematics, where computational studies are used to guide formulation of theorems and provide insights for their proofs. ...
... The first three illustrate our use of GDSC to compute dynamics, which we then used to prove more general results. These theoretical results have been published [8][9][10]. The fourth study demonstrates the wider applicability of GDSC by illustrating how dynamical systems used by other researchers (e.g., [5]) can also be modeled in this framework. ...
... Often, the problem facing the actor is not whether to contribute, but how much to contribute." In [10], we study GDSs where there can be any finite number r of states, that is, K = {0, 1, 2, . . ., r − 1}. ...
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Three studies (N = 1988) describe the development and validation of the Emotional Contagion (EC) Scale, a 15-item unidimensional measure of susceptibility to others'' emotions resulting from afferent feedback generated by mimicry. Study 1 assesses the EC Scale''s reliability (Cronbach''s = .90). Study 2 finds susceptibility (a) positively related to reactivity, emotionality, sensitivity to others, social functioning, self-esteem, and more associated with emotional than cognitive modes of empathy, (b) negatively related to alienation, self-assertiveness, and emotional stability and, (c) unrelated to masculinity and approval motivation. Study 3, an experiment, finds that EC Scale scores reliably predict biases in participants'' evaluations and are correlated with a measure of responsiveness to afferent feedback and self-reports of emotional experience following exposure to emotional expressions.
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We propose a discrete model for how opinions about a given “extreme” subject, about which various groups of a population have different degrees of enthusiasm for or susceptibility to, such as fanaticism, extreme social and political positions, and terrorism, may spread. The model, in a certain limit, is the discrete analogue of a deterministic continuum model suggested by others. We carry out extensive computer simulation of the model by utilizing it on lattices with infinite- or short-range interactions, and on symmetric and hierarchical (or directed) Barabási–Albert scale-free networks. Several interesting features of the model are demonstrated, and comparison is made with the deterministic continuum model.
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Sequential Dynamical Systems (SDSs) are a special type of finite discrete dynamical systems that can be used to model simulation systems. We focus on the computational complexity of testing several phase space properties of SDSs. Our main result is a sharp delineation between classes of SDSs whose behavior is easy to predict and those whose behavior is hard to predict. Specifically, we show the following.1.Several state reachability problems for SDSs are PSPACE-complete, even when restricted to SDSs whose underlying graphs are of bounded bandwidth (and hence of bounded pathwidth and treewidth), and the function associated with each node is symmetric. Moreover, this result holds even when the underlying graph is d-regular for some constant d and all the nodes compute the same symmetric Boolean function. An immediate corollary of this result is a PSPACE-hard lower bound on the complexity of reachability problems for regular generalized 1D-Cellular Automata and undirected systolic networks with Boolean totalistic local transition functions.2.In contrast, the above reachability problems are solvable in polynomial time for SDSs when the Boolean function associated with each node is symmetric and monotone. The PSPACE-completeness results follow as corollaries of simulation results which show for several classes of SDSs, how one class of SDSs can be efficiently simulated by another (more restricted) class of SDSs. We also prove several structural properties concerning the phase space of an SDS. SDSs are closely related to Cellular Automata (CA), concurrent transition systems, discrete Hopfield networks and systolic networks. This observation in conjunction with our lower bounds for SDSs, yields new PSPACE-hard lower bounds on the complexity of state reachability problems for these models, extending some of the earlier results in [K. Culik II, J. Karhumäki, On totalistic systolic networks, Inform. Process. Lett. 26 (5) (1988) 231–236; P. Floréen, E. Goles, G. Weisbuch, Transient length in sequential iterations of threshold functions, Discrete Appl. Math. 6 (1983) 95–98; P. Floréen, P. Orponen, Complexity issues in discrete Hopfield networks, Research Report No. A-1994-4, Department of Computer Science, University of Helsinki, 1994. Also appears in: I. Parberry (Ed.), Comp. and Learning Complexity of Neural Networks: Advanced Topics, 1999; D. Harel, O. Kupferman, M.Y. Vardi, On the complexity of verifying concurrent transition systems, Inform. and Comput. 173 (2002) 143–161; S.K. Shukla, H.B. Hunt III, D.J. Rosenkrantz, R.E. Stearns, On the complexity of relational problems for finite state processes, in: International Colloquium on Automata Programming and Languages, ICALP, 1996, pp. 466–477; A. Rabinovich, Complexity of equivalence problems for concurrent systems of finite agents, Inform. and Comput. 127 (2) (1997) 164–185].
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Granovetter's threshold model of collective action shows how each new participant triggers others until the chain reaction reaches a gap in the distribution of thresholds. Hence outcomes depend on the network of social ties that channel the chain reactions. However, structural analysis is encumbered by the assumption that thresholds derive from changing marginal returns on investments in public goods. A learning-theoretic specification imposes less stringent assumptions about the rationality of the actors and is much better suited to a structural analysis. Computer simulations suggest that threshold effects may be the key to solving the coordination problem: When individual choices are contingent on participation by others, this interdependence facilitates the coordination of contributions needed to shift the bistable system from a noncooperative equilibrium to a cooperative one. Further simulations with low-density networks show that these chain reactions require bridges that link socially distant actors, supporting Granovetter's case for the strength of weak ties.
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Proto-organisms probably were randomly aggregated nets of chemical reactions. The hypothesis that contemporary organisms are also randomly constructed molecular automata is examined by modeling the gene as a binary (on-off) device and studying the behavior of large, randomly constructed nets of these binary “genes”. The results suggest that, if each “gene” is directly affected by two or three other “genes”, then such random nets: behave with great order and stability; undergo behavior cycles whose length predicts cell replication time as a function of the number of genes per cell; possess different modes of behavior whose number per net predicts roughly the number of cell types in an organism as a function of its number of genes; and under the stimulus of noise are capable of differentiating directly from any mode of behavior to at most a few other modes of behavior. Cellular differentation is modeled as a Markov chain among the modes of behavior of a genetic net. The possibility of a general theory of metabolic behavior is suggested.
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The origin of large but rare cascades that are triggered by small initial shocks is a phenomenon that manifests itself as diversely as cultural fads, collective action, the diffusion of norms and innovations, and cascading failures in infrastructure and organizational networks. This paper presents a possible explanation of this phenomenon in terms of a sparse, random network of interacting agents whose decisions are determined by the actions of their neighbors according to a simple threshold rule. Two regimes are identified in which the network is susceptible to very large cascades-herein called global cascades-that occur very rarely. When cascade propagation is limited by the connectivity of the network, a power law distribution of cascade sizes is observed, analogous to the cluster size distribution in standard percolation theory and avalanches in self-organized criticality. But when the network is highly connected, cascade propagation is limited instead by the local stability of the nodes themselves, and the size distribution of cascades is bimodal, implying a more extreme kind of instability that is correspondingly harder to anticipate. In the first regime, where the distribution of network neighbors is highly skewed, it is found that the most connected nodes are far more likely than average nodes to trigger cascades, but not in the second regime. Finally, it is shown that heterogeneity plays an ambiguous role in determining a system's stability: increasingly heterogeneous thresholds make the system more vulnerable to global cascades; but an increasingly heterogeneous degree distribution makes it less vulnerable.
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We consider a rumor transmission model with various contact interactions and explore what effect such interactions have on the spread of a rumor, in particular whether they can explain the rumor recursion. Through mathematical analysis and computer simulations, we conjecture that rumor recursion remains a major challenge to mathematical models of rumors beyond our model proposed here.
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L'int�r�t de l'approche par les jeux globaux ("global games'') est pr�cis�ment d'ancrer les anticipations sur des variables exog�nes r�elles. On peut ainsi garder l'aspect auto-r�alisateur des anticipations mais en restaurant l'unicit� de l'�quilibre et donc un meilleur pouvoir pr�dictif du mod�le. Nous illustrons ces m�canismes sur deux exemples. Le premier a trait au choix r�sidentiel d'agents qui ont une pr�f�rence "identitaire''. Le second a trait � la contagion de paniques bancaires d'un pays � un autre. De mani�re plus g�n�rale, tous les jeux qui pr�sentent des compl�mentarit�s strat�giques sont susceptibles d'�tre analys�s au moyen des techniques des "global games''. Il convient toutefois de rappeler que les techniques utilis�es demeurent assez sp�cifiques: l'incertitude strat�gique porte essentiellement sur les croyances de premier degr� des autres acteurs. Or, si de mani�re plus g�n�rale on suppose que cette incertitude peut porter sur des ordres plus �lev�s, les conclusions des mod�les peuvent changer. Ainsi, Weinstein et Yildiz (2004) montrent que dans un oligopole de Cournot, il y a une tr�s grande multiplicit� d'�quilibres si on suppose que l'incertitude porte sur les croyances de niveaux suffisamment �lev�s.
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