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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}. ...

Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interactions between agents as edges. Agents are in one of a finite set of states at each discrete time step and are assigned functions that describe how their states change based on neighborhood relations. Full characterization of state transitions of one system can give insights into fundamental behaviors of other dynamical systems. In this paper, we describe a discrete graph dynamical systems (GDSs) application called GDSCalc for computing and characterizing system dynamics. It is an open access system that is used through a web interface. We provide an overview of GDS theory. This theory is the basis of the web application; i.e., an understanding of GDS provides an understanding of the software features, while abstracting away implementation details. We present a set of illustrative examples to demonstrate its use in education and research. Finally, we compare GDSCalc with other discrete dynamical system software tools. Our perspective is that no single software tool will perform all computations that may be required by all users; tools typically have particular features that are more suitable for some tasks. We situate GDSCalc within this space of software tools.

... Traditional binary-state threshold models have been extensively studied, and recently multistate threshold models been proposed. Melnik et al. [51] introduced a``progressive"" three-state model, where agents have two thresholds, while Kuhlman and Mortviet [41] extended this formalism, allowing an arbitrary number of states and transitions from each state to every other state. A similar nonlinear model is the multistate majority rule model [13], where every node has a multitude of opinions and nodes change their opinion depending on the majority opinion of groups to which they belong. ...

... Our observations here have important implications for the study of models with kinetic constraints. Threshold models are widely employed models of complex contagion in the social sciences [30,10,52,36], and the AME framework that we have introduced here lends itself to the analysis of the traditional binary-state threshold models [71] as well as more complex multistate threshold models that are the focus of much current attention [51,41]. ...

Multistate dynamical processes on networks, where nodes can occupy one of a multitude of discrete states, are gaining widespread use because of their ability to recreate realistic, complex behaviour that cannot be adequately captured by simpler binary-state models. In epidemiology, multistate models are employed to predict the evolution of real epidemics, while multistate models are used in the social sciences to study diverse opinions and complex phenomena such as segregation. In this paper, we introduce generalized approximation frameworks for the study and analysis of multistate dynamical processes on networks. These frameworks are degree-based, allowing for the analysis of the effect of network connectivity structures on dynamical processes. We illustrate the utility of our approach with the analysis of two specific dynamical processes from the epidemiological and physical sciences. The approximation frameworks that we develop, along with open-source numerical solvers, provide a unifying framework and a valuable suite of tools for the interdisciplinary study of multistate dynamical processes on networks.

... An example of a system characterization is the following: progressive Boolean threshold systems, heavily used in the social sciences 48,71,119 and that are similar to models popularized in 78 , are shown to generate only fixed points as limit cycles 85 . Works with other characterizations include 2,69,83,84,105,121,127 . These types of results, along with those from analysis When two people are co-located, they form a person-person edge in a social network such as that in Fig. 3. Since these interactions give rise to the same person-person edges as in Fig. 3, and the same SIR model vertex functions are used, this network produces the same forward trajectory as that in in Fig. 3. ...

The study of epidemics is useful for not only understanding outbreaks and trying to limit their adverse effects, but also because epidemics are related to social phenomena such as government instability, crime, poverty, and inequality. One approach for studying epidemics is to simulate their spread through populations. In this work, we describe an integrated multi-dimensional approach to epidemic simulation, which encompasses: (1) a theoretical framework for simulation and analysis; (2) synthetic population (digital twin) generation; (3) (social contact) network construction methods from synthetic populations, (4) stylized network construction methods; and (5) simulation of the evolution of a virus or disease through a social network. We describe these aspects and end with a short discussion on simulation results that inform public policy.

... Later, in [40], the authors analyzed how the shape of the dependency graph influences the limit cycle structure of such block-sequential systems, deriving sufficient conditions on this shape which provides only fixed points as limit cycles. Finally, the limit cycle structure for systems whose evolution operator is composed by bi-threshold and multi-threshold functions were considered in [41,42] respectively. ...

In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions.

We study the attractor structure of standard block-sequential threshold dynamical systems. In a block-sequential update, the vertex set of the graph is partitioned into blocks, and the blocks are updated sequentially while the vertices within each block are updated in parallel. There are several notable previous results concerning the two extreme cases of block-sequential update: (i) sequential and (ii) parallel. While parallel threshold systems can have limit cycles of length at most two, sequential systems can have only fixed points. However, Goles and Montealegre [5] showed the existence of block-sequential threshold systems that have arbitrarily long limit cycles. Motivated by this result, we study how the underlying graph structure influences the limit cycle structure of block-sequential systems. We derive a sufficient condition on the graph structure so that the system has only fixed points as limit cycles. We also identify several well-known graph families that satisfy this condition.

A discrete-time dynamical system is proposed to model a class of binary choice games with externalities as those described by Schelling (197320.
Schelling , T. C. ( 1973 ). Hockey helmets, concealed weapons, and daylight saving . Journal of Conflict Resolution , 17 ( 3 ), 381 – 428 . [CrossRef], [Web of Science ®]View all references, 197821.
Schelling , T. C. ( 1978 ). Micromotives and Macrobehavior . New York : W. W. Norton . View all references). In order to analyze some oscillatory time patterns and problems of equilibrium selection that were not considered in the qualitative analysis given by Schelling, we introduce an explicit adjustment mechanism. We perform a global dynamic analysis that allows us to explain the transition toward nonconnected basins of attraction when several coexisting attractors are present. This gives a formal explanation of some overshooting effects in social systems and of the consequent cyclic behaviors qualitatively described in Schelling (197821.
Schelling , T. C. ( 1978 ). Micromotives and Macrobehavior . New York : W. W. Norton . View all references). Moreover, we show how the occurrence of a global bifurcation may lead to the explanation of situations of path dependence and the creation of thresholds observed in real life situations of collective choices, leading to extreme forms of irreversible departure from an equilibrium and uncertainty about the long run evolution of the some social systems.

The overwhelming majority of research on affect and social information processing has focused on the judgments and memories of people in good or bad moods rather than examining more specific kinds of emotional experience within the broad categories of positive and negative affect. Are all varieties of negative affect alike in their impact on social perception? Three experiments were conducted to examine the possibility that different kinds of negative affect (in this case, anger and sadness) can have very different kinds of effects on social information processing. Experiment I showed that angry subjects rendered more stereotypic judgments in a social perception task than did sad subjects, who did not differ from neutral mood subjects. Experiments 2 and 3 similarly revealed a greater reliance upon heuristic cues in a persuasion situation among angry subjects. Specifically, their level of agreement with unpopular positions was guided more by the credibility of the person advocating the position. These findings are discussed in terms of the impact of emotional experience on social information-processing strategies.

Models that provide insight into how extreme positions regarding any social phenomenon may spread in a society or at the global scale are of great current interest. A realistic model must account for the fact that globalization, internet, and other means of mass communications have given rise to scale-free networks of interactions between people. We propose a novel model which takes into account the nature of the interactions network, and provides some key insights into this phenomenon. These include, (1) the existence of a
fundamental difference between a hierarchical network whereby people are influenced by those that are higher in the hierarchy but not by those below them, and a symmetrical network where person-on-person influence works mutually, and (2) that a few “fanatics” can influence a large fraction of the population either temporarily (in the hierarchical networks) or permanently (in symmetrical networks). Even if the “fanatics” disappear, the population may still remain susceptible to the positions originally advocated by them. The
model is, however, general and applicable to any phenomenon for which there is a degree of enthusiasm or susceptibility to in the population.

The spread of information and emotion within groups is studied in models of social diffusion. Evidence has been found that the emotional states of humans affect their information processing abilities, and hence, may affect the spread of information as well. This paper introduces an agent-based model that simulates the spread of information and emotion among a group of agents. The model takes both the influence of emotions upon the spread of information and the influence of information on emotions into account. The approach is exemplified by means of a case study in the domain of emergency evacuation.

The concept of contagion has steadily expanded from its original grounding in epidemic disease to describe a vast array of processes that spread across networks, notably social phenomena such as fads, political opinions, the adoption of new technologies, and financial decisions. Traditional models of social contagion have been based on physical analogies with biological contagion, in which the probability that an individual is affected by the contagion grows monotonically with the size of his or her "contact neighborhood"--the number of affected individuals with whom he or she is in contact. Whereas this contact neighborhood hypothesis has formed the underpinning of essentially all current models, it has been challenging to evaluate it due to the difficulty in obtaining detailed data on individual network neighborhoods during the course of a large-scale contagion process. Here we study this question by analyzing the growth of Facebook, a rare example of a social process with genuinely global adoption. We find that the probability of contagion is tightly controlled by the number of connected components in an individual's contact neighborhood, rather than by the actual size of the neighborhood. Surprisingly, once this "structural diversity" is controlled for, the size of the contact neighborhood is in fact generally a negative predictor of contagion. More broadly, our analysis shows how data at the size and resolution of the Facebook network make possible the identification of subtle structural signals that go undetected at smaller scales yet hold pivotal predictive roles for the outcomes of social processes.

The recent wave of mobilizations in the Arab world and across Western countries has generated much discussion on how digital media is connected to the diffusion of protests. We examine that connection using data from the surge of mobilizations that took place in Spain in May 2011. We study recruitment patterns in the Twitter network and find evidence of social influence and complex contagion. We identify the network position of early participants (i.e. the leaders of the recruitment process) and of the users who acted as seeds of message cascades (i.e. the spreaders of information). We find that early participants cannot be characterized by a typical topological position but spreaders tend to be more central in the network. These findings shed light on the connection between online networks, social contagion, and collective dynamics, and offer an empirical test to the recruitment mechanisms theorized in formal models of collective action.

Project RAP (Risk Avoidance Partnership) trained 112 active drug users to become peer health advocates (PHAs). Six months after baseline survey (N(bl) = 522), 91.6% of PHAs and 56.6% of community drug users adopted the RAP innovation of giving peer intervention, and 59.5% of all participants (N(6m) = 367) were exposed to RAP innovation. Sociometric network analysis shows that adoption of and exposure to RAP innovation was associated with proximity to a PHA or a highly active interventionist (HAI), being directly linked to multiple PHAs/HAIs, and being located in a network sector where multiple PHAs/HAIs were clustered. RAP innovation has diffused into the Hartford drug-using community.

We propose kinetic models for the spread of permanent innovations and
transient fads by the mechanism of social reinforcement. Each individual can be
in one of M+1 states of awareness 0,1,2,...,M, with state M corresponding to
adopting an innovation. An individual with awareness k<M increases to k+1 by
interacting with an adopter. Starting with a single adopter, the time for an
initially unaware population of size N to adopt a permanent innovation grows as
ln(N) for M=1, and as N^{1-1/M} for M>1. The fraction of the population that
remains clueless about a transient fad after it has come and gone changes
discontinuously as a function of the fad abandonment rate lambda for M>1. The
fad dies out completely in a time that varies non-monotonically with lambda.

We introduce the confident voter model, in which each voter can be in one of
two opinions and can additionally have two levels of commitment to an opinion
--- confident and unsure. Upon interacting with an agent of a different
opinion, a confident voter becomes less committed, or unsure, but does not
change opinion. However, an unsure agent changes opinion by interacting with an
agent of a different opinion. In the mean-field limit, a population of size N
is quickly driven to a mixed state and remains close to this state before
consensus is eventually achieved in a time of the order of ln N. In two
dimensions, the distribution of consensus times is characterized by two
distinct times --- one that scales linearly with N and another that appears to
scale as N^{3/2}. The longer time arises from configurations that fall into
long-lived states that consist of two (or more) single-opinion stripes before
consensus is reached. These stripe states arise from an effective surface
tension between domains of different opinions.

This article reviews several classes of theories to elucidate the relationship between adolescent cigarette smoking and friends' cigarette smoking. Perceived influence theories hinge upon an adolescent's perception of friends' smoking behavior. External influence theories are those in which friends' smoking behavior overtly influences adolescent smoking. Group level theories examine how differences at the level of subculture, gender, and race/ethnicity influence the relationship under study. Network theories are also discussed. A model integrating relevant theories into a longitudinal model representing friend influences on adolescent smoking is presented, along with implications of the results presented for adolescent tobacco prevention programs.

We consider the problem of inhibiting undesirable contagions (e.g. rumors, spread of mob behavior) in social networks. Much of the work in this context has been carried out under the 1-threshold model, where diffusion occurs when a node has just one neighbor with the contagion. We study the problem of inhibiting more complex contagions in social networks where nodes may have thresholds larger than 1. The goal is to minimize the propagation of the contagion by removing a small number of nodes (called critical nodes) from the network. We study several versions of this problem and prove that, in general, they cannot even be efficiently approximated to within any factor \(\rho \ge 1\) , unless P = NP. We develop efficient and practical heuristics for these problems and carry out an experimental study of their performance on three well known social networks, namely epinions, wikipedia and slashdot. Our results show that these heuristics perform significantly better than five other known methods. We also establish an efficiently computable upper bound on the number of nodes to which a contagion can spread and evaluate this bound on many real and synthetic networks.

The strength of weak ties is that they tend to be long - they connect socially distant locations, allowing information to diffuse rapidly. The authors test whether this "strength of weak ties" generalizes from simple to complex contagions. Complex contagions require social affirmation from multiple sources. Examples include the spread of high-risk social movements, avant garde fashions, and unproven technologies. Results show that as adoption thresholds increase, long ties can impede diffusion. Complex contagions depend primarily on the width of the bridges across a network, not just their length. Wide bridges are a characteristic feature of many spatial networks, which may account in part for the widely observed tendency for social movements to diffuse spatially.

The cognitive-functional model of discrete negative emotions and attitude change (CFM; Nabi, 1999) attempts to bridge the theoretical gap between "emotional" and "rational" approaches to persuasion by focusing on how emotions motivate attention to and processing of persuasive messages. As a first test of the CFM, this study explored the effects of 2 emotions, anger and fear, and 2 levels of expectation of message reassurance, certainty and uncertainty, on attitudes toward domestic terrorism legislation. Results supported a main effect for emotion type, suggesting that anger promotes deeper information processing than fear, and a main effect for reassurance certainty level, with uncertainty promoting deeper information processing. The expected interaction between emotion type and reassurance expectation level was not found. Implications of these findings for the model and persuasion research generally are discussed.

It is shown that, for a function Δ from {0, 1}n to {0, 1}n whose components from a symmetric set of threshold functions the repeated application of Δ, leads either to a fixed point or to a cycle of length two.

This paper shows how different "network" arguments about how protest spreads imply quite different underlying mechanisms that in turn produce different diffusion processes. There is considerable ambiguity about the relationships among networks, diffusion, and action cycles and the way these can be identified in empirical data. We thus both seek to unpack the "network" concept into different kinds of processes, and then show how these different network processes affect the diffusion processes we are studying. We sketch out some formal models to capture some of these distinctions. This paper extends recent work (Oliver and Myers forthcoming) that develops diffusion models of protest cycles, and focuses on discussing link between network concepts and diffusion concepts in understanding protest cycles. We conceive of social movements as diffuse action fields in which actions affect other actions and the action repertoires of the different actors co- evolve through time and through interaction with each other. Movement activists and regimes engage in strategic interactions, each responding to the actions of the other. Different organizations within a movement respond to the actions of others, as successful tactical innovations and movement frames diffuse to new organizations. News media cover or fail to cover particular protests, and thus encourage or discourage future protests. Each of these processes effects the others, in a complex, multi-faceted set of interactions. Over time, the action set of each actor evolves in response to the actions of the others and, thus, the whole field is one large co-evolving environment in which the characteristics and actions of any actor is constrained and influenced by the characteristics and actions of all other actors in the environment. One central concern about understanding diffusion and networks in protest waves is that

The pace of black insurgency between 1955 and 1970 is analyzed as a function of an ongoing process of tactical interaction between movement forces and southern segregationists. Given a political system vulnerable to challenge and strong internal organization the main challenge confronting insurgents is a preeminently tactical one. Lacking institutionalized power, challengers must devise protest techniques that offset their powerlessness. This is referred to as a process of tactical innovation. Such innovations, however, only temporarily afford challengers increased bargaining leverage. In chess-like fashion, movement opponents can be expected, through effective tactical adaptation, to neutralize the new tactic, thereby reinstituting the power disparity between themselves and the challenger. This perspective is applied to the development of the black movement over the period, 1955-1970. Evidence derived from content-coding all relevant story synopses contained in The New York Times Index for these years is presented showing a strong correspondence between the introduction of new protest techniques and peaks in movement activity. Conversely, lulls in black insurgency reflect the successful efforts of movement opponents to devise effective tactical counters to these innovations.

Objective.
—To address concerns about the effects of weight cycling and to provide guidance on the risk-to-benefit ratio of attempts at weight loss, given current scientific knowledge.Data Sources.
—Original reports obtained through MEDLINE and psychological abstracts searches for 1966 through 1994 on weight cycling, "yo-yo dieting," and weight fluctuation, supplemented by a manual search of bibliographies.Study Selection.
—English-language articles that evaluated the effects of weight change or weight cycling on humans or animals.Data Extraction.
—Studies were reviewed by experts in the fields of nutrition, obesity, and epidemiology to evaluate study design and the validity of the authors' conclusions based on published data.Data Synthesis.
—The majority of studies do not support an adverse effect of weight cycling on metabolism. Many observational studies have shown an association between variation in body weight and increased morbidity and mortality. However, most of these studies did not examine intentional vs unintentional weight loss, nor were they designed to determine the effects of weight cycling in obese, as opposed to normal-weight, individuals.Conclusions.
—The currently available evidence is not sufficiently compelling to override the potential benefits of moderate weight loss in significantly obese patients. Therefore, obese individuals should not allow concerns about hazards of weight cycling to deter them from efforts to control their body weight. Although conclusive data regarding long-term health effects of weight cycling are lacking, non-obese individuals should attempt to maintain a stable weight. Obese individuals who undertake weight loss efforts should be ready to commit to lifelong changes in their behavioral patterns, diet, and physical activity.(JAMA. 1994;272:1196-1202)

Block sequential iterations of threshold networks are studied through the use of a monotonic operator, analogous to the spin glass energy. This allows to characterize the dynamics: transient and fixed points. We then extend this method to networks of generalized majority functions and spin glasses.

The flow of information or influence through a large social network can be thought of as unfolding with the dynamics of an epidemic: as individuals become aware of new ideas, tech-nologies, fads, rumors, or gossip, they have the potential to pass them on to their friends and colleagues, causing the resulting behavior to cascade through the network. We consider a collection of probabilistic and game-theoretic models for such phenomena proposed in the mathematical social sciences, as well as recent algorithmic work on the problem by computer scientists. Building on this, we discuss the implications of cascading behavior in a number of on-line settings, including word-of-mouth effects (also known as "viral marketing") in the success of new products, and the influence of social networks in the growth of on-line communities.

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.

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.

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].

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.

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.

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.

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.

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.

Graph dynamical systems (GDSs) can be used to describe a wide range of distributed, nonlinear phenomena. In this paper we characterize cycle equivalence of a class of finite GDSs called sequential dynamical systems SDSs. In general, two finite GDSs are cycle equivalent if their periodic orbits are isomorphic as directed graphs. Sequential dynamical systems may be thought of as generalized cellular automata, and use an update order to construct the dynamical system map. The main result of this paper is a characterization of cycle equivalence in terms of shifts and reflections of the SDS update order. We construct two graphs C(Y) and D(Y) whose components describe update orders that give rise to cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper bound for the number of cycle equivalence classes one can obtain, and we enumerate these quantities through a recursion relation for several graph classes. The components of these graphs encode dynamical neutrality, the component sizes represent periodic orbit structural stability, and the number of components can be viewed as a system complexity measure.