CHAPTER ELEVEN

Complex Dynamical Systems in Social and Personality Psychology Theory, Modeling, and Analysis

MICHAEL J. RICHARDSON, RICK DALE, AND KERRY L. MARSH

All social processes fundamentally involve change in The scientific study of dynamical systems is con- time: Judgments materialize quickly over milliseconds cerned with understanding, modeling, and predicting or seconds, conversations flow over minutes, and rela- the ways in which the behavior of a system changes tionships evolve across even longer time scales. Put over time. As a formal approach, it has a long his- simply, social systems are dynamical systems. The tory in applied mathematics and physics, and has word “dynamical” simply means time-evolving and been used extensively to understand and model the thus a dynamical system is simply a system whose behavior of many different types of physical systems, behavior evolves or changes over time. Proposing such as the motion (position and velocity) of planets, that social processes are dynamical is not new and mass-spring systems, swinging pendulums, and self- has a long history in social psychology (e.g., Asch, sustained oscillators. In the last few decades, however, 1952; Lewin, 1936; Mead, 1934). Moreover, most an increasing number of researchers have begun to researchers in social-personality psychology would investigate and understand the dynamic behavior of agree that social processes and behavior are dynamical more complex biological, cognitive, and social systems, and change over time. Traditionally, however, social- using the concepts and tools of dynamical systems. personality psychology, like experimental psychology The term “complexity” refers to the fact that in general, has focused on summary statistics, which most biological, cognitive, and social systems typically aggregate over time, such as in the form of magnitudes exhibit behavior that is nonlinear and involves a large (e.g., behavioral frequencies, emotional intensities, number of interacting elements or components. His- and so on). This traditional approach is rooted in the torically, it is the nonlinearity of complex dynamical linear statistical methods developed by Fisher and oth- systems that has largely hindered research on such ers (Meehl, 1978), and is aimed at detecting whether systems, in that the numerical techniques that enable treatments or manipulations, on the whole, affect the one to uncover the dynamics of nonlinear and com- outcome of some measured behavioral state or vari- plex dynamical systems involve an extensive number able. Behavioral change is therefore conceptualized as of computational processes that are impossible to per- the difference between static measures and is mod- form without modern computers. This is true for both eled by covarying responses on such measures. Unfor- abstract nonlinear dynamical models (covered in the tunately, this traditional approach merely describes second section of this chapter) and for the analysis of behavioral change; it does not capture true time evo- behavioral data (discussed in the third section). These lution and so is not always optimal for understand- days, of course, these difficulties of computation no ing the process by which behavioral change occurs. To longer exist, and researchers can formulate and ana- make progress in our understanding of psychological lyze many nonlinear and complex dynamical systems change and process, therefore, researchers need to con- quite easily. Indeed, the fields of nonlinear dynamics sider adopting new tools and methodological concepts, and complex systems, as well as our theoretical under- namely those of dynamical systems. standing of such systems, have grown in parallel with

253 254 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH increases in the availability and computational power covers the basic forms and mathematical properties of of modern computers. dynamical systems models and how dynamical mod- Advances in the modeling and analysis of com- eling can provide insights about the organization or plex dynamical systems have also led to the steady reorganization of stable (or unstable) human and rise of dynamical systems in social-personality psy- social behavior. The final section addresses the basic chology. Vallacher, Read, and Nowak (2002) offer aspects of dynamical systems analysis and focuses on an excellent review of this rise and describe how a number of analysis techniques that can be employed dynamical theory and models can be employed to to uncover the dynamics of a behavioral system from characterize an array of socially relevant behaviors, time-series recordings. such as attitude change, social judgment, and self- perception (for reviews, see Nowak & Vallacher, 1998; DYNAMICAL SYSTEMS THEORY Vallacher & Nowak, 1994). This interest in dynamics The keystone concept throughout this section (and and the dynamical systems approach has continued to the entire chapter) is that the behavior of a complex expand in social-personality psychology, with numer- dynamical system can arise in a self-organized manner ous researchers embracing a dynamical understanding from the free interplay of components and properties of mood (e.g., Gottschalk, Bauer, & Whybrow, 1995; of the system. To unpack what this means, we begin Schuldberg & Gottlieb, 2002), personality expression by briefly defining what a complex dynamical system (Brown & Moskowitrz, 1998; Mischel & Shoda, 1995), is and then go on to describe the abstract properties conformity (Tesser & Achee, 1994), romantic relation- that underlie the behaviors that complex dynamical ships (Gottman, Swanson, & Swanson, 2002), and systems exhibit. person perception and construal (Freeman & Ambady, 2011). Along with more recent advances in fractal What Is a Complex Dynamical System? methods (Correll, 2008; Delignieres,` Fortes, & Ninot, 2004), decision dynamics (Freeman, Dale, & Farmer, The term “complex dynamical system” lacks a con- 2011), cognitive dynamics (Spivey, 2007), coordina- sensus definition, but many researchers agree that tion dynamics (Kelso, 1995; Schmidt & Richardson, complex dynamical systems exhibit three key charac- 2008), and the behavioral dynamics of perception teristics (see Gallagher & Appenzeller, 1999, and arti- and action (Warren, 2006), the concepts and tools cles therein, for a discussion). First, they consist of that have been developed for understanding com- a large number of interacting components or agents. plex dynamical systems appear to provide a promis- This may be said about the behavior of an indi- ing new method for understanding social behavior and vidual person, as in the interacting constraints that cognition. drive social perceptions and decisions (e.g., Read & Here we provide an introductory overview of the Miller, 2002; Smith, 1996). It may also be said about dynamical systems approach and how the theoreti- the behavior of groups of human beings, such as in cal concepts, modeling techniques, and analysis tools dyadic conversation (Buder, 1991) or small work- used to investigate complex dynamical systems can be teams (Arrow, 1997). A second property is that these used to understand social behaviors that emerge and systems exhibit emergence: Their collective behav- change over time. It is by no means a comprehen- ior can be difficult to anticipate from knowledge of sive review of complex dynamical systems and the the individual components that make up the system, dynamical systems approach. Rather, the chapter is and exhibit some coherent pattern or even, in some aimed at providing the reader with the knowledge cases, apparent purposiveness. For example, a person’s needed to seek out more detailed discussions else- social judgment may be a nonobvious consequence of where. Throughout the chapter we include key refer- the interaction among an array of informational con- ences that provide a deeper and more detailed under- straints (e.g., Freeman & Ambady, 2011), and group standing of the concepts and issues discussed. behaviors may be a nonobvious outcome of interac- The chapter is divided into three main sections. tions among group members (Arrow, 1997). Third, The first section covers the basic concepts of dynam- this emergent behavior is self-organized and does not ical systems theory, including their implications for result from a central or external controlling compo- understanding human and social behavior, and how nent process or agent. Although all three characteris- complex ordered behavior emerges from the non- tics are necessary for a system to be considered com- linear interactions that exist between the compo- plex, the appearance of emergent behavior that results nents of a behavioral system. The second section from self-organization is the most distinguishing COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 255 feature of complex systems (Boccara, 2003). Accord- social groups (Arrow, 1997). For instance, the coor- ingly, we turn next to the topic of self-organization. dinated activity observed between pedestrians walk- ing down a crowded sidewalk is the result of self- organizing dynamics (Sumpter, 2010). Goldstone and Self-Organization colleagues have demonstrated that the human-path The term “self-organization” is used to refer to systems that are created between regularly visited behavioral patterns that emerge from the interac- destinations (say, buildings around a university cam- tions that bind the components of a system (social pus) emerge in a self-organized manner and are often or otherwise) into a collective, synergistic system, mutually advantageous to the members of the group while not being dictated a priori by a centralized that created them. Task subroles and divisions of controller. As mentioned earlier, self-organization is labor can also emerge and be spontaneously adopted synonymous with complex systems, and interestingly by individuals during joint action or a social cogni- enough, the most widely used examples of a self- tive task (Egu´ıluz, Zimmermann, Cela-Conde, & San organized and emergent behavioral process are social Miguel, 2005; Goldstone & Gureckis, 2009; Richard- examples, in particular the coordinated activities of son, Marsh, & Baron, 2007; Theiner, Allen, & Gold- social insects such as ants. Take a colony of harvester stone, 2010). Even group dynamics and performance ants, for example, in which different members of the can be self-organized by means of integrative com- ant colony perform one of several different tasks: for- plexity, whereby the individual ideas and opinions aging, patrolling, nest maintenance, or midden work of group members emerge and become dynamically (clearing up debris). Individual ants do not perform integrated over time (Cummings, Schlosser, & Arrow, the same task all the time, but transition between the 1996). different modes of behavior as the need arises, with the appropriate number of ants (workers) engaged in Soft-Assembly a particular task at any given time. That is, patrollers become foragers, foragers become nest maintenance The kinds of self-organized dynamical systems workers, and midden workers becoming patrollers, described in the preceding section, such as social and so forth, based on current conditions (e.g., Gor- insects and sometimes groups of people, are tempo- don, 2007). Task allocation, however, is not achieved rary organizational structures that are put together in by a centralized controller or leader. The queen does a fluid and flexible manner. In the case of the har- not decide who does what, nor does any other ant. vester ants, it does not matter which particular ant In fact, it would be impossible for any ant to over- does which particular job; each ant is capable of tak- see the entire colony. Rather, individual ants can only ing up any job at any point in time. Though obviously detect local tactile and chemical information, with the different in important ways, humans also engage in coordinated behavior of the colony emerging from the modes of behavior flexibly throughout the course of local interactions that constrain the tasks an individual a single day (see Iberall & McCulloch, 1969 for clas- ant should perform (Boccara, 2003). sic a discussion; see Isenhower, Richardson, Marsh, The harvester ant colony highlights how coor- Carello, & Baron, 2010 for a task example). More- dinated social behavior can result spontaneously over, although it seems individuals in a group or social from the physical and informational interactions of network are in complete control of their own behav- perceiving-acting agents. The nest-building behavior ior and are consciously aware of their acts (and could of other social insects, such as termites, bees, and verbalize them if asked), we know that such cen- wasps, is similarly self-organized. So too is the coor- tralized, conscious control is often an illusion: Classic dinated behavior of schools of fish, flocks of birds, and research in social psychology suggests that individuals herds of ungulate mammals: No individual animal has can be unaware of the “true” reasons for their actions precise control over the direction or behavior of the (Nisbett & Wilson, 1977). Indeed, the coordinated group (e.g., Camazine, Deneubourg, Franks, Sneyd, behavior and intentions of socially situated individu- Theraulaz, & Bonabeau, 2001; Theraulaz & Bonabeau, als can be constrained and self-organized by environ- 1995). mental and situational constraints of which we are not Though it is certainly different from that of humans aware, with the unfolding dynamics of human behav- in numerable respects, the self-organization that ants ior reflecting a mutuality of responsiveness between and other social animals promote is directly relevant individuals and the context in which they are embed- to, and sometimes motivates, the study of human ded (Reis, 2008; Richardson, Marsh, & Schmidt, 2010; 256 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

Semin & Smith 2008; Thelen & Smith, 1994). Under- ics of the component elements, situational factors, standing and predicting the time-evolving behavior of and agents themselves (Anderson, Richardson, & an individual or social system is therefore not only Chemero, 2012; Kello, Beltz, Holden, & Van Orden, dependent on identifying the environmental factors or 2007; Van Orden, Kloos, & Wallot, 2011). agents that make up the behavioral system in ques- Thus, if one were to examine the relationship tion, but also on the relevant interagent and agent- between any two levels of an interaction-dominant environment interactions that shape behavior. This dynamical system, one would observe that elements implies that the dynamical behavior of cognitive and or agents at the lower level of the system modulate social systems is highly context dependent. the macroscopic order of the higher level and at the Dynamical systems that exhibit this kind of emer- same time are structured by the macroscopic order gent, context-dependent behavior are often referred of the system. For example, the individuals (micro- to as softly assembled or soft-molded systems, in that level) within a cultural system (macro-level) modu- the behavioral system reflects a temporary coalition of late the behavioral order of the cultural system, with coordinated entities, components, or factors. The term the dynamical organization of the cultural system in “synergy” is sometimes used to refer to softly assem- turn (and at the same time) modulating and structur- bled systems – a functional grouping of structural ing the behavior of the individuals within it. Accord- elements (molecules, genes, neurons, muscles, limbs, ingly, for interaction-dominant systems, it is difficult – individuals, etc.) that are temporarily constrained to and often impossible – to assign precise causal roles to act as a single coherent unit (Kelso, 2009). In contrast, particular components, factors, agents, or system lev- most nonbiological systems or machines are hard- els. Of particular significance for the study of cognitive molded systems. A pendulum clock, for example, is and social behavior is the implication that one can- a hard-molded dynamical system, in that it is com- not hope to appropriately understand behavioral orga- posed of a series of components (parts), each of which nization by attempting to study systems components plays a specific, predetermined, and unchanging role or agents in isolation. For that reason, researchers in shaping the motion of the clock’s pendulum over who have adopted the complex dynamical approach to time. Rigidly organized factory assembly lines, orga- social phenomena have started to conceptualize many nizations, or military structures could also be con- domains of human behavior, from language acquisi- sidered intuitive examples of hard-molded machine- tion (Van Geert, 1991) to group dynamics (Arrow, like social systems. But even within rigid social McGrath, & Berdahl, 2000), as being guided and self- structures one finds differing degrees of fluidity and organized by the dynamic interaction of many con- soft-assembly, both within and across different lev- straints, factors, and processes. els of organization. One could argue that this soft- assembly, present to some degree, is a necessary Nonlinearity requirement for the ongoing existence and success of such social systems and organizations (e.g., Dooley, Anonlinearsystemisoneinwhichthesystem’s 1994; Guastello, 2002). output is not directly proportional to the input, as opposed to a linear system in which the output can be simply represented as a weighted sum of input com- Interaction-Dominant Dynamics ponents. Complex dynamical systems, most notably The key property of softly assembled systems biological and social systems, are nonlinear in this is that they exhibit interaction-dominant dynam- sense. Our attraction to another individual or our self- ics, as opposed to component-dominant dynamics. concept, for instance, may not correspond to a mere For component-dominant dynamical systems, system average or sum of positive and negative attributes behavior is the product of a rigidly delineated archi- (Rinaldi & Gragnani, 1998; Sprott, 2004), but rather tecture of system modules, component elements, or some form of multiplicative combination of attributes agents, each with predetermined functions (i.e., the and situational factors that results in an attraction pendulum clock or a factory assembly line). As noted or self-concept that is more (or less) than the sum earlier, however, for softly assembled interaction- of its parts. The consequence of this principle for dominant dynamical systems, system behavior is understanding human behavior and social systems is the result of interactions between system components, equivalent to the consequence of a system exhibit- agents, and situational factors, with these intercompo- ing interaction-dominant dynamics – a system’s nent and interagent interactions altering the dynam- behavior cannot be reduced to a set of component COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 257 dominant factors that interact in a simple linear A classic example of a simple nonlinear system fashion (Van Orden, Holden & Turvey, 2003). Thus, in that results in chaotic behavior is the logistic map.The the context of complex dynamical systems, the term logistic map is a discrete dynamical system, meaning “nonlinear” also means something more than mul- its behavior changes over discrete rather than con- tiplicative. More generally, it is used to refer to the tinuous time steps (see next section for more details). non-decomposability of a complex dynamic system, More precisely, it is simple dynamical equation of the whereby a nonlinear dynamical system is a system form whose macroscopic behavioral order results from the x(t 1) rx(t)(1 x(t))(11.1) complex interactions of micro-scale components, but + = − via processes of circular causality, also modifies the where x is the behavioral variable, r is a fixed behav- interactions between micro-scale components as well ioral parameter, and t equals time from step 0 to as the behavior of the components themselves. step n (i.e., t 0, 1, 2 . . . , n). To help make this = On the one hand, the nonlinearity of complex equation less abstract and easier to understand, let dynamical systems makes them much more difficult to us assume that this equation is used to model the understand. In fact, the effects of nonlinear processes daily mood of an individual diagnosed with bipolar are typically not known or cannot be known ahead disorder or manic depression, where x represents the of time. On the other hand, it is only because com- individual’s daily mood on a scale of 0 to 1, with plex dynamical systems are nonlinear that they can 1 corresponding to a perfectly positive mood, and r exhibit complex emergent behavior. It is for these rea- representing the severity of the individual’s diagno- sons that an increasing number of researchers and the- sis on a scale of 0 to 4, with 4 corresponding to a orists consider human behavior and social processes severe diagnosis. The predicted daily mood of the indi- to be predominantly nonlinear (see e.g., Guastello, vidual (i.e., x(t 1)) is therefore a simple mathemati- + Koopmans, & Pincus, 2009; Kelso, 1995; Vallacher & cal function of the current day’s mood, x(t),multi- Nowak, 1994; Nowak & Vallacher, 1998 for edited col- plied by the severity of the diagnosis, r,multipliedby lections and reviews). In fact, a defining feature of 1minusthecurrentday’smood,or 1 x( ). For illus- − t human behavior is that it is often unpredictable. It trative purposes, imagine that an individual’s diagno- is this aspect of human behavior that makes the sci- sis was relatively severe, that r equaled 3.8, and the ence of social-personality psychology, as well as the person’s initial mood of x on day zero equaled 0.6. many other fields of psychology (perceptual, cogni- If we then computed or iterated the equation 100 tive, clinical, etc.), so difficult and at the same time times, we would get the time-evolving behavioral pat- so intriguing. In addition, although the word “ran- tern displayed in Figure 11.1, where the value of x, dom” is often used in everyday speech when referring or mood in our example, is plotted as a function of to the unpredictability of human behavior, the gen- time step from 1 to 100. The complex and unpre- eral belief that human behavior is not random, but dictable nature of the behavioral pattern over time rather complex, is also consistent with the notion of is quite evident, with the value of x from one time nonlinearity. step to the next seeming to change in a way that far exceeds the simplicity of the equation that was used to generate it. Chaos It is the latter feature of chaotic systems like The fact that nonlinear dynamical systems can the logistic map (Equation 11.1) that often surprises exhibit complex and unpredictable behavior is inter- researchers. That is, while the behavior of such sys- esting in and of itself and highly relevant for the study tems is completely determined by a set of simple deter- of human behavior. Even more interesting, however, ministic (i.e., nonrandom) equations or rules, can be is one of the key discoveries from the study of nonlin- very complex and very difficult to predict. This is ear dynamical systems: that highly complex behavior because chaotic systems exhibit a sensitive dependence can even emerge from very simple rules or systems on initial conditions: minor differences in starting states so long as the components or agents of the sys- can become amplified as the system evolves over time. tem interact in a nonlinear manner. That is, very Given that measuring or knowing the initial condi- simple deterministic nonlinear systems can produce tions of any natural dynamical system with perfect extremely complex and unpredictable behavior. One precision is impossible, it is therefore equally impossi- notable form of complex behavior they produce is ble to predict the long-term behavior of such systems chaotic behavior. if they are chaotic. 258 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

1

0.9

0.8

0.7

x 0.6

0.5

0.4

0.3

0.2 0 10 20 30 40 50 60 70 80 90 100 time

Figure 11.1. Behavior of the logistic map’s dependent vari- DYNAMICAL SYSTEMS MODELING able, x,duringachaoticregime(i.e.,r 3.8). = The goal of many research endeavors is to effectively model a behavioral system in order to make specific predictions about how the system will behave in the The implication of this principle for understanding future. In personality and social psychology, this has human social behavior is quite profound. Most obvi- typically been done using various forms of regres- ous is that the existence of chaos forces researchers to sion analysis and structural equation modeling (for truly reconsider what it means for a behavioral event instance, see Fabrigar and Wegener, Chapter 19 in to be random (Guastello & Liebovitch, 2009). Perhaps this volume). An advantage of dynamical models is less obvious is that the apparent randomness of vari- that they can handle the time-dependence of behav- able or noisy behavior (i.e., normally distributed ran- ior and are not restricted to making linear assumptions dom noise or variance) that is traditionally assumed about behavioral organization. Accordingly, nonlinear in nearly all psychology studies becomes an empirical dynamical models can provide deep insights about the question. Just because a variable behavior appears to behavior of real-world time-evolving processes and be random does not mean that one should conclude can play a significant role in theorizing about how and that it is random. Chaos also highlights the nonobvi- why certain behavioral processes might emerge. For ous connection between past and future events, and example, Meadows (2008) offers a lucid discussion of that even extremely trivial changes can have a signifi- the benefits of dynamical systems modeling for under- cant effect on the outcomes of time-evolving behav- standing the interactions and behavioral nonlineari- ior. It is for these reasons that chaos has found its ties that anchor social systems, from the parameters way into extensive theoretical discussion of social psy- that lead to stable sustenance of romantic relation- chological systems. A lucid introduction and review of ships (Gottman, Murray, Swanson, Tyson, & Swan- this point is found in Barton (1994). One prominent son, 2003) to the way that societal rules can change application described in Barton’s review is in the clin- patterns of self-organization. ical realm, where chaos has enjoyed a rapid growth of It is important to keep in mind that any model application across a range of traditions. Another popu- of a biological, psychological or social system is at lar, visual introduction to chaos and nonlinear dynam- best an idealization (this is just as true for dynam- ics that utilizes clinical psychology is provided by Abra- ical models as for non-dynamical models) and that ham, Abraham, and Shaw (1990). the goal of a dynamical model is to capture the most COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 259 important features of a system or process (Bertuglia takes the form, & Vaio, 2005). Dynamical modeling involves describ- r1x1(t)(1 x1(t)) αr2x2(t) (1 x2(t)) ing how the behavioral state of a system changes over x1(t 1) − + − (11.2) + = 1 α time. Here the term “state” refers to the current value + r2x2(t)(1 x2(t)) α1r1x1(t)(1 x1(t)) of a variable (or variables) that are used to capture the x2(t 1) − + − + = 1 α system in question. A state variable could be any prop- + erty of a system that might change over time,such where x is a generic variable representing the intensity as the movements of the body, limb, or eyes dur- of some observable, communicative behavior, such as ing social interaction, or the mood, attitudes, or per- approach (or avoidance), and r corresponds to inter- sonality characteristics of a child or adult. One could nal states, such as personality traits, moods, attitudes, even model the change in state of two people, such and values, that shape an individual’s behavior over as the quality of a romantic relationship. Essentially, time. This is a two-dimensional model, in that there a dynamical model describes how the state variables are two state variables, x1 and x2, one state variable of a system evolve over time through rules or equa- and equation for each individual’s behavior. The equa- tions that determine the system’s future state given its tions are also coupled, in that the behavioral state of current state. each individual is dependent on his or her own pre- vious state, as well as the previous state of his or her partner, with the parameter α (alpha) determin- Difference Equations ing the strength of the coupling (i.e., mutual influ- A difference equation is a recursive function some- ence). Although a detailed discussion of this model times called an iterative map and can be used to model and the behaviors it generates is beyond the scope of the behavior of a system at discrete time steps (1, 2, the chapter, its significance is that it predicts increased 3... t, where t equals the number of time steps), with social monitoring and mutual influence (i.e., commu- the state of the system at each time step defined as a nication, mutual reinforcement and self- and other- function of the preceding state. The logistic map, intro- monitoring) when individuals with different internal duced in Equation 11.1 in the previous section, is an states (e.g., personality traits, moods, attitudes) are example of a difference equation. More precisely, the required to synchronize their behavior. This increase logistic map is a one-dimensional nonlinear1 differ- in social monitoring and mutual influence is then pre- ence equation – the dimension of a dynamical system sumed to put greater stress on the interactional sys- equals the number of state variables needed to com- tem by reducing the executive resources. In contrast, pletely describe the system (i.e., one-, two-, three-, . . . when partners have similar internal states, behavioral to n-dimensions) – and is one of the most well-known synchronization can occur with little social monitoring and studied difference equations in the field of non- or mutual influence, leaving more energy and cogni- linear dynamics. This is because despite the simplicity tive resources available for the coupled individuals to of Equation 11.1, it exhibits a wide variety of dynamic pursue common goals. behaviors for different values of the system parame- ter r (specifically for 0 < r < 4). This includes various Differential Equations types of stable fixed point and periodic behaviors, and, as we have already seen, even chaotic behavior (see In contrast to difference equations, which model Figure 11.2). the behavior of state variables across discrete time Of more relevance to understanding social behav- steps, a differential equation is a mathematical equation ior, Nowak and Vallacher have demonstrated how two that models the continuous time evolution of a system coupled logistic equations can be used to model the in terms of the rate of change of state variables over behavioral synchronization of two individuals in social time. As a starting example, consider the simple one- interaction (e.g., Nowak & Vallacher, 1998; Nowak, dimensional nonlinear differential equation Vallacher, & Borkowski, 2002; Vallacher, Nowak & x˙ rx(1 x)(11.3) Zochowski, 2005; also see Buder, 1991). Their model = − where x is the state variable,x ˙ is the rate of change of

1 the x over time, and r is a state parameter. This equa- It is a nonlinear equation because if we expand rx(t)(1 2 − tion is very similar to Equation 11.2 and is the logistic x(t)), we get r(x(t) x ), such that the state of the system − (t) is the product of a constant and the state variable to a power equation in differential form. However, because Equa- 2 3 4 2 greater than one (e.g., x , x , x ...),inthiscase x(t). tion 11.3 models the rate and direction in which x 260 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

0.65 0.67

0.6 0.66 0.55

0.5 0.65

0.45 0.64 x x 0.4 0.63 0.35

0.3 0.62

0.25 0.61 0.2

0.6 0 5 10 15 20 25 0 5 10 15 20 25 me me 1 1

0.9 0.9

0.8 0.8 0.7

0.7 0.6 x x 0.6 0.5

0.4 0.5 0.3

0.4 0.2

0.1 0 5 10 15 20 25 0 5 10 15 20 25 me me

Figure 11.2. Examples of how x changes over time for the same eventual level of attraction. This can be seen logistic map (Equation 11.1) for different values of r,but from an inspection of Figure 11.3, in which the change the same initial condition of x(0) .6. (top left) monotonic = in x over time is presented for four different initial con- approach towards a fixed value of x for r 1.2. (top right) = ditions (x(0) .1,.8, 1.6, 3.9). Notice also that increas- oscillatory approach towards a fixed value of x for r 2.7. = = ing r only changes the rate at which x approaches 1. (bottom left) periodical oscillates between 4 values of x for r 3.2. (bottom right) chaotic behavior for r 3.8. It does not change the dynamics qualitatively, as it = = did for the logistic map in Equation 11.1. Because x is always attracted toward 1, irrespective of initial condi- changes over continuous time, the way the value of tion, x 1 is the stable fixed point for Equation 11.3. = x changes is quite different from that determined by We describe stable fixed point attractors, as well as other Equation 11.2. Imagine that x represents some observ- types of attractors, in more detail later in the chapter. able behavior, such as attraction – in the context of At this stage it is sufficient to say that a stable fixed social interaction, attraction could refer to the pull point is a state toward which the variables of a dynam- toward another person, affiliation, or liking – and r ical system move over time. corresponds to the number of positive attributes. If we Although the simplicity of Equation 11.3 provides restrict our consideration to initial conditions, x(0) > 0, a good introduction to differential equations, the fact and r > 0, then x always approaches x 1, no mat- that its state variable, x, is always attracted toward 1 = ter what initial condition we choose. In other words, means that the degree to which this equation could be setting the parameter r > 0wouldalwayspredictthe used to model complex human and social behavior is COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 261

3 3 3

2 2 2 x x x

1 1 1

0 0 0 me me me extremely limited. An example of a differential equa- Figure 11.3. Examples of how x changes over time for Equa- tion 11.3, using four different initial conditions (x(0) .1, tion that is more relevant to understanding human = .8, 1.6, 3.9). For the left, middle, and right graphs, r .5, 1, and social behavior is = and 2, respectively.

x˙ k x x3 (11.4) = + − more positive information about group X), the individ- ual’s attitude will be predicted to remain negative even where x might represent an individual’s attitude beyond the point (i.e., k > 0) at which the individ- toward a certain political or racial group X and k is ual has more positive than negative experiences with a state parameter that captures the amount of positive group X. The individual will finally transition to hav- or negative experiences or information an individual ing a positive attitude toward group X after a critical has about group X. What is interesting about this sys- number of positive experiences have occurred, that is, tem is that the number and location of its stable fixed at k .35. Conversely, if k is decreased from 1 to –1, = points change as we change the parameter k.Thisis a transition from a positive to a negative attitude will best illustrated by plotting Equation 11.4 as a potential occur at k –.35. = function where the fixed points of the system are rep- The kind of sudden transition predicted by Equa- resented as wells in a one-dimensional landscape, with tion 11.4 is called a phase transition, where phase the depth of the well corresponding to the strength or refers to a qualitative state of the system (i.e., a nega- stability of the stable fixed points or system attractors tive state or phase vs. a positive state or phase). Such (see Figure 11.4). Of particular relevance is that Equation 11.4 pre- dicts that an individual’s attitude toward group X Figure 11.4. Potential functions for Equation 11.4, demon- would remain relatively stable, either negative or pos- strating how the system’s stable fixed points change as k is itive, across a range of k values and then at certain val- scaled from –1 to 1 (left to right, respectively). In these plots, ues of k suddenly transition from a negative to a pos- the x-axis corresponds to the possible values of the state vari- itive state or from a positive to a negative state. More able, x,andthey-axiscorrespondstothepotential(Vx)ofthe system state to move to another state. Wells in the potential specifically, if an individual starts out with a negative function or local minima (minimal potentials) correspond to attitude toward group X, and then k is increased from – stable fixed points, such that the state of the system (i.e., illus- 1 to 1 (i.e., an individual starts to have more and more trated as a small grey ball) is trapped at the bottom of the well positive experiences with group X or receive more and until that state is no longer stable (no longer a minima).

Vx Vx Vx Vx Vx

x - x+ x - x+ x - x+ x - x+ x - x+ 262 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

Figure 11.5. Prototypical examples of (left) a stable fixed attractor is a state or subset of states toward which point attractor, (middle) a limit cycle attractor, and (right) a the dynamical system moves over time (i.e., corre- strange attractor (see text for details). sponds to a final future state or set of states). Attractors are often described geometrically, such as a point or a transitions are a prominent characteristic of complex closed curve, and can be intuitively visualized by plot- dynamical systems, and the fact that nonlinear differ- ting the time-evolving behavior of a dynamical system ential equation models can be used to capture such in its phase space. A phase space is the set of all possible behavior is part of their appeal. As the attitude exam- states of a dynamical system, with each state corre- ple suggests, systems like Equation 11.4 are well suited sponding to a unique point in phase space. A graph- for modeling qualitative changes in human cognition ical depiction of a system’s attractors and the set of and social behavior. For instance, similar nonlinear possible trajectories that can be exhibited by a system systems can be employed to model transitions in cat- given its attractor layout is called a phase portrait (see egorical speech perception (Tuller, 2004; Tuller et al., Figure 11.5). 1994) and between conciliation and aggression dur- The attractor concept has been fruitfully applied ing conflict situations (Colman, Vallacher, Nowak, & in many social contexts. For example, Gottman et al. Bui-Wrzosinska, 2007). (2002) have developed models in which the stabilities Equations 11.2 and 11.3 are examples of one- of intimate relationships can be captured using fixed dimensional differential equations because they have point attractors. As defined earlier, a fixed point attrac- one state variable. Like difference equations, differen- tor, commonly referred to as a stable fixed point (some- tial systems might also require more than one state times called a stable node, equilibrium point, or point variable to describe the behavior of the system. For attractor), is a location in phase space that system tra- instance, the motion of a pendulum requires a two- jectories converge toward as time increases (Figure dimensional or second-order model that determines 11.5 left). A system may have two or more stable both its position and velocity over time. Differential fixed points, in which case the system is considered to systems with two or more state variables can also be be multistable. The models developed by Gottman and coupled. For example, a two-dimensional system of colleagues often have two fixed point attractors: sus- coupled differential equations could be used to rep- tained marriage or divorce. The authors design differ- resent the population of two codependent species of ential equations much like Equation 11.4 and explore animal (i.e., rabbits and foxes). With respect to per- the parameters that govern the dynamic regimes of sonality and social psychology, two-dimensional sys- a married couple into one of these two equilibrium tems have been used to model the coordinated behav- states. Although linear models can be applied in mar- ior of interacting individuals (e.g., Baron et al., 1994; riage research, the goal of the Gottman and colleagues’ Schmidt &; Richardson, 2008; Tesser & Achee, 1994) research agenda is to articulate the dynamic change – and the long-term marital success of a husband and the trajectory that a couple may take – into one or wife (Gottman et al., 2002; Gottman et al., 2003). another stable attractor (divorce, marital misery, hap- piness, etc.). Space limits our discussion of all of the different Attractors types of attractors, but it is important to note two One of the key aims of dynamical modeling is other types of attractor often featured in dynami- to effectively capture the attractors of a system. An cal systems research, namely limit cycle and strange COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 263 attractors. A limit cycle attractor is a subset of states by the system. For example, if one were trying to within a system’s phase space that make up a closed model the behavior of gas molecules within a sealed orbit that system trajectories converge toward as time container, it would be impossible to have a set of increases (see Figure 11.5, middle). Limit cycles are equations specifying the position and velocity of every paths that the system revisits with great regularity – molecule. Likewise, if one were attempting to model such that, after a time, the system’s behavior is fixed the movement synchronization that occurs between along the path (it is limited within that cycle). This the members of an audience at a rock concert, it would kind of attractor is characteristic of periodic behav- be impractical to describe the position and velocity iors that exhibit the same stable spatial-temporal pat- of each individual’s postural and gestural movements tern over time (e.g., cycle with a stable frequency over time. In these and other much simpler cases, and amplitude over time) and, moreover, return to it is often better to devise a dynamical model that the same stable spatial-temporal pattern when per- effectively describes the macroscopic behavior of the turbed. Rhythmic body and gestural movements pro- system as a whole. This involves identifying a state vide classic examples and researchers have modeled variable that is able to capture the global or collective interlimb (Haken, Kelso & Bunz, 1985) and inter- organization of the system. Such variables are referred personal movement dynamics (Schmidt, Carello, & to as collective variables or order parameters. Turvey, 1990), and even complex social behavior, Identifying an appropriate order parameter is not such as speech turns (Buder, 1991), using limit cycle always easy and often requires a significant amount models. of theoretical and empirical work, as well as model A strange attractor is a complex subset of states testing and assessment (for more details see Kelso, within a systems phase space that the state vari- 1995; Nowak & Vallacher, 1998). Once an appropri- able(s) of a dynamical system evolve toward over ate order parameter is identified, however, modeling time. The term “complex” refers to the fact strange and understanding the dynamics of a complex system attractors have a non-integer or fractal dimension. is typically much simpler. For example, in the work of That is, the spatial dimension of the subset of states Vallacher and colleagues reviewed earlier, where the that make up the attractor does not equal the stan- behavioral synchronization of two individuals is mod- dard Euclidean integer dimensions of one, two or eled using coupled logistic equations (Equation 11.2), three (e.g., a fractal attractor might live in a two- the researchers chose a generic variable they called or three-dimensional state space, but is neither two- observable-communicative behavior, and conceived of dimensional nor three-dimensional, but something this as a general patterning of behavior of one person in between). Strange attractors are characteristic of relative to another in interaction. This order parameter chaotic systems (although not exclusively) and in such is simply the intensity of overall behavior during inter- cases are sometimes referred to as chaotic attractors. action, which the authors suppose can be modeled The strange attractor displayed on the right of Figure as the magnitude of a single dynamical system’s state 11.5 is the chaotic attractor of the Lorenz system. variable. In reality, the social agents in this model are The Lorenz attractor is made up of a complex subset surely employing a whole range of observable behav- of states within a three-dimensional space and has a iors, but by distilling these behaviors into one pro- fractal dimension of approximately 2.05. The set of posed order parameter, the model is not only more differential equations that make up the Lorenz sys- tractable and understandable but also more generaliz- tem was originally formulated by Edward Lorenz to able. model atmospheric convection, but it is the strange, Another common example of an order parameter butterfly-like structure of its chaotic attractor that has is relative phase, which captures the location of one made it so famous. periodic or rhythmic behavior in its cycle relative to another. For example, two people walking down the street have the swaying of their arms and legs in a rela- Order and Control Parameters tionship of relative phase. Relative phase is an order Most of the models we have discussed so far have parameter because it quantifies in a single measure involved state variables that represent an individual the spatial-temporal relationship between two peri- behavioral quantity or element. For many complex odic or rhythmic behaviors using a single variable. systems, however, it would be impossible or unfeasi- First employed to capture the stable patterns of syn- ble to have a different state variable and equation for chrony that occur between two mechanical oscilla- every element, agent, or behavioral quantity entailed tors (e.g., coupled pendulum clocks), it has since been 264 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

1

0.9

0.8

0.7

0.6

x 0.5

0.4

0.3

0.2

0.1

0 2.5 33.5 4 r

Figure 11.6. Bifurcation diagram for the logistic map, Equa- time evolution and take place when a system’s control tion 11.1. parameter reaches a particular critical value. Many of the systems described earlier exhibit bifurcations. For instance, the logistic map (Equation 11.2) exhibits a employed to describe numerous biological phenom- series of bifurcations as the parameter r is increased ena, including the behavioral synchrony and social from 0 to 4. The differential system defined by Equa- coordination that occurs between the bodily move- tion 11.4 has two bifurcation points, one when k ments of two interacting individuals (see Schmidt & = .35 and one at k –.35. In fact, nearly every sys- Richardson, 2008 for a review). + = tem described previously exhibits one or more bifur- A control parameter is a system parameter that, when cations as the values of certain control parameters are changed, can significantly influence the dynamical increased or decreased. regime exhibited by a system. These parameters typ- The possible attractors of a dynamical system that ically represent some external force, condition, or fac- emerge or are destroyed as a control parameter is tor that plays an important role in constraining how scaled can be visualized using a bifurcation diagram. a system can evolve over time, and sometimes even The bifurcation diagram for the logistic map (Equation the attractors of a system. This is in contrast to other 11.2) as a function of the parameter r scaled from 2.5 system parameters that are typically fixed and repre- to 4 is displayed in Figure 11.6. The y-axis corresponds sent non-changing constraints or forces. In the dis- to the possible long-term values of x , with r plotted cussion of how an individual’s attitude toward group (t) along the x-axis. It is easy to see from this diagram that X might be modeled using Equation 11.4, parameter the logistic map has a single fixed point for r < 3, with k operated as a control parameter. k represented the the long-term behavior of x equaling a single value amount of positive or negative experiences or infor- (t) of x.Atr 3, however, a bifurcation occurs with the mation an individual had about group X and increas- = possible long-term behavior equaling two fixed point ing or decreasing k influenced attitude strength, with values of x. The number of fixed points then continues the attitude of individuals toward group X eventually to fork or double as r is further increased (e.g., from 2 transitioning from negative to positive or vice versa as to 4, to 8 to 16 . . . fixed points) and eventually exhibits k was scaled past some critical value. chaotic behavior, with the possible long-term behavior of x equaling all possible values of x. Bifurcations (t) There are numerous types of bifurcations and not Bifurcations are changes in the number and/or type all of them need to be described here (for more of fixed points or attractors that constrain a system’s information about the different types of bifurcations, COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 265

Behavioral manifold

Cusp (fold curve) ) x ( Behavior

(High) a (Low) (Low) b

(High) see Strogatz, 1994). One theoretical class of bifurca- Figure 11.7. Behavioral manifold for the cusp catastrophe. tions that are important to mention here are known Each point on the grey surface correspond to the stable fixed as catastrophes. They are called so because they reflect points (behavioral attractors) for different a and b values. The dashed lines illustrate to the different types of behavioral tran- a qualitatively dramatic change in the behavior of a sitions that are possible when b is fixed at some value and a is system. What is particularly interesting about catas- scaled up or down. trophes and the models used to describe them is that they have proven to be theoretically useful for under- friends and family are socially and politically conserva- standing the sudden emergence and or destruction of tive might be pressured not to date an individual who arangeofsocialphenomena,includingattitudes,self- has had a very liberal upbringing and whose friends evaluation, conformity, social relationships, and other and family are all very liberal. Thus, social pressure catastrophic social transitions (e.g., Guastello, 1995; is known as a splitting factor or parameter in Equa- Latane & Nowak, 1994; van der Mass, Kolstein, & van tion 11.5. der Pligt, 2003; Vallacher & Nowak, 1998). The bifurcation diagram for Equation 11.5 is Many of the catastrophic social transitions just displayed in Figure 11.7. The diagram is three- mentioned have been modeled using the cusp catas- dimensional, with dating behavior, x, on the vertical trophe. The cusp model is a two-parameter version of axis and the control parameters for love, a, and social Equation 11.4, namely pressure, b, defining a control surface on the horizon- tal plane. The folded manifold plotted across the three x˙ a bx x3 (11.5) = + − axis is a manifold of the fixed points that exist for dif- where x is the state variable and a and b are systems ferent settings of a and b. That is, each point on the parameters. The benefit of this model over Equation manifold represents a fixed point, with the points on 11.4 is that it can be used to model how the interaction the non-folded (light grey) area of the manifold corre- of two different external forces can influence behav- sponding to stable fixed points and the points on the ior. For instance, this model can be used to describe folded (dark) area corresponding to unstable (repul- the sudden changes in the dating behavior of couples sive) fixed points. A close inspection of Figure 11.7 given two interacting forces: a love and b social reveals when the model predicts that a catastrophic = = pressure (Tesser, 1980; Tesser & Achee, 1994). Here, change in the dating behavior of a coupled would social pressure refers to any family or societal pressure or would not occur. With respect to the latter case, on an individual “not to date” a certain type of indi- in which social pressure, b,isloworzero,thepre- vidual or group of individuals. For instance, an indi- diction (and expectation) is that dating behavior will vidual who has a conservative upbringing and whose change almost linearly with changes in love. The more 266 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH a couple loves each other – the higher the value of behavioral patterns with emergent properties char- a – the more likely a couple is to date. If social pres- acteristic of complex dynamical systems (Wolfram, sure is high, however, increases or decreases in love 2002). The rules that produce such patterns are have little effect on the likelihood that a couple will usually deceptively simple given the complexity of date, unless love increases or decreases past some crit- consequences that can follow from them. Within ical value. At this point the couple will either sud- social psychology, cellular automata have been used denly enter into a strong dating relationship – if love most notably to model the effects of social influence is increased above a critical value – or suddenly stop and public-opinion change based on Latane’s´ social dating altogether – if love is deceased below a critical impact theory (for reviews, see Nowak & Lewenstein, value. 1996; Nowak & Vallacher, 1998). In these models, a two-dimensional cellular automaton is used with cells representing different individuals who possess Cellular Automata, Agent-Based, and an attitude of a certain strength toward a topic. The Artificial Neural Network Models dynamics of an individual’s attitudes are determined Modeling complex nonlinear dynamical systems as a function of the attitude of near neighbors. After using difference and differential equations is a power- a number of iterations, this model eventually results ful way of investigating the stability, dynamic patterns, in a stable pattern of attitudes across individuals (i.e., and self-organization of systems. Using simple deter- cells). Of particular interest is that the time-evolving ministic rules, with well-conceived variables and well- patterns that result from this cellular automaton grounded parameters, can lead to a meaningful and illustrate how key hypotheses from social impact generalizable theoretical understanding of the dynam- theory play out, such as how pockets of minority ics of various types of social behavior. There is, how- opinion emerge over time. ever, another set of methods for modeling the self- Consider an example. In the models presented organizing dynamics of complex social systems, which by Nowak and Lewenstein (1996), local interaction starts from a microscopic level, focusing on how phe- among cells of a cellular automaton can render small nomena emerge from interacting elements or agents pockets or “walls” of resistance, where a cluster of that behave according to simple rules. Such models are cells is mutually supportive in sustaining their opin- cellular automata and agent-based models. ion, despite the onslaught of surrounding opinion. In Acellular automaton is a collection of individual cells additional simulations, they show that such minor- that form an n-dimensional grid of a specified size and ity opinion can grow to become the dominant one. shape, although they are typically restricted to either For this to happen, the minority opinion is sparsely a one-dimensional line (array) or a two-dimensional distributed in a social environment – weak and scat- square lattice of cells.2 Although these cells could rep- tered. Yet, with a small bias to favor that minority resent any component or element, in many social psy- opinion, things quickly change. What emerges in this chological applications a cell represents a person. The model are “clusters” or “bubbles” of minority influ- cells that make up a cellular automaton represent indi- ence that, as they grow, come to interconnect with vidual elements of the collective system (e.g., a com- each other, thus growing in force and slowly domi- munity of individuals), with the behavior or state of nating the once-majority opinion. Importantly, all of each cell at any point in time determined by a set of these simulations depend on simple, local interactions simple rules that define its state based on the state among cells, stretching across space and time. of neighboring cells. In a two-dimensional cellular An extension of classic cellular automata just automaton, for example, a cell might be influenced by described, which involve a collection of fixed cells the four cells directly adjacent to its location (i.e., the whose state changes over time, is to have cells rep- cells above and below and to the left and right) or it resent elemental locations, with the state of a cell cor- might be influenced by all eight cells that surround it responding to whether it is occupied or not. One of (i.e., the previous four, plus those on the diagonal). the earliest applications of cellular automata in the Cellular automata, including one-dimensional social sciences (Shelling, 1971) used this form of cellu- automata, can exhibit a wide variety of complex lar automata and demonstrated how social segregation can result from individuals who are more dissimilar to those around them moving to a different location. 2 Acellularautomatoncouldalsobeathree-dimensionalcube Agent-based modeling is another extension of cellular of cells, for instance. automata models. Such models have more potential COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 267 for complexity, as agents in these models are not con- impression is negative). Sampling information socially fined to a matrix of cells but are able to move around. led to more positive impressions on average, as well One prominent use of agent-based modeling comes as reduced perceiver effects and relationship effects. from the work of Axelrod (1984; Axelrod & Dion, Moreover, the authors found emergent phenomena 1988; Axelrod & Hamilton, 1981) on the emergence that had to do with dyadic reciprocity and the relative of cooperative behavior in games like the prisoner’s accuracy of generalized versus dyadic accuracy (Smith dilemma. Axelrod’s work demonstrates that strategies & Collins, 2009). that are cooperative but punish defections (e.g., the In the Smith and Collins’s (2009) model, although tit-for-tat strategy) are most successful in computer the model is directed toward understanding cognitive simulation tournaments in the long run when pitted processes through transmission of information about against agents using other strategies. Moreover, if the a person (directly or third-person), the interacting program increases the likelihood of an agent taking on agents in the model are individuals. It is important the strategy of an interactant who is more successful to realize, however, that the agents in agent-based in their game, the tit-for-tat strategy spreads in a pop- modeling can be the cognitive elements themselves ulation. Intriguingly for a dynamical perspective, his- (e.g., the interaction between visual perception, infer- tory is crucial for learning such a strategy. If there is a ences, judgments, etc.). Thus, there is a close link reshuffling of agents after each trial so that they have between such models and dynamic models of mem- no continuity in whom they interact with, then there ory. A brand of modeling that has focused entirely is no means for an agent to learn personally the nega- on exploration of the dynamics of social cognitive tive consequences of defecting when the other cooper- processes is artificial neural network models (also called ates (punishment on the next trial) – the agents have connectionist models in the late 1990s). In a classic by then moved on to the next partner. As a result, work in social psychology, Smith and DeCoster (1998) lacking history with an interactant cooperative strate- used a basic associationist neural network model to gies are not learned (Axelrod, Riolo, & Cohen, 2002) explore a wide range of phenomena, including stereo- – the tit-for-tat strategy does not spread. typing and person perception. They argued that neu- One interesting aspect of cellular automaton and ral network models of memory and social judgment – agent-based models more broadly is the emergent which function by integration of basic informational phenomena that result from the structure of linkages cues (much like interaction among agents) – can more between agents. Traditional social psychological meth- parsimoniously account for a range of social phenom- ods do not account well for how being in contact with ena than traditional models of memory (e.g., Wyer & multiple individuals, repeatedly and over time, affects Srull,1989). Modern neural network models can also behavior (Mason, Conrey, & Smith, 2007). Because be used to model cognitive dynamics, including the agent-based models do, they allow a researcher to see dynamics of social cognition; see Smith (1996) for an how novel phenomena may emerge as a consequence excellent early introduction to use of neural network of the interdependencies among agents (Smith & Con- (connectionist) models in social psychology. rey, 2007). A recent agent-based simulation of impres- Such models have also been applied to social sion formation capitalizes on the unique potential for interactive phenomena. Nowak and Vallacher (1998) such models to examine the consequences of flow of review early models of interpersonal dynamics using information across different types of linkages (Smith neural network models. Perhaps the best illustration of & Collins, 2009). In Smith and Collins’s model, par- such models is a very recent model meant to tap into ticipants can sample information about another per- the fast-time-scale dynamics of person perception and son directly or indirectly (e.g., through gossip). They judgment by Freeman and Ambady (2011). They pro- used Kenny’s Social Relations Model (1994) to ana- posed that person construal (such as identifying the lyze key features of impressions – for example, the gender of a person) is constrained by a constellation of degree to which impressions do in fact reflect com- information sources in the environment, such as hair monalities (across perceivers) attributable to the tar- cues, skin cues, facial configural cues, and so on. They get, versus how much variance is owing to perceiver developed an interaction-activation framework (see and specific target-perceiver relationships. The results collection of papers in Rumelhart & McClelland, 1985) of their simulations indicate that the ways of obtain- in which cues are integrated incrementally and proba- ing information matter. The most negative impres- bilistically in time. This provides an array of predictions sion came from one-sided elicitation (because people about how personal construal dynamics is shaped and are likely to cease seeking information if their initial guided by different combinations of cues. The authors 268 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH have explored such dynamics in a variety of exper- of a behavior using various forms of time-series anal- iments on the behavioral dynamics of these percep- ysis. Accordingly, in this final section of the chapter tions using, for example, mouse-tracking methods. In we review some of the tools that can be employed for this behavioral approach, researchers can track par- dynamical analysis of behavioral time series. ticipants as they move their computer mouse toward Before introducing various methods of dynamical varying options on a screen (e.g., during person con- time-series analysis, it is important to appreciate that strual). These mouse movement trajectories can then empirical research and the analysis of behavioral time- be mapped directly onto a computational neural net- series data can be, but is not always, a precursor to work model: The evolution of the computer cursor on modeling. Rather, research and modeling are best con- the screen toward a response box can be captured by a ceptualized as complementary methods of dynami- neural network state, achieving some stable response cal analysis, with researchers often moving back and activation over choices. Accordingly, the dynamical forth between both forms of research (i.e., behavioral model, together with behavioral recordings, makes a research and dynamical modeling), using experimen- tidy dynamical package for exploring the dynamics of tation and time-series analysis to identify key state social decision making, perception, and judgment. For variables, attractors states, and control parameters, a review of the behavioral results, see Freeman et al. and mathematical modeling to better understand and (2011); for software that allows dynamic tracking of test empirical findings and make future predictions. A these social processes, see MouseTracker (Freeman & detailed description of the nuances of how one goes Ambady, 2010). from the dynamical analysis of behavioral time-series data to a dynamical model is well beyond the scope of this chapter. We do wish to emphasize, however, DYNAMICAL SYSTEMS ANALYSIS that building a dynamical model is not always neces- Dynamical modeling is important because it provides sary for understanding the dynamics of behavior. In researchers with a set of tools for understanding in many cases, building a model to simulate the dynam- an abstract and often extremely general way how ics uncovered via behavioral time-series analysis will human and social behavior can emerge. As many of not necessarily provide more insights or additional the examples provided earlier highlight, the power of information about a system’s underlying dynamics. a dynamical model does not always rest on its ability Accordingly, many dynamical systems researchers are to simulate real-world behavior, but rather whether less concerned with building dynamical models and it can generate testable predictions, enhance theoret- instead focus more of their efforts on uncovering the ical development, and motivate research questions. dynamics of behavioral systems via experimentation Unfortunately, there is no step-by-step guide that and the kinds of dynamical time-series analysis tech- one can follow when developing dynamical models niques outlined in the following sections.3 of human and social behavior. Building an effective model requires good understanding of the many dif- Behavioral Measurement ferent types of models and mathematical functions that can be employed to capture differing types of As with any research study, determining valid and dynamics, as well as a significant amount of theo- reliable dependent variables is fundamental. What the rizing and lots of trial and error. A researcher inter- right dependent variable is when investigating the ested in modeling the dynamics of a behavioral system dynamics of social phenomena will of course depend also needs to have a good understanding of the sys- on what behavior is measurable in a given con- tem’s underlying stabilities and its relevant state vari- text, along with a researcher’s theoretical interests. ables and parameters. In many instances, however, researchers in social-personality psychology do not start with sets of state variables, parameters, or mathe- 3 The converse is also true, with some researchers focus- matical functions or equations, and may not know the ing primarily on building abstract dynamical models like nature of a behavioral system’s underlying dynamics. those described in the previous section without collect- In such cases, research typically starts with a tempo- ing real behavioral data (time-series or otherwise). Just as valid as empirical research, this latter “research by model- ral sequence of behavioral measurements or observa- ing” approach is less concerned with simulating real-world tions – a behavioral time series – recorded during exper- behavior and is more concerned with developing a formal yet imental, nonexperimental, or observational research. highly generalizable understanding of how organized behav- A researcher then attempts to uncover the dynamics ior can emerge, change, and dissolve over time. COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 269

Obviously, the dependent variable should capture the respiration patterns assessed at each breath intake state of the agent or system at the time of measure- for dyads involved in lengthy casual conversations ment. As we outlined previously, a behavioral state (McGarva & Warner, 2003; Warner, 1992; Warner, represents a wide array of measures. For example, Waggener, & Kronauer, 1983), an individual’s emo- socially relevant measures may come in the form tional expression twice a minute while watching an of self-esteem, personality characteristics, attitude, or emotive film (e.g., Mauss, Levenson, McCarter, Wil- social dominance measured over time. A measured helm, & Gross, 2005), or an individual’s self-esteem behavioral state could also be a mode or type of or mood every day over the course of two years (e.g., behavior, such as whether an individual carries an Delignieres,` Fortes, & Ninot, 2004). object alone or together with another person, or exclu- Behavioral time series can also be sequences of dis- sionary acts of a pair of individuals with respect crete behavioral events (e.g., occurrence of categorical to some third (perhaps out-group) individual dur- events or coded behaviors) that could be dichotomous ing an interactional game. In social-personality psy- (a single action occurs or not, such that the time series chology, it is also common for researchers to record is a sequence of 1s and 0s) or that involve several dif- other more indirect or implicit measures of a behav- ferent discrete states recorded on a nominal unit scale. ioral state, including physiological measures such as For instance, a researcher could record which object heart rate, cortisol level (Dickerson & Kemeny, 2004), is being looked at and in what sequence during a muscle activity, skin conductance, and neurophysi- joint task (Richardson & Dale, 2005), which words and ological measurement techniques (e.g., using EMG, sequences of words are used by an individual or group EEG, or fMRI; see Berkman, Cunningham, & Lieber- of individuals during a conversation (Louwerse, Dale, man, Chapter 7 in this volume; Cacioppo, Tassi- Bard, & Jeuniaux, 2012), or a dichotomous coding of nary, & Berntson, 2007; Stam, 2005). Overt behaviors when individuals vocalize or not during interactions that have social relevance also include body posi- with someone they believe to be attitudinally simi- tion and movement (for reviews, see Fowler, Richard- lar or dissimilar to themselves (McGarva & Warner, son, Marsh, & Shockley, 2008; Schmidt & Richardson, 2003). 2008) and eye gaze (Richardson & Dale, 2005). Any Irrespective of the type or time scale of the behavior social process likely has a behavioral proxy that can be being measured or recorded, the ordering of observa- tracked, semi-continuously, in time. tions or measurements in the behavioral time series No matter what dependent variable one chooses, must be recorded sequentially in time. Extracting the there are several important requirements when inves- emerging patterns or stable states of behavior from tigating the dynamics of social behavior. First, and data requires historical information about the state of most obviously, the dependent variable must corre- the system preceding its current state. If future obser- spond to a behavioral state measurement that can vations depend on observations that preceded it in be recorded repeatedly over time. This results in time – in other words, that are sequentially dependent a sequence of measurements over time, or more – then data recorded nonsequentially will prevent specifically, a behavioral time series. Here, the term identification of trends, stable states, or reoccurring “time” could refer to “clock” time such as second, patterns that may exist. In many dynamic analy- minute, hour, day, month, etc., or it could refer to ses, an additional constraint is that the time intervals some other time scale, such as trials, sessions, or between sequential measurements must be the same. events. That is, a researcher could record a behavior This is because the dynamic regimes that character- almost continuously, making measurements several ize many continuous behaviors have specific tempo- times a second or after some longer time interval. ral properties that can only be determined if the time For instance, a researcher could record an individ- between measurements is known and equivalent.4 For ual’s postural position 50 times a second during the instance, many human and social behaviors are char- course of a 2-minute conversation (e.g., Schmidt, Fitz- acterized by periodic patterns, from the intrinsic peri- patrick, Caron, & Mergeche, 2011; Shockley, Santana, odicity of brain-wave patterns (e.g., Tognoli, Lagarde, &Folwer,2003),theheartratesofaninfantand DeGuzman, & Kelso, 2007), to the leg and arm move- mother sampled 1,000 times a second during a three- ments of an individual while walking (e.g., Moussa¨ıd, minute face-to-face interaction (Feldman, Magori- Cohen, Galili, Singer, & Louzoun, 2011), vocal activ- 4 Subtle variations in time interval are not always catastrophic, ity assessed at fractions of a second (Warlaumont, especially for longer intervals (i.e., hour or day), but should Oller, Dale, Richards, Gilkerson, & Dongxin, 2010), be minimized as much as possible. 270 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

Perozo, Garnier, Helbing, & Theraulaz, 2010), to the Quera, 2005; Heyman, Lorber, Eddy, & West, Chap- day-to-day mood of an entire population of people ter 14 in this volume; Kreuz & Riordan, 2011). (Dodds, Harris, Kloumann, Bliss, & Danforth, 2011). Such research can be facilitated by powerful anno- In each case, the same or similar behavioral states tation software (Loehr & Harper, 2003). Eye move- recur again and again after a certain period of time ment technologies can use “areas of interest” (AOI) (i.e., with a certain temporal frequency). to transform a sequence of x,y-coordinates on a com- It is also crucial that the measurement device has puter screen to a set of looked-to objects. Researchers enough resolution to reliably capture whether the have also used continuously recorded body move- behavioral state has changed across repeated measure- ments and speech to infer discretely labeled states. For ments. In many instances, traditional methods of mea- example, in the domain of human-computer interac- surement, such as obtaining self-evaluations on a 7- or tion, machine-learning algorithms have been applied 9-point Likert scale, or simply coding whether or not a to extract meaningful states (Castellano, Kessous, & certain behavior or action occurred (i.e., 1 for yes and Caridakis, 2008), such as which emotion is being expe- 0 for no), can provide the resolution needed. In other rienced by a person, given multimodal cues such as cases, such as when recording the subtle gestures, pos- speech, and face and body movements. In addition, tures, or eye movements of conversing individuals, gesture recognition algorithms allow hand gesture pat- or the subtle changes in the body language, emotion, terns (discrete categories) to be obtained by learning and anxiety of an individual during an interpersonal algorithms applied to body movement data (see Mitra confrontation, more advanced methods of measure- & Acharya, 2007 for a review). ment might be required. Thankfully, recent technical progress has facilitated collection of such behavioral Methods of Dynamical Analysis time series. For instance, modern video processing technology enables researchers to acquire objective So what do you do after you have obtained a whole-body activity time series of one or more indi- time-series recording of a behavioral phenomenon? viduals from synchronized multiview video recordings How do you investigate the dynamics present in a (e.g., Kupper et al, 2010; Schmidt, Morr, Fitzpatrick, recorded time series? In general, the dynamical analy- & Richardson, 2012). There are also technologies for sis of a behavioral time series involves qualitative and continuously recording an individual’s movements in graphical assessment of the time-evolving pattern of space and time, such as Polhemus tracking systems behavior and then quantitative linear and/or nonlin- (e.g., Polhemus Liberty and Latus systems, Polhemus, ear time-series analysis. Ltd, Virginaia) and NDI Optorack (Northern Digital Qualitative and graphical assessment.Foranyresearch Inc., Ontario, Canada). There are also now a num- study, knowing what the recorded data “look” like ber of low-cost off-the-shelf gaming systems (e.g., is essential for appropriate understanding and anal- Nintendo Wii remotes and force plates, and the MS ysis. Although visual inspection alone does not typi- Kinect) that can be used to wirelessly record the cally reveal what the underlying dynamics are, it does movements of interacting individuals in real time. As provide a general understanding of the kinds of anal- for physiological measures, Biopac systems are now ysis techniques that will be needed to uncover the widely used for tracking one or more signals, and dynamics. For dynamical analysis, this first and fore- can be used as both an electroencephalogram and most involves graphing the behavioral time series on a electromyogram (Biopac Systems, Inc., Aero Camino time-series plot and visually inspecting the patterns it Goleta, CA). Blascovich (Chapter 6 in this volume) pro- contains. For illustrative purposes, hypothetical exam- vides more details about physiological measurement. ples of some of the different kinds of time series that With regard to recording more discrete behav- might be obtained in social-personality psychology are ior, even for eye movements and gestures, as well displayed in Figure 11.8 (consult sources cited later as language analysis, there are now hardware and in this paragraph for data examples). An inspection software applications that can automatically catego- of different time series highlights just a few of the rize the behaviors being emitted. For language anal- many different types of behavioral time-series patterns ysis, researchers relied on well-developed schemes that could be recorded. In some cases the patterns of in the psychological sciences for coding or transcrib- change over time are relatively simple and regular: the ing social interaction, involving time-intensive prac- monotonic decrease of an individual’s anxiety level tices that require careful selection of units of analy- over the course of 50 therapy session (Heath, 2000) sis, guided by research goals (Bakeman, Deckman, & and the oscillatory movements of an individual’s right COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 271

50 15 40

on 12  30 9 20 Anxiety

Limb Posi Limb 6 10 0 3 05101520253035404550 0123456789101112131415 Session Time (seconds) 9 1500 8 7 1000 6 5 500 4 (msec) RT Self-Esteem 3 2 0 165129193257321385449513577641705769833897961 64 128 192 256 320 384 448 512 Trial Trial 3 6 2 5 1 4 0 3 -1 2

-2 Code Numeric 1

Daily Hedonic Level Hedonic Daily -3 0 1815222936435057647178 0500100015002000250030003500 Time (seconds) Time (msec) arm while walking (Harrison & Richardson, 2009). In Figure 11.8. Hypothetical examples of several types of behav- other cases the patterns of change over time are highly ioral time series. (top left) Change in anxiety level for an indi- complex and appear to be nondeterministic or stochas- vidual over 50 therapy sessions. (middle left) An individual’s self-esteem recorded on a 9-point Likert-scale twice a day for tic (i.e., random): an individual’s self-esteem over the 512 days. (bottom left) An individual’s daily hedonic (mood) course of 1.5 years (see Delignieres` et al., 2004) and level recorded over 12 weeks. (top right) Motion sensor record- the trial-by-trial RT and an individual completing a ing of a individuals right arm movements while walking. (mid- 512 trial lexical decision task (see Holden, 2005). Oth- dle right) Reaction times of a participant completing a 512 ers seem to fall somewhere in between, containing trial lexical decision task. (bottom right) A time series repre- semi-periodic patterns or other complex regularities. senting categorical data obtained from eye movement behav- ior while a person views an array of 6 images. An eye-tracking Two examples are the daily hedonic level or mood system records a numeric identifier 1–6 reflecting which par- of individuals over the course of twelve weeks (see ticular image is being fixated over time (see text for more Larsen & Kasimatis, 1990) and the eye fixations that details). occur when an individual scans the world during a conversation (see Richardson & Dale, 2005). In addition to inspecting time-series plots of one’s reconstruction involves extracting the entire multi- data, plotting a behavioral time series as a trajectory in dimensional dynamics of a system, in all its com- phase space can also provide researchers with a clear plexity, from a one-dimensional time-series recording. qualitative understanding of the attractor(s) that con- Although a detailed description of the steps required to strain the time-evolving behavior. If the state space of complete phase space reconstruction is too involved to a behavioral systems is known a priori (and contains be unpacked here (for a more detailed description and no more than three dimensions), this is often quite tutorial, see Abarbanel, 1996 and Kantz & Schreiber, easy. It is more often the case, however, that the phase 1997), the method itself is powerfully intuitive. For space of a behavioral system is not known a priori. systems that have a phase space with more than In such cases, the process of plotting behavioral time three dimensions, phase space reconstruction also series and a trajectory in phase space requires that provides a quantifiable measure of a system’s dimen- one uncover – or more precisely, recover – the phase sion – an indication of the number of the state vari- space of a behavioral system analytically. Phase space ables required to model the system effectively. Exam- reconstruction involves a number of steps that enable ples of phase space reconstruction applied to a socially a researcher to recover a phase space isomorphic to relevant phenomenon are described in Shockley the system’s real phase space. In short, phase space et al. (2003) and Richardson, Schmidt and Kay (2007), 272 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH who study behavioral synchrony between interacting Spectral analysis and cross-spectral coherence.Oneof partners. In this research, a single bodily movement the first questions commonly asked when analyzing measure – such as postural change – is recorded as a time-series data is whether the data contains any peri- signature of the fluctuations of the total mind/body odic or temporal structure. Consider the limb move- system. To gain access to the higher dimensions of the ment and daily hedonic time series in Figure 11.8. Do system from the (single) one-dimensional time-series the up and down fluctuations occur in a stable peri- measure (i.e., change in posture sway over time), odic manner? If so, after what period of time (i.e., at researchers use phase space reconstruction to discern what frequency) does the pattern repeat itself? Con- how many dimensions best capture the fluctuations ducting a spectral analysis enables one to answer these observed in the movement time series (often as many questions by decomposing a time series into its peri- as 10 dimensions). Once phase space is reconstructed, odic components by estimating how well a set of sine researchers can then mathematically compare how or cosine functions of different frequencies and ampli- two people’s movements change in relation to each tudes fit the data. Performing a spectral analysis is other in this higher-dimensional space. much like conducting a regression analysis in that Linear and nonlinear time-series analysis.Oneofthe you are attempting to decompose the major sources key decisions a researcher must make when inspect- of variation in the data, in this case trying to deter- ing time-series plots is whether the dynamic regime mine which component frequencies account for sig- that characterizes the behavior of interest is simple or nificant amounts of variability in the signal. For highly regular enough to be analyzed using linear methods, stable periodic behavior, like the rhythmic limb move- or whether the behavioral dynamics are sufficiently ments displayed in Figure 11.8, there is usually only complex that one must employ nonlinear methods. one dominant or fundamental frequency component. Unfortunately, there is no definitive rule as to when For less stable periodic data, like the daily hedonic time one should employ linear or nonlinear methods and in series displayed in Figure 11.8, individual frequency many instances, especially when performing a dynam- components will be less powerful. There might also be ical analysis on new phenomena or on behavioral time more than one frequency component in a time series series that have not previously been examined, it is (i.e., multiple frequency components). This is partic- prudent to employ a range of linear and nonlinear ular true for highly complex or semi-periodic time methods in order to determine different aspects of the series. behavioral dynamics recorded. Spectral analysis can also be employed to determine In general terms, however, linear methods of anal- how correlated two time series are by examining, ysis are preferable when the patterning of movement essentially, the similarity of their frequency patterns. or behavior being investigated is highly regular and This comparison is called cross-spectral coherence stationary.Thatis,themeananddispersionofsam- and indexes the correlation between two time series pled values in the time series have a regular pat- on scale of 0 to 1, and is analogous to calcula- tern and remain more or less the same across the tions of the squared correlation coefficient (Gottman, interval of recorded time. The anxiety, daily hedo- 1981; Porges et al., 1980; Warner, 1988). In social- nic level, and limb movement time series in Figure personality research, cross-spectral coherence is com- 11.8 all meet this criteria, as does the RT time series monly employed to examine mutual influence and (although see section on fractal analysis later in the behavioral coordination, that is, the degree to which chapter). For time series data that is nonstationary – the one individual’s behavior is influenced by and/or mean and dispersion of sampled values vary markedly coordinated with the behavior of another. For exam- across the time-series recording – or for behavioral ple, Sadler et al. (2009) used cross-spectral methods time-series that contain a high degree of stochastic to explore the rhythmic relationships between two variability or involve highly complex or aperiodic pat- people’s dominance and affiliative dynamics during terns of change over time, nonlinear methods may be interaction. In this study, coders dynamically tracked more effective. What follows is a brief description of interaction partners using a joystick, thus producing several common and generally applicable linear and dominance/affiliation time series. The authors found nonlinear time-series methods for research in social- that, indeed, pairs of interaction partners exhibit sim- personality psychology (for more detailed discussion ilar affiliation amplitude-frequency patterns. Put dif- and tutorials, see Abarbanel, 1996; Boker & Wenger, ferently, they shared behavioral cycles. 2007; Gottman, 1981; Heath, 2000; Kantz & Schreiber Autocorrelation and cross-correlation.Dynamichuman 1997; Riley & Van Orden, 2005). and social behavior is usually correlated over time. In COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 273 other words, how an individual behaves at any given location of one behavior within its cycle relative to the moment is typically correlated with how the individ- location of the other within its cycle. Thus, if the rela- ual behaved sometime recently. The dependence or tive phase between two behavioral time series remains correlation between future and past behavior can be the same over time, the behavior is said to be coor- determined using autocorrelation. Sometimes called dinated at that relative phase relation. For example, lagged correlation, autocorrelation identifies if future consider two people coordinating their rhythmic gait states are correlated with past states by determining while walking down the street together – their leg the correlation between points in a time series at dif- cycles are in the same place and are cycling together. ferent time lags. A positive autocorrelation indicates Typically, behavioral synchrony is constrained to persistence of behavior after some time lag; the behav- two stable patterns of behavioral coordination over ioral change is similar from one observation to the time, commonly referred to as inphase and antiphase next (e.g., positive changes or state values in both past coordination (Haken, Kelso & Bunz, 1985; Schmidt, and future states). A negative autocorrelation indi- Carello, & Turvey, 1990). Inphase coordination corre- cates anti-persistence of behavior after some time lag; sponds to rhythmic or periodic movements or behav- opposite behavioral change occurs from one observa- iors that move or change in the same direction at the tion to the next (e.g., positive changes or values in the same time (such as the walkers we just mentioned). past state correspond to negative changes or values in Antiphase coordination corresponds to rhythmic or the future states). periodic movements or behaviors that move or change Cross-correlation is a simple extension of autocor- in the opposite direction at the same time. To use the relation and examines the dependence or correlation walking example again, this would mean that indi- between future and past values of different time series. viduals would be continuously moving their legs in It is also commonly used to examine mutual influence opposite patterns – as one person swings her right leg and behavioral coordination, and yields similar results forward, the other would be swinging her right leg to coherence analysis (described earlier), except that back. It is worth noting that these latter descriptions of one can also look at the correlation between individ- inphase and antiphase coordination characterize per- uals’ behaviors at time lags other than zero. Accord- fect or absolute synchrony. During natural social inter- ingly, it can be employed to determine if two behav- action, however, the movements or behavior individ- iors are attracted toward each other, and also whether uals do not usually become coordinated in a perfect one behavior leads or follows another behavior at inphase or antiphase manner, but rather exhibit inter- some specific time lag. mittent periods of inphase and/or antiphase coordina- Relative phase analysis. Another technique for exam- tion (e.g., Richardson, Schmidt, & Kay, 2007; Schmidt ining mutual influence and behavioral coordination & O’Brien, 1997). is relative phase analysis. This technique has been Recurrence analysis. The analysis methods discussed employed most extensively in research examining so far are based primarily on assumptions of linear behavioral synchrony – the rhythmic movement coor- relations that underlie most analyses commonly used dination that occurs between the limb or body move- in psychology (e.g., ANOVA). Accordingly, they are ments of interacting individuals (e.g., Marsh, Richard- only able to capture the linear dynamics of station- son, & Schmidt, 2009; Miles, Lumsden, Richardson, ary time-series data. Recurrence analysis, however, &Macrae,2011;Schmidt&Richardson,2008).Itcan is a nonlinear analysis method and can be employed also be employed to investigate the patterns of coordi- to analyze both stationary and nonstationary data. nation that occur between any set of rhythmic or peri- Indeed, the beauty of recurrence analysis, in com- odic behavior (e.g., the coordinated changes in day-to- parison to other linear time-series methods, is that day mood of husband and wife or mother and child). it does not require assumptions about the structure In short, the technique involves calculating the dif- of the time series being investigated or the underly- ference in the “phase” of two (or more) rhythmic or ing dynamics that shape the recorded structure: The periodic behaviors over time (for details on multivari- behavior can be periodic, nonperiodic, or stochastic, ate relative phase analysis, see Frank and Richardson, even discrete or categorical. 2010; Richardson, Garcia, Frank, Gregor, & Marsh, Although recurrence analysis is still relatively new, 2012). Here the term “phase” refers to the location particularly in psychology (e.g., Riley, Balasubrama- of a system or behavior within its cycle. The relative niam, & Turvey, 1999; Shockley, Santana, & Fowler, phase or “difference in phase” between two rhyth- 2003), there is now substantial evidence that suggests mic or periodic behaviors therefore corresponds to the it is potentially one of the most robust and generally 274 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

Word sequence of Hello Goodbye lyrics 25 1

20 0.8

15 0.6

10 0.4

Numeric code (word) 5 0.2

0 0 0 20 40 60 80 100 120 140 160 180 0 500 1000 1500 Time (word) Time (points) Recurrence Plot

18

16

14

12

10

8 Y(j) points j, time (word)

6

4

2 yousayyesi sayno yousaystopandi saygo go go oh no

0 you say yes i say no you say stopand i say go go go oh no

0 2 4 6 8 10 12 14 16 18 X(i) points i, time (word)

Figure 11.9. (top left) The full time series of words extracted recurrence analysis identifies the dynamics of a sys- from the lyrics of “Hello, Goodbye” by the Beatles. The y-axis tem by discerning (1) whether the states of the system represents the numeric identifier to which a word is assigned, behavior recur over time and, if states are recurrent and the x-axis represents word-by-word unfolding of this “lex- over time, (2) the degree to which the patterning of ical” time series. (bottom left) A recurrence plot of the first 20 words in the lyrics. Each point on the plot represent a relative recurrences are highly regular or repetitive (i.e., deter- point (i,j)inthelyricsatwhichawordisrecurring.The“gogo ministic). Conceptually, performing recurrence analy- go” usage appears as a particular “texture” on the plot, along sis on behavioral data is relatively easy to understand; with the two-word sequence “you say” appearing as a diago- one simply plots whether the recorded points, states, nal line structure. (top right) The anterior-posterior postural events, or categories in a time series or behavioral tra- sway movements of single individual standing and listening jectory are revisited or reoccur over time on a two- to another person speak for 30 seconds, recorded at 50 sam- dimensional plot, called a recurrence plot. This plot ples a second. (bottom right) A recurrence plot of the first 10 seconds of postural data. Despite the nonperiodic and nonsta- provides a visualization of the patterns of revisitations tionary pattern of the postural sway movement, the recurrence in a system’s behavioral state space and can be quan- plot reveals a significant degree of recurrent and deterministic tified in various ways – a process known as recurrence structure, with the patterns of postural behavior reoccurring quantification – in order to identify the structure of the over time. dynamics that exist (see Marwan, 2008 and ; Weber & Zbliut, 2005 for more detailed reviews). The plots in applicable methods for assessing the dynamics of bio- Figure 11.9 are examples of what recurrence plots look logical and human behavior (e.g., Marwan & Meinke, like for a categorical (left plot) and continuous (right 2002; Zbilut, Thomasson, & Webber, 2002), including plot) behavioral time series. social behavior (e.g., Dale & Spivey, 2005; Richard- Like spectral analysis and autocorrelation, recur- son et al., 2008; ; Shockley et al., 2003). Essentially, rence analysis can also be extended to uncover the COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 275 dynamic similarity, mutual influence, or coordinated Figure 11.8 is a good example of a fractal or self-similar structure that exists between two different behav- time-series pattern. This time series is displayed again ioral time series or sequences of behavioral events. in Figure 11.10, with the self-similarity of its temporal This latter form of recurrence analysis is termed cross- fluctuations revealed by zooming in on smaller and recurrence analysis and is performed in much the same smaller sections. At each level of magnification the way as standard (auto) recurrence analysis. The key temporal pattern looks similar (see Bassingthwaighte, difference is that recurrent points in a cross-recurrence Liebovitch, & West, 1994 or Holden, 2005 for a more plot correspond to states, events, or categories in two detailed tutorial). time series or behavioral trajectories that are recurrent A fractal time-series pattern is characterized by an with each other. Cross-recurrence analysis can there- inverse proportional relationship between the power fore be employed to capture and then quantify the co- (P) and frequency (f) of observed variation in a time occurring or coordination dynamics of two behavioral series of measurements.6 That is, for a fractal time- time series or discrete behavioral sequences. Accord- series, there exists a proportional relationship between ingly, some researchers have adopted cross-recurrence the size of a change and how frequently changes of analysis to investigate semantic similarity in conver- that size occur, with this relationship remaining stable sation (Angus, Smith, & Wiles, 2011), perceptual- across changes in scale. It is in this sense that the pat- motor synchrony between people interacting (Shock- tern of variability in a repeatedly measured behavior ley et al., 2003; Richardson & Dale, 2005), and vocal is self-similar; large-scale changes occur with the same dynamics during development (Warlaumont et al., relative frequency as small-scale changes. The degree 2010). to which a dataset approximates this ideal relationship Fractal analysis. Researchers in social-personality between power and frequency, P 1ࢧf α,issumma- = psychology (or in any other field of psychology) com- rized in the scaling exponent, α, with P power, and = monly collapse repeated measurements into summary f frequency. If one plots the power of the differ- = variables, such as the mean and standard deviation, ent spectral frequencies that make up a time series on under the assumption that the measured data contains double-logarithmic axes, α is equivalent to the slope uncorrelated variance or random fluctuations that are of the line that best fits the data (see Figure 11.11). normally distributed. With respect to dynamic behav- That is, α captures the relationship between size and ioral time-series data, however, this is rarely true, and frequency of fluctuations in the time series of behav- thus summary statistics such as the mean and stan- ior. Random fluctuations (i.e., white noise) produce dard deviation often reveal little about how a sys- a flat line in a log-log spectral plot with a slope close tem evolves over time. Indeed, time-series record- to 0, which indicates that changes of all different sizes ings of human performance and behavior typically occur with approximately the same frequency in the contain various levels of correlated variance or data time series. Alternatively, fractal fluctuations, often fluctuations (i.e., nonrandom fluctuations) that are referred to as pink or 1ࢧf noise, produce a line in a log- not normally distributed (Stephen & Mirman, 2010) log spectral plot that has a slope closer to –1, which and, moreover, are structured in a fractal or self-similar indicates the self-similar and scale-invariant scaling manner (Gilden, 2001, 2009; Van Orden, Holden, & relationship characteristic of fractal patterns. Turvey, 2003; Van Orden, Kloos, & Wallot, 2011). It is becoming increasingly clear that the behav- A fractal or self-similar pattern is simply a pat- ior of most natural systems, including human and tern that is composed of copies of itself nested within social systems, exhibit varying degrees of fractal struc- itself. As a result, the structure looks similar at differ- ture (Delignieres` et al., 2006; Gilden, 2009; Holden, ent scales of observation (i.e., magnification). Concep- 2005). Moreover, the degree to which the fluctua- tually similar to geometric fractal patterns (Mandel- tions in a behavioral time series are fractal (i.e., pink) brot, 1982), a fractal time series is therefore a time or not (i.e., white) provides evidence that a system series that contains nested patterns of variability (see Figure 11.10). That is, the patterns of fluctuation and change over time look similar at different scales of self-similar in a statistical sense. Statistical self-similarity sim- magnification or measurement resolution (i.e., as one ply means that a pattern is composed of statistically similar copies of itself (looks, on average, similar at different scales zooms in and out).5 The self-esteem time series in of observation). 6 For an introductory review of the various methods that can 5 In actuality, only ideal mathematical or geometric fractals be employed to measures the fractal structure of time-series are truly self-similar, with real-world fractals considered data, see Delignieres` et al. (2006). 276 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

9 8 7 6 5 4 3 Measurement 2 15121024 Trial 9 8 7 6 5 4 3 Measurement 2 1256512 Trial

9 8 7 6 5 4 3 Measurement 2 1128256 Trial

Figure 11.10. Example geometric and temporal fractal pat- tem. So the behavioral fluctuations a person gives off terns. (left) Koch Snowflake at three levels of magnifica- may hint at social judgment events, such as stereotyp- tion. (right) The repeated self-esteem measurements presented ing or racial bias. This avenue of research still seems in Figure 11.9 at three levels of magnification. The factual relatively unexplored, and surprisingly so, if Correll’s nature of these patterns is revealed by self-similar patterns being observed at smaller and larger magnitudes of observa- (2008) results are robust in multiple contexts. tion (adapted from Holden, 2005).

Further Reading is nonlinear and that its behavior is a consequence Finally, we should note that there are a num- of interaction-dominant dynamics (Van Orden et al., ber of reviews of this material that can be consulted 2003). To this extent, fractal patterns of behavior can in social psychology (e.g., Nowak & Vallacher, 1998; also be a sign of emergence and self-organization. Vallacher & Nowak, 1994) or within cognitive sci- Although these ideas may seem foreign and irrelevant ence more broadly (e.g., Port & Van Gelder, 1995; to the uninitiated, these analyses of fractal fluctua- Spivey, 2007; Warren, 2006). Note as well that there tions have been applied fruitfully in the social domain. are, of course, different levels of theoretical commit- For example, the behavioral waves or periodic flow ment to this agenda. For example, many dynamical of social interaction has a fractal structure (Newtson, systems approaches explicitly avoid the use of men- 1994), as do the dynamics of self-esteem (Delignieres` tal representations, and instead focus on perception- et al., 2004). More recently, Correll (2008) has shown action couplings as a basis for understanding social and that participants who are trying to avoid racial bias human behavior (e.g., Marsh et al., 2009; Richard- show a lesser fractal signature in their response laten- son et al., 2009). Others focus more on the dynam- cies in a video game. Correll discusses these findings ics of internal mental processes or cognitive dynam- in light of characterizing social perception and other ics (e.g., Spivey, 2007). This leads to theoretical processes as a system of many intertwined depen- subtleties that cannot be conveyed here. If readers dencies – as processes of a complex dynamical sys- are intrigued to look further, we would encourage COMPLEX DYNAMICAL SYSTEMS IN SOCIAL AND PERSONALITY PSYCHOLOGY 277

Random Variation (White Noise) Fractal Variation (Pink Noise) Measurement Measurement

Time Time Log Power Log Power

α = 0.02 α = –0.73

Log Frequency Log Frequency them to decide whether the theory, modeling, or Figure 11.11. Examples of time series composed of random analysis of dynamical systems motivates their inter- variation (top left) and fractal variation (top right) and the est. If it is deeply theoretical, for example, then much associated spectral plots with logarithmic axes (bottom left and right, respectively). of the work we review would help tease apart those theoretical nuances. We also encourage readers to read the books of Kelso (1995), Nowak and Val- systems as systems sustained by self-organization, lacher (1998), and Thelen and Smith (1994) for a bringing about soft-assembled processes, through non- more thorough, but still easy-to-read, introduction to linear interaction-dominant dynamics. Our second dynamical systems theory and practice. If one is more key goal was to demonstrate how the dynamics of interested in mathematical modeling, a review of the social processes and behavior can be explored directly primary modeling papers cited will help the researcher using mathematical models. By laying out some of jump straight into the concepts and their practicality the mathematical modeling techniques that can be in modeling. Reviews by Kaplan and Glass (1995) and employed to understand dynamical systems, we show- Strogatz (1994) also provide a good introduction to the cased how the relatively new concepts of dynamical mathematical details of dynamical systems modeling. systems gain concrete manifestation in these explicit With respect to dynamical systems analysis, direct con- models. Our third goal was to provide a brief descrip- sultation with research that has previously employed tion of how the dynamics of behavior can be explored a method of interest is always the best place to start. via the dynamical analyses of recorded data. Thus, in Throughout this chapter we have therefore included a the third section of this chapter we described just a range of relevant research articles that will enable any few of the many linear and nonlinear time-series anal- interested reader to gain a foothold on the relevant yses techniques that a social researcher could employ literature. to investigate the dynamics inherent to his or her own behavioral (time-series) data. Collectively, these sections urge social-personality CONCLUSION psychologists to think of behavior as something that We had three key goals in this chapter. The first was continually changes and, therefore, that must be stud- to lay out some of the basic concepts behind com- ied and modeled as time-evolving. It is often challenging plex dynamical systems. These concepts motivate the- for a researcher to conceptualize his or her context of orists as they seek to understand social and cognitive study in such a way that time series can be collected 278 MICHAELJ.RICHARDSON,RICKDALE,ANDKERRYL.MARSH

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