Social creativity as a function of agent cognition and network properties: A computer model

Social Networks 32 (2010) 263–278

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Social creativity as a function of agent cognition and network properties: A computer model夽 Siddhartha Bhattacharyya a,∗ , Stellan Ohlsson b,1 a b

University of Illinois at Chicago, Department of Information and Decision Sciences,601 S. Morgan Street, Chicago, IL 60607, United States University of Illinois at Chicago, Department of Psychology, 1007 West Harrison Street, Chicago, IL 60607, United States

a r t i c l e

i n f o

Keywords: Agent-based modeling Cognitive capacity Collaboration Creativity Invention Network connectivity Problem solving Social creativity

a b s t r a c t Inventions – concepts, devices, procedures – are often created by networks of interacting agents in which the agents can be individuals (as in a scientific discipline) or they can themselves be collectives (as in firms interacting in a market). Different collectives create and invent at different rates. It is plausible that the rate of invention is jointly determined by properties of the agents (e.g., their cognitive capacity) and by properties of the network of interactions (e.g., the density of the communication links), but little is known about such two-level interactions. We present an agent-based model of social creativity in which the individual agent captures key features of the human cognitive architecture derived from cognitive psychology, and the interactions are modeled by agents exchanging partial results of their symbolic processing of task information. We investigated the effect of agent and network properties on rates of invention and diffusion in the network via systematic parameter variations. Simulation runs show, among other results, that (a) the simulation exhibits network effects, i.e., the model captures the beneficial effect of collaboration; (b) the density of connections produces diminishing returns in term of the benefits on the invention rate; and (c) limits on the cognitive capacity of the individual agents have the counterintuitive consequence of focusing their efforts. Limitations and relations to other computer simulation models of creative collectives are discussed. © 2010 Published by Elsevier B.V.

1. Introduction Artistic, economic, scientific or technological inventions are often created by collectives, a category that includes but is not limited to groups, teams, firms, scientific disciplines and entire communities. We use the term invention in a broad sense that covers the production of novelty in any area of activity, and we distinguish between invention, the creation of something novel for the first time, and dissemination, the spread of something novel through a network. The creative processes of collectives vary with respect to rate. For example, the rate of technological invention in Western Europe from the year 500 AD to the year 1000 AD was lower than the corresponding rate in the period 1500–2000 AD Focusing on individual nations, the increase in economic growth that produced the Industrial Revolution in Britain in the period 1780–1830 was more

夽 The preparation of this article was supported, in part, by a seed grant from the University of Illinois at Chicago. We thank and Poornima Krishnan for assistance in implementing the model and running simulation experiments. ∗ Corresponding author. Tel.: +1 312 996 8794; fax: +1 312 413 0385. E-mail addresses: [email protected] (S. Bhattacharyya), [email protected] (S. Ohlsson). 1 Tel.: +1 312 996 6643; fax: +1 312 413 4122. 0378-8733/$ – see front matter © 2010 Published by Elsevier B.V. doi:10.1016/j.socnet.2010.04.001

rapid than during any comparable preceding period, and the economic policy of New Zealand changed at a more rapid rate in the 1980–2000 period than they did in the 1960–1980 period. The variation in rate of change applies to markets and industries as well as nations. The rate of innovation in the information technology industry in the period 1975–2000 has been higher than the rate of innovation in, say, automobiles, buses and trains over the same period. This year’s car is different from the 1975 car, but the year 2000 laptop computer was not merely different from the computer of 1975. The personal computer did not exist in 1975; we were still working on timesharing machines. In science, we might compare the relative stability of physics in the period of 1790–1830 with the turbulent emergence of relativity and quantum mechanics in 1890–1930. “One of the central historical questions concerning technical progress is its extreme variability over time and place” (Rosenberg, 1982, p. 8). An explanatory model of social creativity should account for variations in the rates of invention and dissemination from one collective to another. In this article we focus on small scientific specialties and medium-sized communities of artists or technologists working at the cutting edge of a new development. Examples include the first generation of French impressionists, the community of World War II scientists who invented radar (Buderi, 1996) and the first generation of cognitive scientists (Gardner, 1985). Such collectives are

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larger than the 2–10 person groups and teams typically studied empirically in experimental social psychology. At the same time, they are smaller in size than the populations that are the focus of interest in studies of, for example, economics of innovation (Galambos and Eliot Sewell, 1997) and the sociology of popular fads (Lynch, 1996). Creative collectives of this type are characterized by three distinctive features: First, there is no division of labor in the classical sense. Instead, all the participating individuals work on the shared problem, and any one of them can accomplish a breakthrough that will affect the future work of the entire collective. Second, it is enough for one member of the collective to have reached a particular conclusion, insight or product for the collective to possess that conclusion, insight or product. This is, in principle, the case in art, science, technology and other endeavors in which results are freely broadcast through a variety of public channels. Third and most important, intermediate and partial results are freely broadcast during ongoing problem solving. Not only the dissemination but also the initial invention of a novelty occurs through interactions among the members. There is a widespread belief that collectives are more inventive than isolated individuals – lone geniuses fiddling in their garages – precisely because their members bring diverse knowledge to bear on the shared task and inventions emerge out of their interactions (John-Steiner, 2000; Paulus and Nijstad, 2003; Sawyer, 2003; West, 2002). If the pooling of knowledge is indeed an important factor, we would expect the rate of invention and dissemination to be a function of properties of the communication network. Knowledge is easier shared, the higher the density of communication links, the higher the probability that one agent decides to communicate with another agent and the probability that a communication is heeded by the recipient. Although a focus on the collective and social nature of invention is a useful corrective to the traditional emphasis on the lone genius model (Hadamard, 1954; Gardner, 1993; Ghiselin, 1952; Gruber, 1974; Koestler, 1964; Simonton, 1999; Wallace and Gruber, 1989), individual cognition is not rendered irrelevant by a focus on social cognition. Empirical studies in the social sciences have documented that the assumed synergy among the members of a collective is not always realized. For example, social psychologists have studied problem solving in groups, and rarely find that a group can do better than its best member (strong synergy), even though the group is often better than or equal to its average member (weak synergy); see Larson (forthcoming) for a review. A number of explanations have been proposed for such process loss: communication difficulties, inappropriate status relations within the collective, affective reaction to group diversity, and others (Basadur and Head, 2001; Levine and Moreland, 2004; Milliken et al., 2003). Some mechanisms of process loss can be formulated with precision. In a series of studies, James R. Larson and coworkers (Larson and Christensen, 1993; Larson et al., 1998, 1994) have shown that groups have a tendency to focus on information that every member knows already, for the statistical reason that a particular concept has a higher probability of being recalled and entered into a discussion, the larger the number of members who know that concept. However, the most obvious source of process loss is the cognitive loads imposed on the individual by the interactions themselves. It requires cognitive processing to participate in a cognitive collective, of whatever size. Communications from others must be attended to and encoded into memory to participate in the individual’s own thinking. There are obvious risks of flooding a capacity limited cognitive system with an overflow of information. When everybody gets 300 email messages a day, nobody has time to create anything. Although the principle of pooling cognitive resources through a communication network is certainly valid, this potential advantage is realized to varying degrees as a function of how the

interactions affect the participating individuals. One goal for the study of social creativity is to clarify the conditions under which the benefits of interaction are greater than the process losses. We conclude that social creativity needs to be described at multiple levels simultaneously (Cowan and Jonard, 2003; Cowan et al., 2004; Schilling and Phelps, 2007; Fischer et al., 2005; Larson, 2007; Pirola-Merlo and Mann, 2004). At the level of individual cognition, memories, thoughts, decisions, concepts and ideas combine and interact to produce the individual’s contribution to the collective effort. At the level of the collective, the individuals interact to produce the output of the collective. To explain collective creation is not to choose between these two levels of analysis, but to clarify the relation between them. After all, the network can only support the sharing of ideas if the individual agents have any ideas to share. Hence, the rate of invention and dissemination is likely to be a function of both network parameters (density of communication links, disposition to communicate, the probability that a communication is heeded, etc.) and properties of the cognitive architecture of the participating individuals (capacity limits, problem solving strategies, etc.). A model of rate of invention and dissemination must draw upon both network theory and cognitive psychology. In this paper, we describe the initial version of a computer simulation model of a creative collective and present some results regarding the effect of agent and network parameters on the model’s rate of invention and dissemination. Our approach draws upon prior work in a variety of disciplines, but exhibits some important differences. There are three key issues in building a model of a social system: How are the individuals conceptualized and implemented? How is the social system implemented in the model? How is the relation between the simulated individual and the simulated social system represented? That is, what process in the model corresponds to the types of interactions that occur in the simulated social system? Different disciplines handle these issues in different ways. One approach to the first issue is to simplify the model of the individual by removing all internal processing. An individual is then a mere way station, a node in the network that receives messages and passes them on unchanged. This is approach has been used extensively in the recent wave of network models in which the individual nodes in the network lack any internal structure (Cowan and Jonard, 2004; Schilling and Phelps, 2007; Watts, 2003). Because they do not take the cognitive work of the individual into account, this type of model provide few resources for accounting for process loss. They are primarily models of dissemination rather than models of invention. At the opposite end of the scale are models of the human cognitive architecture as formulated in cognitive psychology (Anderson, 1983, 2007; Newell, 1990; Sun et al., 2005). Symbolic artificial intelligence models of human cognition attempt to duplicate the full range of human cognitive abilities (Ohlsson, 2008). Models of this sort make detailed assumptions about each component of cognitive functioning, e.g., memory retrieval and capacity limits. But most of the details have little or no impact on the functioning of the social system in which an individual participates. For example, it may not matter at the social level exactly how cognitive processing is limited, it only matters that it is limited. Models of the cognitive architecture also require extensive cognitive task analysis and the formal representation of the content of the knowledge human beings bring to bear in the simulated interactions. To date, such models have been used to simulate invention by individuals but not dissemination. Between these two extremes are models that capture certain gross features of individual cognition that are hypothesized to impact the functioning of the social system of interest, but abstract over other features. The model of the individual varies widely from model to model. Schilling and Phelps (2004), building on the network model of insight by Schilling (2005), has described

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a two-level model of social creativity based on the assumption of level-invariance: Both the individual and the collective can be modeled as networks; moreover, as the same kind of networks. Our approach starts from the opposite assumption: The individual and the collective are two qualitatively different types of systems, each characterized by a distinct set of processes, functions and quantitative parameters, and a theory needs to describe both separately and then specify how the properties of the individual scale up to determine the properties of the collective. Cowan et al. (2004) have presented a two-level model of knowledge growth, in which the individual is represented by a measure of amount of knowledge, but nothing is said about how knowledge is processed in invention and dissemination. Our approach differs in that we model explicitly the symbol structures that embody the task related knowledge of the individual agent, as opposed to representing that knowledge solely by a quantitative measure of its amount. In yet a third approach, March (1991) in an influential paper modeled the development and diffusion of organizational knowledge by representing each individual as a vector of beliefs about m dimensions of reality, each belief being either 1 (adopted), 0 (uncertainty) or −1 (rejected). The vector values either correspond to or deviate from the true values, and they change as a function of interactions with other individuals and with the culture of their organization, the latter also represented as a belief vector (Miller et al., 2006; Fang et al., in press). Although the line of modeling initiated by March’s (1991) paper represents the knowledge of the individual, it abstracts over the processes by which knowledge changes and hence does not represent capacity limitations. Although we also represent knowledge as vectors of symbols, our approach is closer to that of Larson (2007) and of Saunders and Gero (2001), in which the problem solving strategies of group members are interlinked to explain the unfolding problem solving behavior of the group, but we specify the internal structure of individual cognition differently. We include sufficient structure in the model of the individual to capture the dynamics between memory, thinking and communication. Knowledge elements are represented by symbol vectors, and the individual is represented by a long-term memory, a capacity-limited working memory, processes for storing knowledge elements in long-term memory and retrieving them into working memory, and a set of operations by which the individual creates new knowledge elements. The purpose of this model is not to make novel claims about the cognitive architecture, but to capture the gross features of individual cognition that might influence functionality at the collective level. Those features include the capacity to generate new ideas and to search a space of possibilities, but also capacity limitations that bound the rationality of the individual. In short, our model includes enough cognitive machinery to allow us to capture the internal dynamics of individual cognition and explore its impact on social processing. The second and third of the three issues – how the social system itself and the interaction between the individual and system are represented a – also exhibit a variety of approaches. In models of formal organizations, the question arises how to represent the organization itself, as opposed to the set of participating individuals. March (1991) introduced the concept of a belief vector that represents the belief state of the organization, corresponding to, for example, the beliefs held by the leadership of the organization or explicitly codified in organizational documents. In this line of modeling, the interaction is represented by the fact that individual belief vectors have a certain probability of changing in the direction of greater similarity with the organizational belief vector. Similarly, Siggelkow and Levinthal, 2003) and Siggelkow and Rivkin (2005) represented formal organizations as activity vectors, where each activity can take two values. Decision making at the social system consists of choosing a particular activity configuration with a view towards maximizing a value function. Although

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the activity vectors can be partitioned into subsets that are under the control of different decision makers, there is no explicit representation of the individual in this model, and hence none of the interaction between the individual and the firm. Our approach differs form these modeling efforts in the choice of social system. We focus on collectives that do not have a formal organizational structure, but in which individuals interact in the course of their problem solving efforts. When the main focus of modeling is on the structure of the network, the social system is typically modeled by communication links between pairs of individuals. In this type of model, the individual interacts with the system by broadcasting and receiving messages. We have adopted this approach, but with the difference that we model messages as symbol vectors in which each value represents a knowledge element. The purpose of this network aspect of the model is to capture the gross features of loose collaboration that characterizes creative collectives. Importantly, the simulated individuals communicate intermediate and partial results before they have found a complete solution to the shared task, so our model is a model of collective invention as well as dissemination. Our hypothesis is that the density of communications in creative collectives interacts with the capacity limitations on individual cognition. We intend to study this interaction by systematically varying the density of communication connections and how much information flows along those connections. This raises the question of what kind of basic network structure to assume. Small-world networks are by definition networks in which there are only a few long-range links, so increasing the density of connections could only apply to the local connections. Similarly, a top-down, formal organizational network, by definition, only has a links from the top node to the next level, and so on. There are obvious possibilities for interaction between density and networks structure. To isolate the effect of communication density, we therefore study a more uniform sort of network, in which each individual communicates with a randomly chosen subset of other individuals. We note that this type of interaction structure appears to fit well to the types of creative collectives that we have in mind, e.g., scientific disciplines, communities of independent inventors, communities of artists, etc. In scientific communities, each individual researchers interacts with others via email, conference presentations, publications, journal and grant reviewing, and in other ways. The question of interest is how the rates of invention and dissemination are impacted by the two key sets of properties, the parameters of the cognitive architecture and the properties of the network itself. Our initial hypotheses were the following: (a) A network of interacting agents invents at a different rate than the set of those same individuals in a situation without interaction. (b) Network properties, especially the density of connections, will affect rates of invention and dissemination, with higher connectivity associated with higher rates of invention and dissemination. (c) Properties of the individual agent’s cognitive architecture, especially cognitive capacity, will affect the rates of invention and dissemination, with higher capacity associated with higher rates. (d) Connection density and cognitive capacity interact. In particular, higher cognitive capacity will off-set the process losses associated with higher connection densities, so that increases in connection density is more beneficial for high-capacity than for low-capacity agents. To anticipate, the interactions between network parameters and parameters of the cognitive architecture turn out to be more complicated than this hypothesis states. We describe our model at the conceptual level, provide implementation details and then report our simulations. The discussion section interprets the results and relates them to other models of social creativity. We end by discussing limitations and future research.

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2. A two-level model of social creativity The cognitive architecture of an individual is complex and there is no possibility of modeling everything that happens when a person participates in a creative process (perception, attention allocation, memory, language understanding, decision making, reasoning, learning, etc.). But every aspect of individual cognition is not equally important for the individual’s participation in a creative collective. For example, object recognition is a research topic in cognitive psychology that has generated hundreds of research studies. For present purposes we do not need to be concerned with this process, because its properties do not scale up to the collective level. It does not matter for the functioning of, for example, a research and development team exactly how its members are able to recognize a pair of pliers as a pair of pliers; what matters is only that they are so capable. If they were not, their interactions would have to adjust to this fact, but because it is a reasonable assumption that they have this capability, we can abstract from the internal structure of the object recognition process, complex though it might be. In general, only a few gross features of the cognitive architecture impact the collective level. Our model of individual cognition is designed to capture those, and only those features. This is why we refer to it as a coarse-grained model. We assume that cognitive processes are operations on knowledge representations, and that this fact is essential for understanding the functioning of the individual. We model knowledge elements as simple lists of symbols. For purposes of the simulation, it does not matter what we take the elements to stand for. They can interpreted as ideas, concepts, data, questions, theories, propositions, schemas, beliefs, blue prints, tactics, plans, goals, subgoals and so on. The cognitive architecture is a system for processing knowledge elements, and thought operations are modeled as operations that transform such elements into new elements. The consequence of explicitly modeling the internal cognitive processing of individuals is that each simulated individual – each agent – in our model is capable of solving the shared task in isolation, without any communicating with other agents. We use the zerocommunication (lone geniuses) scenario as a base line condition for our simulations. Some simulations of networks lack any base line condition against which network effects can be compared. The coarse-grained model of the cognitive architecture incorporates hypotheses that are now standard in cognitive psychology. There is a long-term memory (a.k.a. the stock) that holds the agent’s knowledge elements; a small subset of knowledge elements are heeded at any moment in time and are held in working memory, also called the active list. The active list represents the information that is currently being actively considered and processed, either as inputs into cognitive operations or as content to be communicated to other agents. In keeping with a central principle of cognitive psychology, we assume a limited capacity for the active list. Empirical studies of the working memory of people provide varying estimates of the average human working memory capacity, but those estimates typically fall in the 5–9 range. The small capacity of working memory strongly constrains the possible solution paths. New elements can enter the active list in three ways: through retrieval from long-term memory, as results from cognitive operations, or via communication from other agents. The entry of new elements in the active list can displace current elements, and displaced elements might have to be re-created for later use if needed. Certain active list elements are also stored into the long-term memory. Which elements are retrieved from long term memory and saved into long term memory in any one cognitive step obviously has the potential to strongly impact a creative process; this is based on a measure of utility of elements, as described below. The particular repertoire of cognitive operations included in our simulation is derived from the operations explored in research

on genetic algorithms (Holland, 1975). The use of such operations is common in various complex adaptive systems (Holland, 1995) and agent-based models including studies of organizations, society and networks (Axelrod, 1997; Bruderer and Singh, 1996; Macy and Willer, 2002). New elements can be obtained through exchange of component symbols from current elements in active list, or through random changes to symbols in a current element; each operation is described in detail in the next section on implementation. The processing is guided by some goal, an element that represents a desirable effect, product or state of affairs that is the target of the creative effort. This target element can be thought of as a description of a desirable end state (e.g., “a battery with half the weight and twice the life time of current batteries”). To solve the current problem is to produce an instance of the target element. Creative individuals often pursue multiple goals in parallel, and our model has that capability as well; agents in the model can thus simultaneously search for instances of more than one target element. The model assigns each knowledge element a value that measures its promise, i.e., how useful that element is likely to be in pursuing a target. The value is used to guide all cognitive processing. The values are assigned by an evaluation function, i.e., a function from knowledge elements to numeric values. This function can be conceptualized in different ways in different domains. For example, chess players have a well-developed system for evaluating the strength of chess positions. This is based on the power of the individual pieces, the number of pieces left for each side and so on. In other domains, the evaluation function needs to be conceptualized differently. We model the existence of some evaluation function generically, by assigning knowledge elements a value that measures their similarity to the target in terms of overlapping symbols. Notice that a very accurate evaluation function might provide an unrealistically focused problem solving effort. In reality, the accuracy of evaluation functions – that is, how predictive they are of distance to a person’s goal – is likely to vary. Noise in the evaluation function is thus a parameter of potential importance for the individual’s contribution to the collective effort. The cognitive architecture iterates through a standard operating cycle: At the beginning of a new cycle, it receives elements communicated to it by other individuals and inserts these into its working memory. Second, the architecture selects elements to be retrieved from long-term memory, if any; these, too, are inserted into its working memory, possibly displacing other elements. One or more cognitive operations are then selected, together with a subset of the active elements as input to each operation. The operations create new elements that appear in working memory, once again possibly displacing elements that were already there. Certain working memory elements are next selected for saving into the long-term memory. Finally, one or more working memory elements are communicated to other agents. The cycle then starts over. The selection of elements for retrieval, storing, communication and displacement is probabilistic and based on the relative values of the elements. Higher valued elements are more likely to be selected for processing, and less likely to be displaced from memory. Many details are intentionally left out of this coarse-grained model of individual cognition. What is the nature of the capacity limitation on working memory? How are knowledge elements encoded into memory? Should the elements be understood as schemas, propositions, rules or something else? Is memory retrieval implemented as spread of activation or in some other way? These questions, and dozens of other, related questions, are the subject of research in cognitive psychology. However, most details of individual cognition have little impact at the level of interaction with a collective. For example, it does not matter at the collective level exactly how working memory capacity is limited; it only matters that it is limited. The coarse-grained model is

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an attempt to capture those features of the cognitive architecture that are likely to have an impact at the collective level. The collective is simulated by a set of agents working in parallel. Each agent pursues the same target (or list of targets). This is the sense in which the agents have a shared task. Each agent pursues the shared target, but also communicates the results of its efforts to the other agents in the network. This is the sense in which the agents collaborate. Our model is not a model of division of labor. There is no breakdown of the shared task into component tasks and no center that allocates these subtasks to separate agents. Instead, each agent does its best to reach the shared target, as is the case when multiple research laboratories pursue the answer to the same research question or when multiple inventors pursue the first working prototype of a new type of device. Each agent has a set of communication links. The outgoing links designate those agents to which an agent communicates its partial results; the incoming links designate those agents from which an agent receives communications. Communicated elements enter working memory and exist there on the same conditions as elements retrieved from memory or created by thought operations. They can be encoded into memory, used as inputs to cognitive operations, broadcast in turn or be displaced by the arrival of additional elements in working memory. In each cycle of operation, each agent makes a probabilistic decision as to which, if any, of the knowledge elements currently in its working memory should be broadcast to other agents. All working memory elements are potential candidates for broadcasting, regardless of origin. It is important that the agents communicate content to one another. That is, they do not merely pass on a quantity of activation or knowledge, but information that can help bring the problem solving process of another agent closer to the shared target. Every agent is working on the whole task, but the agents communicate intermediate and partial results, in addition to complete solutions. The decision to broadcast is taken with a certain probability. The elements to be broadcast are selected on the basis of their value. The reception of a communication is also probabilistic, to model the possibility that one agent is not heeding all communications it receives from other agents. Each agent has a subset of agents to which that agent broadcasts. We distinguish between the density of the connections and the topology of the connections. The density of connections is indicated by the proportion of all agents to which any given agent communicates. Zero density corresponds to the lone genius scenario, each agent working away in isolation. Beyond that, density can in principle vary from sparse, each agent communicating to a few, perhaps even a single, recipient, to complete, in which case each agent communicates to everyone else. Real collectives tend to fall somewhere between these extremes. The topology of the network has received much attention in classical social psychology and also in recent network studies, especially studies of the so-called small world effect. In the simulations reported in this article, we did not systematically vary the topology. Instead, each agent broadcasts to a randomly chosen subset of the network. We focus on what we think of as intermediate size collectives, larger than the 2–10 member groups typically studied in social psychology experiments but smaller than the large populations often studied in sociology, network models or internet studies. In the simulations reported in this paper, the size of the network was set at 50 agents. This type of collective correspond to, for example, a small community of researchers all working on a shared problem and communicating along the way through E-mail, conferences and publications, and to a market with number of research and development firms racing to invent the next product. We distinguish invention, the first creation of a knowledge element that matches a given target, from dissemination, the spread of that knowledge element through the network. The process of con-

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Fig. 1. Schematic diagram of agent and a portion of a network.

structing an element that instantiates a target models the process of invention. The endpoint of this process is the first cycle in which some agent creates a knowledge element that matches the target. If we continue to run the model past that point, the element that instantiates the target will spread through the collective, in part because it is (re-) created by other agents and in part because it has high probability of being communicated from agent to another. Agents do not directly recognize a target as such, but only perceive the (noisy) value of the matching knowledge element. Given noisy values and probabilistic communication, a target found by one agent will not spread instantaneously across the network but will require multiple cycles. The end point of the dissemination process is the cycle in which every agent in the collective (or some specified proportion of them) posses a copy of that element. The two-level structure of our model is illustrated in Fig. 1. The top part of the figure represents the cognitive architecture of a single agent, while the bottom part illustrates a portion of a network of such individuals. The details of the implementation are given in the next section.

3. Model details and implementation Agent-based models (ABM) have been used to study a wide range of social, economic and natural phenomena caused by processes that are inherently parallel and distributed across multiple, autonomous but interacting entities (Epstein and Axtell, 1996; Holland, 1995; Bonabeau, 2002; Sun and Naveh, 2007). An ABM specifies the properties and behaviors of individual agents, how agents interact with each other, and provide a distributed parallel computational model comprising the collection of agents. Such models enable the study of higher-level system properties that cannot be derived analytically or by aggregating statistically over the agents. Because the agents communicate knowledge structures, the latter is the case for our model.

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The ABM presented in this article is implemented with the Swarm modeling tool kit (Stefansson, 1997; Terna, 1998). Swarm is an object-oriented platform for agent-based modeling and simulation. Agents in the model are implemented as software objects, and agents are endowed with specified properties and rules of behavior and interaction. Swarm includes simulation tools needed to carry out and coordinate the activities of the agents, schedule events, manage agent interactions and so on. Cognitive processes are operations on knowledge elements. Knowledge elements are represented as lists of symbols. Symbols generically denote knowledge primitives (concepts), and the lists represent complex knowledge structures (facts, beliefs, principles, rules, schemas, blue prints, etc.) formed by combining the primitives. In the simulations reported in this article, the vocabulary of symbols was just the ten digits 0–9 and the maximum size of a knowledge element was set to 10 symbols (with repetitions allowed). That is, the agents faced a search space of 1010 problem states, which is large enough to correspond to search spaces in creative enterprises. Two knowledge elements are identical (“match”) if and only if all their symbols are identical at each list position, with one exception: The symbol “0” is used as a so-called wild card symbol, i.e., it represents “don’t care”. According to these matching rules, the two lists 1, 2 and 1, 2 match, 1, 2 and 2, 1 do not, but 1, 2 and 1, 0 do. As noted above, the cognitive processing by agents is goal driven. Goals are modeled as target knowledge elements that reflect some desirable end states. Each such target is a list of the same sort as a knowledge element. To achieve or obtain a target means to produce (through operations defined below) a knowledge element that matches that target. Target elements can provide partial specifications of a desired end state. Such targets carry the wild card symbol in certain positions, reflecting the lack of specific knowledge primitives at these positions. A target with fewer specified symbols is easier to achieve since multiple knowledge elements can obtain a match with this target. An agent’s long-term memory is called the stock and its working memory is called its active list. Both are defined as sets of knowledge elements. The maximum size of the stock and active list are parameters in the model. In the simulations reported in this article, the stock size was set to 50, and the size of the active list was systematically varied. Agents send and receive knowledge elements along communication links. Agents can be connected to other agents in different ways to implement different network topologies. In the simulations reported in this paper, we considered random networks in which an agent is connected to a certain proportion of other agents, as specified by the connection density parameter, cd . When an agent communicates a knowledge element, it is broadcast to all the agents to which it is connected, as would be the case, for example, in sending an E-mail message to a distribution list. The connection density is a model parameter that was systematically varied in the simulations. The selection of knowledge elements for various cognitive processes is based on their values. The value of a knowledge element is defined as its similarity to the target, measured as the number of matching symbols in specific locations in the two lists. Noise is added to the values to reflect the uncertainty that characterizes creative problem solving. In simulation runs with multiple targets, the value of an element is its highest overlap with any target (not the average across targets).The value for an entire memory (stock or active list) is defined as the average of the values of the knowledge elements in that memory. Selection of a knowledge element from a memory is probabilistic, with higher valued elements more likely to be selected. The insertion of a new knowledge element into a memory is also guided by its value, relative to the average memory value. When a memory is filled to capacity, a new knowledge ele-

ment replaces a lower valued element currently in memory, again selected probabilistically. The model works by iterating through operating cycles. In an operating cycle, each agent receives elements broadcast from other agents, decides whether to heed them, decides whether to retrieve elements from the stock, selects elements in the active list for processing, selects operations to execute, decides whether to store any of the new elements in the stock and, finally, decides whether and which elements to broadcast. The next cycle then commences. The model keeps cycling until it has reached its target or targets, or until the maximum number of cycles has been carried out. These operations are implemented through the following set of methods defined at the agent level: • Receive. Knowledge elements received from other agents may or may not be entered into the active list. The attention parameter, pr gives the probability of a communicated element being heeded (as opposed to ignored). Heeded elements have a certain probability of being entered into the active list if their values are higher than the current active list value. • StockToActiveList. Knowledge elements currently in the stock are retrieved, i.e., entered into the active list, probabilistically based on their values. This method models the agents’ capability of making use of prior knowledge. • ProcessActiveList. Knowledge elements in the active list are processed to produce new knowledge elements. The operations are similar to those employed in genetic algorithms (see below for further details). This method models the agents’ ability to infer new conclusions. • ActiveListToStock. Knowledge elements currently in the active list are transferred to the stock. This is once again a probabilistic operation, with elements from the active list replacing elements in the in stock that have lower values. This method models the agents’ capability of learning, i.e., encoding information into long-term memory for future use. • Send. Knowledge elements from the active list can be broadcast to other connected agents. The verbosity parameter, pc , specifies the probability that an agent decides to communicate with other agents on any one cycle of operation. Together, Receive and Send model the agents’ ability to interact with other agents. The order in which the methods are described above corresponds to the order in which they execute during an operational cycle. Agents have a finite processing ability, and the operations defined above are not performed on all memory elements in a cycle. The operating intensity parameter specifies the number of elements processed through each of the operations. This was set to 4 for the simulations reported here. The attention and verbosity parameters, pr and pc were set to 0.5 throughout; on average, an agent will thus heed half the communications from other agents, and communicate half of its own partial results. New knowledge elements are obtained in the active list by processing its current elements using operators that were based on genetic algorithms (Holland, 1975). The crossover operator takes two knowledge elements and interchanges their symbols in randomly chosen locations to obtain two new elements. Suppose that an agent possesses two knowledge elements, K1 and K2 , defined as the two lists K1 : 4 5 2 1 6 7 8 1 9 1 and K2 : 5 1 2 3 8 7 1 1 1 3.

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Table 1 Explanation and symbols for model parameters. Parameter

Explanation

Symbol

Density Verbosity Attention Capacity Memory Size Creativity

The proportion of agents to which an agent broadcasts. The probability that an agent will broadcast a result. The probability that an agent will heed a communication. Size of the active list. Size of the stock. The number of agents. The probabilities of using cross-over, mutation, add p(cross), and drop operations.

cd pc pr Cap(AL) Cap(Stock) N p(mutate), p(add), p(drop)

If the crossover locations are randomly chosen as 3, 5 and 9, two new knowledge elements are produced (elements at the switch positions are underlined): K1 : 4 5 2 1 8 7 8 1 1 1 and K2 : 5 1 2 3 6 7 1 1 9 3. Notice that the crossover at the third position causes no change, because the symbol (“2”) at that position is the same in both lists. A single knowledge element can also undergo random changes through the mutation, add and drop operators. The mutation operator replaces a symbol at a (randomly chosen) location with a randomly chosen symbol. The add operator adds a random symbol to a knowledge element (if the latter is not filled to the maximum length of 10 symbols). The drop operator removes a symbol at a random location in a knowledge element; all symbols are then adjusted to the left. Knowledge elements in an agent’s active list are selected for these operations probabilistically, based on their values. The different operators are chosen based on parameters defining the probabilities for crossover, mutation, add and drop. The probability for crossover was set to 0.7 and the probabilities for mutation, add and drop were each set to 0.1. Crossover effects exploitation of currently known elements, while the random change operators effect exploration over the space of knowledge elements. Agents undertake operations at every cycle of the simulation. Each simulation run starts with an initial random set of elements in the agents’ active lists and stocks. Every simulation reported here was run for a maximum of 1000 cycles. Results for each setting of the model parameters are taken as averages over 10 simulation runs, each with a different random seed of initial knowledge elements. Table 1 lists the parameters in the model. 4. Experiments and results

ber of wild card (“don’t care”) symbols. A hard target is a vector in which all the positions are filled with specified symbols. Targets with larger number of specified symbols will, on average, take the collective longer to discover. A second set of experiments examined the case in which multiple targets were being pursued simultaneously. Connectivity and capacity were varied in the same way as in the single-target experiments. For each combination of parameter values – connectivity, active list size and target difficulty – we ran 10 simulations and the values displayed are averages across the 10 runs. Each simulation was run for a maximum of 1000 cycles. We assessed the performance of the simulation with three dependent measures. The main outcome measure was the rate of invention or the first-cycle-to-target, defined as the first operating cycle in a run in which any agent in the network achieves a target; that is, in which a matching knowledge element appears in some agent’s active list. A second outcome measure pertains to the dissemination of a target. The dissemination rate was defined as the number of operating cycles required before a target has spread to a given proportion (75%) of the agents. For the experiments with multiple targets, we also quantified the breadth of dissemination as the number of agents achieved by at least one agent in the course of the 1000 cycles of a simulation run. 4.2. Basic network effect We first examine how the network compares with a group of unconnected agents. Fig. 2 shows the first-cycle-to-target for unconnected agents and for low (10%), medium (50%) and high (90%) levels of connectivity. The results are shown for two separate targets that differed in difficulty. The easy target was the list (1,2,3,4,5,6,7,0,0) and the hard target was (1,2,3,4,5,6,7,8,9). The former target has fewer specified vector positions and is thus expected to be easier to discover. The targets were considered in separate runs, so the data in Fig. 2 are from single-target runs. The interactions among the agents had a large effect. Interacting agents, even at low connectivity, show better performance than the unconnected case in which the agents do not communicate

4.1. Overview We conducted simulation experiments to examine the effects of communication density, modeled as network connectivity, and the cognitive capacity of the agents, modeled as active list size, on the performance of the collective as a whole. We varied connectivity through four levels. The first level was 0%, i.e., no connections, corresponding to a population of lone geniuses all working on the same task without interacting. This served as a control and comparison condition that allowed us to verify that our model exhibits network effects. The other levels of connectivity were 10%, 50% and 90%. The effect of cognitive capacity was examined by varying the active list size through the values 5, 10, 20 and, in some simulations, 50 elements. We first examine the case in which the collective pursues a single, shared target. Performance was measured separately for easy and hard targets. An easy target is a symbol vector with some num-

Fig. 2. First cycle to target when the active list is set to 10, for four levels of connectivity and for one easy and one hard target. The values are means across 10 simulation runs.

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across multiple runs are greater for hard than for easy targets and this effect dominates the small benefits of higher connectivity. The main lesson is that connectivity interacts with target difficulty. As the next section shows, so does cognitive capacity. 4.3. Effects of cognitive capacity

Fig. 3. The number of cycles required to disseminate one easy and one hard target to 75% of the agents, for three levels of connectivity. The values are means across 10 simulation runs.

but operate independently. Even low (10%) connectivity lowers the number of cycles to target from 31 to 12 for the easy target and from 85 to 17 for the hard target; recall that these values are averages across 10 simulation runs. While yet greater connectivity leads to quicker discovery of the target, the difference between medium (50%) and high (90%) connectivity is small. As expected, the hard target takes longer to discover, especially for the unconnected agents, less so when the agents interact. With increasing connectivity, the difference between easy and hard targets decreases and at 90% connectivity, it has disappeared. A different pattern was observed with respect to the dissemination measure; see Fig. 3. Recall that the dissemination rate is the number of cycles required for the target to spread to 75% of the agents. As expected, increasing connectivity leads to faster dissemination. However, the effect was small and it was only observed for the easy target. The hard target followed the opposite trend. The number of cycles to disseminate to 75% of the agents is higher for the hard target at low connectivity, and higher still for medium and high connectivity. Inspection of the results for individual runs indicated that model performance was more variable across the 10 simulation runs for the hard than for the easy target. The standard deviations for the easy target across the three levels of connectivity were 2.6, 1.6 and 1.4, indicating high similarity across 10 runs. But the corresponding values for the hard target were 52.8, 63.2 and 34.7, indicating large variations across runs. When some agent in the network hits upon the right track early in a run, as is likely with an easy target, communication helps moving the network as a whole in the direction of the target, but if all agents are on the wrong track early on, as might happen with a difficult target, their interactions will instead make the agents persevere on the wrong track. The latter case is more probable for a hard target, so averages

Common sense suggests that because individuals have limited cognitive capacity, there can be such a thing as too much networking. There is process loss in terms of the need to allocate memory space and processing capacity to the communications. In our model, this is represented by the possibility for displacement of potentially useful elements from the active list as a consequence of the reception of communications from other agents, and by the upper limit on how many operators that can be executed within a single cycle of operation. Intuition suggests that any negative effect of communication will be greater at higher levels of communication but lesser with greater cognitive capacity. To investigate the relation between capacity and performance, we systematically varied connectivity through the same four levels as above. In addition, we varied the active list size through 5, 10 and 20 elements. As before, we measured rate of invention by the first cycle in which the target is discovered by any one agent. The results are shown in Fig. 4. Once again, we observe a strong network effect. For all capacity levels, low connectivity leads to earlier achievement of targets than the unconnected case. This effect is particularly striking for the difficult target (see panel b). The targets are achieved sooner as connectivity is increased, but the differences between low and medium connectivity are smaller in magnitude for both easy and hard targets. At medium connectivity, there are no additional benefits of yet higher connectivity. Contrary to our prediction, increasing active list capacity does not lead to significantly fewer cycles to target. On the contrary, increasing the size of the active list to 20 elements leads to small increases in the number of cycles required to achieve the target as compared to active lists of 5 or 10 elements. This is true for both easy and hard targets. While counter-intuitive, this effect is understandable in terms of the focus on a single target. A smaller active list enables narrow focus on one target, while a larger active list tends to distribute the processing efforts across a broader range of elements, some of which are not on the path to the target but nevertheless occupy memory space and processing cycles. This interpretation suggests that increasing active list capacity might yield improved performance when the agents are pursuing multiple targets in parallel; we investigate this issue in the next section. We next examine the dissemination of the targets through the network in terms of the number of cycles required for a target to be achieved by 75% of the agents. The results are shown in Fig. 5 for

Fig. 4. The first-cycle-to-target for three levels of cognitive capacity and four levels of connectivity. Panel a shows the results for an easy target, while panel b shows the results for a difficult target. The values are means across 10 simulation runs.

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Fig. 5. The dissemination rate, measured in terms of the number of cycles required for a target to spread to 75% of the agents, for three levels of connectivity and three levels of cognitive capacity. Panel a shows the results for an easy target, while panel b shows results for a hard target. Values are means across 10 simulation runs. Table 2 Seven symbol vectors that served as targets in the multiple-target simulations. Target no.

Difficulty level

Content

1 2 3 4 5 6 7

Easy Easy Moderate Moderate Hard Hard Hard

123456700 987654300 123456789 987654321 123454321 123654789 987456321

three capacity levels and three levels of connectivity. For the easy target, medium and high connectivity yields quicker dissemination than low connectivity, with a slight advantage for high connectivity. Contrary to expectation, increasing active list from 5 to 20 elements causes slightly slower dissemination rates. The results for the hard targets show the opposite pattern. Increasing connectivity slows down the dissemination rate, but increasing the active list causes a strong and consistent beneficial effect. In short, for easy targets, dissemination is fastest with a small active list and high connectivity, while for hard targets, dissemination is fastest with a large active list and low connectivity. Although both active list capacity and connectivity interact with the difficulty of the target, there is no evidence in Fig. 5 of any interaction between capacity and connectivity. 4.4. Searching for multiple targets in parallel The single target case provides an essential evaluation of the effects of communication density and cognitive capacity, and it applies to contexts in which the members of a creative collective pursue a single, specific invention or discovery. But members of creative communities often work on multiple projects in parallel. For example, a scientist might run multiple projects, a mathematician might be working on multiple proofs and an inventor might consider several potential inventions or discoveries. We next explore connectivity and cognitive capacity in the presence of multiple targets. In this case, the individual elements in memory are valued by matching them to all the targets and the value of the element is taken to be its highest similarity to any target. We consider seven targets, two easy, two moderately hard and three hard; see Table 2. The moderately hard targets specify a few additional elements more than the easy ones, while the hard targets involve a mix of symbols in specific locales taken from the moderately hard ideas.2

2 Since the don’t care (0) is considered to match any defined symbol at a locale, a target with trailing 0s would be equivalent to the shorter string that does not specify any symbol in these positions.

Fig. 6. The first-cycle-to-target for four levels of connectivity and four active list sizes, averaged over 10 runs for each of two easy and three hard targets.

The easy targets, with fewer specified elements, are expected to be achieved sooner than the others. While both the moderate and hard targets specify the same number of defined elements, the moderate ones can be obtained by adding two elements to the easy targets; the hard targets, on the other hand, will require recombination of sets of elements from the moderate targets, and can generally require greater effort to achieve. As in the single-target simulations, we varied connectivity across four levels: unconnected (0%), low (10%), medium (50%) and high (90%), where the percentages refer to the proportion of agents in the network with which each agent communicates. The stock size was kept at 50, but the active list size was varied through 5, 10, 20 or 50 elements. In evaluating the performance of the model, we once again consider the first cycle to achievement of a target (by any agent), the number of cycles required to spread a target to 75% of all agents and the number of targets found in a simulation run. Each value is an average over 10 runs. In the multiple-target runs, we also averaged over the targets at each level of difficulty. We present the results for the two easy and three hard targets. We do not include results for the moderately hard targets since they were similar to the findings for the easy targets. The network did not achieve all seven targets in every run. When reporting the average over 10 simulation runs, we use a value of 1000 cycles (corresponding to the maximum number of cycles) for runs in which a target was not achieved before the end of the run. This underestimates the value we would have obtained, had the simulation run without limit. These results are presented in Fig. 6. Once again the low connectivity networks exhibit better performance than the unconnected case. However, the effect is not linear. Increasing connectivity above 10% leads to larger number of cycles being required to find the targets. Once again, there is process loss when there are more interactions. The beneficial effect of a large active list is clear and particularly apparent at the highest con-

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Fig. 7. Number of targets achieved in a run as a function of four levels of connectivity and three levels of capacity.

nectivity level. When the active list has room for 50 elements, the facilitating effect of cognitive capacity is so great that it completely counteracts the process less caused by higher connectivity. The results in Fig. 6 were achieved by including all runs in the analysis. In cases where a target was not obtained before the end of the run, we used 1,000, the maximum number of cycles as an estimate of the first-cycle-to-target for that target. The question arises whether the number of targets actually found interacts with the model parameters. The number of targets found is shown in Fig. 7 as a function of connectivity and capacity. The number of targets found decreases with increasing connectivity, but increases with increasing active list size. These interactions means that the estimate of

the first-cycle-to-target that we used in cases in which some target or targets were not found will affect different simulation runs differently. This complicates the interpretation of Fig. 6. Given the result in Fig. 7, we decided to also study the relations between connectivity, capacity and target difficulty in those situations in which the targets were actually found. Inspection of the simulation runs revealed that there were at least four runs in which every target was found within the cycle limit of 1000 cycles for every combination of parameter settings, so we next consider the first-cycle-to-target, averaged over the four smallest values selected among those runs in which all targets were found. The results are displayed in Figs. 8 and 9. For both easy (panel 8a) and hard (panel 8b) targets, the number of cycles to target drops sharply from no connectivity to low and medium connectivity. There is no additional improvement in performance as we increase connectivity from median (50%) to high (90%) connectivity. The effect of capacity was smaller in magnitude than the effect of connectivity. Furthermore, the effect was not in the expected direction. In both panels of Fig. 8, the curve for the active list capacity of 50 elements is higher than the other curves, indicating that higher capacity begins to have detrimental effects eventually. The non-linear effect becomes evident if the data are re-plotted as in Fig. 9. For both easy and hard targets, the number of cycles to the first discovery of a target drops sharply as active list capacity is increased from 5 to 10 for the no connections case, but then

Fig. 8. The first-cycle-to-target, averaged over the four smallest values selected among those runs in which all targets were found, for two easy (panel a) and two hard (panel b) targets.

Fig. 9. First cycle to target as a function of four levels of connectivity and four levels of capacity, averaged over four runs in which all targets were found. The data are shown separately for easy targets (panel a) and hard targets (panel b).

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Fig. 10. Dissemination of targets through the network in terms of the percentage of agents who possess a target at the end of a simulation run as a function of connectivity and capacity. The data are shown separately for easy (panel a) and hard (panel b) targets.

increases again. A population of isolated agents reaches a target fastest when the agents operate with an intermediate active list size. For the connected agents, active lists of 5 and 10 elements produce approximately the same performance, but larger active lists slow down discovery. The beneficial effect of connectivity is also clear in Fig. 9, with low connectivity being worse than medium and high connectivity, but there was no consistent difference between the latter two conditions. Finally, we investigate dissemination in the multiple-target case. Not every multi-target simulation resulted in 75% of the agents possessing a target before the end of the 1000 cycles. Hence, we cannot use the number of cycles until 75% of agents posses a target as the measure of dissemination rate. Instead, for this analysis, we measure the dissemination rate by the percentage of agents who possess a target at the end of a run. The results are plotted in Fig. 10 as a function of connectivity and capacity, and we separate the results for the two easy and three hard targets. Once again, we see a clear and strong network effect. Compared with unconnected agents, the low, medium and high connectivity conditions all produce faster dissemination. Among the latter three levels, low connectivity consistently does better than the medium and high, while the latter two show no consistent difference. At all connectivity levels, larger capacity results in faster dissemination. The effect does not appear for the unconnected case, but it is monotonic in the other conditions. In this case, dissemination does not interact with target difficulty, as evidenced by the similarity of the results for easy and hard targets. The multi-target case is similar to the single-target case for hard targets (Fig. 5, panel b) in that higher connectivity leads to slower dissemination but higher capacity leads to faster dissemination. (In comparing Figs. 5 and 10, it is useful to remember that on the measure plotted in Fig. 4, lower values mean faster dissemination, while on the measure plotted in Fig. 10, higher values means faster dissemination.) 5. Conclusions, discussion and future work 5.1. Summary and interpretation The simulations confirmed some of our expectations but not all. We observed a consistent network effect. The rate of invention was strongly affected by the difference between the lone genius case – a set of unconnected individuals – and the low connectivity case in which each individual communicates with a randomly chosen subset of 10% of the other individuals in the network. This basic network effect was present in both the single-target and multiple-target conditions and for both easy and hard targets; see

Figs. 2, 4 and 8. Interaction speeds up the rate of invention. The explanation is obvious: When an individual discovers a partial result that is close to a target and broadcasts it to other agents, it saves cognitive work for those other agents and moves the entire collective forward in the search space. The contrast between the zero connectivity and low connectivity cases proves that our model captures some aspect of true collaboration. As we increase the communication density from low to medium and high, the rate of invention continues to speed up but at a negatively accelerated rate. The advantage of increasing communication density from medium to high is either small in magnitude or nonexistent. This is so in the single-target and multiple-target cases and for both easy and hard targets; see Figs. 2, 4 and 8. That is, the benefits of communication are almost completely realized by a modest level of interaction. The explanation why there are smaller and smaller benefits at yet higher level of communication is that communication entails process losses of various kinds: Most obviously, numerous communications occupy working memory space and threaten to displace from working memory an individual’s own partial results before the latter have been protected from loss by being moved into long-term memory or by being communicated to other agents. Multiple communications from connected agents also compete for attention in an agent’s active memory, and potentially useful ideas from other agents may hereby be lost. An illustrative example of such process loss is given in Appendix A. A simple simulation with 5 agents, and considering a low connectivity setting where each agent connects with two others and a high connectivity setting where each agent connects with all four others also helps illustrates this: the number of cycles for 1, 2 and 3 agents to achieve a target are found to be {78, 79, 80} and {74, 143, 298} for the low and high connectivity settings, respectively. While the earliest times to target are similar, the times taken by the second and third agents to achieve the target are higher with greater connectivity, and is indicative of the process losses that can occur. In general, too much interaction distracts. In contrast, the effect of cognitive capacity on the rate of invention departed from our expectations. Cognitive psychologists discuss the capacity of human working memory in terms of limitations (on storage space, processing resources, etc.), and some have hypothesized that what psychologists measure with intelligence tests is nothing but working memory capacity. If so, one would expect greater capacity to facilitate cognitive processing. We also expected larger capacity to help disproportionately more at higher connectivity levels by counteracting the expected process losses at the highest levels of communication density. Neither expectation was verified. In the single-target case, the curves for the three

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levels of capacity are superimposed on each other, with a slight tendency for the curve for the largest capacity to exhibit the slowest rate of invention. This was so for both easy and hard targets; see Fig. 4. In the multiple-target case, we see that same pattern across four levels of capacity, with the curve for the largest capacity once again exhibiting the slowest rate of invention, for both easy and hard targets; see Fig. 8. There is no sign in either the single- or the multiple-target case of any interaction with communication density; the curves for the different capacity levels are nearly parallel in both Figs. 4 and 8. In short, increasing cognitive capacity has no beneficial effect on the rate of invention in our model; in particular, it is not the case that higher cognitive capacity counteracts the process losses that reduce the benefits of interaction at the higher communication densities. With hindsight, we explain this pattern in terms of focus. Recall that the simulated individuals are guided in their search for a target by an evaluation function that measures the promise or potential usefulness of each knowledge element. Greater cognitive capacity lowers the probability that a high-value element will be displaced from working memory, but it also enables lower-value elements to remain in working memory for longer periods of time. Any element in working memory has some probability of being chosen as input for further operations on any one operating cycle. The larger the working memory, the lower the value of the least valuable elements and the greater the probability that the individual will waste cycles on those less valuable elements. As capacity is increased, this lack of focus counteracts the advantage of having a lower probability of losing the highest-valued elements. A simple simulation with 5 agents and 5 interconnecting links (each agent connected with two others), and agents’ active memory sizes set at 5, 10 and 20 helps illustrates this: the number of cycles for 1, 2 and 3 agents to achieve a target are found to be {78, 79, 80}, {70, 135, 144} and {115, 141, 183} for active memory sizes 5, 10, and 20, respectively; the longer times to achieve the target with increasing memory levels seen here arises from the aforementioned lack of focus with higher memory. With greater connectivity, higher memory capacity can enable more of the received ideas to enter and agent’s memory, replacing lower values ideas. For further processing of these ideas in memory, however, the agent has this larger set of ideas to consider, and attention thereby tends to get diffused. Since targets will be achieved only through recombinations of particular ideas, selection of these specific ideas for processing can be delayed. The upshot is that there is no net effect of increasing working memory capacity beyond 5 or 10 elements. We note that this working memory size coincides with the typical mean for empirical measures of human working memory size. Perhaps working memory is limited in humans because there is no net advantage to having a greater one, the advantage of being able to retain what is important being cancelled by the disadvantage of being able to retain what is not so important. The rate of dissemination was also strongly affected by our model parameters, but the effects exhibit a different pattern. In the single-target case, increasing connectivity improves the rate of dissemination for easy targets. The effect is small in Fig. 2 (see the bottom curve), but rather greater in Fig. 5 (panel a). However, for hard targets, increasing connectivity slows down dissemination; see Fig. 3 (top curve) and Fig. 5 (panel b). The multi-target case resembles the hard single-target case in this respect: The rate of dissemination is fastest at low connectivity, for both easy and hard targets (see Fig. 10, both panels). This is a clear but counterintuitive case of process loss. As more knowledge elements are shipped around the network, each individual receives more communications in each cycle of operation, which in turn raises the probability that some high-value elements close to a target – indeed, the target itself – will be displaced before it is communicated. With respect to dissemination, the expected counteracting effect of greater cog-

nitive capacity does appear. In the single-target case, the benefit of higher capacity for dissemination is consistent across three capacity levels, for each connectivity level; see Fig. 5, panel b. Likewise, for the multi-target case, the benefit of greater capacity is consistent across levels of communication density. The application of these conclusions to particular examples of creative networks depends on what we take the agents in the model to symbolize. In the above paragraphs, we have assumed that an agent is an individual person, and that hence cognitive capacity can be identified with working memory capacity. In this case, cognitive capacity cannot be deliberately manipulated, and the only prescriptive conclusion is that a modest level of interaction is likely to be sufficient to realize the benefits of collaboration However, there is nothing in the structure of the model that enforces the interpretation of agents as persons. The agents can themselves be collectives, such as research groups or research and development teams that operate in the context of a population of similar groups or teams. Under this interpretation, the nature of communication links only change a little. An E-mail message sent to a distribution list of interested colleagues or a conference presentation to a highly specialized audience can still serve as prototypical examples of broadcasting partial results to a subset of likeminded colleagues. But cognitive capacity must be re-interpreted if agents are themselves collectives. What is the cognitive capacity of a group or a research team? This question would be more effectively pursued by cognitive anthropologists than by cognitive modelers like ourselves, but a rough cut might go as follows: The ‘working memory’ of a group encompasses all the relevant ideas, theories, findings and other materials that the group members can draw upon in their work without conducting extensive search, either because they know them well enough to be able to retrieve them from memory (“didn’t Smith and Jones in their 2001 article find that. . .”) or because they have them ready at hand (“my copy of the Smith and Jones article should be in the pile here, let’s look it up”). The question of capacity, and hence focus, applies to this extended set of immediately available materials. In this context, the agents-asgroups interpretation of our modeling results imply that invention or discovery is faster if each group has a narrow definition of what counts as “relevant material”, but dissemination is faster if the such groups have a broader definition of relevance. This makes excellent sense: Narrow focus increases the probability of making a discovery, while broad receptivity on the part of each group helps a new idea or result to get around. The general lesson is that there is no best configuration of a creative collective, because the usefulness of a configuration depends on the outcome measure employed, which in turn reflects the purpose of the collective. 5.2. Relations to other models Although the topic of collective or social creativity has been the subject of much attention in as diverse disciplines as social psychology, sociology, economics, artificial intelligence, mathematical network studies and the history of technology (Abrahamson and Rosenkopf, 1997; Cowan et al., 2004; Fischer et al., 2005; Fisher and Ellis, 1990; Fountain, 1998; Latané and L’Herrou, 1995; Loch and Huberman, 1999; Lynch, 1996; Mowery and Rosenberg, 1979; Paulus and Nijstad, 2003; Williams and Yang, 1999). Computer modeling is one of the most commonly used tools in this field (Berdahl, 1998; Bonabeau, 2002; Cowan and Jonard, 2003; Saunders and Gero, 2001; Schilling and Phelps, 2007). Our approach makes four contributions. First, we cast the problem as one of understanding the interactions between two system levels, the individual agent and the collective in which he or she participates. Unlike network models in which the nodes are mere way stations for messages or quantities of activation, and also unlike models based on the principle of level invariance, we start with

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the principle that units at both levels have internal structure, but of different kinds. The description of individuals needs to include those gross features of their cognitive architecture that affect how they interact during collaborative problem solving (and only those features). The relevant features are abstracted from contemporary research in cognitive psychology. Thus, this part of our model is solidly grounded in the empirical research that supports contemporary models of the cognitive architecture. Our goal was not to contribute to the theory of the cognitive architecture, but to build on it by including in our simulation those features that are likely to affect the functioning of a creative collective, most importantly the dynamics between short- and long-term memory, thinking and communication. A second and closely related feature of our approach is that the agents communicate content. They remember and operate upon symbol structures, creating new symbol structures that represent the conclusions of inferences or other cognitive products. The latter are represented as symbol vectors that can stand for any type of cognitive entity – concept, proposition, schema, mental model, rule, etc. – and an agent’s goal is defined in terms of such a vector and its thinking capabilities are defined in terms of operations on such vectors. The agents in our model are thus faced with a real, and realistically huge, search space. The agents communicate by passing symbol structures – not abstract quantities – to each other, possibly helping each other by passing along a result that speeds up the search by another agent. In this way, we model the passing back and forth of content that is the core of real communications without tying the model to any particular hypothesis about mental representation. This use of generic knowledge representations and generic processes enables us to model the processing of content without having to implement a symbolic artificial intelligence system that actually performs some creative task. Third, we focus on the case of agents who are loosely connected in the sense that they interact primarily by communicating partial results (ideas) but nevertheless share in the same creative endeavor. Our model is not a model of the division of labor. There is no central agency that distributes subtasks to agents and then integrates their returns. Instead, each agent is working on the complete, shared task, as would be the case in a race among research groups for a particular discovery. Also, the agents in our simulations broadcast their partial results to subsets of other agents before they have found a solution to the shared task. Thus, our model is not a model of diffusion only, but actually models the prior process of invention. Examples of this type of collective are found throughout the sciences and among communities of artists and technologists, and they differ from formal organizations, especially those based on the division of labor. The comparison of different levels of connectivity with the no-connection case, a control condition we have not noticed in other network simulations, turned out to be very informative. The no-connection condition is meaningful in our case precisely because the agents have internal structure and are capable of reaching the shared goal on their own. Fourth, the agents in our model can share a single goal (target) or simultaneously pursue multiple shared targets. The latter case is a realistic representation of how creative agents (individuals or teams) function. They seldom have a single, well-defined target but pursue a repertoire of distinct but related enterprises, and the work performed in the pursuit of one enterprise is simultaneously considered with respect to its contributions to the others. A successful researcher seldom works on a single question, an inventor is likely to pursue multiple projects in parallel and the tendency of artists to have multiple canvasses going at any one time is proverbial. Our simulation results show some differences in outcomes between the single and multiple target cases, indicating that there is reason to distinguish between them. We do not know of another simulation of creative networks that covers both

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the single and the multiple target cases within the same processing structure. 5.3. Limitations and future work The current version of our model exhibits several limits and simplifying assumptions, some of which might have effects on the outcome measures. A fundamental boundary on the present work is its focus on collectives in which the participants collaborate by sharing information. We do not model physical interactions of the hand-me-the-hammer variety, nor do we model collaboration through division of labor. Each agent works independently on the common problem, and the agents collaborate only by sharing partial results. There is no mechanism in our model for analyzing the shared problem into subproblems, each of which gets assigned to a different agent. How limiting is this focus? First, there are creative collectives that instantiate this model. Scientists who work on the same research problem but at different research institutions approximate this model. They collaborate primarily by sharing partial results via conference presentation, panels, publications, manuscript and proposal reviews and face-to-face interactions. The members of a large research and development team might also approximate this type of collaboration, as do, for periods of time, communities of painters, designers and inventors. These cases are interesting enough in their own right to motivate efforts to understand them better. Second, the model is relatively insensitive to how its knowledge elements are interpreted. In the application reported in this article, we primarily interpreted them as knowledge representations in the minds of individuals and the operations as thought operations, but they could be reinterpreted as (representations of) physical objects and the operations as corresponding to physical actions. The process that we here have called communication would then be reinterpreted as the sharing of physical products. Hence, the scope of the model is wider then it first seems. Another limitation is that the simulated collectives reported in this article were all homogenous; that is, each agent had the same parameter values as every other agent. This strengthens the finding that there is a basic network effect – if every agent is the same, the effect of communication is necessarily due to the communication itself, not to a best member effect – but it is nevertheless unrealistic. In real collectives, agents are likely to function somewhat differently. In particular, they are likely to possess somewhat different cognitive capacities and to use somewhat different evaluation functions. This limitation is not inherent in the model but a simplification for the sake of the initial exploration of the model. There is nothing in the principles behind the model or in its implementation that prevents the agents from being heterogeneous in one or more of the key parameters. In future work, we intend to explore the effect of heterogeneity in the evaluation function. By endowing some agents with a greater noise in their evaluation function, we can give them a heightened ability to depart from common wisdom (as encoded by the evaluation function). The proportion of noisy agents in the collective then becomes an additional parameter in the model. The effect of this parameter is the subject of ongoing investigations. A third simplification in the current version of the model is the topology and nature of the communications network. In the simulations reported in this article, each agent was assigned communication links to a randomly selected subset of other agents, with the size of the subset being the same for each agent but varying from simulation to simulation. However, there is a longstanding hypothesis in research on collective cognition that the topology of the network matters. The particular topology known as “small world structure”– many local connections augmented

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with a few long-distance connections between the local clusters – has been shown to have strong effects on the behavior of activation networks (Cowan and Jonard, 2004; Schilling and Phelps, 2007; Uzzi and Spiro, 2005; Watts, 2003). The small-world topology contrasts both with the random topology used in our simulation and the hierarchical topology of formal organizations based on the division of labor. In the organizational theory literature, various models explore different organization structures that facilitate learning (Siggelkow and Rivkin, 2005; Siggelkow and Levinthal, 2003); Fang et al. (in press) consider subgroups with varying levels of cross-group linkages and find such structures to outperform random network structures in organizational learning. The question whether such differences matter in a creative collective can be explored in our model by assigning links according to some definite topological scheme instead of assigning them randomly. Our models allows consideration of different network topologies and in future work, we plan to use random networks as a baseline to which to compare the functioning of other network topologies. In addition, real agents are likely to vary with respect to how many contacts they maintain. Although the agents all had the same number of connections to other agents in the simulations reported in this article, there is nothing in the structure of our model that prevents us from exploring heterogeneity in the disposition to communicate, with the distribution of this disposition over the agents being yet another system parameter. In an interesting study, Ziherl et al. (2006) consider social capital of research groups in terms of networks amongst researchers, and find that networks with moderate ties and having diversity among members correspond to better productivity of junior researchers. Examining such findings using our model, with heterogeneous agents and network properties, is another useful area for future research. The emphasis on the social side of creativity brings with it the question of how individual and social creativity are related. Our approach assumes that both levels of analysis are important and contributes to the functioning of creative collectives. Individual cognition is best described as a small number of unique components linked via unique types of links, while social networks are best described as in terms of large numbers of similar components,

all interacting according to the same rules. The model presented in this article is a first attempt to formalize this view. In constructing this particular formalization, we have drawn upon work in cognitive psychology, in contrast to the many network models of “innovation” that ignore the potential contributions of this discipline for understanding something so human and so psychological as creativity. Future work will show whether this approach can provide theoretical explanations for observed empirical effects and effective prescriptions for the organization of creative collectives. Appendix A. Illustrative example of process loss with increased connectivity The following example helps illustrate how process losses can arise from higher connectivity between agents. Consider an agent receiving ideas from other connected agents, adding them to its active memory based on perceived value, and processing ideas in memory to generate new ideas. Assume agents A, B, C, D, each with four ideas in memory. For the purpose of illustration, the ideas are labeled a1, a2, a3, a4 for agent A; b1, b2, b3, b4 for agent B, and so on, with values decreasing in order of higher digits (value (a1) > value (a2) > value (a3) > value (a4)), and equivalently numbered ideas from different agents having similar values (value (a3) ≈ value (b3), etc.). Assume a single target idea, T, which is achievable through recombination of ideas a1 and b2. These ideas thus need to cooccur in memory for potential recombination in order to achieve the target T. Take A to be our focal agent of interest, receiving ideas from other agents in each step. At consecutive steps, the ideas generated by agents are of increasing value; consequently, the ideas received by A are of higher value, and will replace lower-valued ideas currently in memory. Two scenarios are examined: (i) low connectivity, where A is connected to and receives ideas from a single agent B, and (ii) high connectivity, where A is connected to and receives ideas from three other agents, B, C, and D. These are shown below, with content of agent A’s memory in{. .} and received ideas [.] at each step; some of the received ideas replace existing ideas in memory, which gives a modified set of ideas in memory at the beginning of the next step. (i) Low connectivity case

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(ii) High connectivity case

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