How Network Structure Affects Social Creativity

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Social Networks 32 (2010) 263–278

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Social Networks

journa l homepage: www.e lsev ier .com/ locate /socnet

ocial creativity as a function of agent cognition and network properties: Aomputer model�

iddhartha Bhattacharyyaa,∗, Stellan Ohlssonb,1

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

r t i c l e i n f o

eywords:gent-based modelingognitive capacityollaborationreativity

nventionetwork connectivityroblem solvingocial creativity

a b s t r a c t

Inventions – concepts, devices, procedures – are often created by networks of interacting agents in whichthe agents can be individuals (as in a scientific discipline) or they can themselves be collectives (as infirms interacting in a market). Different collectives create and invent at different rates. It is plausible thatthe rate of invention is jointly determined by properties of the agents (e.g., their cognitive capacity) andby properties of the network of interactions (e.g., the density of the communication links), but little isknown about such two-level interactions. We present an agent-based model of social creativity in whichthe individual agent captures key features of the human cognitive architecture derived from cognitivepsychology, 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 ofinvention and diffusion in the network via systematic parameter variations. Simulation runs show, amongother results, that (a) the simulation exhibits network effects, i.e., the model captures the beneficial effectof collaboration; (b) the density of connections produces diminishing returns in term of the benefits on theinvention rate; and (c) limits on the cognitive capacity of the individual agents have the counterintuitiveconsequence of focusing their efforts. Limitations and relations to other computer simulation models of
scuss
creative collectives are di
. Introduction

Artistic, economic, scientific or technological inventions areften created by collectives, a category that includes but is notimited to groups, teams, firms, scientific disciplines and entireommunities. We use the term invention in a broad sense thatovers the production of novelty in any area of activity, and weistinguish between invention, the creation of something novel forhe first time, and dissemination, the spread of something novelhrough a network.

The creative processes of collectives vary with respect to rate.or example, the rate of technological invention in Western Europe
rom the year 500 AD to the year 1000 AD was lower than theorresponding rate in the period 1500–2000 AD Focusing on indi-idual nations, the increase in economic growth that produced thendustrial Revolution in Britain in the period 1780–1830 was more
� The preparation of this article was supported, in part, by a seed grant from theniversity of Illinois at Chicago. We thank and Poornima Krishnan for assistance in

mplementing 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.

378-8733/$ – see front matter © 2010 Published by Elsevier B.V.oi:10.1016/j.socnet.2010.04.001

ed.© 2010 Published by Elsevier B.V.

rapid than during any comparable preceding period, and the eco-nomic policy of New Zealand changed at a more rapid rate in the1980–2000 period than they did in the 1960–1980 period. The vari-ation in rate of change applies to markets and industries as wellas nations. The rate of innovation in the information technologyindustry in the period 1975–2000 has been higher than the rateof innovation in, say, automobiles, buses and trains over the sameperiod. This year’s car is different from the 1975 car, but the year2000 laptop computer was not merely different from the computerof 1975. The personal computer did not exist in 1975; we werestill working on timesharing machines. In science, we might com-pare the relative stability of physics in the period of 1790–1830with the turbulent emergence of relativity and quantum mechanicsin 1890–1930. “One of the central historical questions concerningtechnical progress is its extreme variability over time and place”(Rosenberg, 1982, p. 8). An explanatory model of social creativityshould account for variations in the rates of invention and dissem-ination from one collective to another.

In this article we focus on small scientific specialties and
medium-sized communities of artists or technologists working atthe cutting edge of a new development. Examples include the firstgeneration of French impressionists, the community of World WarII scientists who invented radar (Buderi, 1996) and the first gen-eration of cognitive scientists (Gardner, 1985). Such collectives are
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64 S. Bhattacharyya, S. Ohlsson /

arger than the 2–10 person groups and teams typically studiedmpirically in experimental social psychology. At the same time,hey are smaller in size than the populations that are the focusf interest in studies of, for example, economics of innovationGalambos and Eliot Sewell, 1997) and the sociology of popularads (Lynch, 1996). Creative collectives of this type are character-zed by three distinctive features: First, there is no division of laborn the classical sense. Instead, all the participating individuals workn the shared problem, and any one of them can accomplish areakthrough that will affect the future work of the entire col-

ective. Second, it is enough for one member of the collective toave reached a particular conclusion, insight or product for theollective to possess that conclusion, insight or product. This is, inrinciple, the case in art, science, technology and other endeavors inhich results are freely broadcast through a variety of public chan-els. Third and most important, intermediate and partial results are

reely broadcast during ongoing problem solving. Not only the dis-emination but also the initial invention of a novelty occurs throughnteractions among the members.

There is a widespread belief that collectives are more inventivehan isolated individuals – lone geniuses fiddling in their garages –recisely because their members bring diverse knowledge to bearn the shared task and inventions emerge out of their interactionsJohn-Steiner, 2000; Paulus and Nijstad, 2003; Sawyer, 2003; West,002). If the pooling of knowledge is indeed an important factor,e would expect the rate of invention and dissemination to be a

unction of properties of the communication network. Knowledges easier shared, the higher the density of communication links, theigher the probability that one agent decides to communicate withnother agent and the probability that a communication is heededy the recipient.

Although a focus on the collective and social nature of inventions a useful corrective to the traditional emphasis on the lone genius

odel (Hadamard, 1954; Gardner, 1993; Ghiselin, 1952; Gruber,974; Koestler, 1964; Simonton, 1999; Wallace and Gruber, 1989),

ndividual cognition is not rendered irrelevant by a focus on socialognition. Empirical studies in the social sciences have documentedhat the assumed synergy among the members of a collective isot always realized. For example, social psychologists have studiedroblem solving in groups, and rarely find that a group can do betterhan its best member (strong synergy), even though the group isften better than or equal to its average member (weak synergy);ee Larson (forthcoming) for a review.

A number of explanations have been proposed for such pro-ess loss: communication difficulties, inappropriate status relationsithin the collective, affective reaction to group diversity, and oth-

rs (Basadur and Head, 2001; Levine and Moreland, 2004; Millikent al., 2003). Some mechanisms of process loss can be formulatedith precision. In a series of studies, James R. Larson and co-orkers (Larson and Christensen, 1993; Larson et al., 1998, 1994)ave shown that groups have a tendency to focus on informationhat every member knows already, for the statistical reason thatparticular concept has a higher probability of being recalled andntered into a discussion, the larger the number of members whonow that concept.

However, the most obvious source of process loss is the cogni-ive loads imposed on the individual by the interactions themselves.t requires cognitive processing to participate in a cognitive col-ective, of whatever size. Communications from others must bettended to and encoded into memory to participate in the individ-al’s own thinking. There are obvious risks of flooding a capacity

imited cognitive system with an overflow of information. Whenverybody gets 300 email messages a day, nobody has time to cre-te anything. Although the principle of pooling cognitive resourceshrough a communication network is certainly valid, this potentialdvantage is realized to varying degrees as a function of how the

Networks 32 (2010) 263–278

interactions affect the participating individuals. One goal for thestudy of social creativity is to clarify the conditions under whichthe benefits of interaction are greater than the process losses.

We conclude that social creativity needs to be described at mul-tiple 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 andinteract to produce the individual’s contribution to the collectiveeffort. At the level of the collective, the individuals interact to pro-duce the output of the collective. To explain collective creation isnot to choose between these two levels of analysis, but to clarifythe relation between them. After all, the network can only supportthe sharing of ideas if the individual agents have any ideas to share.Hence, the rate of invention and dissemination is likely to be a func-tion of both network parameters (density of communication links,disposition to communicate, the probability that a communicationis heeded, etc.) and properties of the cognitive architecture of theparticipating individuals (capacity limits, problem solving strate-gies, etc.). A model of rate of invention and dissemination mustdraw upon both network theory and cognitive psychology.

In this paper, we describe the initial version of a computer sim-ulation model of a creative collective and present some resultsregarding the effect of agent and network parameters on themodel’s rate of invention and dissemination. Our approach drawsupon prior work in a variety of disciplines, but exhibits some impor-tant differences. There are three key issues in building a model of asocial system: How are the individuals conceptualized and imple-mented? How is the social system implemented in the model? Howis the relation between the simulated individual and the simulatedsocial system represented? That is, what process in the model cor-responds to the types of interactions that occur in the simulatedsocial system? Different disciplines handle these issues in differentways.

One approach to the first issue is to simplify the model of theindividual by removing all internal processing. An individual is thena mere way station, a node in the network that receives messagesand passes them on unchanged. This is approach has been usedextensively in the recent wave of network models in which theindividual nodes in the network lack any internal structure (Cowanand Jonard, 2004; Schilling and Phelps, 2007; Watts, 2003). Becausethey do not take the cognitive work of the individual into account,this type of model provide few resources for accounting for pro-cess loss. They are primarily models of dissemination rather thanmodels of invention. At the opposite end of the scale are models ofthe human cognitive architecture as formulated in cognitive psy-chology (Anderson, 1983, 2007; Newell, 1990; Sun et al., 2005).Symbolic artificial intelligence models of human cognition attemptto duplicate the full range of human cognitive abilities (Ohlsson,2008). Models of this sort make detailed assumptions about eachcomponent of cognitive functioning, e.g., memory retrieval andcapacity limits. But most of the details have little or no impact onthe functioning of the social system in which an individual par-ticipates. For example, it may not matter at the social level exactlyhow cognitive processing is limited, it only matters that it is limited.Models of the cognitive architecture also require extensive cogni-tive task analysis and the formal representation of the content ofthe knowledge human beings bring to bear in the simulated inter-actions. To date, such models have been used to simulate inventionby individuals but not dissemination.

Between these two extremes are models that capture certain
gross features of individual cognition that are hypothesized toimpact the functioning of the social system of interest, but abstractover other features. The model of the individual varies widelyfrom model to model. Schilling and Phelps (2004), building onthe network model of insight by Schilling (2005), has described

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two-level model of social creativity based on the assumption ofevel-invariance: Both the individual and the collective can be mod-led as networks; moreover, as the same kind of networks. Ourpproach starts from the opposite assumption: The individual andhe collective are two qualitatively different types of systems, eachharacterized by a distinct set of processes, functions and quanti-ative parameters, and a theory needs to describe both separatelynd then specify how the properties of the individual scale up toetermine the properties of the collective. Cowan et al. (2004) haveresented a two-level model of knowledge growth, in which the

ndividual is represented by a measure of amount of knowledge, butothing is said about how knowledge is processed in invention andissemination. Our approach differs in that we model explicitly theymbol structures that embody the task related knowledge of thendividual agent, as opposed to representing that knowledge solelyy a quantitative measure of its amount. In yet a third approach,arch (1991) in an influential paper modeled the development and

iffusion of organizational knowledge by representing each indi-idual as a vector of beliefs about m dimensions of reality, eachelief being either 1 (adopted), 0 (uncertainty) or −1 (rejected).he vector values either correspond to or deviate from the truealues, and they change as a function of interactions with otherndividuals and with the culture of their organization, the latterlso represented as a belief vector (Miller et al., 2006; Fang et al., inress). Although the line of modeling initiated by March’s (1991)aper represents the knowledge of the individual, it abstracts overhe processes by which knowledge changes and hence does notepresent capacity limitations.

Although we also represent knowledge as vectors of symbols,ur approach is closer to that of Larson (2007) and of Saundersnd Gero (2001), in which the problem solving strategies of groupembers are interlinked to explain the unfolding problem solv-

ng behavior of the group, but we specify the internal structuref individual cognition differently. We include sufficient structuren the model of the individual to capture the dynamics between

emory, thinking and communication. Knowledge elements areepresented by symbol vectors, and the individual is representedy a long-term memory, a capacity-limited working memory, pro-esses for storing knowledge elements in long-term memory andetrieving them into working memory, and a set of operations byhich the individual creates new knowledge elements. The pur-ose of this model is not to make novel claims about the cognitiverchitecture, but to capture the gross features of individual cogni-ion that might influence functionality at the collective level. Thoseeatures include the capacity to generate new ideas and to search apace of possibilities, but also capacity limitations that bound theationality of the individual. In short, our model includes enoughognitive machinery to allow us to capture the internal dynamicsf individual cognition and explore its impact on social processing.

The second and third of the three issues – how the social sys-em itself and the interaction between the individual and systemre represented a – also exhibit a variety of approaches. In modelsf formal organizations, the question arises how to represent therganization itself, as opposed to the set of participating individ-als. March (1991) introduced the concept of a belief vector thatepresents the belief state of the organization, corresponding to,or example, the beliefs held by the leadership of the organiza-ion or explicitly codified in organizational documents. In this linef modeling, the interaction is represented by the fact that indi-idual belief vectors have a certain probability of changing in theirection of greater similarity with the organizational belief vec-
or. Similarly, Siggelkow and Levinthal, 2003) and Siggelkow andivkin (2005) represented formal organizations as activity vectors,here each activity can take two values. Decision making at the
ocial system consists of choosing a particular activity configura-ion with a view towards maximizing a value function. Although

Networks 32 (2010) 263–278 265

the activity vectors can be partitioned into subsets that are underthe control of different decision makers, there is no explicit rep-resentation of the individual in this model, and hence none of theinteraction between the individual and the firm. Our approach dif-fers form these modeling efforts in the choice of social system.We focus on collectives that do not have a formal organizationalstructure, but in which individuals interact in the course of theirproblem solving efforts. When the main focus of modeling is on thestructure of the network, the social system is typically modeledby communication links between pairs of individuals. In this typeof model, the individual interacts with the system by broadcast-ing and receiving messages. We have adopted this approach, butwith the difference that we model messages as symbol vectors inwhich each value represents a knowledge element. The purpose ofthis network aspect of the model is to capture the gross features ofloose collaboration that characterizes creative collectives. Impor-tantly, the simulated individuals communicate intermediate andpartial results before they have found a complete solution to theshared task, so our model is a model of collective invention as wellas dissemination.

Our hypothesis is that the density of communications in creativecollectives interacts with the capacity limitations on individual cog-nition. We intend to study this interaction by systematically varyingthe density of communication connections and how much informa-tion flows along those connections. This raises the question of whatkind of basic network structure to assume. Small-world networksare by definition networks in which there are only a few long-rangelinks, so increasing the density of connections could only apply tothe local connections. Similarly, a top-down, formal organizationalnetwork, by definition, only has a links from the top node to thenext level, and so on. There are obvious possibilities for interactionbetween density and networks structure. To isolate the effect ofcommunication density, we therefore study a more uniform sortof network, in which each individual communicates with a ran-domly chosen subset of other individuals. We note that this typeof interaction structure appears to fit well to the types of creativecollectives that we have in mind, e.g., scientific disciplines, com-munities of independent inventors, communities of artists, etc. Inscientific communities, each individual researchers interacts withothers via email, conference presentations, publications, journaland grant reviewing, and in other ways.

The question of interest is how the rates of invention anddissemination are impacted by the two key sets of properties,the parameters of the cognitive architecture and the propertiesof the network itself. Our initial hypotheses were the follow-ing: (a) A network of interacting agents invents at a differentrate than the set of those same individuals in a situation with-out interaction. (b) Network properties, especially the densityof connections, will affect rates of invention and dissemination,with higher connectivity associated with higher rates of inven-tion and dissemination. (c) Properties of the individual agent’scognitive architecture, especially cognitive capacity, will affectthe rates of invention and dissemination, with higher capacityassociated with higher rates. (d) Connection density and cogni-tive capacity interact. In particular, higher cognitive capacity willoff-set the process losses associated with higher connection den-sities, so that increases in connection density is more beneficialfor high-capacity than for low-capacity agents. To anticipate, theinteractions between network parameters and parameters of thecognitive architecture turn out to be more complicated than thishypothesis states.

We describe our model at the conceptual level, provide imple-mentation details and then report our simulations. The discussionsection interprets the results and relates them to other modelsof social creativity. We end by discussing limitations and futureresearch.

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. A two-level model of social creativity

The cognitive architecture of an individual is complex and theres no possibility of modeling everything that happens when aerson participates in a creative process (perception, attentionllocation, memory, language understanding, decision making, rea-oning, learning, etc.). But every aspect of individual cognition isot equally important for the individual’s participation in a creativeollective. For example, object recognition is a research topic in cog-itive psychology that has generated hundreds of research studies.or present purposes we do not need to be concerned with this pro-ess, because its properties do not scale up to the collective level. Itoes not matter for the functioning of, for example, a research andevelopment team exactly how its members are able to recognize aair of pliers as a pair of pliers; what matters is only that they are soapable. If they were not, their interactions would have to adjust tohis fact, but because it is a reasonable assumption that they havehis capability, we can abstract from the internal structure of thebject recognition process, complex though it might be. In general,nly a few gross features of the cognitive architecture impact theollective level. Our model of individual cognition is designed toapture those, and only those features. This is why we refer to it ascoarse-grained model.

We assume that cognitive processes are operations onnowledge representations, and that this fact is essential for under-tanding the functioning of the individual. We model knowledgelements as simple lists of symbols. For purposes of the simula-ion, it does not matter what we take the elements to stand for.hey can interpreted as ideas, concepts, data, questions, theories,ropositions, schemas, beliefs, blue prints, tactics, plans, goals,ubgoals and so on. The cognitive architecture is a system for pro-essing knowledge elements, and thought operations are modeleds operations that transform such elements into new elements. Theonsequence of explicitly modeling the internal cognitive process-ng of individuals is that each simulated individual – each agent

in our model is capable of solving the shared task in isolation,ithout any communicating with other agents. We use the zero-

ommunication (lone geniuses) scenario as a base line conditionor our simulations. Some simulations of networks lack any baseine condition against which network effects can be compared.

The coarse-grained model of the cognitive architecture incor-orates hypotheses that are now standard in cognitive psychology.here is a long-term memory (a.k.a. the stock) that holds the agent’snowledge elements; a small subset of knowledge elements areeeded at any moment in time and are held in working memory,lso called the active list. The active list represents the informationhat is currently being actively considered and processed, either asnputs into cognitive operations or as content to be communicatedo other agents. In keeping with a central principle of cognitive psy-hology, we assume a limited capacity for the active list. Empiricaltudies of the working memory of people provide varying estimatesf the average human working memory capacity, but those esti-ates typically fall in the 5–9 range. The small capacity of workingemory strongly constrains the possible solution paths.New elements can enter the active list in three ways: through

etrieval from long-term memory, as results from cognitive oper-tions, or via communication from other agents. The entry of newlements in the active list can displace current elements, and dis-laced elements might have to be re-created for later use if needed.ertain active list elements are also stored into the long-term mem-ry. Which elements are retrieved from long term memory and
aved into long term memory in any one cognitive step obviouslyas the potential to strongly impact a creative process; this is basedn a measure of utility of elements, as described below.
The particular repertoire of cognitive operations included inur simulation is derived from the operations explored in research

Networks 32 (2010) 263–278

on genetic algorithms (Holland, 1975). The use of such operationsis common in various complex adaptive systems (Holland, 1995)and agent-based models including studies of organizations, societyand networks (Axelrod, 1997; Bruderer and Singh, 1996; Macy andWiller, 2002). New elements can be obtained through exchange ofcomponent symbols from current elements in active list, or throughrandom changes to symbols in a current element; each operationis described in detail in the next section on implementation.

The processing is guided by some goal, an element that repre-sents a desirable effect, product or state of affairs that is the targetof the creative effort. This target element can be thought of as adescription of a desirable end state (e.g., “a battery with half theweight and twice the life time of current batteries”). To solve thecurrent problem is to produce an instance of the target element.Creative individuals often pursue multiple goals in parallel, andour model has that capability as well; agents in the model canthus simultaneously search for instances of more than one targetelement.

The model assigns each knowledge element a value that mea-sures its promise, i.e., how useful that element is likely to be inpursuing a target. The value is used to guide all cognitive process-ing. The values are assigned by an evaluation function, i.e., a functionfrom knowledge elements to numeric values. This function can beconceptualized in different ways in different domains. For exam-ple, chess players have a well-developed system for evaluating thestrength of chess positions. This is based on the power of the indi-vidual pieces, the number of pieces left for each side and so on. Inother domains, the evaluation function needs to be conceptualizeddifferently. We model the existence of some evaluation functiongenerically, by assigning knowledge elements a value that mea-sures their similarity to the target in terms of overlapping symbols.Notice that a very accurate evaluation function might provide anunrealistically focused problem solving effort. In reality, the accu-racy of evaluation functions – that is, how predictive they are ofdistance to a person’s goal – is likely to vary. Noise in the evalu-ation function is thus a parameter of potential importance for theindividual’s contribution to the collective effort.

The cognitive architecture iterates through a standard operatingcycle: At the beginning of a new cycle, it receives elements commu-nicated to it by other individuals and inserts these into its workingmemory. Second, the architecture selects elements to be retrievedfrom long-term memory, if any; these, too, are inserted into itsworking memory, possibly displacing other elements. One or morecognitive operations are then selected, together with a subset of theactive elements as input to each operation. The operations createnew elements that appear in working memory, once again possi-bly displacing elements that were already there. Certain workingmemory elements are next selected for saving into the long-termmemory. Finally, one or more working memory elements are com-municated to other agents. The cycle then starts over. The selectionof elements for retrieval, storing, communication and displacementis probabilistic and based on the relative values of the elements.Higher valued elements are more likely to be selected for process-ing, and less likely to be displaced from memory.

Many details are intentionally left out of this coarse-grainedmodel of individual cognition. What is the nature of the capac-ity limitation on working memory? How are knowledge elementsencoded into memory? Should the elements be understoodas schemas, propositions, rules or something else? Is memoryretrieval implemented as spread of activation or in some otherway? These questions, and dozens of other, related questions, are
the subject of research in cognitive psychology. However, mostdetails of individual cognition have little impact at the level ofinteraction with a collective. For example, it does not matter atthe collective level exactly how working memory capacity is lim-ited; it only matters that it is limited. The coarse-grained model is

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n attempt to capture those features of the cognitive architecturehat are likely to have an impact at the collective level.

The collective is simulated by a set of agents working in par-llel. Each agent pursues the same target (or list of targets). Thiss the sense in which the agents have a shared task. Each agentursues the shared target, but also communicates the results of

ts efforts to the other agents in the network. This is the sense inhich the agents collaborate. Our model is not a model of division

f labor. There is no breakdown of the shared task into componentasks and no center that allocates these subtasks to separate agents.nstead, each agent does its best to reach the shared target, as is thease when multiple research laboratories pursue the answer to theame research question or when multiple inventors pursue the firstorking prototype of a new type of device.

Each agent has a set of communication links. The outgoing linksesignate those agents to which an agent communicates its par-ial results; the incoming links designate those agents from whichn agent receives communications. Communicated elements enterorking memory and exist there on the same conditions as ele-ents retrieved from memory or created by thought operations.

hey can be encoded into memory, used as inputs to cognitive oper-tions, broadcast in turn or be displaced by the arrival of additionallements in working memory.

In each cycle of operation, each agent makes a probabilistic deci-ion as to which, if any, of the knowledge elements currently in itsorking memory should be broadcast to other agents. All work-

ng memory elements are potential candidates for broadcasting,egardless of origin. It is important that the agents communicateontent to one another. That is, they do not merely pass on a quan-ity of activation or knowledge, but information that can help bringhe problem solving process of another agent closer to the sharedarget. Every agent is working on the whole task, but the agentsommunicate intermediate and partial results, in addition to com-lete solutions. The decision to broadcast is taken with a certainrobability. The elements to be broadcast are selected on the basisf their value. The reception of a communication is also proba-ilistic, to model the possibility that one agent is not heeding allommunications it receives from other agents.

Each agent has a subset of agents to which that agent broadcasts.e distinguish between the density of the connections and the

opology of the connections. The density of connections is indicatedy the proportion of all agents to which any given agent commu-icates. Zero density corresponds to the lone genius scenario, eachgent working away in isolation. Beyond that, density can in princi-le vary from sparse, each agent communicating to a few, perhapsven a single, recipient, to complete, in which case each agent com-unicates to everyone else. Real collectives tend to fall somewhere

etween these extremes. The topology of the network has receiveduch attention in classical social psychology and also in recent net-ork studies, especially studies of the so-called small world effect.

n the simulations reported in this article, we did not systematicallyary the topology. Instead, each agent broadcasts to a randomlyhosen subset of the network.

We focus on what we think of as intermediate size collectives,arger than the 2–10 member groups typically studied in social psy-hology experiments but smaller than the large populations oftentudied in sociology, network models or internet studies. In theimulations reported in this paper, the size of the network was sett 50 agents. This type of collective correspond to, for example, amall community of researchers all working on a shared problemnd communicating along the way through E-mail, conferences
nd publications, and to a market with number of research andevelopment firms racing to invent the next product.
We distinguish invention, the first creation of a knowledge ele-ent that matches a given target, from dissemination, the spread of

hat knowledge element through the network. The process of con-

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

structing an element that instantiates a target models the processof invention. The endpoint of this process is the first cycle in whichsome agent creates a knowledge element that matches the target.If we continue to run the model past that point, the element thatinstantiates the target will spread through the collective, in partbecause it is (re-) created by other agents and in part because ithas high probability of being communicated from agent to another.Agents do not directly recognize a target as such, but only per-ceive the (noisy) value of the matching knowledge element. Givennoisy values and probabilistic communication, a target found byone agent will not spread instantaneously across the network butwill require multiple cycles. The end point of the disseminationprocess is the cycle in which every agent in the collective (or somespecified proportion of them) posses a copy of that element.

The two-level structure of our model is illustrated in Fig. 1. Thetop part of the figure represents the cognitive architecture of a sin-gle agent, while the bottom part illustrates a portion of a networkof such individuals. The details of the implementation are given inthe next section.

3. Model details and implementation

Agent-based models (ABM) have been used to study a widerange of social, economic and natural phenomena caused by pro-cesses 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 ABMspecifies the properties and behaviors of individual agents, howagents interact with each other, and provide a distributed paral-lel computational model comprising the collection of agents. Such
models enable the study of higher-level system properties that can-not be derived analytically or by aggregating statistically over theagents. Because the agents communicate knowledge structures, thelatter is the case for our model.

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The ABM presented in this article is implemented with thewarm modeling tool kit (Stefansson, 1997; Terna, 1998). Swarm isn object-oriented platform for agent-based modeling and simula-ion. Agents in the model are implemented as software objects, andgents are endowed with specified properties and rules of behaviornd interaction. Swarm includes simulation tools needed to carryut and coordinate the activities of the agents, schedule events,anage agent interactions and so on.Cognitive processes are operations on knowledge elements.

nowledge elements are represented as lists of symbols. Symbolsenerically denote knowledge primitives (concepts), and the listsepresent complex knowledge structures (facts, beliefs, principles,ules, schemas, blue prints, etc.) formed by combining the prim-tives. In the simulations reported in this article, the vocabularyf symbols was just the ten digits 0–9 and the maximum sizef a knowledge element was set to 10 symbols (with repetitionsllowed). That is, the agents faced a search space of 1010 problemtates, which is large enough to correspond to search spaces in cre-tive enterprises. Two knowledge elements are identical (“match”)f and only if all their symbols are identical at each list position,

ith one exception: The symbol “0” is used as a so-called wild cardymbol, i.e., it represents “don’t care”. According to these matchingules, the two lists 〈1, 2〉 and 〈1, 2〉 match, 〈1, 2〉 and 〈2, 1〉 do not,ut 〈1, 2〉 and 〈1, 0〉 do.

As noted above, the cognitive processing by agents is goal driven.oals are modeled as target knowledge elements that reflect someesirable end states. Each such target is a list of the same sort asknowledge element. To achieve or obtain a target means to pro-uce (through operations defined below) a knowledge element thatatches that target. Target elements can provide partial specifica-

ions of a desired end state. Such targets carry the wild card symboln certain positions, reflecting the lack of specific knowledge prim-tives at these positions. A target with fewer specified symbols isasier to achieve since multiple knowledge elements can obtain aatch with this target.An agent’s long-term memory is called the stock and its working

emory is called its active list. Both are defined as sets of knowl-dge elements. The maximum size of the stock and active list arearameters in the model. In the simulations reported in this arti-le, the stock size was set to 50, and the size of the active list wasystematically varied.

Agents send and receive knowledge elements along communi-ation links. Agents can be connected to other agents in differentays to implement different network topologies. In the simulations

eported in this paper, we considered random networks in whichn agent is connected to a certain proportion of other agents, aspecified by the connection density parameter, cd. When an agentommunicates a knowledge element, it is broadcast to all the agentso which it is connected, as would be the case, for example, inending an E-mail message to a distribution list. The connectionensity is a model parameter that was systematically varied in theimulations.

The selection of knowledge elements for various cognitive pro-esses is based on their values. The value of a knowledge elements defined as its similarity to the target, measured as the numberf matching symbols in specific locations in the two lists. Noise isdded to the values to reflect the uncertainty that characterizes cre-tive problem solving. In simulation runs with multiple targets, thealue of an element is its highest overlap with any target (not theverage across targets).The value for an entire memory (stock orctive list) is defined as the average of the values of the knowledge
lements in that memory. Selection of a knowledge element frommemory is probabilistic, with higher valued elements more likely
o be selected. The insertion of a new knowledge element into aemory is also guided by its value, relative to the average memory

alue. When a memory is filled to capacity, a new knowledge ele-

Networks 32 (2010) 263–278

ment replaces a lower valued element currently in memory, againselected probabilistically.

The model works by iterating through operating cycles. In anoperating cycle, each agent receives elements broadcast from otheragents, decides whether to heed them, decides whether to retrieveelements from the stock, selects elements in the active list for pro-cessing, selects operations to execute, decides whether to store anyof the new elements in the stock and, finally, decides whether andwhich elements to broadcast. The next cycle then commences. Themodel keeps cycling until it has reached its target or targets, oruntil the maximum number of cycles has been carried out. Theseoperations are implemented through the following set of methodsdefined at the agent level:

• Receive. Knowledge elements received from other agents may ormay 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 proba-bility of being entered into the active list if their values are higherthan the current active list value.

• StockToActiveList. Knowledge elements currently in the stock areretrieved, i.e., entered into the active list, probabilistically basedon their values. This method models the agents’ capability ofmaking use of prior knowledge.

• ProcessActiveList. Knowledge elements in the active list are pro-cessed to produce new knowledge elements. The operations aresimilar to those employed in genetic algorithms (see below forfurther details). This method models the agents’ ability to infernew conclusions.

• ActiveListToStock. Knowledge elements currently in the active listare transferred to the stock. This is once again a probabilistic oper-ation, with elements from the active list replacing elements in thein stock that have lower values. This method models the agents’capability of learning, i.e., encoding information into long-termmemory for future use.

• Send. Knowledge elements from the active list can be broadcastto other connected agents. The verbosity parameter, pc, specifiesthe probability that an agent decides to communicate with otheragents on any one cycle of operation. Together, Receive and Sendmodel the agents’ ability to interact with other agents.

The order in which the methods are described above corre-sponds to the order in which they execute during an operationalcycle.

Agents have a finite processing ability, and the operationsdefined above are not performed on all memory elements in acycle. The operating intensity parameter specifies the number ofelements processed through each of the operations. This was setto 4 for the simulations reported here. The attention and verbosityparameters, pr and pc were set to 0.5 throughout; on average, anagent 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 pro-cessing its current elements using operators that were based ongenetic algorithms (Holland, 1975). The crossover operator takestwo knowledge elements and interchanges their symbols in ran-domly chosen locations to obtain two new elements. Suppose thatan agent possesses two knowledge elements, K1 and K2, defined asthe two lists

K : 4 5 2 1 6 7 8 1 9 1
1
and

K2: 5 1 2 3 8 7 1 1 1 3.

S. Bhattacharyya, S. Ohlsson / Social Networks 32 (2010) 263–278 269

Table 1Explanation and symbols for model parameters.

Parameter Explanation Symbol

Density The proportion of agents to which an agent broadcasts. cd

Verbosity The probability that an agent will broadcast a result. pc

Attention The probability that an agent will heed a communication. pr

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expected to be easier to discover. The targets were considered inseparate runs, so the data in Fig. 2 are from single-target runs.

The interactions among the agents had a large effect. Interact-ing agents, even at low connectivity, show better performance thanthe unconnected case in which the agents do not communicate

Capacity Size of the active list.Memory Size of the stock.Size The number of agents.Creativity The probabilities of using cross-over, mutation

If the crossover locations are randomly chosen as 3, 5 and 9, twoew knowledge elements are produced (elements at the switchositions are underlined):

K ′1: 4 5 2 1 8 7 8 1 1 1

nd

K ′2: 5 1 2 3 6 7 1 1 9 3.

Notice that the crossover at the third position causes no change,ecause the symbol (“2”) at that position is the same in both lists.

A single knowledge element can also undergo random changeshrough the mutation, add and drop operators. The mutation oper-tor replaces a symbol at a (randomly chosen) location with aandomly chosen symbol. The add operator adds a random symbolo a knowledge element (if the latter is not filled to the maximumength of 10 symbols). The drop operator removes a symbol at a ran-om location in a knowledge element; all symbols are then adjustedo the left.

Knowledge elements in an agent’s active list are selected forhese operations probabilistically, based on their values. The dif-erent operators are chosen based on parameters defining therobabilities for crossover, mutation, add and drop. The probabil-

ty for crossover was set to 0.7 and the probabilities for mutation,dd and drop were each set to 0.1. Crossover effects exploitationf currently known elements, while the random change operatorsffect exploration over the space of knowledge elements.

Agents undertake operations at every cycle of the simulation.ach simulation run starts with an initial random set of elements inhe agents’ active lists and stocks. Every simulation reported hereas run for a maximum of 1000 cycles. Results for each setting

f the model parameters are taken as averages over 10 simula-ion runs, each with a different random seed of initial knowledgelements. Table 1 lists the parameters in the model.

. Experiments and results

.1. Overview

We conducted simulation experiments to examine the effectsf communication density, modeled as network connectivity, andhe cognitive capacity of the agents, modeled as active list size, onhe performance of the collective as a whole. We varied connectiv-ty through four levels. The first level was 0%, i.e., no connections,orresponding to a population of lone geniuses all working on theame task without interacting. This served as a control and com-arison condition that allowed us to verify that our model exhibitsetwork effects. The other levels of connectivity were 10%, 50% and0%. The effect of cognitive capacity was examined by varying the
ctive list size through the values 5, 10, 20 and, in some simulations,0 elements.
We first examine the case in which the collective pursues a sin-le, shared target. Performance was measured separately for easynd hard targets. An easy target is a symbol vector with some num-

Cap(AL)Cap(Stock)N

p(cross), and drop operations. p(mutate), p(add), p(drop)

ber of wild card (“don’t care”) symbols. A hard target is a vector inwhich all the positions are filled with specified symbols. Targetswith larger number of specified symbols will, on average, take thecollective longer to discover. A second set of experiments examinedthe case in which multiple targets were being pursued simultane-ously. Connectivity and capacity were varied in the same way as inthe single-target experiments. For each combination of parametervalues – connectivity, active list size and target difficulty – we ran10 simulations and the values displayed are averages across the 10runs. Each simulation was run for a maximum of 1000 cycles.

We assessed the performance of the simulation with threedependent measures. The main outcome measure was the rate ofinvention or the first-cycle-to-target, defined as the first operatingcycle in a run in which any agent in the network achieves a target;that is, in which a matching knowledge element appears in someagent’s active list. A second outcome measure pertains to the dis-semination of a target. The dissemination rate was defined as thenumber of operating cycles required before a target has spread toa given proportion (75%) of the agents. For the experiments withmultiple targets, we also quantified the breadth of disseminationas the number of agents achieved by at least one agent in the courseof the 1000 cycles of a simulation run.

4.2. Basic network effect

We first examine how the network compares with a groupof unconnected agents. Fig. 2 shows the first-cycle-to-target forunconnected agents and for low (10%), medium (50%) and high(90%) levels of connectivity. The results are shown for two sep-arate targets that differed in difficulty. The easy target was thelist (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

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

270 S. Bhattacharyya, S. Ohlsson / Social

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performance when the agents are pursuing multiple targets in par-

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

ut operate independently. Even low (10%) connectivity lowershe number of cycles to target from 31 to 12 for the easy targetnd from 85 to 17 for the hard target; recall that these values areverages across 10 simulation runs. While yet greater connectiv-ty leads to quicker discovery of the target, the difference between

edium (50%) and high (90%) connectivity is small. As expected, theard target takes longer to discover, especially for the unconnectedgents, less so when the agents interact. With increasing connec-ivity, the difference between easy and hard targets decreases andt 90% connectivity, it has disappeared.

A different pattern was observed with respect to the dissem-nation measure; see Fig. 3. Recall that the dissemination rate ishe number of cycles required for the target to spread to 75% of thegents. As expected, increasing connectivity leads to faster dissem-nation. However, the effect was small and it was only observed forhe easy target. The hard target followed the opposite trend. Theumber of cycles to disseminate to 75% of the agents is higher forhe hard target at low connectivity, and higher still for mediumnd high connectivity. Inspection of the results for individual runsndicated that model performance was more variable across the 10imulation runs for the hard than for the easy target. The standardeviations for the easy target across the three levels of connectiv-

ty were 2.6, 1.6 and 1.4, indicating high similarity across 10 runs.ut the corresponding values for the hard target were 52.8, 63.2nd 34.7, indicating large variations across runs. When some agentn the network hits upon the right track early in a run, as is likely

ith an easy target, communication helps moving the network as
whole in the direction of the target, but if all agents are on therong track early on, as might happen with a difficult target, their
nteractions will instead make the agents persevere on the wrongrack. The latter case is more probable for a hard target, so averages

ig. 4. The first-cycle-to-target for three levels of cognitive capacity and four levels of cesults for a difficult target. The values are means across 10 simulation runs.

Networks 32 (2010) 263–278

across multiple runs are greater for hard than for easy targets andthis effect dominates the small benefits of higher connectivity. Themain lesson is that connectivity interacts with target difficulty. Asthe next section shows, so does cognitive capacity.

4.3. Effects of cognitive capacity

Common sense suggests that because individuals have limitedcognitive capacity, there can be such a thing as too much network-ing. There is process loss in terms of the need to allocate memoryspace and processing capacity to the communications. In our model,this is represented by the possibility for displacement of potentiallyuseful elements from the active list as a consequence of the recep-tion of communications from other agents, and by the upper limiton how many operators that can be executed within a single cycleof operation. Intuition suggests that any negative effect of com-munication will be greater at higher levels of communication butlesser with greater cognitive capacity.

To investigate the relation between capacity and performance,we systematically varied connectivity through the same four levelsas above. In addition, we varied the active list size through 5, 10and 20 elements. As before, we measured rate of invention by thefirst cycle in which the target is discovered by any one agent. Theresults are shown in Fig. 4.

Once again, we observe a strong network effect. For all capac-ity levels, low connectivity leads to earlier achievement of targetsthan the unconnected case. This effect is particularly striking for thedifficult target (see panel b). The targets are achieved sooner as con-nectivity is increased, but the differences between low and mediumconnectivity are smaller in magnitude for both easy and hard tar-gets. At medium connectivity, there are no additional benefits ofyet higher connectivity.

Contrary to our prediction, increasing active list capacity doesnot lead to significantly fewer cycles to target. On the contrary,increasing the size of the active list to 20 elements leads to smallincreases in the number of cycles required to achieve the target ascompared to active lists of 5 or 10 elements. This is true for both easyand hard targets. While counter-intuitive, this effect is understand-able in terms of the focus on a single target. A smaller active listenables narrow focus on one target, while a larger active list tends todistribute the processing efforts across a broader range of elements,some of which are not on the path to the target but neverthelessoccupy memory space and processing cycles. This interpretationsuggests that increasing active list capacity might yield improved

allel; 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 tobe achieved by 75% of the agents. The results are shown in Fig. 5 for

onnectivity. Panel a shows the results for an easy target, while panel b shows the


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 levelsof 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 2Seven symbol vectors that served as targets in the multiple-target simulations.

Target no. Difficulty level Content

1 Easy 1234567002 Easy 9876543003 Moderate 1234567894 Moderate 987654321

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5 Hard 1234543216 Hard 1236547897 Hard 987456321

hree capacity levels and three levels of connectivity. For the easyarget, medium and high connectivity yields quicker disseminationhan low connectivity, with a slight advantage for high connectivity.ontrary to expectation, increasing active list from 5 to 20 elementsauses slightly slower dissemination rates. The results for the hardargets show the opposite pattern. Increasing connectivity slowsown the dissemination rate, but increasing the active list causesstrong and consistent beneficial effect. In short, for easy targets,issemination is fastest with a small active list and high connec-ivity, while for hard targets, dissemination is fastest with a largective list and low connectivity. Although both active list capac-ty and connectivity interact with the difficulty of the target, theres no evidence in Fig. 5 of any interaction between capacity andonnectivity.

.4. Searching for multiple targets in parallel

The single target case provides an essential evaluation of theffects of communication density and cognitive capacity, and itpplies to contexts in which the members of a creative collectiveursue a single, specific invention or discovery. But members of cre-tive communities often work on multiple projects in parallel. Forxample, a scientist might run multiple projects, a mathematicianight be working on multiple proofs and an inventor might con-

ider several potential inventions or discoveries. We next exploreonnectivity and cognitive capacity in the presence of multiple tar-ets. In this case, the individual elements in memory are valued byatching them to all the targets and the value of the element is

aken to be its highest similarity to any target. We consider seven
argets, two easy, two moderately hard and three hard; see Table 2.he moderately hard targets specify a few additional elementsore than the easy ones, while the hard targets involve a mix of
ymbols 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, aarget with trailing 0s would be equivalent to the shorter string that does not specifyny symbol in these positions.

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

The easy targets, with fewer specified elements, are expected to beachieved sooner than the others. While both the moderate and hardtargets specify the same number of defined elements, the moderateones can be obtained by adding two elements to the easy targets;the hard targets, on the other hand, will require recombinationof sets of elements from the moderate targets, and can generallyrequire greater effort to achieve.

As in the single-target simulations, we varied connectivityacross four levels: unconnected (0%), low (10%), medium (50%) andhigh (90%), where the percentages refer to the proportion of agentsin the network with which each agent communicates. The stocksize 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 to75% of all agents and the number of targets found in a simulationrun. Each value is an average over 10 runs. In the multiple-targetruns, we also averaged over the targets at each level of difficulty.We present the results for the two easy and three hard targets. Wedo not include results for the moderately hard targets since theywere similar to the findings for the easy targets.

The network did not achieve all seven targets in every run. Whenreporting the average over 10 simulation runs, we use a value of1000 cycles (corresponding to the maximum number of cycles) forruns in which a target was not achieved before the end of the run.This underestimates the value we would have obtained, had thesimulation run without limit. These results are presented in Fig. 6.

Once again the low connectivity networks exhibit better per-formance than the unconnected case. However, the effect is not
linear. Increasing connectivity above 10% leads to larger number ofcycles being required to find the targets. Once again, there is pro-cess loss when there are more interactions. The beneficial effect of alarge active list is clear and particularly apparent at the highest con-

272 S. Bhattacharyya, S. Ohlsson / Social

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ig. 7. Number of targets achieved in a run as a function of four levels of connectivitynd three levels of capacity.

ectivity level. When the active list has room for 50 elements, theacilitating effect of cognitive capacity is so great that it completelyounteracts the process less caused by higher connectivity.

The results in Fig. 6 were achieved by including all runs in thenalysis. In cases where a target was not obtained before the end ofhe run, we used 1,000, the maximum number of cycles as an esti-

ate of the first-cycle-to-target for that target. The question ariseshether the number of targets actually found interacts with the
odel parameters. The number of targets found is shown in Fig. 7 asfunction of connectivity and capacity. The number of targets foundecreases with increasing connectivity, but increases with increas-
ng active list size. These interactions means that the estimate of

ig. 8. The first-cycle-to-target, averaged over the four smallest values selected among th) targets.

ig. 9. First cycle to target as a function of four levels of connectivity and four levels of capeparately for easy targets (panel a) and hard targets (panel b).

Networks 32 (2010) 263–278

the first-cycle-to-target that we used in cases in which some tar-get or targets were not found will affect different simulation runsdifferently. This complicates the interpretation of Fig. 6.

Given the result in Fig. 7, we decided to also study the rela-tions between connectivity, capacity and target difficulty in thosesituations in which the targets were actually found. Inspection ofthe simulation runs revealed that there were at least four runs inwhich every target was found within the cycle limit of 1000 cyclesfor every combination of parameter settings, so we next considerthe first-cycle-to-target, averaged over the four smallest valuesselected among those runs in which all targets were found. Theresults are displayed in Figs. 8 and 9.

For both easy (panel 8a) and hard (panel 8b) targets, the numberof cycles to target drops sharply from no connectivity to low andmedium connectivity. There is no additional improvement in per-formance as we increase connectivity from median (50%) to high(90%) connectivity.

The effect of capacity was smaller in magnitude than the effectof connectivity. Furthermore, the effect was not in the expecteddirection. In both panels of Fig. 8, the curve for the active listcapacity of 50 elements is higher than the other curves, indicatingthat higher capacity begins to have detrimental effects eventu-ally.

as in Fig. 9. For both easy and hard targets, the number of cyclesto the first discovery of a target drops sharply as active list capac-ity is increased from 5 to 10 for the no connections case, but then

ose runs in which all targets were found, for two easy (panel a) and two hard (panel

acity, averaged over four runs in which all targets were found. The data are shown


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ig. 10. Dissemination of targets through the network in terms of the percentage ond capacity. The data are shown separately for easy (panel a) and hard (panel b) ta

ncreases again. A population of isolated agents reaches a targetastest when the agents operate with an intermediate active listize. For the connected agents, active lists of 5 and 10 elements pro-uce approximately the same performance, but larger active listslow down discovery. The beneficial effect of connectivity is alsolear in Fig. 9, with low connectivity being worse than medium andigh connectivity, but there was no consistent difference betweenhe latter two conditions.

Finally, we investigate dissemination in the multiple-targetase. Not every multi-target simulation resulted in 75% of the agentsossessing a target before the end of the 1000 cycles. Hence, weannot use the number of cycles until 75% of agents posses a tar-et as the measure of dissemination rate. Instead, for this analysis,e measure the dissemination rate by the percentage of agentsho possess a target at the end of a run. The results are plotted

n Fig. 10 as a function of connectivity and capacity, and we sep-rate the results for the two easy and three hard targets. Oncegain, we see a clear and strong network effect. Compared withnconnected agents, the low, medium and high connectivity con-itions all produce faster dissemination. Among the latter three

evels, low connectivity consistently does better than the mediumnd high, while the latter two show no consistent difference. Atll connectivity levels, larger capacity results in faster dissemina-ion. The effect does not appear for the unconnected case, but it is

onotonic in the other conditions. In this case, dissemination doesot interact with target difficulty, as evidenced by the similarityf the results for easy and hard targets. The multi-target case isimilar to the single-target case for hard targets (Fig. 5, panel b) inhat higher connectivity leads to slower dissemination but higherapacity leads to faster dissemination. (In comparing Figs. 5 and 10,t is useful to remember that on the measure plotted in Fig. 4, loweralues mean faster dissemination, while on the measure plotted inig. 10, higher values means faster dissemination.)

. Conclusions, discussion and future work

.1. Summary and interpretation

The simulations confirmed some of our expectations but notll. We observed a consistent network effect. The rate of inventionas strongly affected by the difference between the lone genius

ase – a set of unconnected individuals – and the low connectiv-ty case in which each individual communicates with a randomlyhosen subset of 10% of the other individuals in the network. Thisasic network effect was present in both the single-target andultiple-target conditions and for both easy and hard targets; see

ts who possess a target at the end of a simulation run as a function of connectivity

Figs. 2, 4 and 8. Interaction speeds up the rate of invention. Theexplanation is obvious: When an individual discovers a partialresult that is close to a target and broadcasts it to other agents,it saves cognitive work for those other agents and moves the entirecollective forward in the search space. The contrast between thezero connectivity and low connectivity cases proves that our modelcaptures some aspect of true collaboration.

As we increase the communication density from low to mediumand high, the rate of invention continues to speed up but at a nega-tively accelerated rate. The advantage of increasing communicationdensity from medium to high is either small in magnitude or non-existent. This is so in the single-target and multiple-target casesand for both easy and hard targets; see Figs. 2, 4 and 8. That is,the benefits of communication are almost completely realized by amodest level of interaction. The explanation why there are smallerand smaller benefits at yet higher level of communication is thatcommunication entails process losses of various kinds: Most obvi-ously, numerous communications occupy working memory spaceand threaten to displace from working memory an individual’s ownpartial results before the latter have been protected from loss bybeing moved into long-term memory or by being communicated toother agents. Multiple communications from connected agents alsocompete for attention in an agent’s active memory, and potentiallyuseful ideas from other agents may hereby be lost. An illustrativeexample of such process loss is given in Appendix A. A simple sim-ulation with 5 agents, and considering a low connectivity settingwhere each agent connects with two others and a high connec-tivity setting where each agent connects with all four others alsohelps illustrates this: the number of cycles for 1, 2 and 3 agents toachieve a target are found to be {78, 79, 80} and {74, 143, 298}for the low and high connectivity settings, respectively. While theearliest times to target are similar, the times taken by the secondand third agents to achieve the target are higher with greater con-nectivity, and is indicative of the process losses that can occur. Ingeneral, too much interaction distracts.

In contrast, the effect of cognitive capacity on the rate ofinvention departed from our expectations. Cognitive psychologistsdiscuss the capacity of human working memory in terms of limita-tions (on storage space, processing resources, etc.), and some havehypothesized that what psychologists measure with intelligencetests is nothing but working memory capacity. If so, one would
expect greater capacity to facilitate cognitive processing. We alsoexpected larger capacity to help disproportionately more at higherconnectivity levels by counteracting the expected process lossesat the highest levels of communication density. Neither expecta-tion was verified. In the single-target case, the curves for the three

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evels of capacity are superimposed on each other, with a slight ten-ency for the curve for the largest capacity to exhibit the slowestate of invention. This was so for both easy and hard targets; seeig. 4. In the multiple-target case, we see that same pattern acrossour levels of capacity, with the curve for the largest capacity oncegain exhibiting the slowest rate of invention, for both easy andard targets; see Fig. 8. There is no sign in either the single- or theultiple-target case of any interaction with communication den-

ity; the curves for the different capacity levels are nearly paralleln both Figs. 4 and 8. In short, increasing cognitive capacity has noeneficial effect on the rate of invention in our model; in particu-

ar, it is not the case that higher cognitive capacity counteracts therocess losses that reduce the benefits of interaction at the higherommunication densities.

With hindsight, we explain this pattern in terms of focus. Recallhat the simulated individuals are guided in their search for a targety an evaluation function that measures the promise or potentialsefulness of each knowledge element. Greater cognitive capacity

owers the probability that a high-value element will be displacedrom working memory, but it also enables lower-value elements toemain in working memory for longer periods of time. Any elementn working memory has some probability of being chosen as inputor further operations on any one operating cycle. The larger theorking memory, the lower the value of the least valuable elements

nd the greater the probability that the individual will waste cyclesn those less valuable elements. As capacity is increased, this lackf focus counteracts the advantage of having a lower probabilityf losing the highest-valued elements. A simple simulation withagents and 5 interconnecting links (each agent connected with

wo others), and agents’ active memory sizes set at 5, 10 and 20elps illustrates this: the number of cycles for 1, 2 and 3 agents tochieve a target are found to be {78, 79, 80}, {70, 135, 144}and {115,41, 183} for active memory sizes 5, 10, and 20, respectively; the

onger times to achieve the target with increasing memory levelseen here arises from the aforementioned lack of focus with higheremory. With greater connectivity, higher memory capacity can

nable more of the received ideas to enter and agent’s memory,eplacing lower values ideas. For further processing of these ideas inemory, however, the agent has this larger set of ideas to consider,

nd attention thereby tends to get diffused. Since targets will bechieved only through recombinations of particular ideas, selectionf these specific ideas for processing can be delayed.

The upshot is that there is no net effect of increasing work-ng memory capacity beyond 5 or 10 elements. We note that this

orking memory size coincides with the typical mean for empir-cal measures of human working memory size. Perhaps working

emory is limited in humans because there is no net advantageo having a greater one, the advantage of being able to retain whats important being cancelled by the disadvantage of being able toetain what is not so important.

The rate of dissemination was also strongly affected by ourodel parameters, but the effects exhibit a different pattern. In

he single-target case, increasing connectivity improves the rate ofissemination for easy targets. The effect is small in Fig. 2 (see theottom curve), but rather greater in Fig. 5 (panel a). However, forard targets, increasing connectivity slows down dissemination;ee Fig. 3 (top curve) and Fig. 5 (panel b). The multi-target caseesembles the hard single-target case in this respect: The rate ofissemination is fastest at low connectivity, for both easy and hardargets (see Fig. 10, both panels). This is a clear but counterintu-tive case of process loss. As more knowledge elements are shipped
round the network, each individual receives more communica-ions in each cycle of operation, which in turn raises the probabilityhat some high-value elements close to a target – indeed, the targettself – will be displaced before it is communicated. With respecto dissemination, the expected counteracting effect of greater cog-
Networks 32 (2010) 263–278

nitive capacity does appear. In the single-target case, the benefit ofhigher capacity for dissemination is consistent across three capac-ity levels, for each connectivity level; see Fig. 5, panel b. Likewise, forthe multi-target case, the benefit of greater capacity is consistentacross levels of communication density.

The application of these conclusions to particular examples ofcreative networks depends on what we take the agents in the modelto symbolize. In the above paragraphs, we have assumed that anagent is an individual person, and that hence cognitive capacitycan be identified with working memory capacity. In this case, cog-nitive capacity cannot be deliberately manipulated, and the onlyprescriptive conclusion is that a modest level of interaction is likelyto be sufficient to realize the benefits of collaboration However,there is nothing in the structure of the model that enforces theinterpretation of agents as persons. The agents can themselves becollectives, such as research groups or research and developmentteams that operate in the context of a population of similar groupsor teams. Under this interpretation, the nature of communicationlinks only change a little. An E-mail message sent to a distribu-tion list of interested colleagues or a conference presentation to ahighly specialized audience can still serve as prototypical examplesof broadcasting partial results to a subset of likeminded colleagues.

But cognitive capacity must be re-interpreted if agents are them-selves collectives. What is the cognitive capacity of a group or aresearch team? This question would be more effectively pursuedby cognitive anthropologists than by cognitive modelers like our-selves, but a rough cut might go as follows: The ‘working memory’of a group encompasses all the relevant ideas, theories, findingsand other materials that the group members can draw upon intheir work without conducting extensive search, either becausethey know them well enough to be able to retrieve them frommemory (“didn’t Smith and Jones in their 2001 article find that. . .”)or because they have them ready at hand (“my copy of the Smithand Jones article should be in the pile here, let’s look it up”). Thequestion of capacity, and hence focus, applies to this extended setof immediately available materials. In this context, the agents-as-groups interpretation of our modeling results imply that inventionor discovery is faster if each group has a narrow definition of whatcounts as “relevant material”, but dissemination is faster if the suchgroups have a broader definition of relevance. This makes excellentsense: Narrow focus increases the probability of making a discov-ery, while broad receptivity on the part of each group helps a newidea or result to get around. The general lesson is that there is nobest configuration of a creative collective, because the usefulness ofa configuration depends on the outcome measure employed, whichin turn reflects the purpose of the collective.

5.2. Relations to other models

Although the topic of collective or social creativity has been thesubject of much attention in as diverse disciplines as social psy-chology, sociology, economics, artificial intelligence, mathematicalnetwork studies and the history of technology (Abrahamsonand 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). Com-puter modeling is one of the most commonly used tools in thisfield (Berdahl, 1998; Bonabeau, 2002; Cowan and Jonard, 2003;Saunders and Gero, 2001; Schilling and Phelps, 2007). Our approachmakes four contributions. First, we cast the problem as one of
understanding the interactions between two system levels, theindividual agent and the collective in which he or she partici-pates. Unlike network models in which the nodes are mere waystations for messages or quantities of activation, and also unlikemodels based on the principle of level invariance, we start with

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he principle that units at both levels have internal structure, butf different kinds. The description of individuals needs to includehose gross features of their cognitive architecture that affect howhey interact during collaborative problem solving (and only thoseeatures). The relevant features are abstracted from contemporaryesearch in cognitive psychology. Thus, this part of our model isolidly grounded in the empirical research that supports contem-orary models of the cognitive architecture. Our goal was not toontribute to the theory of the cognitive architecture, but to buildn it by including in our simulation those features that are likelyo affect the functioning of a creative collective, most importantlyhe dynamics between short- and long-term memory, thinking andommunication.

A second and closely related feature of our approach is that thegents communicate content. They remember and operate uponymbol structures, creating new symbol structures that representhe conclusions of inferences or other cognitive products. The lat-er are represented as symbol vectors that can stand for any typef cognitive entity – concept, proposition, schema, mental model,ule, etc. – and an agent’s goal is defined in terms of such a vectornd its thinking capabilities are defined in terms of operations onuch vectors. The agents in our model are thus faced with a real,nd realistically huge, search space. The agents communicate byassing symbol structures – not abstract quantities – to each other,ossibly helping each other by passing along a result that speedsp the search by another agent. In this way, we model the passingack and forth of content that is the core of real communicationsithout tying the model to any particular hypothesis about men-

al representation. This use of generic knowledge representationsnd generic processes enables us to model the processing of con-ent without having to implement a symbolic artificial intelligenceystem that actually performs some creative task.

Third, we focus on the case of agents who are loosely con-ected in the sense that they interact primarily by communicatingartial results (ideas) but nevertheless share in the same creativendeavor. Our model is not a model of the division of labor. Theres no central agency that distributes subtasks to agents and thenntegrates their returns. Instead, each agent is working on the com-lete, shared task, as would be the case in a race among researchroups for a particular discovery. Also, the agents in our simulationsroadcast their partial results to subsets of other agents before theyave found a solution to the shared task. Thus, our model is not aodel of diffusion only, but actually models the prior process of

nvention. Examples of this type of collective are found throughouthe sciences and among communities of artists and technologists,nd they differ from formal organizations, especially those basedn the division of labor. The comparison of different levels of con-ectivity with the no-connection case, a control condition we haveot noticed in other network simulations, turned out to be very

nformative. The no-connection condition is meaningful in our caserecisely because the agents have internal structure and are capa-le of reaching the shared goal on their own.

Fourth, the agents in our model can share a single goal (tar-et) or simultaneously pursue multiple shared targets. The latterase is a realistic representation of how creative agents (individ-als or teams) function. They seldom have a single, well-definedarget but pursue a repertoire of distinct but related enterprises,nd the work performed in the pursuit of one enterprise is simul-aneously considered with respect to its contributions to the others.

successful researcher seldom works on a single question, annventor is likely to pursue multiple projects in parallel and the
endency of artists to have multiple canvasses going at any oneime is proverbial. Our simulation results show some differencesn outcomes between the single and multiple target cases, indicat-ng that there is reason to distinguish between them. We do notnow of another simulation of creative networks that covers both
Networks 32 (2010) 263–278 275

the single and the multiple target cases within the same processingstructure.

5.3. Limitations and future work

The current version of our model exhibits several limits andsimplifying assumptions, some of which might have effects onthe outcome measures. A fundamental boundary on the presentwork is its focus on collectives in which the participants collabo-rate by sharing information. We do not model physical interactionsof the hand-me-the-hammer variety, nor do we model collabora-tion through division of labor. Each agent works independently onthe common problem, and the agents collaborate only by sharingpartial results. There is no mechanism in our model for analyzingthe shared problem into subproblems, each of which gets assignedto a different agent.

How limiting is this focus? First, there are creative collec-tives that instantiate this model. Scientists who work on thesame research problem but at different research institutionsapproximate this model. They collaborate primarily by sharingpartial results via conference presentation, panels, publications,manuscript and proposal reviews and face-to-face interactions. Themembers of a large research and development team might alsoapproximate this type of collaboration, as do, for periods of time,communities of painters, designers and inventors. These cases areinteresting enough in their own right to motivate efforts to under-stand them better. Second, the model is relatively insensitive tohow its knowledge elements are interpreted. In the applicationreported in this article, we primarily interpreted them as knowl-edge representations in the minds of individuals and the operationsas thought operations, but they could be reinterpreted as (repre-sentations of) physical objects and the operations as correspondingto physical actions. The process that we here have called commu-nication would then be reinterpreted as the sharing of physicalproducts. Hence, the scope of the model is wider then it first seems.

Another limitation is that the simulated collectives reported inthis article were all homogenous; that is, each agent had the sameparameter values as every other agent. This strengthens the findingthat there is a basic network effect – if every agent is the same, theeffect of communication is necessarily due to the communicationitself, not to a best member effect – but it is nevertheless unre-alistic. In real collectives, agents are likely to function somewhatdifferently. In particular, they are likely to possess somewhat differ-ent cognitive capacities and to use somewhat different evaluationfunctions.

This limitation is not inherent in the model but a simplifica-tion for the sake of the initial exploration of the model. There isnothing in the principles behind the model or in its implementa-tion that prevents the agents from being heterogeneous in one ormore of the key parameters. In future work, we intend to explorethe effect of heterogeneity in the evaluation function. By endowingsome agents with a greater noise in their evaluation function, wecan give them a heightened ability to depart from common wisdom(as encoded by the evaluation function). The proportion of noisyagents in the collective then becomes an additional parameter inthe model. The effect of this parameter is the subject of ongoinginvestigations.

A third simplification in the current version of the model isthe topology and nature of the communications network. In thesimulations reported in this article, each agent was assigned com-munication links to a randomly selected subset of other agents,
with the size of the subset being the same for each agent butvarying from simulation to simulation. However, there is a long-standing hypothesis in research on collective cognition that thetopology of the network matters. The particular topology knownas “small world structure”– many local connections augmented

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ith a few long-distance connections between the local clustershas been shown to have strong effects on the behavior of acti-

ation networks (Cowan and Jonard, 2004; Schilling and Phelps,007; Uzzi and Spiro, 2005; Watts, 2003). The small-world topologyontrasts both with the random topology used in our simulationnd the hierarchical topology of formal organizations based onhe division of labor. In the organizational theory literature, vari-us models explore different organization structures that facilitateearning (Siggelkow and Rivkin, 2005; Siggelkow and Levinthal,003); Fang et al. (in press) consider subgroups with varying lev-ls of cross-group linkages and find such structures to outperformandom network structures in organizational learning. The ques-ion whether such differences matter in a creative collective cane explored in our model by assigning links according to someefinite topological scheme instead of assigning them randomly.ur models allows consideration of different network topologiesnd in future work, we plan to use random networks as a baselineo which to compare the functioning of other network topolo-ies. In addition, real agents are likely to vary with respect toow many contacts they maintain. Although the agents all hadhe same number of connections to other agents in the simula-ions reported in this article, there is nothing in the structure ofur model that prevents us from exploring heterogeneity in theisposition to communicate, with the distribution of this disposi-ion over the agents being yet another system parameter. In annteresting study, Ziherl et al. (2006) consider social capital ofesearch groups in terms of networks amongst researchers, andnd that networks with moderate ties and having diversity amongembers correspond to better productivity of junior researchers.

xamining such findings using our model, with heterogeneousgents and network properties, is another useful area for futureesearch.

The emphasis on the social side of creativity brings with it theuestion of how individual and social creativity are related. Ourpproach assumes that both levels of analysis are important and
ontributes to the functioning of creative collectives. Individualognition is best described as a small number of unique compo-ents linked via unique types of links, while social networks areest described as in terms of large numbers of similar components,
Networks 32 (2010) 263–278

all interacting according to the same rules. The model presented inthis article is a first attempt to formalize this view. In construct-ing this particular formalization, we have drawn upon work incognitive psychology, in contrast to the many network models of“innovation” that ignore the potential contributions of this disci-pline for understanding something so human and so psychologicalas creativity. Future work will show whether this approach canprovide theoretical explanations for observed empirical effects andeffective prescriptions for the organization of creative collectives.

Appendix A. Illustrative example of process loss withincreased connectivity

The following example helps illustrate how process losses canarise from higher connectivity between agents. Consider an agentreceiving ideas from other connected agents, adding them to itsactive memory based on perceived value, and processing ideas inmemory to generate new ideas. Assume agents A, B, C, D, each withfour ideas in memory. For the purpose of illustration, the ideas arelabeled 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 ideasfrom different agents having similar values (value (a3) ≈ value (b3),etc.). Assume a single target idea, T, which is achievable throughrecombination of ideas a1 and b2. These ideas thus need to co-occur in memory for potential recombination in order to achievethe target T.

Take A to be our focal agent of interest, receiving ideas fromother agents in each step. At consecutive steps, the ideas gener-ated by agents are of increasing value; consequently, the ideasreceived by A are of higher value, and will replace lower-valuedideas currently in memory. Two scenarios are examined: (i) lowconnectivity, where A is connected to and receives ideas from asingle agent B, and (ii) high connectivity, where A is connected toand receives ideas from three other agents, B, C, and D. These areshown below, with content of agent A’s memory in{. .} and receivedideas [.] at each step; some of the received ideas replace existingideas in memory, which gives a modified set of ideas in memory atthe beginning of the next step.

(i) Low connectivity case

Social Networks 32 (2010) 263–278 277

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ii) High connectivity case

With greater connectivity, multiple higher valued ideas receivedrom other agents can bring about rapid changes in an agent’s mem-ry, thereby diminishing opportunities for generation of certainewer ideas. The example above assumes that the agent A accu-ately recognizes values of different ideas in determining whicheceived ideas to take into memory. When agents’ perceived val-es of different ideas is inexact, process losses as shown above cane more severe.

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How Network Structure Affects Social Creativity

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