Social Cognition in Complex Team Networks

Samuel M. D. Seaver�, R. Dean Malmgren

�, Andre A. Moreira

�,

Marta Sales-Pardo�, Daniel Diermeier

�, Luıs A. Nunes Amaral

�����

�Department of Chemical and Biological Engineering

Northwestern University, Evanston, IL 60208, USA�Departamento de F ısica

Universidade Federal do Cear a, Cear a, BrazilManagerial Economics and Decision Sciences, Kellogg School of Management

Northwestern University, Evanston, IL 60208, USATo whom correspondence should be addressed; E-mail: amaral@northwestern.edu.

March 14, 2006

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Abstract

We use the density classification task to model social cognition in complex networks.Agents are embedded in a team network (1) and use simple social learning heuristics toadjust their current state. We investige various agent-specific biases and system perfor-mance measures. We show that the performance of team networks perform less well thansmall-world networks.

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1 Knowledge Networks

Innovation and problem solving is almost always the consequence of collaboration in teams

or large groups. Here we model creative interaction in organizations as a complex network

(e.g. (1–3)) and systematically investigate whether and how the structure of networks affects an

organization’s ability to solve problems and find innovative solutions.

Recent research has begun to uncover the factors that lead to successful team networks.

An important example is Uzzi and Spiro’s (2005) empirical study of the Broadway industry.

To conduct their analysis Uzzi and Spiro use a newly created data set on creative networks in

Broadway musicals (all 2258 Broadway productions from 1877-1990). This industry is a good

choice to investigate this question for two reasons. First, there are no firms. Teams consist of

producers, directors, lyricists, composers, etc. and form on a project-by-project basis. Dense

social clusters form if the same people repeatedly collaborate, new links form as new-comers

join the system or as incumbents collaborate on a project for the first time. The second reason

is that success in the Broadway industry can easily be measured by box-office revenues, critical

acclaim or the length of runs.

Uzzi and Spiro show that the connectedness of the network (measured by clustering coef-

ficients) has a strong effect on the average yearly success of the industry, even if controlled

for various other independent variables such as market characteristics etc. Surprisingly, the

relationship is curvi-linear. Uzzi and Spiro suggest that if networks are too dense, too much im-

itation may lead to the dominance of conventional ideas. On the other hand, if networks are not

sufficiently connected, fresh ideas may emerge, but they remain isolated. From a managerial

point of view this means that there is a “sweet-spot” in the structure of the networks. An orga-

nization could potentially measure the connectivity of their internal network and then attempt

to increase or decrease its connectivity.

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Influenced by this line of work, Guimera et al. (2005) study a model of team formation

characterized by the propensities that incumbents continue to collaborate on a new project or

are matched with other incumbents or with newcomers. They show that different matching

probabilities lead to different network topologies and therefore different expected performance.

Surprisingly, the model is able to replicate the structure of various collaboration networks from

different scientific communities.

What is missing from these papers is a behavioral model of collective problem solving that

then can be used to examine the effects of network topologies on system performance in more

detail.

2 A Model of Collective Problem Solving

In a recent series of papers (2,5) we demonstrated how a complex network topology, combined

with noise can alter the efficiency with which a system of decentralized units performs a global

cognition task. The idea is to consider networks of agents who can be in one of two states

denoted here ��������� . Moreover, to capture the effects of misunderstandings and other forms

of miscommunication there is a probability of miscommunication. That is, with probability agents perceive the state of any connected agent to be randomly ��� . Collective problem solving

is modeled as a density classification task, a widely used measure of coordination and global

information processing (4). For a system comprised of units whose state is a binary variable,

the density classification task is completed successfully if all units converge to the same state

and the coordinated state is identical to the majority state in the initial configuration. The initial

state of the system can be interpreted as receiving a partial signal about the correct solution.

Agent may use various social learning heuristics. Specifically, we consider a population

comprised of independent agents and assign to each agent a bias ���� ������������������������ with �!����"�"�"#�$ . If �%�&�'�����(��� , then the agent is “conservative.” A conservative agent requires

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a “qualified” majority of her neighbors to convince her to change her state. If � � � ��� , the

agent is ”partisan” towards ��� . A partisan agent will “stick” to his preferred state until con-

vinced otherwise by a qualified majority of his neighbors. The update rule for an arbitrary agent

implementing a qualified majority heuristic is:

�����(� � � � ���� �� � � � ��������� � � �$���������� � ������� � � � �$�� � � ��������� � � �$� � (1)

where � ���(��� is the average noisy “signal” an agent receives from his ��� neighbors

We further assume that agents can have different levels of bias strength � � �� � ��� , which

measures the fraction of an agents’ neighbors that must be in the state opposing the state pre-

ferred by the agent in order to “force” him to change state. If #� ��� the agent uses a majority

rule, whereas if �� � � the agent is frozen and will never change his state.

The efficiency ��� ��� �� �$ � of an updating rule � , is a function of noise intensity , rewiring

probability � , and system size . Specifically, we have:

� �(����� �� �(��� � � �(��� � (2)

where �� is the number of agents that are in state “� � ” and !� is the number of agents that

are in state “ � � ”. For each realization, we let the system evolve for "� timesteps. We define

the efficiency � of a single realization as

� #%$'& �)(

�)( � (+*-, � �����/.�� (3)

The efficiency ��� for each set of parameter values is the average of � over 1000 realizations.

Moreira et al. (2004) consider small world networks (6) with � �10 neighbors, rewired with

a probability � (where a small world topology occurs at �32 � ) and a probability of miscom-

munication. They show that if agents use the majority heuristic, that is, set � �4� , the system

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exhibits rapid and robust convergence to the correct state, provided the interaction structure is

characterized by moderate noise and constitutes a small world-network. Importantly, both con-

ditions are necessary for effective problem solving. Moreover, more complicated decision-rules

that work well in the case of � � and �&� � , for example the Gacs-Kurdyumov-Levin (GKL)

rule (4) fail to function when communications is noisy or interaction occurs in an asynchronous

fashion. Intuitively, this implies that random connections to other members of the network not

only are important for problem solving, but that if they exist even extremely simple decision

heuristics can be successful. Surprisingly, for � � 0 , the case for which the GKL rule was opti-

mized, the efficiency of the majority rule reaches a value of 0.85, which is greater than what is

obtained with the GKL rule under the idealized conditions &��� and � ��� . Strikingly, unlike

the GKL rule, the majority rule yields efficient coordination even for asynchronous updating.

One can thus understand the role of each of the components in this condition. The noise enables

the system to escape conformations with multiple domains. The rewiring of the connections al-

lows fast access to information from across the system—i.e., the long-range connections make

the system a small-world. Importantly, the small-world phenomena is not enough to ensure the

convergence to the correct classification. Only with the combined effect of the noise and the

small-world topology can the system reach a consensus within the permitted evolution time.

Moreira et al. originally used a value of � � � " � , however with the use of agents that have a

preferred state, we need to ensure that the majority state, chosen to be “+1”, is always the same.

To find the ’best’ value for � we explore, for a variety of system sizes, the resulting efficiency

with which a system reaches consensus within "� time steps, as a function of � (Fig. 1A). We

choose a combination of a system size that is computationally inexpensive ( � $ � � ) and the

lowest value of � that will lead to an average efficiency 2 � "�� ��� of the time ( � � � " ��� ). The

resulting efficiency calculated by the original and new methods is compared as a function of

noise in a homogenous population and is found to be similar (Fig. 1B).

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Seaver et al (2006) consider more general interaction structures with � � � . In this case

agents may exhibit various forms decision biases, as discussed above. Adding partisans to

the model (even if they are distributed evenly between partisans for� � or ��� ) dramatically

decreases the performance of the system. In the case of conservative agents the model yields

a surprising conclusion. For moderate levels of bias ( #� � � # � ) the system not only shows

remarkable levels of efficiency, but system performance actually increases as the fraction of

conservatives increases, provided the noise level is sufficiently high. However, there is an im-

portant trade-off with the speed which a solution is reached. That is, the time to reach consensus

grows in the fraction of conservations. If the fraction is larger than 30%, consensus cannot be

reached within "� time steps.

To estimate the “typical” number of time steps it takes for a system to reach the steady

state we use the following method: We generate 5000 time series of � time steps with different

initial conditions using the same parameters. (Fig. 2). We compute the average time series,� �(��� � ���������� ���������� � � � �(��� . Then for each time step � we compute the average efficiency for the

remaining time steps ��� �(��� � �� ����� � �(��� � �� � . We fit the resulting curve (Fig. 2B) to a stretched

exponential

� ��� � �����%� ����� ���! "$#�% � (4)

We estimate the convergence time � as the time step at which the efficiency variation equals

95% of the total variation of �'& ������ �)( � �+*�, � " � �� ��- (5)

We explore a system with �4$ � ��� ��4� "�$ �-� �4� " � � � �. �4� " $ , and � $ # � . We obtain

the convergence time � for the system at � � "�$ and an increasing�'.

. It can be seen from Fig.

3A that the time needed for a system to reach a steady state increases with�/.

. The steady state

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is dominated by agents in the state “+1” meaning that a system is highly efficient at reaching a

consensus state of “+1” provided that it is given enough time to evolve.

Finally, from Moreira et al. we know that a larger � will make a system more robust to

noise (2). Our results have shown that the presence of conservatives will also make a system

more robust to noise. Therefore, to observe whether conservatives would make a system more

robust to noise independently of � we explore the efficiency for � � $ , � $ � � , � � � " � � ,���" # � , and ��� # � (Fig. 4). Though these results are not directly comparable to the results

for � � 0 , the same trend can be seen with � " # � where the presence of conservatives still

makes the system more robust to noise.

3 Team Networks

In addition to considering a small-world topology, we will also consider more realistic social

networks generated according to the model of Guimera et al. (1) The model is built on the

concept that collaboration usually occurs in teams. A network is built using teams of a fixed

number of agents ( � ) as building blocks. For each agent in a new team there is a probability �

that it will be an incumbent, that is, an agent already present in the network, and a probability

� ��� that the agent will be a newcomer. If there are more than one incumbents in a team, there

is a probability � that an extra incumbent will be a previous collaborator of the incumbents

already in the team. Otherwise, with a probability � ��� , the extra incumbent is any incumbent

in the network. Finally, agents that do not participate in any team for a long time are considered

to have ’retired’ and are removed from the network, allowing the size of the network to reach a

steady state.

Almost every network generated using this model is fragmented and thus for a more reliable

comparison to idealized small-world networks we use the giant components of the networks.

For one set of giant components we use � ��� "�� 0 $ and a retirement rate (���" � � to reach both

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a mean network size �� � � and an average giant component size that is �

� � � of the network.

For the second set of giant components we use � � � " � and ( � � $ � to reach a mean network

size of �4$ $ � and an average giant component size that is � � � � of the network. For both set

of giant components, the average size is � $ � � agents. Fig. 5 contain examples of the giant

components we use displaying the differences in the network organisation. In order to quantify

the differences in sparsity we measure the average degree of every node in these two sets—in the

giant components that only make up 50% of their networks, their average degree is � $ " � 0 and

in the giant components that make up � � � of their networks, the average degree is � $ " $ � . We

explore the efficiency of these networks at reaching consensus as a function of the fraction of

conservatives and of the noise intensity. The results are present in Fig. 6. The average degree of

each agent in the small-world networks is 0 simply because of the number of neighbors �&� 0 ,but this measure does not seem to be the dominant reason that the overall efficiency in the team

networks is much smaller. Compare these results with the results for small-world networks

where � � $ (Fig. 4). The dominant factor seems to be that these networks are not small-world

networks.

4 Conclusion

We provide a model of social cognition in complex networks. We show that very simple rules

can perform very well in problem solving tasks such as the density classification system pro-

vided the networks satisfy the small-worls property and interaction between agents is subjecto

moderate noise. In the case where interaction between agents is given by a self-organizing team

network estimated from the data, the system performs moderately well, but not as well as in the

case of small-world networks. This suggests the room for managerial intervention to improve

sytem performance to increase the connectedness of the network.

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References and Notes

1. Guimera, R, Uzzi, B, Spiro, J & Amaral, L.A.N (2005) Science 308 697.

2. Moreira A. A, Mathur, A, Diermeier, D & Amaral, L.A.N (2004) Proc. Nat. Acad. Sci 101

12083.

3. Uzzi, B & Spiro, J (2005) Am. J. Sociol. 111 447.

4. Crutchfield, J. P & Mitchell, M. (1995) Proc. Natl. Acad. Sci. U.S.A. 92, 10742.

5. Seaver, S. M. D, Malmgren, R. D, Moreira A. A, Sales-Pardo, M, Diermeier, D & Amaral,

L.A.N (2006) Science under review.

6. Watts, D. J.& Strogatz, S. H. (1998) Nature. 393 440.

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0.5 0.52 0.54 0.56 0.58 0.6

Probability that εj(t=0)=+1 , c

0.4

0.5

0.6

0.7

0.8

0.9

1.0

E

1012014018011601

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Noise

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

E

B

A

Figure 1: Exploration of initial distribution of state “+1”—The probability that an agents’ statewould initially be “+1” ( � ) in Moreira et al. (2) is � " � . The initial majority was always tiny, andthe final consensus was either 1 or -1. In order to force a consistent consensus in the contextof which we could explore biased agents � was chosen to be � " ��� . Fig. 1A highlights why� � � " ��� and � $ � � was chosen. Fig. 1A shows the response of systems to the increase inthe starting fraction of agents with a positive state (� �1� " � � and � � " " ). A system size of 401was chosen as a compromise due to the performance time and thus � " ��� was used as the startingfraction where each realisation would, on average, achieve an efficiency 2 � " � � . By using theparameter in this manner the majority will always be “+1” and will almost always become thefinal consensus of the system, and the dynamics that resist the consensus will be more obviousand easier to understand. Fig 1b shows a comparison between the original (’old’) method ofdetermining efficiency (2), and the new method where �1$ � � . In addition � �1� " � for the oldmethod and � � � " ��� for the new method.

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0 2000 4000 6000 8000 10000Time

0

0.2

0.4

0.6

0.8

1

ε(t)

,

0

0.2

0.4

0.6

0.8

1

ε(t)

0 2000 4000 6000 8000 10000Time

0.8600.8620.8640.8660.8680.8700.8720.8740.8760.8780.880

ε av(t

)

A B

Figure 2: Method used to estimate ��� —(A) Examples of one of� � � � time series generated

from a system where � $ � � , � � "�$ , � � � " � � ,�. � � " $ , and � $ # � and the average

time series � �(��� (B) The resulting curve when, for each time step � in � ����� we compute theaverage efficiency for the remaining time steps � �� ����� � � � �� � ��� � � �� � . This curve is fitted to a

stretched exponential� ��� � � ��� � � ��� ��� " # % � . � �)( � �+*�, ��� ���� � - which is an estimation of the

time it takes for the efficiency to vary by � ��� of the total variation at � ��� .

0 0.2 0.4 0.6 0.8 1Fraction of Conservatives

101

102

103

104

t*

0 0.2 0.4 0.6 0.8 1Fraction of Conservatives

0.6

0.7

0.8

0.9

1

Eas

,E

BA

Figure 3: (A) Time � for a population of agents to reach the stationary state. (B) Comparisonof the asymptotic efficiency ( � �� ) and the efficiency � at � � "� as a function of the fractionof conservatives. Note that for

�. � � "�� , the system cannot reach the stationary state in the "�timesteps used in the simulations.

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0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.2 0.4 0.6−1.0−0.8−0.6−0.4−0.2 0.0 0.2 0.4 0.6 0.8 1.0

Co

nse

rvat

ives

Fra

ctio

n o

f

s = 2/5 s = 3/5

Noise

Bias Strength

Efficien

cy

Figure 4: Efficiency of the system at reaching consensus as a function of the fraction of con-servatives and of the noise intensity. The parameters are � $ � � , � � � " ��� , � � � " � � , and�&� $ . Note that the change in � means a change in the range of . It is clear that the presenceof conservatives will increase the robustness of the system.

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AA

B

Figure 5: Examples of initial configurations and final states of “Teams” networks—The net-works here are built according to the model of Guimera et al, being giant components thatconsist of (A)

� � � or (B) � � � of the number of nodes. For each set, two different seedswere used to create the different distribution of agent types and states. Note that the final local-ized regions of blue (“-1”) nodes are clustered around conservatives of the same state, which isindicative of the short-range influence that conservatives have.

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0.0 0.2 0.4 0.6

0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6

0.2 0.4 0.6−1.0−0.8−0.6−0.4−0.2 0.0 0.2 0.4 0.6 0.8 1.0

Fra

ctio

n o

f C

on

serv

ativ

es

Noise

s = 2/7 s = 4/7 s = 6/7Bias Strength

Efficien

cy

A

B

Figure 6: Efficiency of the ’teams’ model—Efficiency of the system, built according to the’teams’ model of Guimera et al., at reaching consensus as a function of the fraction of conser-vatives and of the noise intensity. These results are for systems with 57% of the agents beinginitially in state +1 and (A) a mean number of 398 agents, � � � "�� 0 $ , � � � " � , ( � " � � (B)a mean number of 363 agents, � � � " � , � � � " � , ( � ��$ � . The value used for � in A meansthat the average fraction of the total number of nodes that are in the giant component used isaround $ � � and the average degree is $ " � 0 . In B they are

� � � and $ " $ � respectively. The otherparameter value used is � � � " ��� .

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