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Cognitive Systems Research 21 (2013) 7–21 www.elsevier.com/locate/cogsys

Cognitive stigmergy: A study of emergence in small-group social networks Action editor: Margery Doyle and Leslie Marsh Ted G. Lewis Center for Homeland Defense and Security, Naval Postgraduate School, Monterey, CA 93943, United States Available online 27 June 2012

Abstract This paper proposes a model and theory of leadership emergence whereby (1) small social groups are modeled as small world networks and a betweeness metric is shown to be a property of networks with strong leadership, and (2) a theory of group formation based on stigmergy explains how such networks evolve and form. Specifically, dominant actors are observed to emerge from simulations of artificial termites constructing a wood chip network in a random walk, suggesting a correlation between various preferential attachment rules and emergent network topologies. Three attachment rules are studied: maximizing node betweeness (intermediary power), maximizing node degree (node connectivity), and limiting radius (size of the network in terms of network distance). The simulation results suggest that a preference for maximizing betweeness produces networks with structure similar to the 62-node 9-11 terrorist network. Further simulations of emergent networks with small world properties (small radius) and high betweeness centrality (strong leader) are shown to match the topological structure of the 9-11 terrorist network, also. Interestingly, the same properties are not found in a small sampling of human made physical infrastructure networks such as power grids, transportation systems, water and pipeline networks, suggesting a difference between social network emergence and physical infrastructure emergence. Additionally, a contagion model is applied to random and structured networks to understand the dynamics of anti-leader sentiment (uprisings and counter-movements that challenge the status quo). For random networks, simulated pro-leader (pro-government) and anti-leader (pro-rebel) sentiments are propagated throughout a social network like opposing diseases to determine which sentiment eventually prevails. Simulations of the rise of rebel sentiment versus the ratio of rebel to government sentiment show that rebel sentiment rises on less than 100% rebel/government sentiment when government sentiment is high (strong leadership), but requires greater than 100% rebel/government sentiment when government sentiment is low (weak leadership). However, when applied to the structured 9-11 terrorist network, rebel sentiment is slow to rise against strong leadership, because of the high betweeness structure of the 9-11 network. These results suggest a theory of how and why human stigmergy evolves networks with strong leaders, and why successful social networks are resilient against anti-leader sentiment. The author concludes that a combination of small world and high betweeness structure explain how social networks emerge strong leadership structure and why the resulting networks are resilient against being overthrown by a dissenting majority. Published by Elsevier B.V. Keywords: Stigmergy; Self-organization; Emergence; Social network analysis; Betweeness; Small world

1. Introduction Stigmergy is a well-known behavior observed in simple animals such as wasps, termites, and certain species of ants. These organisms build relatively complex structures by following rudimentary rules. For example, ants build complex

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transportation networks connecting nests to food by secreting pheromones as they return to their nest. Even though the ant mechanism is simple, the resulting “ant highway” network is surprisingly efficient. In fact, artificial ant algorithms have been used to solve the Traveling Salesman Problem (Colorni, Dorigo, & Maniezzo, 1991). More surprising are the bridges, tunnels, and pillars constructed by termites. These structures are constructed by a series of stimulus–response stages whereby completion of

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one stage stimulates construction of the next stage. Significantly, the progression from one construction stage to another is stimulated by the work, and not the cooperation and collaboration of the termites. One thing leads to another without the benefit of a plan or supervision. The termite’s stimulus–response behavior is modified as the project progresses by the work itself. Without a finished blueprint in their heads, or a coordinated group effort, simple termites are able to construct complex structures. This is called stigmergy – a series of behaviors driven by repeated stimulus–response cycles. These simple animals instinctively construct a physical structure such as a pile of mud, the results of which stimulate a different response in other termites (from a pile to a pillar or arch, for example), which in turn leads to another stimulus, and so forth. Rather complex structures emerge in this way, through a series of behavior-changing stimulus–response pairs. In a sense, the state of the construction project is transmitted to other termites though the construction, itself. French biologist Pierre-Paul Grasse´ coined the term stigmergy in 1959 to mean, “Stimulation of workers by the performance they have achieved” (Bonabeau, 1999). He combined the Greek equivalent of “mark” and “action”, to form “stigmergy”, after observing termites alternating between marking and building increasingly complex structures. Stigmergy has been observed in ants, bees, fish, birds, artificial life boids, swarms, migrations, and robots equipped with flocking and foraging software (Theraulaz & Bonabeau, 1999). 1.1. Cognitive stigmergy Stigmergy is sometimes related to foraging and flocking behaviors observed in biological and artificial life forms, i.e. humans. For example, human stigmergy has been observed in a variety of activities, including traffic flow, elections, document editing of joint publications in Wikipedia.com, viral marketing, web site ranking by Google.com, collaborative filtering on Amazon.com, peer-to-peer sharing of music files, coordinated combat operations, and scheduling and planning (Parunak, 2005). Elliot speculates that human stigmergy, or group collaboration, is the fundamental principle underlying much of the social networking emerging in today’s “Web 2.0” culture (Elliott, 2006). Stigmergy explains how animals construct complex structures, but it does not explain why. In this article the author claims that key social networking behaviors in humans are in fact forms of stigmergy, and suggests that the basic organizing principle of social networks is a network science property akin to betweeness.1 Specifically, this paper attempts to explain why leadership, influence, and power emerge within social groups as a byproduct of “power grabbing” through control of the flow of informa-

1 Betweeness is defined as the number of paths through a node from all other nodes to all other nodes.

tion within a group. Furthermore, this paper suggests that “power grabbing” produces resilient networks that are capable of surviving disruptions in authority and control simply because of their topological structure – by maximizing a network science property related to betweeness.2 The paper begins with a review of a few useful metrics used in network science. Then the 9/11 terrorist social network, containing the 9/11 hijackers, is analyzed to determine its topological properties. Most notably, the leader of the 9/11 hijackers (Atta) is shown to hold the number one position with respect to his connectivity (degree) and betweeness (control of the flow of information) properties. In addition, the betweeness rankings of the six most influential members of the 9/11 terrorist network are shown to dramatically decline according to a power law – a sign of hierarchical structure within the network. The author claims that the power law signature relates to social power and explains why betweeness is an appropriate property to use to study this form of human cognitive stigmergy. Why does betweeness hierarchical structure lead to strong social networks? The author challenges the notion that social network resilience – as measured by leadership strength in the face of anti-leader sentiment – has real-world meaning. The author challenges his own hypothesis by simulating the disruption of small group social networks through the spread of anti-leadership sentiment. If a network has poor leadership or lacks cohesion, it should be easy to wrest control from its leaders. Specifically, the spread of anti-leader sentiment – like a contagion – can lead to disruption of leadership and thus control of a social network. Strength of leadership, when challenged by a contagious idea, can be used as a measure of resiliency. The more resilient a network, the more difficult it is for anti-leader sentiment to achieve a majority. On the contrary, resistance to a challenging idea or theme shows support for the status quo. The author claims that hierarchical betweeness increases the resilience of social networks, because it establishes a hierarchy that makes it difficult for anti-leadership sentiment to spread and achieve a majority position within the network. First, a social network is defined as a collection of nodes (actors), links (relationships), and a rule for connecting nodes and links together (topology). Then we simulate the formation of social networks using simulated termites that performs network “wiring”, or linking of nodes, by visiting each node once. This simulation produces various topologies depending on the self-organizing rule applied by a few randomly wandering termites. High betweeness is observed to be the self-organizing rule that most-closely matches the social structure of the 9/11 terrorist network. Thus, maximizing betweeness is chosen as the fundamental organizing principle for further study. A betweeness maximizing “follow-the-leader” organizing rule is proposed as an even better approximation to the topology emergent

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In graph theory, betweeness is usually defined as the index of a node.

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in real-world networks (9/11 terrorist). Finally, the hypothesis that betweeness emergence in small group structure adequately explains cognitive stigmergy (as defined here) is further challenged by two additional experiments: (1) comparison with physical structures created by cooperating humans, and (2) simulation of the spread of anti-leader sentiment in social networks and its impact on network resilience. We find that cognitive stigmergy does not produce physical structures like those found in the real world, but that the proposed “follow-the-leader” organizing rule simulated here supports the claim that “follow-the-leader” self-organization produces resilient networks. While the results of these simulations are non-conclusive, they do suggest a direction for further research. 1.2. Simulation approach The proposed theory is developed by performing a series of simulations – first on a handful of artificial termites, and then on random social networks. The goal is to observe which factors affect social network structure the most and then formulate a theory of cognitive stigmergy based on these observations. A handful of artificial termites are assigned the task of constructing a network by connecting pairs of simulated wood chips (nodes) together (links) as the termites randomly traverse a 2D space littered with wood chips. Each time a termite encounters a wood chip, it drops the one it is carrying and picks up the new chip if a certain stigmergic rule is satisfied. The previous and next wood chips are linked together. After hundreds of thousands of iterations, the network containing chips as nodes, linked together according to a stigmergic rule, exhibits a non-random structure – the topology of the social network. The question we address with this simulation is fundamental: what stigmergic rule produces network structure similar to the 9/11 terrorist network? Three self-organization rules (algorithms) are tested to determine which rules produce a network with topological structure similar to the 9/11 terrorist social network. One rule increases betweeness; another increases node degree; and the third rule limits the overall network radius (small world property). The betweeness rule was chosen because it is a measure of information flow through a network: higher betweeness equates with more flow. The degree rule approximates “popularity” – the tendency of highly connected nodes to attract additional links. The radius-limiting rule produces a small world, because no node is very far from any other node. 1.3. Results We show that an algorithm that combines directed and undirected betweeness, called the “follow-the-leader” rule, produces networks that best match the structure of the 9/11 terrorist network. Directed betweeness pays attention to the direction of social links between actor (node) pairs, and establishes a hierarchical relationship among actors.

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Undirected betweeness does not establish a hierarchy, but measures the influence an actor has on other actors. Additionally, we test this rule against physical network systems produced by humans and find no similarity. Thus, networks formed by the follow-the-leader cognitive stigmergy exhibit structure similar to the 9/11 terrorist social network topology, but the network structure of human-made networks do not. Furthermore, follow-the-leader stigmergy appears to provide greater resiliency against attacks on the integrity and leadership of networks studied here. Resiliency may be a reason why such networks survive and persist in the real world. The author conjectures that directed betweeness self-organization tends to build hierarchical relationships embedded within a social network while undirected betweeness tends to decrease the radius (size) of the network. The author’s follow-the-leader algorithm does both – increases directed and undirected betweeness. Furthermore, the emergence of betweeness structure from random networks shows remarkable agreement with the betweeness structure of the 9/11 terrorist’s social network. This suggests a theory of how, and why, small group social networks tend to be small world with high-betweeness: dominant actors attempt to reduce the distance between actor-pairs and build hierarchies at the same time – because an actor that can control information flow (betweeness) and also reduce the diameter of the network (small world effect) gains power over other actors. The simulations are not a complete explanation of why high betweeness leads to social structure within human groups, but it does suggest a theory that needs additional testing. According to the author’s theory, a combination of hierarchical and strong intermediary power guides the evolution of small group network topology. These social networks tend to survive better than less structured networks. Performing additional simulations, described in detail, below, tests this assertion. A further claim is that the self-organizing rule described here is a kind of cognitive stigmergy may be questioned. Why is “follow-the-leader” self-organization a form of stigmergy? Follow-the-leader progresses in stages as described below in terms of an algorithm. Each stage stimulates additional work, which forms a tight feedback loop: Cognitive Stigmergy Algorithm: 1. Initially create a random network. 2. Repeat until there are no more changes: 3. For each (independent) FROM–LINK–TO: 2.1. Randomly select a set: FROM–LINK–TO. 2.2. Select another actor at random, RANDOM. 2.3. Rewire the LINK only if self-organization is improved: FROM–LINK–RANDOM.

An increase in the self-organization property (degree, betweeness, etc.) stimulates further evolution toward a

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more stable structure. Actors are not coordinated, nor do they have a plan, but instead, they each act independently and are stimulated to continue to act as long as links may be switched. A new network topology emerges that is stronger and more resilient as defined by the organizing rule. 2. Structure in social networks If cognitive stigmergy explains how social network structure emerges, then betweeness explains why. The author hypothesizes that human collaboration is driven by betweeness maximization. The most powerful actor in the social network is the one with the largest betweeness value – in terms of both directed and undirected links. Much has been written about social network analysis, SNA. A comprehensive survey of SNA can be found in Wasserman and Faust (1994). SNA uses network theory to explain human behavior. A social network is represented as a graph containing nodes (actors) representing individuals or organizations, links between pairs of nodes representing some kind of relationship between node pairs, and a mapping function that associates links with node pairs. Think of the mapping function as a “wiring diagram” or topological map of the network. 2.1. Social network formulation More formally, a social network G is a 3-tuple, G = {N, L, f}, where N is a set of n nodes – also called actors in a social network; L is a set of m links, and f is a mapping function f: N  N, defining the network’s topology or “wiring diagram”. If node x € N is connected directly to node y € N, then x is adjacent to y and y is adjacent to x. Nodes x and y are neighbors. If x is connected to y, but not the reverse, the network is directed (links have a direction). Define the degree of a node in x € N as the number of links connecting the node to other nodes. Let the degree sequence of a network G be d = {d1, d2, . . . , dn} where di is the degree of node i. Finally, let the degree sequence distribution of network G be defined as g = {g1, g2, . . . , gk}, where gi is the fraction of nodes in G with degree equal to i. Each gi is obtained by counting the number of nodes with degree i, and then dividing by n. 2.2. Types of networks Networks are generally classified according to their degree sequence distribution. If g obeys a Poisson distribution, we say the network is random. If it obeys a power law, we say it is scale-free. If a network has high cluster coefficient, we consider it a clustered network (Holme & Kim, 2002). In this paper we deal with random and scale-free networks, only. The length of a path from node x to node y is measured in hops – the number of links traversed along a path from node x to node y. Since it is possible for more than one

path to exist between any node-pair x and y, we are concerned only with the shortest path unless otherwise specified. The shortest path between nodes x and y is also known as a direct path. A strongly connected network is one in which a path exists between all pairs of nodes. That is, all nodes are reachable from all other nodes. Let ri be the radius of node i, defined as the length of the longest direct path from node i to all other reachable nodes in the network. Radius is calculated by tracing all shortest paths from node x to all other reachable nodes, and then selecting the longest path. If the largest ri over all n nodes is small, we classify the network as small world. Smallworld networks have relatively small radii, meaning that all nodes are relatively near to one another. If a random network is rewired in certain ways, it may be transformed into another random network, or a network with scale-free, clustered, or small-world structure. In fact, clustered and scale-free networks are common forms of structured networks derived from random networks. How structured networks are formed from random networks is the subject of numerous articles and books on network science (Adamic, 1999; Adamic & Huberman, 2000; Adamic, Lukose, Puniyani, & Huberman, 2001; Albert, 2005; Albert & Barabasi, 2002; Albert, Jeong, & Barabasi, 1999; Barabasi, 2002, 2003; Holme & Kim, 2002; Lewis, 2009; Newman, Watts, & Strogatz, 2002). Structured scale-free, clustered, and small-world networks can be artificially created through a variety of algorithms as described in Lewis (2009). These algorithms produce networks with a fixed topology. They are static. But in this research, the author is interested in network topologies that change topology g as a byproduct of stigmergy. These are dynamic networks, because topology changes over time. The transformation of a network over time is called emergence. The question is, “What dynamic processes cause topological structures to emerge over time, and what is the relationship between emergence and social network structure?” 2.3. Actor power Social network analysts often use betweeness as a measure of the influence one actor has on all others. Betweeness is calculated by counting the number of paths passing through an actor from all actors to all other actors. That is, the betweeness of actor x is equal to the number of (shortest) paths from actor u – x to actor v – x, for all u, v actor-pairs in N. If links are directed, the betweeness measure is also directed, meaning that paths flow in one direction, only. Thus, directed and undirected betweeness are two measures of flow attributable to network structure. The actor with the highest betweeness is a powerful broker, because it controls more of the flow of information between all other actor pairs than any other actor. In this paper, normalized betweeness is obtained by dividing path counts by the largest count associated with the highest

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Fig. 1a. The 9/11 terrorist network containing the terrorists who planned and carried out the multiple attacks on the United States on September 11, 2001, laid out according to betweeness.

“betweener” actor. Normalized betweeness is a fraction in (0, 1), and equated with “relative power” in this paper. 2.4. The 9/11 terrorist network The idea of describing terrorist groups as social networks isn’t new. Arquilla et al. used social network theory to express and analyze terrorist groups even before the horrific attacks of 9/11 (Arquilla, 2007; Arquilla & Ronfeldt, 2001). They asked, how does the structure of a terrorist network affect its capabilities? Jackson quantified and classified various network topologies and suggested that ‘function follows form’, asserting that the capabilities of a terrorist network relate to the topological structure of the network. But neither of these authors explained how terrorist networks (or groups in general) create social networks with discernable leaders – dominant actors that exercise great influence over the group. As a cogent example of ‘terrorists as networks’, consider the 9/11 terrorist group whose members attacked the United States in 2001 (Krebs, 2002). This social network contains 62 nodes (individual actors) and 150 links (relationships) as shown in Fig. 1. It contained several of the 9/11 hijackers as well as actors involved in a number of other attacks. The leaders of this group are easily identified by simply ranking actors according to their betweeness values. For example, Fig. 1a is arranged in a concentric circle so that the highest betweeness actors are drawn in the center, while lower betweeness-valued actors radiate outward. The top 5 betweeners and their normalized betweeness values are Atta (100%), Moussaoui (42%), B. Khemais (32%), N. al-Hazmi (28%), and Hanjour (18%), where  indicates participation in the 9/11 attacks. Mohammad Atta is widely recognized as the leader of the hijackers. Moussaoui was arrested prior to the 9/11 attacks, and is often considered to be the 20th hijacker. Ben Khemais was not part of the 9/11 attack,

but was subsequently arrested and convicted for plotting to bomb the US Embassy in Rome. Al-Hazmi and Hani Hanjour died on Flight 77 as it crashed into the Pentagon. Simple network analysis confirms the prominent role of these actors in the 9/11 attacks. Fig. 1b is arranged concentrically with the smallest radius-valued nodes located in the center of the layout, with larger-radius nodes radiating outward from the center. A low radius value indicates closeness to all other nodes in the network. Atta, Moussaoui, Yarkas, and alShibh are no more than 3 hops from all other nodes in the network. N. al-Hazmi, al-Shehhi, and Hanjour are 4 hops. Some analysts believe al-Shehhi was the next in command after Atta. It is noteworthy that the 9/11 terrorist network was a small world, because all actors are within 4 hops of one another. Fig. 1c shows the network arranged according to degree3 – the higher the degree of an actor, the nearer it is to the center. The top 5 terrorists in terms of degree are Atta (22), al-Shehhi (18), Hanjour (13), al-Hazmi (11), and Khemais (11). Perhaps the fact that al-Shehhi is second in degree only to Atta is why he is presumed to be second in command. This presumption begs a key question addressed by this paper, “what determines leadership in a social network?” Fig. 1d shows the degree sequence distribution. The network is far from random, because it does not follow a Poisson distribution. But it is not entirely scale-free, because it does not strictly follow a power law, either. However, it is a long-tailed histogram similar to a power law, with one hub with 22 links (Atta). From this histogram we can conclude that the 9/11 terrorist network has structure lying somewhere between random and scale-free topologies. 3 Degree is defined as the number of links connecting an actor to other actors.

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Fig. 1b. The 9/11 terrorist network containing the terrorists who planned and carried out the multiple attacks on the United States on September 11, 2001, laid out according to radius.

Fig. 1c. The 9/11 terrorist network containing the terrorists who planned and carried out the multiple attacks on the United States on September 11, 2001, laid out according to degree.

Fig. 1d. The degree sequence distribution of the 9/11 terrorist network. A log–log plot is shown in the upper left-hand portion of the graph, along with a straight-line fit to the data. A perfect fit to the log–log plot suggests a scale-free network structure is present.

From Fig. 1 we might conclude that Atta is the lead terrorist because he ranks first on all three lists. He ranked first with respect to betweeness, radius, and degree. Moussaoui (degree = 8) places second in two lists, but does not appear at all in the top 5 of the degree list. The third place position is held by different terrorists in each ranking. Yarkas only appears in one list – he ties with three other terrorists for being closest to all others. What forces contributed to the 9/11 terrorist’s network structure and can we say anything meaningful about social structure from this network? According to Jackson (2006), the strength of terrorist networks is determined by the strength of the ties or links binding the social network together. He places such social networks into three broad categories: tightly coupled,

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coupled, and loosely coupled. But he does not relate the ‘strength’ of terrorist networks to the topological structure of the network, itself. The author argues that what Jackson calls “control and influence” relates to the structure of the network more than the strength of its links (this assertion does not minimize the importance of link strength, but rather enhances Jackson’s theory). The following sections attempt to further correlate the ‘structural strength’ of a social network with its leadership and resilience. 3. Social stigmergy Analysis of the 9/11 terrorist network raises a number of questions about social organization. For example, does ranking according to betweeness, radius, or degree, as shown in Fig. 1a, have any meaning at all? Second, does the structure of a network, according to betweeness, radius, or degree, emerge from some hidden order? That is, can the social structure of networks be explained by self-organizing behavior akin to stigmergy? The following experiments were performed to find out. 3.1. Artificial termites Consider the following artificial termite experiment based on Mitchel Resnick’s work and inspired by Craig Reynold’s flocking algorithm (Resnick, 1997; Reynolds, 1987). In Resnick’s simulation, artificial wood chips are randomly spread over a 2-dimensional space. Artificial termites randomly walk through the space picking up adjacent chips whenever they are empty-handed. If a wandering termite already has a chip in tow, it drops the one it is holding and wanders off, looking for another chip to collect. As shown in Fig. 2, termites tend to herd the chips into a pile. Resnick’s simulation was modified to create a network by adding links between dropped and picked up chips. In addition, links are removed if they become too long, roughly approximating the evaporation of a pheromone link. The shape or topological structure of the termite-constructed “social network” depends on the stigmergic action performed each time a link is inserted between chip-pairs. Links are inserted between chips x and y according to one of three simple self-organization rules, as follows: Rule 1: Maximum Betweeness. Chips x and y are connected via a link if the betweeness of picked-up chip y is greater than or equal to the betweeness of dropped chip x. Links prefer to attach to higher intermediary chips. This rule simulates a preference for “power” as defined by the number of paths controlled by the chip node. “Power-seeking” termites tend to build hierarchies – high betweeness nodes increase their “betweeness power” as less powerful nodes connect to them. Rule 2: Maximum Degree. Chips x and y are connected via a link if the degree of picked-up chip y is greater than

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or equal to the degree of dropped chip x. Links prefer to attach to highly linked chips. This rule simulates a preference for “popularity” as defined by the degree of wood chips. “Popularity-seeking” termites tend to build highdegreed hubs as lower-degreed nodes give up links and higher-degreed nodes gain links. Rule 3: Limited Radius. Chips x and y are connected if the largest radius over all nodes in the network is bounded by some maximum value. A link is not allowed if it would increase the largest radius beyond a set maximum value – 5 in the experiments performed here. This rule tends to form small world networks because no chip node is more than a maximum number of hops from all other nodes (Adamic, 1999; Mathias, 2001). This rule is intended to test the hypothesis that small world emergence leads to discernable leadership structure in a social network. Rule 1 suggests that networks form because termites gravitate toward highly influential actors more than lessinfluential actors, because betweeness is interpreted as a measure of intermediary or broker power. In a social network, the actor with the highest betweeness value controls the flow of information passing through it on its way to other actors. Intermediaries exercise power through control of the message. Rule 2 suggests that networks form because termites build structures that gravitate toward highly connected actors more than less-connected actors. Actor degree is interpreted as a measure of popularity, and the desire to be connected to the most popular actor. This is the wellknown self-organization rule for creating scale-free networks. Group power is equated with degree, because the highest-degreed actor, the hub, is connected to more actors than any other actor. Rule 3 suggests that networks form because termites build networks that gravitate toward small worlds. That is, leaders try to position themselves in the center of the network as defined by radius. The smaller the maximum radius, the smaller the world, and the more influence a leader has over all others. Small world structure will play a major role in the simulations to follow, but other rules may also lead to small world structure. In a way, this rule produces an extreme small world network to be compared with other structured networks. 3.2. Emergence of structure Fig. 2 illustrates the emergence of social structure in a very simple society, and may tell us something about the emergence of power in social networks. All three of the simulations produce networks with discernable structure after 200,000 iterations, as indicated by the degree sequence distribution shown in the lower left-hand corner of each plot. (A random network would produce a symmet-

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Fig. 2. Ten artificial termites collecting wood chips in a field of 62 randomly distributed wood chips. Termites are shown with X-shaped “legs”, and wood chips are shown as solid ovals. Links are shown as straight lines. (a) Initial configuration showing a few chips that have been moved, dropped, and linked to their predecessor. (b) Emergence of wood chip network using betweeness-maximizing self-organization, after 200,000 iterations. The degree sequence distribution is shown in the lower left-hand corner, suggesting a scale-free-like structure. (c) Emergence of wood chip network using degree-maximizing self-organization, after 200,000 iterations. The degree sequence distribution shown in the lower left-hand corner suggests a long-tail scale-free structure. (d) Emergence of wood chip network using radius-minimizing self-organization, after 200,000 iterations. The largest allowed radius is 5 in this simulation. The degree sequence distribution shown in the lower left-hand corner shows little structure.

ric Poisson distribution.) Structure increases as the degree sequence distribution becomes longer-tailed as shown in Fig. 2c. The author claims that power in human social networks derives from control of information flow (betweeness), connectivity (degree), and network “size” as measured by the maximum radius of the network. This hypothesis is tested by observing network topologies that emerge from the termite experiments. Topology (structure) is determined by fitting the distribution of interest to a power law, H(d)  1/dq, with exponent q. Thus, the value of q can be used as a measure of structure – large values of q imply more structure. (Randomness or Poisson distribution are indications of non-structure.) Using the exponent as a measure of “structuredness”, it is determined that rules 1 and 2 produce networks with the most structure. That is, betweeness and degree maximizing stigmergy tend to produce topologies analogous to human social networks (to be shown later). But, Fig. 2b and d

shows similar structures, so additional simulations are needed to drill deeper into the structure produced by degree maximization versus radius limiting. The degree sequence distribution of Fig. 2b is similar to the distribution for Fig. 2d, but rule 1 and 3 use different metrics. Instead of fitting the degree distribution to a power law, Fig. 2b should fit a betweeness distribution to a power law. This was done, and the results are shown in Table 1, which lists the results of averaging the rankings of the top 6 nodes emerged from termite experiments over five samples, for each rule described above. A power law is fit to the leading 6 nodes, yielding the power law exponent, q, for comparison. Table 1a shows little difference among the three structuring algorithms. But, Table 1b shows a major difference between Rule 1: Maximum Betweeness and all other rules. Maximum betweeness affects network structure more than the other self-organization rules examined here. Therefore, this metric is explored in greater detail in the remainder of this paper.

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Is betweeness an adequate measure of social network structure? Is stigmergy in artificial termites relevant to emergence of social network structure in human organizations? To verify that betweeness emergence is correlated with structure in social networks, a series of simulations were performed as described below, with emphasis on the emergence of social network “leaders”.

Directed betweeness counts the number of directed links from actors to all other actors along directed paths. Assuming that links point in the direction of actors with higher betweeness, directed betweeness is a measure of how high up in the social hierarchy an actor is. Higher directed betweeness indicates more power. (Note: undirected betweeness ignores the direction of links and so counts all paths through an actor. Thus, directed betweeness establishes a hierarchy.) But, undirected paths through an actor, indicate how influential an actor is within the entire network, because undirected links are also bi-directional. Therefore, a follow-the-leader algorithm uses both directed and undirected betweeness to evolve power relationships within the network. Actors in the social network attempt to increase both bi-directional (undirected) and directional betweeness, because bi-directional betweeness determines overall influence, and directional betweeness determines an actor’s rank within the social hierarchy. Actor nodes are assumed to try to increase their power by alternating between applying undirected betweeness maximization (for many iterations) and directed betweeness (for many iterations). In this way actors try to increase both their influence over the entire network and consolidate their power over the actors beneath them in the hierarchy.

4.1. Stigmergic emergence

4.2. Follow-the-leader emergence

Suppose two very simple rules of stigmergy are applied to a directed social network, as follows. Each person or actor is represented by a node, and directed links represent the influence one person has on another. This social network is directed because a link points to an actor that it influences. Actors labeled ‘from’ influence ‘to’ actors, but not the reverse. Now suppose a very simple rule is repeatedly applied to an initially random social network as follows: A link and an actor are randomly selected. The link is switched to the randomly chosen actor if it has a higher value than the actor currently pointed to, see Cognitive Stigmergy Algorithm, above. In other words, a random link is switched to a random actor if the value of the random actor is greater than the value of the “to” actor. This process is repeated forever (remember stimulus–response?). Eventually, a new network structure emerges. That is, the network self-organizes into a topology that increases each actor’s influence over other actors, according to one of the rules discussed here. Further suppose we define power in a social network as a value proportional to how much information flow an actor controls. In other words, power is proportional to betweeness. The actor with the highest number of paths passing through it is the most powerful actor, because it exerts the most influence over its neighbors (adjacent actors). If a betweener actor cancels or enhances messages passing through it, it also negates or enhances the possible behaviors of the group.

The follow-the-leader algorithm combines directed and undirected betweeness emergence. First, undirected betweeness is applied N times to randomly selected links and actors, followed by applying directed betweeness N times. This pattern is repeated forever, or until there is very little change in the topology of the network. Undirected betweeness emergence is thought to reduce the size of the network, where size is defined by the network’s mean radius. Directed betweeness is thought to increase the dominant actor’s power within the network, by increasing the number of betweeness paths the dominant actor controls. Here is a pseudo-code version of the algorithm.

Table 1 Power law exponent for top 6 nodes of networks created by artificial termite stigmergy. Stigmergy rule

Degree sequence exponent

R-square

(a) Degree sequence distribution power law exponent and R-squared value Increase degree 1.39 0.712 Increase betweeness 1.41 0.785 Limit radius to 5 1.48 0.820 Random 0.952 0.658 Betweeness ranked exponent

R-square

(b) Betweeness ranking power law exponent and R-squared value Increase degree 2.59 0.874 Increase betweeness 5.67 0.976 Limit radius to 5 2.31 0.839 Random 3.07 0.985

4. Emergence of leaders from random networks

Follow-the-leader Algorithm: Repeat forever(N): 1. Repeat N times with Direction = undirected: REWIRE (Direction). 2. Repeat N times with Direction = directed: REWIRE (Direction). REWIRE(Direction): 1. Randomly select a link, random_link that points to head_node. 2. Calculate betweeness(Direction) for all nodes 3. Randomly select a node, random_node 4. Rewire link to point to the random_node

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5. Re-calculate betweeness(Direction) for all nodes 6. If random_node.betweeness < head_node.betweeness revert: switch link back to head_node. Otherwise, accept the rewiring. Note that the follow-the-leader algorithm transforms networks into small worlds – networks with smaller radii. Thus, the follow-the-leader algorithm decreases a network’s radius, resulting in fewer hops from the leader actor to all other actors. This property may explain why betweeness self-organization produces more resilient networks, a topic we take up next. (Alternatively, we might claim that resiliency is the cause, and small world properties are the result.) The follow-the-leader algorithm was applied to random networks with 62 nodes and 150 links. N ranged from 50 to 200 and instead of running the simulations forever, rewiring ceased after approximately 1000 (N = 50) to 5000 (N = 200) iterations. The normalized betweeness values of the top 6 actors were averaged over 5 samples to obtain the data shown in Table 2 and plotted in Fig. 4. The ranked betweeness values were fit to a power law as shown in Fig. 4a. Table 2 Ranking by betweeness from emerged follow-theleader networks containing 62 nodes and 150 links. These numbers were obtained by averaging over five samples each. Rank

Betweeness (%)

1 2 3 4 5 6

100 56 48 46 44 43

The number of paths running through the highest-ranking node increases as emergence shapes the network. This is not surprising, because the algorithm aims to increase betweeness. In one test, the highest-ranking nodes that produced Table 2 increased their betweeness from an average of 514–827 undirected paths. This is an expected result, as the dominant actor consolidates power by restricting flow of information to other actors. Further evidence of power consolidation is evident in the sharp drop in betweeness rankings of the top 5 nodes. This is shown in Table 2 and Fig. 4a as a precipitous drop in normalized betweeness values. That is, the dominant nodes are the bottlenecks of the network. They control the paths passing throughout the entire network via a funneling effect. 4.3. Shape of the 9/11 terrorist network Now suppose the simulation results are compared with the betweeness topology of the 9/11 terrorist network. It is clear from Fig. 3 that the so-called ringleaders of the 9/11 terrorist network were al-Shehhi, al-Shibh, Atta, Hanjour, Moussaoui, and S. Al-Hazmi, in that order. This can be easily deduced by visual inspection, because the network is laid out so that higher betweeness nodes appear in the center of the network. These actors are at the center of the social network in terms of betweeness and also in terms of their role in the 9/11 attacks. 5. Rank and leadership Does betweeness predict leadership in a social network? There is no other empirical study to support this conjecture, but the results shown in Fig. 4a obtained from applying the follow-the-leader algorithm to a random network of equal size strongly support this claim. The betweeness rank

Fig. 3. The 9/11 terrorist social network containing 62 actors (nodes) and 150 links (relationships), is shown here with the directed betweeness value of the top six terrorists and their names (Krebs, 2002).

T.G. Lewis / Cognitive Systems Research 21 (2013) 7–21

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Fig. 4. Comparisons of betweeness ranking of follow-the-leader emerged networks and human-made infrastructure networks with 9/11 Terrorist network. (a) Comparison of betweeness ranking of three different emergence algorithms versus the 9/11 terrorist network shows follow-the-leader emergence is a better fit to the 9/11 terrorist network. Correlation coefficient is 99.2%. (b) Log(Betweeness) versus log(Rank) power law comparison of random networks, follow-the-leader emerged networks, and 9/11 terrorist network. Follow-the-leader emergence matches the 9/11 terrorist network structure with power law exponents of .38 (emerged, R-squared = .98) and .40 (9/11 terrorist, R-squared = .93), and coefficient of correlation of 0.98. (c) Comparison of undirected between ranking of human-made infrastructure networks versus the 9/11 terrorist network shows the 9/11 terrorist network to be more structured with respect to undirected betweeness ranking than infrastructure networks.

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order of nodes roughly matches the 9/11 terrorist network rank order. The coefficient of correlation between emerged random network and the terrorist network ranking is high (99.2%), although the fit is far from perfect. Specifically, the best-fit power law exponents are 0.46 (emerged random) and 0.40 (9/11 network), respectively with R-squared values of .90 and .93, respectively. Follow-the-leader betweeness self-organizes random networks into structured networks similar to the real 9/11 terrorist’s social network. But the self-organized network is not quite as structured as the actual 9/11 network, perhaps because the emergence takes much longer than 200 iterations, each. 5.1. Does degree determine power? As a sanity check, the author also simulated the formation of self-organized networks using the maximum degree algorithm. Of course this algorithm creates a hub, and other highly connected nodes – much like the 9/11 terrorist network. (Atta is the hub of the 9/11 terrorist network.) But maximum degree emerged networks do not fit the real network as well as follow-the-leader emerged networks. The data plotted in Fig. 4a was obtained by averaging over five networks, in each case. Additional simulations produced the comparisons shown in Fig. 4b. Random networks were compared with follow-the-leader emerged networks versus undirected and directed betweeness measures. Fig. 4b shows that undirected betweeness is a better fit to the 9/11 terrorist network than directed betweeness. Furthermore, when the ratio of emerged versus random network metrics were compared, follow-the-leader emergence shrunk the radius of the initial random network to 90% of its original size; doubled the number of maximum undirected paths passing through the betweeness-dominant node; and reduced the average undirected betweeness to 60% of the initial random network. The follow-the-leader algorithm shrinks the network and increases the dominant actor’s betweeness power. While not perfect, these results suggest a correlation between follow-the-leader emergence and real small-group network structure. 5.2. Comparison with physical infrastructure The author was curious whether the 9/11 social network topology matched other human-made networks such as telecommunications, gas and oil pipelines, power grids, transportation networks, and drinking water systems. Physical structures may not relate to unconscious behavior in social networks, but it does relate more closely with animal stigmergy. Parunak (2005) lists a number of human–human stigmergy constructions that mimic termite and ant behavior. Adamatzky et al. showed that the Mexican federal highway system connecting major cities in Mexico is easily approximated by slime mold (Adamatzky, Martınez, Chapa-Vergara, Asomoza-Palacio, & Stephens, 2010). Could it be that human-made physical infrastructure

produces topologies similar to the follow-the-leader emerged structure? This conjecture proved to be false for the physical structures examined here. Fig. 4c shows the results of comparing the 9/11 terrorist network undirected betweeness structure with the undirected betweeness structure of a variety of human-made infrastructure networks. The sample infrastructures analyzed here are similar in topology to the artificially generated networks of Fig. 4a, but much less structured than the 9/11 terrorist social network. Why the difference? Physical infrastructure is governed in part by the landscape, regulations, and economics, while social network structure is apparently governed (in part) by betweeness power – a property immune to physical constraints. 6. Resiliency High betweeness and small world topology brought on by increasing betweeness argues for resiliency in social networks, where resiliency is defined as the ability of the status quo structure to prevent a take-over by a rogue actor or actors. In fact, it may be argued that the follow-the-leader self-organization rule produces real networks because they survive in the face of challenges. In other words, they exist in nature because they are the survivors of Darwinian processes. If the policies of leadership are carried out without question, even in the face of opposition, the network is considered highly resilient. If, however, an opposition sentiment is able to gain a majority foothold after some time, the network is considered fragile. Majority sentiment is defined here as a position held by 50% or more of the actors in the network. An example of this occurred in February 2011 when anti-government sentiment in Egypt swept away the ruling party. Is it possible to quantify this resilience and determine if structure plays a role in resilience? 6.1. A model of sentiment To test the hypothesis that the 9/11 terrorist network, and networks with similar topology are extremely resilient against dissent, the following simulation was performed on artificially generated networks and the 9/11 terrorist network. Let sentiment be defined as a number in [0, 1], where zero represents no sentiment and one represents complete agreement with either a pro- or anti-leadership position. The status quo or governing body of a social network is assigned sentiment GOV, say 0.50, and the opposition or rebel body is assigned sentiment REBEL, say of 0.20. These numbers are treated as probabilities in the following analysis. A higher GOV sentiment means it is more likely that an actor will side with the GOV sentiment than REBEL sentiment. Sentiment is assumed to have a viral effect – it spreads like a contagion from actor to actor through links in the social network. In this model, sentiment acts like an infectious disease. If node x is connected to node y and x sides

T.G. Lewis / Cognitive Systems Research 21 (2013) 7–21

with the rebels, then y is persuaded to also side with the rebels with probability REBEL, unless it is already decided. Similarly, if x has already sided with the government, then it persuades y with probability GOV. Sentiment spreads throughout the network by infecting undecided actors. Both government and rebel sentiments cease spreading when all actors have been contaminated. Initially, all nodes are undecided, except for one randomly selected rebel and the network leader (the leader is the actor with the highest betweeness value). Thus, a rebel contagion is started at some random node while a government contagion is started at the leader node. At each step in the simulation, neighboring nodes are infected with one sentiment or the other, according to corresponding probabilities. The government sentiment spreads from the leader with probability GOV and the rebel sentiment spreads from the random actor with probability REBEL. Once infected (decided), nodes remain infected (decided) – they do not change their position. The question is, “what happens to the social network as these two competing sentiments spread throughout the network?” Of particular interest is the survival of the network in the face of a dissenting rebel opinion. Suppose it takes a majority to overthrow the current leader. That is, the rebels overthrow current leadership of the social network if rebel sentiment infects 50% or more of the nodes. Otherwise the status quo leadership survives the threat. Fig. 5 shows the results of many simulations on random networks with 62 nodes and 150 links, as well as the 9/11 terrorist network described earlier. In all cases the probability of a dissenting majority (rebel sentiment) overthrowing the leadership (government sentiment) grows logistically as the sentiment ratio REBEL/GOV rises. The probability of rebel sentiment growing to 50% or more of the actors is shown in Fig. 5 for two random networks: one with government sentiment set at 10% (weak government) and the other with government sentiment set at 50% (strong government). Clearly, the likelihood of overthrow is greater when the ratio of rebel-to-government sentiment is lower than 100% for a strong government and much lower for a weak government. This counter-intuitive result is partially due to the high ini-

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tial sentiment of the government (50%): rebel sentiment must be correspondingly high to achieve a ratio of 70%, say. But the counter-intuitive result also says something about the contagiousness of sentiment in small world networks – small-diameter means faster spread of the contagion, which in turn increases the likelihood that an upstart rebel contagion spreads to the far reaches of the network – reaching undecided actors before the pro-leadership contagion reaches them. 6.2. Resilience of the 9/11 terrorist network Results of applying the contagion model to the 9/11 terrorist network show extremely resilient behavior assuming pro-leadership sentiment is high (50%). Fig. 5 suggests that it would be very difficult to overthrown the 9/11 terrorist leadership even when rebel sentiment is twice that of the status quo. The 9/11 network is much more resilient than the corresponding random networks. The reason is simple: the 9/11 terrorist network is much more structured than its random network equivalent. In particular, the leadership actors of the 9/11 network are “betweeness bottlenecks” through which most “contagion vectors” must pass to reach other actors. The leadership actors block access to other actors because of their high betweeness value. The 9/11 terrorist network is both highly structured and resilient. It is the betweeness structure than makes it resilient. Note: in these simulations, weak government is defined as one with a low sentiment value (10%), and strong government is defined as one with a high sentiment value (50%). The simulations showed that a strong government can be overcome by relatively smaller rebel sentiment. In fact, the probability of overthrowing the governing leaders exceeds 50% even when rebel sentiment is less than 50% that of the government. Strong governments are relatively easier to overthrow, based on random social network topology, but the definition of strong is arbitrary, here. Second, note that the 9/11 terrorist network is far from random. It has pronounced structure as defined by

Fig. 5. Probability of majority dissent within a social network rises logistically with ratio of sentiment.

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the betweeness property described earlier. It is highly unlikely to be overthrown by a dissenting majority even when rebel sentiment is twice that of government sentiment. This is due to the topology of the 9/11 network. Because of its span of control (high betweeness of its leaders) and the small world property (leaders are fewer hops away from all actors), the 9/11 terrorist network is able to fend off challenges from (random) actors. It is more resilient. 7. Conclusion The follow-the-leader algorithm studied here is a rather surprising predictor of social network structure. At least it did a good job of modeling the 9/11 terrorist social network. This suggests that leaders consolidate power through control of messaging: they attempt to maximize betweeness by building hierarchies and small world structures. (Small world structure is the result of undirected betweeness maximization). Additionally the simple contagion model proposed here may explain why high betweeness-structured social networks enhance resilience. High betweeness nodes create choke points that limit the spread of anti-leadership sentiment. This is a consequence of both hierarchical ordering and small world self-organization. This study is not conclusive enough to claim a general result. Rather, the author proposes the mixture of betweeness and small world emergence as a theory for why social networks such as the 9/11 terrorists emerge as they do. The theory passes a “reasonableness test”, but more simulations and testing against real-world networks is needed. Furthermore, the lack of similarity with human-made infrastructures such as power grids, pipelines, and transportation systems suggest that something different operates in cognitive stigmergy than physical stigmergy. It is reasonable to assume that social networks are formed on the basis of betweeness maximization, because betweeness measures “intermediary power” – the power of a broker. Actors derive their power through the information flow they broker. It is also reasonable to assume that a strong leader desires to exercise control over as many subordinates as possible. This is facilitated by being “close” to all subordinates. The small world property of a social network guarantees this span of control. A leader’s control diminishes as the distance between leader and subordinate increases. There are (at least) several arguments why it is premature to generally accept this theory. First, the betweeness self-organization rule is one of many rules that might explain why a high-betweeness leader emerges from random networks. Only two other rules (plus random attachment) were tested. More self-organizing rules need to be studied. Second, this study examined and compared the results with only one real-world social network – the 9/11 terrorist network. One sample is inadequate to substantiate a general theory. And the 9/11 terrorist network is small – only 62 nodes and 150 links. This theory may not scale to

larger networks. However, the theory may partially support Dunbar’s Number (Dunbar, 1992), which suggests a tipping point around networks with 130–150 actors. Crenshaw (2011) argues that social network theory is inadequate for describing the temporal nature of terrorist networks. She says, “How to treat change over time is problematic”, which cannot be disputed here. However, social network analysis provides an explanation for ‘function follows form’, assuming a snapshot in time is representative of the constantly evolving and changing terrorist group. References Adamatzky, A., Martınez, G. J., Chapa-Vergara, S. V., Asomoza-Palacio, R., & Stephens, C. R. (2010). Approximating Mexican highways with slime mould. Natural Computing, 10(3), 1–22. Adamic, L. A. (1999). The small world web. In S. Abiteboul & A.-M. Vercoustre (Eds.), Research and advanced technology for digital libraries. Lecture notes in computer science 1696 (pp. 443–452). New York: Springer-Verlag. Adamic, L. A., & Huberman, B. A. (2000). Power-law distribution of the world wide web. Science, 287, 2115. Adamic, L. A., Lukose, R. M., Puniyani, A. R., & Huberman, B. A. (2001). Search in power-law networks. Physical Review E, 64(4), 046135. Albert, R. (2005). Scale-free networks in cell biology. Journal of Cell Science, 118, 4947–4957. Albert, R., & Barabasi, A.-L. (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74, 47–97. Albert, R., Jeong, H., & Barabasi, A.-L. (1999). Diameter of the world wide web. Nature, 401, 130–131. Albert, R., Jeong, H., & Barabasi, A.-L. (2000). The Internet’s Achilles’ Heel: Error and attack tolerance of complex networks. Nature, 406, 378–382. Arquilla, J. (2007). The end of war as we knew it? Insurgency, counterinsurgency and lessons from the forgotten history of early terror networks. Third World Quarterly, 28(2), 369–386. Arquilla, J., & Ronfeldt, D. F. (2001). Networks and netwars: The future of terror, crime, and militancy. Santa Monica: Rand. Barabasi, A.-L. (2002). Linked: The new science of networks. Cambridge, MA: Perseus Publishing. Barabasi, A.-L. (2003). Scale-free networks. Scientific American, 288(5), 60–69. Bonabeau, E. (1999). Editor’s introduction: Stigmergy. Special issue of Artificial Life on Stigmergy, 5(2), 95–96. Colorni, A., Dorigo, M., & Maniezzo, V. (1991). Distributed optimization by ant colonies. In Proceedings of ECAL91 (pp. 134–142). Crenshaw, M. (2011). Mapping terrorist organizations. Center for International Security and Cooperation, Stanford University (February). Dunbar, R. I. M. (1992). Neocortex size as a constraint on group size in primates. Journal of Human Evolution, 22(6), 469–493. Elliott, M. (2006). Stigmergic collaboration: The evolution of group work. M/C Journal, 9(2). Retrieved 13.02.11 (May 2006). Holme, P., & Kim, B. J. (2002). Growing scale-free networks with tunable clustering. Physical Review E, 65(2), 026107. Jackson, B. A. (2006). Groups, networks, or movements: A commandand-control-driven approach to classifying terrorist organizations and its application to Al Qaeda. Studies in Conflict and Terrorism, 29(3), 241–262. Krebs, V. E. (2002). Uncloaking terrorist networks. First Monday, 7(4). (April 2002) Lewis, T. G. (2009). Network science: Theory and applications. Hoboken, NJ: John Wiley & Sons.

T.G. Lewis / Cognitive Systems Research 21 (2013) 7–21 Newman, M. E. J., Watts, D. J., & Strogatz, S. H. (2002). Random graph models of social networks. Proceedings of the National Academy of Sciences, 99(Suppl. 1), 2566–2572. Nisha, M., & Gopai, V. (2001). Small worlds: How and why. Physical Review E, 63. Parunak, H. V. D. (2005). A survey of environment and mechanisms for human–human stigmergy. E4MAS, 163–186. Resnick, M. (1997). Turtles, termites, and traffic jams: Explorations in massively parallel microworlds. Cambridge, MA: MIT Press.

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Reynolds, C. (1987). Flocks, herds and schools: A distributed behavioral model, In SIGGRAPH ‘87: Proceedings of the 14th annual conference on computer graphics and interactive techniques (pp. 25–34). Association for Computing Machinery. Theraulaz, G., & Bonabeau, E. (1999). A brief history of stigmergy. Artificial Life, 5(2), 97–116 (Spring 1999). Wasserman, S., & Faust, K. (1994). Social networks analysis: Methods and applications. Cambridge: Cambridge University Press.

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