Running head: INDEX OF FLOCKING BEHAVIOR An index for quantifying flocking behavior

One of the classic research topics in adaptive behavior is the collective displacement of groups of organisms such as flocks of birds, schools of fish, herds of mammals and crowds of people. However, most agent-based simulations of group behavior do not provide a quantitative index for determining the point at which the flock emerges. We have developed an index of the aggregation of moving individuals in a flock and have provided an example of how it can be used to quantify the degree to which a group of moving individuals actually forms a flock. Index of flocking behavior 3 Measuring flocking behavior: An index for quantifying the coordinated movement of individuals Moving in a coordinated way is common behavior throughout nature. Flocks of birds, schools of fish, herds of mammals and even crowds of people are systems composed of a certain number of individual entities that coordinate their movements in order to achieve coherent displacement. All these systems share common properties that have been studied using mathematical models (Okubo, 1986; Tanner, Jadbabaie & Pappas, 2003) and rule-based models (Aoki, 1982; Huth & Wissel, 1994). The latter implement rules that generate complex behavior and use agent-based computer simulation to create an artificial world where virtual agents react in accordance with the signals of the environment and act by following simple low-level rules in order to achieve an established goal (Maes, 1997). In accordance with the framework proposed by the adaptive-behavior approach (Meyer & Guillot, 1991), the complex behavior patterns observed in organisms are the result of those organisms’ reaction to changes in their environment. Such reactions are guided by sets of simple low-level rules that result in the emergence of complex higherlevel behavior (Beer, 1990; Brooks, 1991; Holland, 1995). In a seminal paper, Reynolds (1987) postulated an explanation for steering in artificial birds (named boids) guided by three rules applied to each individual: (a) each boid attempted to avoid collisions with its neighbors, (b) each boid attempted to stay close to its neighbors, and (c) each boid attempted to match the velocity of its neighbors. These rules were based on the neighbors’ local behavior and did not include centralized coordination, but produced collective motion along a common heading (Werner & Dyer, 1992; Zaera, Cliff & Bruten, 1996; Camazine et al, 2001). Index of flocking behavior 4 Based on Reynolds’ approach, agent-based simulation has been used to study the collective displacement patterns of a wide range of organisms, exploring factors such as the relationship between group behavior and low-level rules, body characteristics and the group’s environmental setting (Inada, 2001; Kunz & Hemelrik, 2003; Oboshi, Kato, Mutoh & Hito, 2002). Moreover, this approach is also applied to human displacements in order to understand and predict pedestrian flows and the movement of crowds of people (Schreckenberg & Sharma, 2002). However, when flocking behavior is studied using agent-based simulation, flock detection is sometimes carried out merely by observing the changes in the agents’ locations over time on the computer screen. Indicators based on the degree of parallelism of the agents’ orientation (e.g., polarity, often calculated as an aggregation of the deviation of each agent’s orientation from the average orientation) or those based on measures of inter-agent distance (often aggregations of nearest neighbor distance or distance to the flock center) have been used to analyze flock behavior (Kunz & Hemelrik, 2003; Parrish & Viscido, 2005). Other proposed indicators show properties of flock stability, shape or trajectory, but there is no simple index integrating different measures that indicates the degree of flocking behavior as a whole and that is easily applicable to agent-based simulations (Zaera, Cliff & Bruten, 1996). Therefore, in order to objectively measure flocking behavior, we defined an index of the degree to which a set of agents actually forms a flock. A moving group as a whole is considered a flock when all the agents have similar headings and the distance between them is low enough; the heading of an agent at time t is defined as the vector connecting its location at t-1 with its location at t. For agents i and j at time t, we define an aggregation index as follows: ( ) ( ( )) ( ( )) ij ij ij f t H t Z d t α = ∆ ⋅ Index of flocking behavior 5 Their aggregation is the product of H, a function of the difference between the agents’ headings at t (∆αij (t)), i.e., the difference, in degrees, between the vectors defining their headings, and Z, a function of their distance at t (dij (t)): ( ) ( ( )) 1 180 ij ij t H t α α ∆ ∆ = − ° 1 ( ( )) 1 1 exp ( ( ) ) / ij ij Z d t d t m m γ δ = −   + − ⋅ − ⋅   Both functions can yield values of between 0 and 1. Function H is equal to 1 when the agents’ headings are identical, i.e., ∆αij (t) = 0o, and is equal to 0 when the two agents face opposite directions, i.e., ∆αij (t) = 180o. Function Z is the inverse logistic function (γ > 0, 0 0); it tends toward 1 when the distance between the two agents is close to 0 (in which case the exponential function yields a high value), and it tends toward 0 when the distance is great (in which case the exponential function yields a value close to 0). Thus, at time t, the aggregation index for agents i and j approaches 1 only when both H and Z approach 1, and approaches 0 when either H or Z approaches 0. Therefore, the more agents face similar directions and the closer they are, the greater their aggregation index; on the other hand, if the agents have identical headings but are far away from each other, or if they are close to each other but their headings are opposite, their aggregation index is low. The inverse logistic function is used to ensure that, when the distance between agents is either small or great, smooth changes in distance cause smooth changes in function Z; however, when the distance reaches a critical value, a smooth change in distance causes a big change in Z. Figure 1 shows three inverse logistic functions for specific parameters γ = 5 (flattest curve), γ = 10 and γ = 20 (steepest curve); for the three curves, δ = 0.5, m = 20. Note that Z decreases as the distance increases, and that Index of flocking behavior 6 the sharpest descent occurs for distances of around δ·m = 10. The greater the γ value the more abrupt the descent. On the other hand, the distance at which the abrupt change occurs can be adjusted by changing the δ value; e.g., if δ = 0.8 and m = 20, the critical distance is 16. Parameter m is the maximum interagent distance that is judged to define them as a group, given the dimensions of the world. Then given m, setting δ at 0.5 sets the changing point at m/2. Given that extreme γ values produce radical discrimination or no discrimination at all, it would seem reasonable to assign it medium-range values. A global aggregation index or flocking index for all the agents present is defined as the arithmetic mean of the fij(t) indices, N being the total number of agents: 1 ( ) ( ) ( 1) / 2 ij i j F t f t N N 0. Therefore, in order to evaluate F(t) appropriately, we need to know the distribution function of the index for specific N, m, γ, and δ values in case the agents have random locations and headings. The distribution function can be estimated by assigning random coordinates and headings to the N agents repeatedly and independently many times (say, 10,000 times), and by calculating F each time; by Index of flocking behavior 7 averaging the Fs, an estimate of their mathematical expectancy (E[F]) can be obtained. Thus, the index can be converted into a Cohen’s (1960) kappa coefficient: [ ] [ ] ( ) ( ) 1 F F t E F t E F κ − = − In other words, at each time step, the difference between the obtained F(t) and its expected value in the case of random movement is divided by the maximum possible difference. Therefore, kappa can be viewed as the degree to which agent interaction actually causes a flock: a flock exists when ( ) 0 F t κ > (i.e., the agents’ headings are similar and their distances are shorter than in the random case), and the closer ( ) F t κ is to 1, the more defined the flock is (i.e., the more similar the agents’ headings are and the shorter the distance between them). Converting F into a kappa coefficient makes it easier to interpret, because while actual values of F depend on the number of agents and the size of the world in which they move, kappa can be viewed as a sort of standardized index, making it possible to compare flocks with different group and world sizes. We will show that ( ) F t κ makes it possible to distinguish between different kinds of flocking behavior (i.e., compact, disperse, etc.). Using agent-based simulation, we will generate groups of agents that act according to specific low-level rules of interaction (see below), and will check if the apparent flocks shown on the computer screen match the results indicated by ( ) F t κ over time.