Simple Heuristics and the Modelling of Crowd Behaviours

2.1 Force-Based Modeling Framework

The so-called social force model was probably the first individual pedestrian-based model that was capable of predicting the emergence of collective, self-organized crowd patterns out of local interactions among individuals [11, 12]. Since its first publication in 1995, there has been important development of this model and new models based on similar concepts [13, 14]. Therefore, the original social force model can be seen today as a general framework commonly used in the community of crowd modelers.

While the very first specifications of the social force model had only qualitative connections to empirical observations, many recent studies made use of tracking algorithms to reconstruct trajectories of interacting pedestrians from video recordings taken in streets, train stations, or highly crowded areas, and to elaborate and calibrate increasingly precise specifications of the model [14, 15, 16, 17]. Today, the most recent specifications are able to generate realistic crowd behavior, and to reproduce – at least qualitatively – every known crowd behavior, including lane formation, oscillation at bottlenecks [11], social groups’ behavior [8], stop-and-go waves [18], and crowd turbulence [19].

2.2 Strength and Limitations of Physics-Based Approaches

One of the greatest advantages of force-based models is their versatility, which probably explains the large variety of existing specifications in the literature. In fact, the social force framework is very easy to adapt by adding new parameters, changing their values, modifying the shape of the equation the forces involved, or adding new attractive or repulsive components. For example, many different specifications of the model only differ in the formulation of the equation of the main repulsive force \( {\overrightarrow{R}}_{\it ij} \). Therefore, it is not difficult to extend the model to new kinds of interactions by adding a new term to the final equation. Thus, social interactions can be implemented by adding attraction forces among group members [8], indirect interactions with the active walker model [10], or imitation by means of a herding force in situations of panics [9]. Another important strength of the social force framework is that it benefits from all the existing tools of statistical physics, which favors its development in the literature. For example, the transformation of the individual-based model into a macroscopic description based on partial differential equations is facilitated.

Nevertheless, the force-based approach exhibits some theoretical limitations. First, although force-based models can reproduce any empirical observation, it is important to notice that this often requires a modification, an adjustment, or a recalibration of the model. In fact, force-based models have excellent fitting capabilities, in the sense that a model can reproduce almost any observation after adjusting the equation of motion accordingly. However, they offer relatively low predictive power: a model that has been previously calibrated on a given dataset often fails to quantitatively predict another set of observations. One example is the experimental specification of the model presented in Ref. [15]. This force-based model was specifically designed to reproduce individual avoidance trajectories as observed under experimental conditions (we call it dataset A). However, it failed to quantitatively reproduce the complex dynamics of lane formation observed in Ref. [3] (dataset B). Nevertheless, a recalibrated model finally allowed to reproduce correctly the dataset B, but did not work any longer for dataset A. Therefore, this particular specification was not able to reproduce the features of dataset A and dataset B at the same time. Likewise, the recent simulation of crowd turbulence required the use of very strong inter-individual repulsive forces that would probably not be adapted for the simulation of pedestrian movements at low density [19]. Thus, even through this specification is an extremely valuable tool for the planning of mass events where crowd turbulence often occurs, it implies using different specifications of the model depending on the expected crowd density level.

Another issue with force-based models is that quantitative agreement with empirical data often requires rather sophisticated movement equations. In particular, it seems from the past decade that crowd models have tended to become increasingly more complex in an attempt to reproduce the growing amount of new empirical observations. The specification presented in Ref. [15], for instance, is a good illustration of how sophisticated the model equations have to be to precisely reproduce individual avoidance trajectories. In fact, it is not a trivial problem to find a proper balance between attractive and repulsive effects that would correctly reproduce the speed and direction changes during an avoidance manoeuver. In Ref. [15], this was achieved at the cost of using sophisticated equations of motion, which is also the case for many other force-based specifications. Although the high complexity of a model does not undermine its simulation capabilities, it might reduce the model’s usefulness for deciphering the dynamics of the system.

2.3 Outcome Models and Process Models

One possible reason explaining the above limitations is that force-based models aim at describing directly the observed movements of pedestrians, rather than the internal cognitive processes leading to the observed movement. In other words, force-based models are excellent at reproducing observations under specific conditions, but do not capture the intrinsic cognitive mechanisms operating at the level of the individual. It is very unlikely that pedestrians would be really under the influence of attractive and repulsive forces. Instead, it seems more plausible that they rely on simple and unconscious rules and navigation strategies, which let them act as if they were subject to external forces.

This is the distinction between so-called “outcome models” describing directly the observed behaviour, and “process models” describing the cognitive processes that give rise to the observed behaviour [20, 21]. To illustrate the difference between outcome models and process models, consider the following example: Imagine the problem of modelling the movements of a baseball outfielder trying to catch a ball in the air. One can imagine two distinct modelling approaches. The first consists in measuring the expected landing point of the ball according to its current trajectory and other environmental variables, and to use an attraction force to describe the player’s motion toward that point. Even through this modelling approach would require sophisticated equations and the use of many parameters, it would probably reproduce the movements of the player in a realistic manner. The second method is to elaborate a process model that relies on the so-called gaze heuristic – a simple rule-of-thumb that is used by people and animals to catch a moving target [20, 22]. The gaze heuristic consists of three steps: Fix your gaze on the ball, start running, and adjust your running speed and direction so that the angle of gaze remains constant. A simple model implementing these three rules would also reproduce the movements of the player, but it can ignore all causal variables necessary to compute the landing point of the ball, such as the velocity, angle, air resistance, or speed and direction of wind among others. Interestingly, both methods would probably predict a similar trajectory. Nevertheless, the process model generates the motion from the bottom-up, and is easier to formulate than the outcome model.

The above example can be easily put in parallel with the modelling of pedestrian movements. Existing force-based models are outcome models. Yet, a consistent framework for the development of a process model is still lacking. In the following, we describe a first attempt to elaborate such a framework based on Ref. [23], and discuss its strength and limitations.

2.4 A Process Model of Pedestrian Behaviour

To answer the first question, there is no doubt that a pedestrian relies on visual cues as primary source of information to decide where to walk [24, 25]. Similar vision-based approaches to describe people’s movement in complex buildings have been undertaken in the past [26, 27, 28, 29]. While interactions among individuals are the most important for the study of crowd dynamics, these studies only concern interactions between the individual and the physical environment. Recently, there have been several attempts to elaborate models of pedestrians’ behaviour in crowds based on visual cues [30], among which is the heuristic model that we will present in the next section.

2.4.1 Formalization of Visual Cues

What would be the simplest way to describe the visual cues that a pedestrian uses to navigate in a dynamic environment? Some simplifications have to be made here, as it would be impossible to describe the entire complexity of the human visual system. Therefore, we assume that the relevant information is the distance before collision with surrounding people, objects and obstacles. Therefore, we describe the input information of a pedestrian i by a function f(α) describing the distance that the pedestrian can walk toward the direction α at his or her comfortable walking speed S_i⁰ before a collision occurs. Because the function f(α) describes the visual information of the pedestrian, it is only defined for values of α in the interval [−ϕ, ϕ] delimiting the vision range of the individual. Here, the value α = 0 corresponds to the looking direction represented by a unit vector \( {\overrightarrow{H}}_i \) and the vision field of the pedestrian is ϕ degrees to the left and to the right of the direction \( {\overrightarrow{H}}_i \). If no collision is found toward the direction α, then the value of f(α) is set to a default maximum value d_max, which represents the horizon distance of the pedestrian as sketched in Fig. 1. Even though the computation of the collision distances with other people requires complex geometrical calculations, it remains a reasonable assumption as collision anticipation is a core capacity of the human brain [24, 31]. It is important to note here that this formulation requires some other simplifying assumptions, such as the fact that pedestrians have a circular body of radius r_i (although it is possible to adapt the model to elliptical body shapes as we will mention later in this article, in the ‘Possible improvements’ section).

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2.5 Benefits and Drawbacks of the Heuristic Model

Beside the fact that the heuristic model is based on plausible hypotheses and offers a new perspective on existing literature, it is interesting to note that it actually solves a long-standing open issue about how to combine multiple simultaneous interactions [35]. For example, in a situation where an individual A is facing three other individuals B, C, and D, force-based models typically assume that A sums up the repulsive effects that B, C and D would have separately in the absence of the others. However, this approach raises several theoretical issues. First, it is not obvious that simultaneous influences are additive. One can also integrate them in different ways such as averaging over them, or combining them nonlinearly. Second, it is unclear how to determine the influential neighbors and how to weight their influence. For example, it has been suggested that only the k closest individuals should be taken into account, or only those located in a certain interaction radius R [35, 36, 37]. But the values of k and R remain unclear and seem to vary with increasing density level. Another subsequent problem is that the repulsive effect in force-based models is necessarily proportional to the number of interacting partners. Consider the example in which a person is avoiding a group of standing pedestrians. Because repulsive forces typically cumulate, the shape of the avoidance trajectory will change according to the number of people in the standing group, which often generates inconsistent simulations. In contrast, in the heuristic model, interactions cannot be reduced to the superimposition of pair interactions because individuals react to a visual pattern in an integrative way. Therefore, rather than balancing multiple repulsion effects originating from other individuals, the simulated pedestrian detects an efficient free way to the destination and follows it, which generates realistic behaviour from the bottom-up. Therefore, it is not necessary to define explicit rules for combining simultaneous interactions or setting a fixed number of interacting partners in advance. These problems vanish naturally when using an integrated visual input, such as the pedestrian’s visual field.

Nevertheless, the cognitive model exhibits technical limitations due to the difficulty of its implementation. In fact, the calculation of the function f(α) describing the visual pattern of the pedestrian is a complex algorithm that requires heavy computational power and is prone to programming errors. The program typically needs to identify the first collision point between one moving object (the focus pedestrian) and many others (the other individuals), and to repeat this step for all directions α of the vision field, for all pedestrians i, and at every time step t. Therefore, simulations of the cognitive model are systematically longer than simulations of a force-based model, and may reach unreasonable computation time for simulations of large crowds (e.g. higher than 1,000 individuals). Even through this technical limitation does not directly challenge the validity of the model, it can hinder or slow down its future development, and dissuade the use of the model in commercial software. In the next section, we will suggest the search-and-stop rule as a future improvement that is likely to reduce the needed computational power. Advances in programming techniques and particularly in ray-casting algorithms may also help in resolving this issue.

Another issue with the cognitive model is that it is more difficult to extend than force-based models. With the social force approach, modeling a new component of pedestrians’ behavior, such as the side preference, the herding behavior during panics, or the movements of pedestrians in social groups, always consists of adding a new force component and calibrating this force with empirical data [8, 9, 10, 15]. This systematic methodology facilitates the development of the model. Updating the cognitive model, however, always requires some deep questioning about the cognitive processes involved in the situation being studied. For example, the cognitive model fails to correctly reproduce the situation where two pedestrians are facing each other at a narrow bottleneck, such as a door. In this situation, the visual field of each individual indicates that there is no empty space where it is possible to move, and therefore both pedestrians stop moving and wait endlessly for an opportunity to move ahead. Of course, this situation would never happen in real life because people have learnt that one has to step back and let the other one go first. But implementing this improvement in the model is more complicated, as simulated pedestrians need to identify the problem first, and solve it afterward.

2.6 Possible Improvements

As we mentioned earlier in this article, the cognitive approach described above should be considered as an initial basic framework rather than a final model. We have deliberately kept the model in its simplest form so far, to highlight the most basic heuristics needed to reproduce the fundamental features of crowd dynamics. But this first attempt certainly requires many improvements. In this section, we describe some possible improvements that could be considered in the future.

The first modification that could be undertaken is the implementation of elliptical body shape. Although the circular body shape seems to be a reasonable simplification in the first place (as demonstrated by the accurate predictions of the model), it may be relevant to use an elliptical body shape instead in the future [38]. In fact, at high densities, elliptical bodies may reduce the collision frequency and therefore affect the density threshold at which stop-and-go waves and crowd turbulence form. The implementation of elliptical body shape would also constitute a first step for modeling movements of pedestrians’ shoulders, which is known to facilitate avoidance maneuvers in crowded areas [39]. This improvement is expected to be important even at low density because it may help unblocking some face-to-face situations, such as the one described in the previous section. However, this update remains challenging, as it will require even more sophisticated algorithms for the detection of collisions.

Another important improvement would be the elaboration of a search-and-stopping rule [20, 40]. The current version of the model presumes that pedestrians select the best walking direction among all directions within their field of view (i.e. the direction that minimizes the distance to the destination point). However, it might be more realistic to assume that the pedestrian instead searches a good walking direction in his visual pattern and stops searching when a good enough solution is found, without exploring the complete vision field. This would require the implementation of a search rule (what is the starting point of the searching? What is the exploration sequence?), and a stopping rule (what is a good enough solution? When to stop searching?). Ideally, this rule would predict the kinds of eye movements that pedestrians engage in, based on theoretical principles and empirical data [41]. For example, one may assume that the individual starts searching in front of him (around α = 0), explores the visual pattern toward the sides, and stops when a satisficing walking direction is found. In addition, biased exploration patterns toward the right-hand side could possibly reproduce the features of the side preference [15]. The search-and-stop rule could also drastically reduce the computational power, as the complete field of vision would not need to be computed in advance, and might produce approximate decisions in a way that is closer to the pedestrian’s actual decision process.

Finally, it could also be interesting to include noise and fluctuations in the simulated behavior of the pedestrian. In particular, most existing modeling approaches rely on an “extrinsic” source of fluctuations, in the sense where random fluctuations of the movement are added after the deterministic behavior has been calculated. In force-based models, for instance, the equation of movement typically includes a random component that causes the pedestrian to deviate randomly from a pre-calculated position (e.g. the term \( \overrightarrow{\varepsilon}(t) \) in the original equation of the social force model).

When using a heuristic-based approach, however, it becomes possible to inject an intrinsic source of stochasticity. In such a way, the noise is not added after the movement is computed, but directly included in the cognitive process that generates the movement. For example, one may add random errors when the pedestrian estimates the expected collision points with other pedestrians, with less precise estimations on the sides than in the center of the visual field. The precision of the estimates in the model could even be based on data on the relative area that the visual cortex devotes to processing stimuli at a particular distance from the center of the visual field [42]. When the pedestrian makes eye movements (fixations) to particular parts of the visual field, the precision of their estimates of the location and behavior of stimuli near that part of the visual field would increase the most.