The Inflection Point of the Speed-Density Relation and the Social Force Model

Abstract

It has been argued that the speed-density diagram of pedestrian movement has an inflection point [1] (p. 3, "Domain I: ... At low densities there is a small and increasing decline of the velocity ... Domain III: ... For growing density the velocity remains nearly constant."). This inflection point was found empirically in investigations of closed-loop single-file pedestrian movement.The reduced complexity of single-file movement does not only allow a higher precision for the evaluation of empirical data, but it also significantly simplifies analytical considerations. This is especially true if one assumes homogeneous conditions, i.e. neglects temporal variations (consider time averages, neglect stop-and-go waves), individual differences of pedestrians (all simulated pedestrians have identical parameters) and investigates only steady-state (not the initial phase). As will be shown in this contribution one then can make a transition from the microscopic to a continuous and macroscopic perspective.Building on that it will be shown that certain (common) variants of the Social Force Model (SFM) do not produce an inflection point in the speed-density diagram if -assuming periodic boundary conditions -infinitely many pedestrians contribute to the force computed for one pedestrian. It will furthermore be shown that if -in said 1d movement situation -one only considers nearest neighbors for the computation of the interpedestrian forces the Social Force Model in the continuous description results in the so called Kladek formula for the speed-density relation. Since the Kladek formula exhibits the desired inflection point this observation is used as a motivation for an extension of the Social Force Model which allows to transform the continuous description of the SFM continuously to the Kladek formula and which also exhibits the inflection point in the speed density relation. It will be shown then, that this extended SFM yields astonishingly similar speed density relations as the original SFM when only a fixed limited number of (nearest) pedestrians are considered in the computation of the inter-pedestrian force. Finally it will be discussed, if also the description of the speed-density diagram for (motorized, four-wheel) vehicular and/or bicycle traffic could benefit from these measures.

Some Indications on How to Calibrate the Social Force Model of Pedestrian Dynamics

“…The reduction to nearest neighbor interactions made the analytical investigation much easier, but obviously this step implies that results cannot be exact. It may be interesting in this regard that it was found that the approach of considering all other pedestrians as having an impact on a certain pedestrian does not lead to the most realistic speed–density relation, but that cutting forces at a certain number or at least suppressing them with more remote neighborhood degree is beneficial ( 26 , 27 ). Thus the approximation of nearest neighbor interaction can also be interpreted as an improvement of the model.…”

An analytical solution of the Social Force Model for uni-directional flow

“…Second, it was possible before to solve the SFM analytically for single file movement [13,14] which allowed to derive equations with which model parameter values can be computed from observables [15]. The latter is a decidedly valuable result since it allows direct calibration of the model (with all limitations following from approximations made as part of the analysis).…”