Collective traffic of agents that remember

The motion of agents is modelled using a social force model, similar to Helbing et al [7, 8, 9]. The velocity of an agent evolves in time based on Eq 1, where: i) the first term in the RHS is the restitution force that restores the agent to its desired direction and speed $\mathbf{v}_{0}$, ii) the second term is the force due to the effect of the memory of an agent and, iii) the third term $\mathbf{I}_{j,b}^{i}$ is the net social interaction of the agent to avoid collision with other agents and the boundary.

$$m\frac{d\mathbf{v}_{i}}{dt}=\frac{m}{\tau}(\mathbf{v}_{0}-\mathbf{v}_{i})+\tilde{\beta}\mathbf{M}_{i}+I_{j,b}^{i}$$

(1)

Here, the memory $\mathbf{M}_{i}(t)$ of an agent $i$ at time $t$, is defined as the total deviation of the velocity of the agent from its desired velocity $\mathbf{v}_{0}$ in the time window $[t-T,t]$ of the agent’s recent past (see Eq 2).

$$\mathbf{M}_{i}=\int_{t-T}^{t}\big{(}\mathbf{v}_{0}-\mathbf{v}_{i}(t^{\prime})\big{)}dt^{\prime}$$

(2)

When the agent is unable to travel at its desired speed and direction, the memory term takes a positive value and offers an additional force to help restore the agent’s motion sooner.

$$I_{j,b}^{i}=\sum_{\forall j\in\mathcal{N}_{i}}\Big{[}\mathbf{F}_{i,j}^{n}+\mathbf{F}_{i,j}^{t}\Big{]}+\mathbf{F}_{b}^{n}+\mathbf{F}_{b}^{t}$$

(3)

The term $\mathbf{I}_{j,b}^{i}$ can take different forms depending on the context of the traffic problem. For pedestrian dynamics [8], Helbing and co-workers considered interaction forces similar to that used in granular flows: a sum of the total normal and tangential (frictional) forces due to contact. However, $\mathbf{I}_{j,b}^{i}$ can also include small-ranged forces that prevent collisions [9], which are more suitable for traffic flow problems. The qualitative results shown in the paper do not depend on the exact choice of the models for interactions.

If one were to imagine $\mathbf{v}_{0}$ as the set-point for a given agent, i.e., the desired state that the agent wants to achieve, then the restitution force and the memory term in Eqs 1 and 2, exactly resemble a proportional and an integral parts of a controller (PI), respectively. Presence of an integral component can result in overshoots and oscillations before the agent reaches its set-point. In the absence of memory and when there are continuous collisions with obstacles as agents move, we can expect the dynamics of the agent to show an offset (not reach $\|\mathbf{v}_{0}\|$)—since, it is well known that a proportional controller cannot take the system exactly to its set-point. Addition of memory could guarantee reaching the set-point, even in the presence of obstacles.

One could also conceive a dynamics resembling a PID controller. This modification does not qualitatively change the dynamics of the system. Since, adding a derivative term for the deviation $\mathbf{v}_{0}-\mathbf{v}_{i}$, gets absorbed into the inertia (LHS term of Eq 1). In other words, the derivative term acts like an effective mass term that slows the response of the agent to any social force.

The memory term in Eq 2 is computed within a time window of $[t-T,t]$. Any information from outside the time window is not used and all the information within are equally weighted. A simpler and more elegant formulation can be arrived at, if we weight the information with an exponential weighting as shown in Eq 4.

$$\mathbf{M}_{i}=\int_{0}^{t}e^{\frac{t^{\prime}-t}{\tilde{\alpha}}}\big{(}\mathbf{v}_{0}-\mathbf{v}_{i}(t^{\prime})\big{)}dt^{\prime}$$

(4)

An exponential weighting prioritises information close to the current time instant than from a distant past. This allows us to move away from a discontinuous time window set by $T$ and towards a time scale $\tilde{\alpha}$. Then, we can differentiate Eq 4 with time, using the Leibniz rule for differentiating under the integral sign to get Eq 5.

$$\frac{d\mathbf{M}_{i}}{dt}=\mathbf{v}_{0}-\mathbf{v}_{i}-\frac{\mathbf{M}_{i}}{\tilde{\alpha}}$$

(5)

This allows us to convert the memory term, which was previously an integral, into a dynamic state of the agent. Eq 5 describes the evolution of this memory-state of the agent: $\mathbf{v}_{0}-\mathbf{v}_{i}$ serves as the instantaneous source for the memory while $-\frac{\mathbf{M}_{i}}{\tilde{\alpha}}$ is the rate at which memory decays.

The equations for the dynamics of an isolated agent, far away from the boundary, can be written compactly as in Eq 6. This is after: i) the governing equations in Eq 1 are scaled ($t$ with $\tau$, $\|\mathbf{v}\|$ with $\|\mathbf{v}_{0}\|$, $\|\mathbf{M}\|$ with $\|\mathbf{v}_{0}\|/\tau$), ii) the interaction terms are dropped and, iii) the scaled memory parameters become $\alpha=\tilde{\alpha}/\tau$ and $\beta=\tilde{\beta}/m$.

$$\frac{d}{dt}\begin{pmatrix}v_{x}\\ v_{y}\\ M_{x}\\ M_{y}\end{pmatrix}=\begin{pmatrix}-1&0&\beta&0\\ 0&-1&0&\beta\\ -1&0&-\frac{1}{\alpha}&0\\ 0&-1&0&-\frac{1}{\alpha}\end{pmatrix}\times\begin{pmatrix}v_{x}\\ v_{y}\\ M_{x}\\ M_{y}\end{pmatrix}+\begin{pmatrix}1\\ 0\\ 1\\ 0\end{pmatrix}$$

(6)

To understand the effect of memory ($\alpha$ and $\beta$) on the movement characteristics of a single agent, it is enough to look at the eigen values of the matrix in Eq 6. The eigen values are $\frac{1}{2\alpha}\times\big{[}-(\alpha+1)\pm\sqrt{-4\alpha^{2}\beta+\alpha^{2}-2\alpha+1}~{}\big{]}$, repeated twice. Analysing the eigen values helps us partition the $\beta-\alpha$ space into four regions (see Fig 1, i).

Stability: When $\beta>0$, the memory term aids in restoring the agent to its desired state at steady state (see Fig 1, ii). And when $\beta<0$, memory opposes the agent’s attempt to reach its desired state. However, since the restitution force $(\mathbf{v}_{0}-\mathbf{v}_{i})$ always acts to restore the agent’s dynamics, a small negative $\beta$ memory does not immediately destabilise the dynamics and only increases the time taken to reach steady state (see Fig 1, vi). However, when $\beta<-\frac{1}{\alpha}$, the eigen values begin to have a positive real part and the memory term overpowers the restitution force leading to unstable dynamics (see Fig 1, vii).

Overshoots & oscillations: When $\beta>\frac{1}{4}-\frac{1}{2\alpha}+\frac{1}{4\alpha^{2}}$, the eigen values become complex conjugates and the dynamics become under-damped which results in oscillations in both the memory and the velocity of the agent (Fig 1, iv and v). Also, when $\alpha=1$, any $\beta$ greater than $0$ results in oscillations. This divides the region exhibiting non-oscillatory dynamics into two. The high-$\alpha$ part of this region exhibits dynamics where we observe overshoots; i.e. the velocity takes a larger value before asymptotically tapering to the desired, steady state value (see Fig 1, ii and iii to compare the dynamics in the low-$\alpha$ and high-$\alpha$ regions).

To understand the effect of memory on the dynamics of the collective, we consider a well known system: agents exiting a room via a narrow door (See inset of figure 2 i). We use the conditions similar to that in ref [8]. The governing equations in Eq 1 and 5 are simulated where every agent has a $\mathbf{v}_{0}$, directed along the line joining the agent centre and the mid-point of the exit of the door. The inter-agent interactions lead to collisions between agents as they crowd near the exit, slowing them down and giving rise to temporary clogging. Now as agents slow down, memory force increases since the source for the memory is $\mathbf{v}_{0}-\mathbf{v}_{i}$. This makes the agents push harder as they attempt to exit the room. To understand how these forces experienced by the agents lead to the collective escape of these agents, we introduce an order parameter $\mathcal{C}$ that characterises the clogging propensity. Here, $\mathcal{C}$ is simply the fraction of the total time when the agent-number in the room remains a constant. With this definition, a high value of $\mathcal{C}$ would correspond to the prevalence of a large number of clogging events during which agent number in the room remains unchanged.

Figure 2 i, shows the heat map of $\mathcal{C}$ in the $\beta-\alpha$ space. We find the landscape of $\mathcal{C}$ to be non-monotonic with respect to both the principle axes: i.e., an increase in the memory does not necessarily reduce the propensity to form clogs. This observation is similar to the so-called faster-is-slower effect (FIS), reported by Helbing et al [8, 10, 11], where agents trying harder do not necessarily lead to more efficient escape. In addition, we find that the observed non-monotonic effect is more pronounced in certain regions of the $\alpha-\beta$ space. For instance, when either $\alpha$ or $\beta$ are high (see figure 2 i), the non-monotonicity with respect to the other parameter is well pronounced in comparison to when they are low. We believe this feature arises due to the intrinsic dependence of the memory force on the parameters $\alpha$ and $\beta$ as seen in figure 1. When an agent remembers only its recent past, i.e. when $\alpha$ is small, a high value of $\beta$ is required to produce a force of a similar magnitude to transition from non-oscillatory to oscillatory behaviour and vice versa.

We partition the $\alpha-\beta$ space in figure 2 i into four regions while preserving the $\alpha-\beta$ relationship discussed previously. Region I is monotonic with respect to $\mathcal{C}$ and completely covers and extends over the region in the $\alpha-\beta$ space corresponding to non-oscillatory behaviour at the agent-level (compare with figure 1). Region II corresponds to the ideal amount of memory that agents can possess to efficiently escape through the exit. Region III corresponds to the FIS phenomenon and it is sandwiched between regions II and IV which exhibit efficient collective escape.

To understand how memory affects the escape dynamics and the formation of clogs, we look at the averaged dynamics of the memory term for the agents within the room. Figure 2 ii shows the time evolution of the mean and standard deviation of the memory $\|M_{i}(t)\|$ in system, averaged over all the independent realisations of the escape dynamics. The initial transient dynamics show that the effect of memory is more pronounced for the agents with lesser strength of memory (i.e., regions $II>III>IV$). This is because memory increases when the agents begin to slow down and agents with a higher strength of memory reach their desired state faster. This trend disappears as the agents clog near the entrance: both the mean and standard deviation of memory of the agents within the room increases rapidly in time. The dynamics of agents with memory in region III begin to have a higher overall memory with smaller variation between the agents within the room in comparison to II and IV. In other words, the strength and time scale of memory is such that all the agents try hard to exit by a similar amount which favours the formation of clogs and gives rise to a high value of $\mathcal{C}$. However, the higher variation and the lower mean memory of agents with memory in region IV implies that the some agents have more memory than the rest, which favours a more efficient exit strategy giving rise to lower values for $\mathcal{C}$.