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We apply measures of complexity, emergence, and selforganization to an urban traffic model for comparing a traditional trafficlight coordination method with a selforganizing method in two scenarios: cyclic boundaries and nonorientable boundaries. We show that the measures are useful to identify and characterize different dynamical phases. It becomes clear that different operation regimes are required for different traffic demands. Thus, not only is traffic a nonstationary problem, requiring controllers to adapt constantly; controllers must also change drastically the complexity of their behavior depending on the demand. Based on our measures and extending Ashby’s law of requisite variety, we can say that the selforganizing method achieves an adaptability level comparable to that of a living system.
We live in an increasingly urbanized world [
Given its inherent complexity, traffic varies constantly [
Nonstationary problems change constantly. Which should be their desired regime? Would this also change? How can these be measured? We explore these questions in the context of traffic light coordination, applying recently proposed measures of complexity, emergence and selforganization based on information theory [
In the next section, we present our working framework: a city traffic model, the traffic light coordination methods compared, and the proposed measures. The city traffic model is based on elementary cellular automata, allowing us to detect precisely several phase transitions. The traffic light coordination methods studied are “greenwave” and “selforganizing”. The former attempts to optimize expected flows and is used in many cities. The latter has been recently proposed and adapts to immediate traffic demands. The measures are based on information theory and aim at being general and simple. Results of applying our measures to both traffic light coordination methods in our city traffic model follow under different boundary conditions: cyclic and nonorientable. The discussion focusses on the usefulness of our measures and the insights they give in the system studied. A future work section outlines possible improvements, refinements and extensions. Conclusions close the paper. Appendixes include axioms of the proposed measures, details of how data is processed, and preliminary results using a realistic open source traffic simulator.
We performed our study on a simulation developed in NetLogo [
We used a previously proposed city traffic model [
Each street is represented as an elementary cellular automaton (ECA), coupled at intersections. Each ECA contains a number of cells which can take values of zero (empty) or one (vehicle). The state of cells is updated synchronously taking into consideration their previous state and the previous state of their closest neighbors. Most cells simply allow a vehicle to advance if there is a free space ahead. This behavior is modeled with ECA rule 184, which is one of the simplest highway traffic models (See
This city traffic model is conservative,
The average velocity
Flow
Theoretically, the optimum υ and
Our previous studies have assumed cyclic boundaries, as is usual with ECA. However, this can lead to a quick stabilization of the system for certain densities, as vehicle trajectories become repetitive (assuming no turning probabilities, the model is deterministic).
Following a suggestion by Masahiro Kanai [
Kanai [
The problem of coordinating traffic lights is EXPTIMEcomplete [
In this work, we compare a traditional method which tries to optimize expected flows with a selforganizing method. These are described in detail in [
Many cities employ the “greenwave” method to coordinate traffic lights [
In [
We have proposed and refined a selforganizing method for traffic light coordination where each intersection follows simple local rules, reaching close to optimal global coordination [
Rule 1 gives preference to streets with higher demand (few vehicles wait more than several) and promotes the formation of platoons (few vehicles waiting may be joined by more to form larger groups, which reaching further intersections can trigger the green light before decreasing their speed). Rule 2 maintains a minimum green time to prevent fast switching of traffic lights in high densities. Rule 3 maintains platoons together, although allowing the splitting of large platoons. Rule 4 is useful for low densities, allowing few vehicles to trigger a green light if there is no vehicle approaching the current green. Rules 5 and 6 are useful for high densities, switching lights to red if there are cars stopped downstream of the intersection, preventing its blockage. The pseduocode of the algorithm extended for multiple directions can be found in [
In [
We recently proposed measures of emergence, selforganization, and complexity based on information theory in [
The concept of emergence has been controversial for centuries (for a comprehensive overview, see [
where
In its most general form, emergence can be understood as information produced by a process or system [
Minimum
where
since
Having the same equation to measure emergence, information, and entropy could be questioned. However, different phenomena such as gravity and electrostatic force are also described with the same equation. Still, this does not mean that gravitation and charge are the same. In the same fashion, there are differences between emergence, information, and entropy, which depend more on their usage and interpretation than on the equation describing them. Thus, it is justified to use the same expression to measure different phenomena.
For example, in cellular automata, it is considered that patterns such as gliders (spatial dynamic structures) are emergent [
Selforganization, just like emergence, has many notions, definitions, and measures. In its most general form, it can be seen as a reduction of entropy [
Since
which satisfies the axioms listed in Appendix A.1. With this measure of selforganization, static patterns have the highest
It might seem counterintuitive to define selforganization as the opposite of emergence, being both properties of complex systems. However, when taken to their limits, emergence is maximal in chaotic systems (
where the 4 is included as a normalizing constant, bounding
Historically, Shannon defined information as entropy [
In this work, we measure the
Results in this section were obtained simulating a tenbyten street grid, each equidistant street representing 800m,
Simulations were performed for one hundred densities
All figures in the Results and Discussion sections show phase transitions of the selforganizing method for the cyclic boundary scenario with vertical dotted lines for comparison. Preliminary results of a more realistic simulation are presented in Appendix A.3.
As reported in [
It can be seen that the selforganizing method goes above the optimality curve in the fullcapacityintermittent phase (
Since the greenwave method has periodic cycles, for their switching
The selforganizing method adapts constantly to changes in demand, as it can be seen from the measures variation for different densities. The switching is most irregular (greatest
The measures of vehicles crossing at an intersection, as seen in
Vehicle intervals are measured at different street locations (not intersections) chosen randomly each run. These measures are similar to those from intersections described above, although they reach maximum
We performed similar simulations as the ones described in the previous subsection but with nonorientable boundaries. Results for υ and
Compared to the cyclic boundaries scenario, the greenwave method has a lower υ for its intermittent phase and transitions into the gridlock phase at a lower density.
For the selforganizing method, there is no freeflow phase, as this depends on the correlations formed by the looping platoons which are destroyed with the nonorientable boundaries. The transition between the quasifreeflow and underutilizedintermittent phases occurs at a lower density. The fullcapacityintermittent phase is expanded for higher densities, transitioning later into the quasigridlock phase. The transition between the quasigridlock and gridlock phases occurs at a slightly lower density.
In general, the selforganizing method maintains itself at or near the optimality curves, showing that its performance is not strongly dependent on the boundary conditions studied here.
As for the measures of
Not only do traffic lights have to adapt their duration at the seconds scale to match the scale at which traffic demand changes; traffic lights must also adapt their regime at the scale at which density shifts from phase to phase. Extending Ashby’s law of requisite variety [
It might be possible to define measures which capture this requisite complexity and how it should match the complexity of the controlled/environment at different scales. One possibility can be with a measure of
As an initial exploration, we compared the
From this study, we can generalize the use of our measures for guiding selforganization of other systems. We have seen that a desired value of
The results presented in this paper are promising, but there are several points that should be further explored.
The relationship between dynamical phases and measures should be studied in greater detail.
The phase transitions in the nonorientable scenario can be further investigated, relating the performance measures such as υ and
We plan to perform similar studies with a more realistic traffic model [
Massive simulations with hundreds of streets and thousands of intersections are intended to better observe the dynamical phases and their transitions.
We have calculated the optimality curves for υ and
Can our measures be used to guide the selforganization of traffic lights even closer towards optimality?
Can our measures be used to guide the selforganization of other systems?
In this work we applied measures of emergence, selforganization, and complexity to an abstract city traffic model and compared two traffic light coordination methods. Varying boundary conditions yielded similar results. The measures reflect why the selforganizing method is much better than the greenwave method: for certain dynamical phases regular behavior is required, while for others adaptive, complex behavior is most efficient. The greenwave method by definition has only regular behavior, and thus is unable to adapt to constantly shifting demands. The selforganizing method has enough “requisite complexity” to cope with different behaviors required by changes in demand. Having an autopoiesis greater than one, it can be said that the selforganizing traffic lights have (computationally) the properties of a living system [
This work shows an example of the usefulness of our proposed measures to guide the selforganization of systems, which are being applied in other areas as well [
C.G. was partially supported by SNI membership 47907 of CONACyT, Mexico. We are grateful to three anonymous reviewers who provided useful comments.
D.Z., N.F., J.A., D.A.R. and C.G. designed research; D.Z., G.C. L.D.A., J.Z., N.F., and C.G. performed research and analyzed data; D.Z., N.F., J.A., D.A.R. and C.G. wrote the paper.
The authors declare no conflicts of interest.
Further discussion and justification of these axioms can be found in [
Emergence has the same axioms as Shannon’s information. Quoting Shannon [
Suppose we have a set of possible events whose probabilities of occurrence are
If all the
If a choice be broken down into two successive choices, the original
with these few axioms, Shannon demonstrates that the only function
A measure of selforganization
The range of
It is straightforward to check that
The complexity function
The range is the real interval [0, 1].
It is natural to consider the product of
For calculating
For example, if the maximum value is 121 and the minimum value is 11, then all points between 11 and 21 are mapped to the symbol ‘0’, points between 22 and 32 are mapped to the symbol ‘1’, and so on until the bin between 111 and 121, corresponding to the symbol ‘9’.
We present here preliminary results of simulations performed in SUMO (Simulation of Urban MObility) [
A grid of tenbyten streets was also used in this simulation. However, each street had two lanes with opposite directions. The distance between intersections was of 150 meters. The boundary conditions of these simulation were open. This is different from the ECA model, where density could be conserved. Each street was injected with traffic from the incoming lanes with a different rate of seconds between cars. The values in the simulations were from two to 40 s between vehicles, using only even numbers, and one car per second for the highest density achieved. This was performed eight times with 4000 cars in each simulation for each method. Since the number of cars was fixed, the total time for the simulations was different. For the higher densities, the duration was around 400 s and approximately 4000 s for the lower densities. The density was measured by the average number of cars in the simulation, between the exit of the first car from the network and the entrance of the last car to the network, divided by the maximum number of cars that could be in the network. High densities could not be reached because traffic in entry lanes does not allow more vehicles to flow to the central streets of the network. In future work, we will increase the density by starting the simulation with cars inside the network instead of only injecting traffic.
Results for velocity and flow are shown in
The greenwave method has only an intermittent phase, which offers almost the same υ for all densities. There is no gridlock phase because SUMO prevents intersections from being blocked (vehicles stop before the intersection—even if they have a green light—if they are unable to cross), a feature which is not realistic for cities such as Mexico.
Since the preliminary results presented in this Appendix correspond only to densities 0
Measures of intervals of vehicles at an intersection are shown in
Further work with SUMO will involve studying higher densities, calculating optimality curves, removing the blockage restriction, and evaluation in simulations based on real cities.
ECA rules used in city traffic model as traffic lights switch [
Cyclic and nonorientable boundaries of a fourbyfour street grid.
Results for cyclic boundaries: average velocity υ and average flux
Results for cyclic boundaries:
Results for cyclic boundaries:
Results for cyclic boundaries:
Results for nonorientable boundaries: average velocity υ and average flux
Results for nonorientable boundaries:
Results for nonorientable boundaries:
Results for nonorientable boundaries:
Results for SUMO: average velocity
Results for SUMO:
Results for SUMO:
Results for SUMO:
ECA rules used in city traffic model [
000  0  0  0 
001  0  0  0 
010  0  1  0 
011  1  1  1 
100  1  1  0 
101  1  1  0 
110  0  1  0 
111  1  1  1 
Six rules of the selforganizing method [

1. On every tick, add to a counter the number of vehicles approaching or waiting at a red light within distance 
2. Lights must remain green for a minimum time 
3. If a few vehicles ( 
4. If no vehicle is approaching a green light within a distance 
5. If there is a vehicle stopped in the street a short distance 
6. If there are vehicles stopped on both directions at a short distance 