The Peak Stability Analysis through Hysteresis Phenomenon on Heterogeneous Networks

,


Introduction
Due to the increasing number of vehicles in urban city, the requirement for accessibility and efciency of road networks remains an essential question.However, the contradiction between the limited resources and the continuous growth of trafc demand results in the instability of the peak performances.Hence, it is important to identify the period and severity of road congestion based on corresponding indexes and to analyze the stability performances for diferent peaks.
Empirical and real data studies show that well-defned macroscopic relations exist in urban trafc networks rather than irregular dispersive scatter.In the urban network, instability is manifested in the inability to maintain average fow at a sustained level over the peak period.Te performance level fuctuates frequently in the corresponding period and great diferences in density between diferent road sections exist.In macroscopic trafc fow, instability usually refers to the formation of stop-and-go waves and congestions presented a subject-specifc stability.
In this paper, we analyzed the network peak stability by focusing on the hysteresis phenomenon in the MFD.To focus on hysteresis, the data collected in January 2020, in Atlanta, Georgia, were used.Te properties of hysteresis are discussed through real data.Trough the comparison between morning peak and evening peak, the precondition of hysteresis is discussed and the network stability is analyzed.Meanwhile, trafc indexes including occupancy, fow, and speed are cross-compared to explain the diference between morning and evening peaks.Accordingly, we provide a discussion on network peak stability based on real data and present the guiding value of hysteresis to the peak stability.Te main contributions include the following points: (1) In this paper, the mechanism of hysteresis is summarized and its existence is illustrated with a real case.Trough summarizing relevant studies, we mathematically deduced the process of hysteresis and recognized heterogeneity as a factor leading to network instability.(2) A 3D MFD with time series is introduced to further observe the changing patterns of hysteresis.Combined with the 3D coordinate system of variances of trafc indexes, it brought a diferent perspective to comprehending the network status at diferent moments by integrating the characteristics of the time-varying graph with the MFD.Trough the comparison of morning and evening peaks, the infuencing factor to hysteresis is discussed.Te diferences of hysteresis in the two peaks are analyzed as well.(3) To interpret the properties of the hysteresis loop, we divided it into three stages based on diferent change trends of network fow and occupancy.Each stage presents a diferent trend.Te various degrees of spatial heterogeneity in occupancy in the onset and ofset of the peak period are compared.Te diferent performances between the morning and evening peaks are exhibited.Te corresponding stability level is discussed.

Literature Review
Early studies focused on macroscale trafc patterns with data of lightly congested real-world networks [1] or simulations with artifcial routing rules and static demand [2].In these studies, the existence of an invariant macroscopic relation for urban networks was initially investigated.Te studies of Daganzo [3] made a clear distinction between free-fow and congested network states.Te empirical analysis of congestion patterns also revealed additional complexity and non-steady-state conditions of trafc states.
Based on the traditional trafc fow basic model and empirical data observed in Yokohama, a new concept of the macroscopic relationship between space-mean fow and density was proposed by Geroliminis and Daganzo [4].Tis is called the macroscopic fundamental diagram (MFD), which revealed a clear relationship.Teir feld experiments revealed that there was MFD within a large-scale urban area [4,5].In several studies, the existing conditions of MFD and its corresponding infuence factors including road infrastructure, trafc demand, signal control strategy, fow distribution, and the driving behaviours were investigated [6][7][8][9].It was found that a homogeneous network will maintain a well-defned MFD [3].In other words, the network with even and consistently distributed density will get a low-scattered MFD.Buisson and Ladier [7] showed that heterogeneity has a strong impact on the shape/scatter of MFD by analyzing the real data from a medium-sized French city.Moreover, the spatial distribution of road network density is the key component afecting the scatter of an MFD and its shape [10].When analyzing the impacts of diferent transport modes on trafc performance, Loder et al. [11] extended MFD into the 3D-MFD so as to ofer a novel framework at the urban scale.It gives us inspiration to observe the MFD characteristics of time-varying trafc fow and its stability.
Meanwhile, a phenomenon called hysteresis exists in some network MFD with further research.Treiterer and Myers [12] frst observed a trafc loop as separation in speed-density diagrams between the acceleration and deceleration curves.Since then, hysteresis has been observed more frequently.In trafc systems, the concept of hysteresis refers to the MFD graph indicating the ratio of trafc outfow and accumulation, which forms a closed rather than a linear curve.Initially, sudden fow drop, which can also be recognized as "capacity drop" and the predecessor of trafc hysteresis, was identifed disputing with the fowdensity curve for a road segment.It was addressed by the discontinuity between congested and free-fow regions.Numerous researchers indicated a downstream around road capacity [13].Te hysteresis phenomenon was proposed with more studies focusing on the whole process of congestion to describe such transitions in discontinuous fow.It extended the fow drop to the evolution of network statue during the congestion period.Te mechanisms of hysteresis were proposed including a mathematical theory based on acceleration, deceleration, and equilibrium fow [6,14].Daganzo et al. [15] mentioned that the hysteresis loop may disappear when lane-observed data were aggregated in 1-5 min intervals.Te clockwise hysteresis loop was mentioned as a symbol of trafc incidents.Amini et al. [16] discussed the infuence of incidents on the MFD pattern and the possibility of hysteresis loop appears around incidents.Te MFD hysteresis phenomena were further explored by Geroliminis and Sun [17] in the freeway network systems.Tey found that hysteresis was related to the distribution of fows on urban networks.Laval [18] suggested that driver behaviour is also an infuential factor of hysteresis apart from trafc fow status.Te empirical implications on travel time variance yielding the hysteresis phenomenon in day-to-day travel were discussed by Yildirimoglu et al. [19,20].Knoop et al. [21] proposed a continuous function called GMFD relating the average fow to both the average density and the (spatial) inhomogeneity of density.It can describe hysteresis patterns in the MFD.Kieran and Connaughton [22] described and validated a data-driven method based on identifying atypical fuctuations in the relationship between density and fow, quantitatively separating atypical fuctuations from typical trafc states.Te degree of fuctuations generated by network fow can be analyzed with this method.It can help to evaluate the stability level when hysteresis occurs.Raju et al. [23] developed a relationship between relative distances versus relative velocity among the leaderfollower vehicles and examined the hysteresis phenomenon for vehicles under corresponding behaviours.Johari et al. [24] reviewed the 50-year history of macroscopic modelling of urban networks.Te lack of empirical studies on the hysteresis phenomena was mentioned when it comes to the topics of network equilibrium relations.Te multimodal NMFD in diferent trafc conditions, which contains the existence of hysteresis, might afect the NMFD shape.
Hysteresis is a unique phenomenon that most appears with congestion periods and relates to the low level of network stability.Piu and Puppo [25] investigate the mathematical modelling and the stability of multilane trafc.Te stability of the steady states in the multilane system is discussed, which has a reference value for the analysis of network stability.Wang et al. [26] designed a memory feedback control signal based on the historical trafc information of the vehicle itself to improve the intelligent driver model.In the verifcation part, the stability condition in the heterogeneous network is discussed.It shows the distribution of network fow and the strength of memory feedback control signal afect the stability level.Control strategies that optimize the network heterogeneity can improve its stability.
However, few studies focus specifcally on the network instability when hysteresis occurs.As a manifestation of the instability in a specifc period, the premises and properties of hysteresis are worth concern.Te stability level in diferent peak periods can be analyzed through hysteresis.Te relative equilibrium analysis has not been explored including the diferences and variances of main trafc indexes in diferent periods.Tese problems will be discussed in this paper.Based on the analysis in this paper, the correlation network indexes can be used as the measure of stability level when hysteresis occurs.Tis has been discussed in another prior work [27].

Mathematical Deduction.
For trafc fow, by rapid changing in fow experiences, the trafc status will have an incoherent transition.Under trafc conditions of constant travel speed, the equilibrium fow is equivalent to the fow increasing linearly with the change of average density.Meanwhile, it is often observed that a rapid decrease is experienced by travel speed at a particular density level.Te notion of discontinuous phase trajectories in trafc dynamics is not exotic since system theory reveals that sudden phase transitions often occur in complex nonlinear systems.Tis phenomenon has been extensively noticed in the macroscopic fundamental diagram (MFD) curve of urban networks relating to the onset and ofset of congestion, which is regarded as a unique efect caused by discontinuous nonequilibrium trafc status.
A well-defned MFD, which can be divided into two parts by the critical accumulative vehicles and form a quadratic curve, refects the relationship between network-weighted fow and network accumulative vehicles.For a road network, the general equation of the MFD curve can be expressed as in equations ( 1)-( 4) for the network-weighted fow, density, speed, and occupancy, respectively.When we focus on a time period t, we can obtain equations ( 5) and ( 6): Since this paper discusses the network inhomogeneity in MFD, the inhomogeneity h i considers all road segments at one moment and determines the standard deviation.Terefore, it can be calculated from the standard deviation of the probability density function shown as follows: From Knoop et al. [21], the network inhomogeneity h i can be derived from the known standard deviation of the uniform distribution function.Combining with the function ∅(k i ), we aggregate all segments in the network and obtain the inhomogeneity for time t from the standard deviation as follows: It can also be simply expressed as the following function: Densities on road segments satisfy k i ∼ (k m − x, k m + x), with a variable x for the fuctuations of density.In the MFD curve, we regard it as a negative conic and have the following expression:

Journal of Advanced Transportation
Accordingly, when there exists h i , which makes MFD shape does not form a linear curve with time, network fow can be calculated as equation (11): We defne upper density boundary and lower density boundary as It can be observed that network fow is changing with diferent conditions and variables with the inhomogeneity.Tis process is expressed as the hysteresis in the MFD curve.It is a phenomenon presented by network status by time series, which indicates the beginning of a negative impact of congestion on the road network.When the MFD curve is chronologically described, a clip curve appears around the congestion in some networks.
Meanwhile, we can consider the tendency of diferent variables in hysteresis.According to Zhang [6], hysteresis owns similar changing process to "shockwave," but in the network level.Zhang uses acceleration and deceleration branches to refer to the shockwave process.As the similarity to hysteresis, we also use acceleration and deceleration phases to analyze the change trend of network fow during hysteresis.Te switch position of acceleration and deceleration branches can be regarded as the max fow in hysteresis (top point in the clip shape).Te trafc fow during some time period t is in the acceleration phase if the following expression holds: and deceleration phase if the following expression holds: Te acceleration and deceleration phases switched at the extreme point which is also the critical density, and there should satisfy v a � v e � v d .So we have the following equation: When it comes to the MFD curve, strong travelling/ shock or rarefaction waves can be regarded as congestion.Network fow is most prominent under acceleration and deceleration phases, which should in turn highlight the fundamental structure of nonequilibrium fow-density relationships.
As the accumulative vehicles can hardly reach the fully congested number in MFD.Trafc status with irregular changes is exhibited by the uneven distribution of fows.In the urban network, with the increasing accumulated number of vehicles, the exorbitant network density will result in bottlenecks at some segments of the network and the fow on those roads will gradually reach their capacity.When link density variance is low, the average network fow is consistently higher for the same network density.Te performance of the road network has not changed linearly with the trafc load.If occupancy keeps increasing until density exceeds a critical capacity, the arrival rates and network fow decrease.Hence, because of the inhomogeneous distribution of vehicular density in the network, there is a possibility for some segments of the network to become congested when the remaining parts stay noncongested or even in free fow status.
It can be regarded as, when going through a trafc congestion, there are three network stages [6]: anticipation dominant phase (stage 1); balanced anticipation and relaxation phase (stage 2); and relaxation dominant phase (stage 3).Tis is similar to the shockwave process.Te diference is that stage 3 in hysteresis is not as stable as stages 1 and 2. Tere are more fuctuations and the details will be analyzed with real data in the case study.
Te aforementioned conjectures lead to the following micro-macro models for vehicle n at a road segment i n : where γ denotes the relaxation time constant and ∆ > 0 is the distance vehicle n travels in time period t.
In this process, network in stage 1 with free fow state and going into trafc congestion.Suppose that network fows travel into a denser trafc region.Te slope of the MFD curve on the left side is positive but decreasing.Together with the consistency requirement, from equation for stage 1, we can conclude that network satisfes Similarly, we can obtain the equations of network state for stages 2 and 3: Specifc change rules and properties will be analyzed in the Case Study with real data.
Generally, the hysteresis in the MFD represents the result of macroscopic queueing and spillback and the evolution of regional congestion.It is an important property in the time dimension of the macroscopic trafc status of the road network.Te properties of hysteresis are related to the onset and ofset of congestion.By observing the variances of occupancy, network fow, and average speed in hysteresis, it is reasonable to analyze the change rules of regional congestion.
Te conventional MFD is a 2D curve not containing the time series.It cannot be identifed when hysteresis occurs beside the clip-shape.Trough such a curve, the fuctuation and time-varying conditions of the network cannot be revealed.Tus, we extended it to a 3D version.Tere were some previous studies that used 3D MFD as well; however, the timer shaft was not introduced as the unique axial.
Furthermore, we proposed a 3-stage division for hysteresis to depict the diferent features and changing rules during peak hours.Te basis of division between stages can correspond to the analysis above.To further describe the properties of hysteresis, a three-dimensional coordinate system on variances of trafc indexes is established.By crosscomparing, the distribution of variances, the diferences between morning and evening peaks can be explained.

Case Study.
In this study, we applied the data collected from 352 detectors across 9 urban roads in Atlanta, Georgia, from GDOT in January 2020.Occupancy, volume, and speed were available on average every 5 min.Considering the integrity of the data, some invalid data have been fltered.Te MFD acquired by diferent detectors on each road is presented in Figure 1.
If we regard the ftted curve as the quadratic function of occupancy, it means q � f(o i ).Ten, the ftting degree of MFD on 9 roads with m detectors can be calculated as follows: It can be observed that the relation of occupancy and fow appears in diferent shapes on diferent road segments.Both the occupancy and fow have various value ranges on 9 road segments.R 2 values are diferent for all 9 roads.Te diferent dispersions indicate diferent distributions of vehicles.Terefore, heterogeneity is clear for the trafc fow when considering these road segments as a macroscopic network.Te selected network and location of detectors are shown in Figure 2(a).
Since this paper studies the network peak stability through the hysteresis phenomenon, we focus on analyzing the corresponding trafc characteristics when hysteresis occurs, including its premises, the change trend of diferent indexes, and the comparison of network stability in the morning and evening peaks.Terefore, in the period with complete data, we selected Jan 15 th with relatively obvious hysteresis of the network MFD as the focus date.So, as to verify the change trend in diferent hysteresis stages, we proposed in the mathematical deduction above.After removing the duplicated data and the noise points, we aggregated the values of 9 road segments to describe the relation of weighted fow vs. occupancy and acquired the network MFD based on equations ( 1)-( 6) as shown in Figure 2(b).
As the data were collected every 5 min, there were 288 scatter points distributed in the MFD curve.Te ftted line conforms to a quadratic curve.When the occupancy was about 14, the average network fow reached the maximum value.When occupancy exceeded this threshold, average weighted fow decreased and network performance would relatively decline.Notice that, although we focus on the data on 15 th in this part, there are still several other days that have occurrences of hysteresis in the period we collected (Figure 3).All these days share similar network MFD scatter pattern.Te variation of weighted fow and occupancy before peak hours is higher when hysteresis occurred, which is refected in the network MFD; that is, the scatter span of the period between free fow and peak is larger and the distribution is uneven.
Accordingly, it is worth concerning when network performance experienced such changes and how the hysteresis occurs.Based on the foregoing discussion and Journal of Advanced Transportation analysis, hysteresis stands for the process of macroscopic queueing and spillback.It is an important feature in the time dimension, while the properties of time series are not completely expressed in the ordinary MFD curve like in Figure 2(b).Terefore, the hysteresis loop is not obvious.
To observe the timeline, the time-varying graph is addressed in Figure 4(a).It is observed that the morning and evening peaks occur at approximately 6:00-9:30 and 16: 00-19:30, respectively.Te maximum value of occupancy and network fow reaches at 17:05-17:10 and 7:25−7:30.On account of this, the MFD of the network is described by time series and the curve is ftted as shown in Figure 4(b).
Tere are two hysteresis loops in the MFD curve of Jan 15 th .We marked 0:00-12:00 (including morning peak) with the blue line and the red line of 12:00-23:55 (including evening peak).According to Figure 4(b), the same occupancy has diferent corresponding values of fow representing diferent time points conforming to the analysis of hysteresis in the last part.For instance, when occupancy was 15, the corresponding fow was at 7:30, 8:00, and 16:30.It is also worth mentioning that there is a larger value range of weighted fow in the morning peak than in the evening peak.
We have recognized through mathematical deduction that network inhomogeneity is an important factor to the existence of hysteresis.Te initial state of network in the morning and evening peaks is diferent.In other words, the inhomogeneity is diferent when diferent peak periods began.Corresponding to such diference, the hysteresis phenomenon of the morning peak is more obvious compared to the evening peak, and the hysteresis loop is more homogeneous in MFD ftting curve.

Journal of Advanced Transportation
It is reasonable that, on one hand, the initial distribution of vehicles on the network has a higher degree of heterogeneity in the morning peak than that in the evening peak due to the diferent travel demands on diferent road segments.On the other hand, the network initial state in the morning peak is closer to the empty load, which means there is more space for loading the critical fow.Te variation rate of network fow has a greater diference in each segment.It will be compared with the evening peak through data analysis in the following section.Furthermore, the performance of the network in diferent trafc indexes in both the morning and evening peaks should be considered along with the distinction between them.However, we divide hysteresis into stages according to the aforementioned mathematical analysis so as to better observe its features.

Tree Stages for Hysteresis.
A clear hysteresis loop can be observed by focusing on the blue half.However, the red half is more concentrated indicating the irregular distribution of MFD points and the stronger fuctuation of network fow in the afternoon.Terefore, we frst focus on the blue half.Comparing a common simulated MFD curve (Figure 5(a)) to this real data based on the MFD curve with time series, the diference of the peak can be seen.Meanwhile, the span of the hysteresis loop is not limited to peak hours.To better understand the evolution of hysteresis, we divided it into three stages as shown in Figure 5(b).Te division is based on the relative change rate of network-weighted fow and occupancy in the MFD curve.
Stage 1: 6:00-7:25.From the beginning of morning peak to the maximum weighted fow.Both fow and occupancy change positively over time in this stage.Generally, it can be considered as the evolution of free fow to the capacity of the network.Due to the network heterogeneity brought by diferent change rates of fows on diferent road segments, this stage possesses a larger growth span of occupancy than that of fow compared to the ideal state of the network (e.g., in simulation), which comes as an important premise for the form of the hysteresis loop.In this stage, the network satisfes the following expression: Stage 2: 7:25−8:10.From the max network-weighted fow to the highest occupancy.Te slope of the MFD curve gradually declines to 0 at the beginning of this stage.Te network-weighted fow starts to experience negative change after reaching the capacity while network occupancy is still increasing.Compared to the duration of this stage, its span on MFD is smaller than other stages.Despite that, the network becomes congested at this stage.Te MFD evolves from the maximum value of fow to the maximum value of occupancy in a single peak period.In this stage, the network satisfes the following expression: Stage 3: 8:10−9:30.After the occupancy climaxes at about 8:10, hysteresis enters stage 3, and both fow and occupancy of the network decrease.Tere is the main diference between the real data MFD and the welldefned MFD without hysteresis.Te curve did not go back linearly and rather formed the half bottom of the clip shape of hysteresis instead.As the fow starts to decline earlier in stage 2 than occupancy, there is a larger reduction span of occupancy compared to the fow in stage 3. Te ofset of congestion mainly lies in this period.Similarly, the network satisfes the following expression: It is worth mentioning that when hysteresis enters Stage 3, network returns to a low load status gradually and cumulative vehicles gradually get back to a conventional level.A wavelike rise is experienced after a period of decline by network fow and occupancy.Te regain of fow comes faster than occupancy, which is not similar to the variation trend in stage 1.As a result, the hysteresis loop does not appear to be a well-formed closed loop in the MFD curve.It forms a clip shape with a regular upper part and a fuctuating lower part.
Because of the limitations of the 2D perspective, the fuctuation of indexes over time in each stage is not always clear.It will be clearer in the 3D version with individual time coordinates.Especially for the evening peak, hysteresis is not as obvious as that in the morning.Terefore, we extend it to a 3D MFD to better understand each stage and analyze the changing process.

Peak Stability Analysis.
Although there are some fuctuations of fow or occupancy in the morning peak, the hysteresis loop and its properties in diferent stages can still be recognized through the curve.On the other hand, it is difcult to identify the hysteresis in the evening peak when the fuctuations become more frequent and irregular as the red line presents for the afternoon.It represents the stability level in diferent period changes with time.During hysteresis, the change interval is not as large as the blue half.However, there are always points of diferent periods distributed in the same area on this kind of 2-dimensional MFD.To observe the existence of hysteresis in the evening peak and identify the variance of the MFD curve in time series, we extended the MFD curve to three dimensions.Te time series was considered as the Z-axis.Network MFD was developed from Figures 4(b In such a 3-dimensional MFD, the semitransparent surface represents the originally ftted curve under 2D.Te blue and red half-lines correspond to the time half of the day in Figure 4(b).Specifcally, the timeline of hysteresis in the morning peak in Figure 6(a) conforms to the three stages divided in Figure 5. Relevant transformation rules are revealed by both fow and occupancy in diferent stages.When it comes to the evening peak presented in Figure 6(b), a diferent changing process is observed.Te change interval of occupancy remains considerable.By contrast, the value interval of fow is limited to a small range.Tere are similar stages 1 and 2 to morning peak for 16:00-17:05 and 17: 05-17:30, respectively.However, stage 3 is not present at the evening peak.To better understand the reason, the diference between the morning and evening peaks should be considered.Te network is under low load status before the morning peak starts.Te accumulative vehicles are low in the early morning.However, for the evening peak, it comes to the opposite situation.Te network accumulative vehicles remain at a medium level in the afternoon leading to the smaller interval change than in the morning.In addition, medium-level initial vehicles also bring trip diversity to the network.Terefore, the fuctuation and irregularity become more notable in the evening peak.
By extending MFD to a three-dimensional form, it can conveniently observe the relevant indexes and trafc status of the road network at any time of the day.For instance, when the maximum fow of the day appears at 7:25 in the morning peak for about 1400 veh/5 min, the corresponding occupancy is around 14.5%.Moreover, when the maximum occupancy appears at 17:05 in the evening peak for 18.5%, the corresponding fow is about 1280 veh/5 min.Meanwhile, based on the data, it is verifed that not only exorbitant vehicles in the network in total restrict the performance but they are also assembled at some shorter jams at parts of the road networks.It can be explained that the probability of spillover is increased by the inhomogeneity in spatial distribution, which continuously decreases the network fow.Combined with the analysis above, the change trend of the network in the morning peak is more fxed than that in the evening peak, with a higher network max fow.However, though the fow fuctuated more frequently in the evening peak, the network maintains average fow at a sustained level of around 1280 veh/5 min.Terefore, the stability level in the evening peak comes higher than that in the morning peak.
Te peak stability is diferent from the network capacity.It indicates the ability to maintain average fow at a sustained level.In the following section, we confrmed the division of three stages with the variances of diferent trafc indicators in each period and observed the variation trend of network trafc status through the comparison between morning and evening peaks.

Comparison of Morning and Evening Peaks.
Tree-dimensional MFD provides the trafc status of the road network at any time point and also locates congestion on the corresponding period when the hysteresis appears.Furthermore, it brings a kind of angle to comprehend the change rule of the network in diferent periods through the comparison of trafc index variances.
To be specifc, we selected the morning and evening peaks as the study objects and found a corresponding threedimensional coordinate system with three network indicators, including occupancy variance, fow variance, and speed variance every 5 min.In order to show the completeness of the change rules at the peak, 1 hour before and after the peak is also considered in this case.Tat means the period of 5:00-10:30 and 15:00-20:30 with 66 counted points.Similarly, blue and red points are marked as morning and evening peaks in Figures 7 and 8, respectively.
Figure 7 illustrates the scatter of the variances of three indexes from 5:00 to 10:30, which contains the morning peak.As mentioned in Figure 4(a), the maximum value of Journal of Advanced Transportation network fow reached at 7:25−7:30; therefore, we regard 7:30 as the demarcation point of the morning peak.In this case, "o" and "×" represent two diferent halves, which can be also considered as the onset and ofset of congestion, respectively.Generally, most points approach the origin point and the dispersion is low in this three-dimensional coordinate.It can be observed that the distribution of speed variances is more uneven compared to the fow and occupancy.Te dispersion of speed variances is higher both in speed-fow and speed-occupancy subgraphs.Meanwhile, we corresponded this three-dimensional coordinate to the 3 stages for hysteresis introduced in the last section and further verifed those properties in the pairwise comparison of the three indicators.Compared to  the fow, the scatter of occupancy variance is more concentrated through the occupancy-fow subgraph.Tis is consistent with the causes of hysteresis analyzed in stage 1.In the occupancy-fow subgraph, numerous scatter points are located in the frst and third quadrants, a small number in the fourth quadrant.Te second quadrant has the fewest points.It corresponds to the changing trend in stages 1 and 2. With the onset of congestion in the network, the average fow starts to decline before occupancy, while the variances of occupancy are relatively small.Moreover, the distribution of "×" points is more irregular in all three subgraphs, which aligned with the properties of stage 3. Te points in the third quadrant are mostly related to stage 3, representing negative changes in both occupancy and fow.However, there are both "×" points distributed in the second and third quadrants for stage 3.It is in the period that the network is negatively afected but not blocked by reaching maximum capacity.Few points were scattered around other quadrants, for each stage, the volatility of diferent trafc indexes was proved during the whole hysteresis.Trough speed-fow and speedoccupancy subgraph, the distribution of speed variances is more uneven.It makes sense that more "o" points are distributed in the second and third quadrants for the speedoccupancy subgraph, especially in the third quadrant.Furthermore, "o" points were distributed in the frst and fourth quadrants for the speed-fow subgraph, especially in the fourth quadrant.It means that the fuctuation of the speed of vehicles in the network is more obvious compared to the occupancy and fow in the morning peak, while such fuctuation has less impact on network efciency within a certain range.
Figure 8 shows the variances of occupancy, fow, and speed from 15:00 to 20:30, including the evening peak with red "o" and "×," respectively.Similarly, we regarded 17:10 as the demarcation point.Compared to Figure 7, the regularities of distribution of each index variance are not exactly the same.Trough the occupancy-fow subgraph, most points were distributed in the second, third, and fourth quadrants.Tere were only 8 points located in the frst quadrant opposite the morning peak, because of the difference of the initial accumulative vehicles in the network analyzed in the last section.Both "o" and "×" points in the second and fourth quadrants represent persistent volatility around the evening peak.It is consistent with the previous discussion on the properties of network performance and confrms the reason for no hysteresis loop existing in the evening peak.In addition, the dispersion of speed variances in speed-fow and speed-occupancy subgraphs is smaller than that in the morning peak.It represents a gentle change during this whole period.Te heterogeneity of the network is an important condition for the hysteresis phenomena with higher initial accumulative vehicles.
In general, there is a more obvious hysteresis in the morning peak and higher stability in the evening peak.Te trend of network is constant in the morning peak, but the cumulative fow change is more signifcant.Although more frequent fuctuations exist in the evening peak, the network maintains average fow at a sustained level.Combining the 3 stages analyzed above, the initial network status and heterogeneous process of change are both premises of hysteresis in MFD.Precisely, comparing the evening peak with the morning peak, the heterogeneity in the growth process of network fow is the key factor.When hysteresis exists in the MFD curve, the upper part of the loop is more regular than that of the lower part.Tere are more fuctuations after the max network occupancy was reached.

Conclusions
In this paper, hysteresis is studied as a representative phenomenon of network in low stability level.Te mechanism of hysteresis is summarized and its existence was illustrated with real cases while discussing its basis in heterogeneity.Trough the mathematical deduction, the network inhomogeneity is recognized as a precondition of hysteresis.Te way in which the formation of hysteresis causes changes in network stability is discussed.Real data are collected from a road network in Atlanta to exemplify the analysis.We also extended the MFD to a 3D version with time series to further observe the change rules of hysteresis.Trough the comparison of morning and evening peaks, we revealed that the diferent initial network status is also an infuencing factor of hysteresis.Terefore, the heterogeneity in the growth process of network fow is the key factor to the network peak stability.To interpret the properties of the hysteresis loop, we divided it into three stages based on diferent change trends of network fow and occupancy.Each stage presents a different trend.Te various degrees of spatial heterogeneity in occupancy in the onset and ofset of the peak period are compared.
Furthermore, to verify the properties of hysteresis and compare the stability level in diferent peaks, a 3D coordinate system of variances of trafc indexes is introduced.Tese 3D graphs can contribute to intuitively comprehend the network status at diferent moments through combining the characteristics of the time-varying graph with the MFD curve.Trough the cross-comparison of occupancy, average fow, and speed variances, we explained the diferent performances between the morning and evening peaks.Notice that hysteresis does not always occur in a network.Te morning peak has a higher possibility to display signifcant hysteresis than the evening peak caused by the diferent heterogeneity and initial status, which indicates a higher peak fow does not represent a high stability level.Te factors that cause the possibility of hysteresis occurring in diferent networks (network structure, fow environment, control strategy, etc.) deserve further comparative study.How to establish a mathematical expression of network peak stability to refect the equilibrium status through the proposed coordinate system of variances will be investigated in the following study.

Notations
Variables and Functions q i : Te fow on the road segment i k i : Te density on the road segment i o i : Te occupancy on the road segment i q w : Network-weighted fow k w : Network-weighted density o w : Network-weighted occupancy v: Vehicle speed on road segment and network l i : Te length of each road segment N: Te number of detectors in the network h i : Te network inhomogeneity k m : Mean value of all road segments density q max : Maximum network fow k c : Network critical density k j : Network jam density ∅(k i ): Te probability density function on road segment i v e : Network equilibrium speed

Figure 4 :
Figure 4: (a) Time series of network fow and occupancy; (b) MFD as time series in 2-dimensional vision.
) to 6, which combines the characterization indexes of Figures 4(a) and 4(b).