How many people can simultaneously move through a pedestrian space? The impact of complex flow situations on the shape of the fundamental diagram

Dorine C. Duives; Martijn Sparnaaij; Serge P. Hoogendoorn; Winnie Daamen

REVIEW 3 major objections 4 minor 10 references

How many people can simultaneously move through a pedestrian space? The impact of complex flow situations on the shape of the fundamental diagram

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Pedestrian capacity drops as movement complexity rises

desk verdict A genuinely new high-density pedestrian experiment with a plausible central pattern, but the capacity-decreases-with-complexity claim needs a stationarity check and error bars before the numbers can be taken quantitatively. read the letter →

arxiv 1908.07208 v1 pith:OI5HSRBV submitted 2019-08-20 physics.soc-ph physics.data-an

Dorine C. Duives , Martijn Sparnaaij , Winnie Daamen , Serge P. Hoogendoorn This is my paper · ORCID

classification physics.soc-phphysics.data-an

keywords pedestrianmovementdynamicsfundamentaldiagramlaboratoryexperimentheterogeneouscrowdcapacityflowratiobidirectionalintersecting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the capacity of a pedestrian space—the maximum sustainable flow through a cross-section—shrinks as the movement task becomes more complex. Using a laboratory crowd of 130 to 140 heterogeneous pedestrians, the authors measured global maximum flow rates of 0.65, 0.86, 0.23, and 0.54 people per metre per second for a bidirectional 50-50 corridor, a bidirectional 80-20 corridor, an intersecting 50-50 crossing, and an intersecting 80-20 crossing. They conclude that capacity decreases when flows must cross at right angles and when counterflows are more balanced, because both conditions reduce the chances to avoid collisions. The result matters because most existing capacity values come from homogeneous, unidirectional crowds at lower densities, whereas real crowds are heterogeneous, dense, and often move in crossing patterns.

What carries the argument

The central object is the fundamental diagram built from individual pedestrian trajectories captured by ceiling cameras. Density and speed are computed with the Voronoi method: each pedestrian's local density is the inverse of the area of their Voronoi cell, and global density and speed are averages over the measurement area. The experiment manipulates two factors—movement base case (bidirectional corridor vs. 90-degree intersection) and flow ratio (50-50 vs. 80-20)—while raising the inflow every minute. The load-bearing comparison is the maximum of the global flow rate, taken as the 99th percentile, across the four scenarios; that maximum is the paper's operational definition of capacity.

What would settle it

A replication that holds each inflow level constant for several minutes and checks whether density and flow stabilise within the first minute would settle whether the capacity differences are steady-state phenomena; if the maxima shift with the step duration, the complexity conclusion would need revision.

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Extended reading notes

Core claim

The central discovery is that the fundamental diagram, the density-flow relation for pedestrian movement, is not one universal curve: its shape and its peak depend on the movement base case and the flow ratio. The maximum sustainable global flow rate—the 99th percentile of the global flow—was highest in the bidirectional 80-20 corridor (0.86 P/m/s) and lowest in the intersecting 50-50 crossing (0.23 P/m/s), with bidirectional 50-50 at 0.65 and intersecting 80-20 at 0.54. This ordering supports the paper's hypothesis that capacity is governed by the scope for collision avoidance: the more demanding the geometry (crossing vs. opposing streams) and the more evenly the flows are split, the more avoidance manoeuvres are needed and the less throughput the space sustains. The paper also reports that local fundamental diagrams lack the classic single-peaked shape, while global diagrams show it, and that flow continues even at densities around 7 people per square metre, reproducing in a controlled European heterogeneous crowd a phenomenon previously seen only in field observations.

Load-bearing premise

The experiment steps the inflow up every minute and treats the resulting density-flow pairs as points on a fundamental diagram, which assumes the crowd reaches a quasi-stationary state within each minute; if one minute is too short, the measured maxima capture transients rather than sustainable capacity.

Editorial extensions

If this is right

Facility capacity guidelines that quote a single number for corridors and crossings will overestimate throughput for intersecting layouts and balanced counterflows.
Since local fundamental diagrams lack a clean peak, capacity estimates should be based on global, area-averaged quantities rather than on single-pedestrian measurements.
Very high densities (above 6 P/m²) do not force a full stop in forward movement over short periods, so crowd-safety and evacuation models should not assume a hard jam density.
The density at which capacity is reached differs by scenario, so demand-capacity checks should use scenario-specific fundamental diagrams rather than a universal curve.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The four capacity points suggest a rough 'complexity penalty' could be quantified—an inference the authors do not draw—allowing planners to discount corridor capacities when crossings or balanced counterflows are introduced.
Because inflow was increased every minute, the measured maxima may mix transient and quasi-stationary states; if longer dwell times shift the maxima, the capacity ordering could change, so this is a natural robustness check.
The absence of instructions in these four scenarios means the crowd was relatively 'normal'; the assignment conditions (crossing, fast walking) that the authors plan to analyse would test whether goal-orientation and unpredictability amplify the complexity effect.
The local-versus-global discrepancy hints that pedestrians may temporarily move faster through dense patches to maintain global throughput; a microscopic model reproducing this behaviour would explain why local scatter is large while global flow remains smooth.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Short version: the CrowdLimits data are real and the experiment is the first to combine a heterogeneous European sample, densities above 6 P/m2, and bidirectional and intersecting base cases in one controlled setup. That part is worth your time. The paper's headline claim, that capacity decreases with complexity, is directionally consistent with Table 4, but the support is thinner than the abstract suggests.

What's genuinely good: the trajectory data appear carefully produced, with manual correction to 100% of tracks, and the authors are appropriately circumspect in places, calling the results 'preliminary' and noting that the local fundamental diagrams do not show the classic mountain shape. The finding that flow continues at very high densities in a Western population is a useful replication of Helbing's Hajj results and goes beyond the Asian student samples in Lian and Cao.

Where the soft spots are: first, no statistical support for the ordering of the four qmax values. Two repetitions per scenario, no error bars, no significance test, and a percentile mismatch—Section 5.2 says 99th, Table 4 is labeled 95th. That alone prevents a quantitative reading. Second, and more load-bearing, the stationarity assumption. Inflow is stepped every minute, and the paper never checks that density and flow stabilize within that interval. If relaxation takes longer than a minute, the qmax values are transient loading peaks, not sustainable capacities. That concern bites hardest on the intersecting 50-50 scenario, whose 0.23 P/m/s could be an unfinished build-up rather than a genuinely low capacity. The paper's own description of runs stopped early because outflow dropped and queues could not be replenished adds to the worry.

I don't think the central argument collapses—the pattern across all four scenarios is consistent with complexity increasing friction—but the evidence as presented does not yet support the strong conclusion in Section 7. The fix is straightforward: report per-replicate values, add a stationarity diagnostic (e.g., density and flow as a function of time within each minute), and settle the percentile question.

Who should read this: anyone working on pedestrian fundamental diagrams or crowd safety will want to know this dataset exists. It deserves a serious referee, not a desk reject, but the referee should be instructed to ask for the missing checks before publication.

Best, [Name]

Referee Report

3 major / 4 minor

Summary. The paper reports the CrowdLimits laboratory experiment, in which 130-140 heterogeneous participants performed bidirectional and 90-degree intersecting pedestrian flows at 50-50 and 80-20 directional splits, with inflow stepped upward every minute. From automatically tracked and manually corrected trajectory data, the authors compute local and global Voronoi-based densities, speeds, and flow rates, and summarize the fundamental diagrams through 95th/99th percentile maxima. They report that global maximum flow is highest for bidirectional 80-20 (0.86 P/m/s), lower for bidirectional 50-50 (0.65 P/m/s), lower still for intersecting 80-20 (0.54 P/m/s), and lowest for intersecting 50-50 (0.23 P/m/s), and conclude that the capacity of pedestrian infrastructures decreases with increasing complexity.

Significance. If the directional ordering of these maxima is reliable, the paper would provide rare empirical evidence on how movement-base-case complexity degrades the capacity of pedestrian infrastructure under high densities, extending earlier homogeneous-population studies to a heterogeneous European population. Strengths of the manuscript include the deliberate heterogeneity of the participant pool, the use of the Voronoi method for density and speed, and the manual verification/correction of all trajectories, which gives high-quality trajectory data. The claimed conclusion, however, rests on a small number of runs and on an untested quasi-stationarity assumption, so the quantitative claims should be regarded as provisional rather than established.

major comments (3)

[Section 3.3 and Section 6.2] The central load-bearing assumption is that each one-minute inflow step allows the crowd to reach a quasi-stationary state, so that density-flow pairs lie on a fundamental diagram rather than on a transient loading path. The paper states in Section 3.3 that inflow was increased every minute and in Section 3.2 that some runs were stopped early when outflow dropped and queues could not be replenished, but no check is reported that density and flow stabilized within each minute. This is especially relevant for the intersecting 50-50 scenario, whose global qmax of 0.23 P/m/s may reflect an unfinished build-up rather than a genuinely lower sustainable capacity. The authors should either provide evidence of stabilization (e.g., time series within each minute, or a comparison of the first and second half of each interval) or reinterpret the reported maxima as transient loading values, which would weaken the capacity-decrease conclusion.
[Section 5.2 and Table 4] There is a direct inconsistency between the text and the results table: Section 5.2 states that rho_max, q_max, and v_max are computed as the 99th percentile, while Table 4 is labeled as the 95th percentile. Since the reported global maxima are the primary evidence for the paper's central claim, the authors must state which percentile was actually used and recompute the table consistently. The ordering in Table 4 could change if the two percentile definitions are mixed across scenarios or if the 95th and 99th percentiles behave differently across the four scenarios.
[Table 3 and Section 6] Each of the four base scenarios has only two experimental repetitions (e.g., bidirectional 50-50 is run twice on Day 1 and intersecting 50-50 twice on Day 2), yet Section 6 presents the results without error bars, replicate-level ranges, or significance tests. With n=2 per scenario and no measure of run-to-run variability, the quantitative ordering in Table 4 (0.65, 0.86, 0.23, 0.54) is not robust. The paper itself call the results preliminary in Section 7. The authors should report per-run maxima and, at minimum, bootstrap or resampling intervals, or explicitly downgrade the conclusion to a directional trend pending a larger sample.

minor comments (4)

[Figure 5 and Figure 6 captions] The captions of Figures 5 and 6 are identical, but Figure 6 shows global fundamental diagrams while Figure 5 shows local ones; the caption for Figure 6 should say 'global' instead of 'local'.
[Section 3.2] The text contains an empty table reference ("TABLE ") after the sentence about increasing flow rates every minute; the intended table appears to be Table 2, and the cross-reference should be added.
[References] The Munkres reference in the reference list is dated 1997, but the text cites it as 1957; please verify the correct publication year for the assignment problem algorithm.
[Throughout] There are several typographical issues, including 'increasi ng' in the abstract, 'emprical' in the Helbing et al. reference, and inconsistent spacing around table numbers; a careful language edit would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the fundamental-diagram maxima are measured directly from trajectory data via q = v·rho, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claim—that the maximum global flow rate decreases with increasing movement complexity—rests on directly measured quantities, not on a derivation that folds in its own conclusion. Section 5.1 defines speed (eq. 3-4) and density (eq. 1-2) from tracked pedestrian trajectories, and flow as the product of speed and density (eq. 5-6). The maximum flow rates in Table 4 are empirical percentiles of these measured time series, read off from the four scenarios; no model is fitted, no parameter is calibrated to a subset and then renamed as a prediction, and no prior uniqueness theorem is invoked to force an ansatz. The authors cite earlier work for measurement conventions (Steffen & Seyfried 2009 for Voronoi density), for scenario choice (Kretz et al. 2006 for 50-50 vs 80-20 flow ratios), and for inflow calibration (Zhang et al. 2014, Wong et al. 2010), but these citations do not determine the reported qmax ordering or the complexity-capacity conclusion. The only substantive concern—whether one-minute inflow steps allow quasi-stationary states—is a validity/measurement issue about transient versus sustainable capacity, not a circularity in which a result is equivalent to its own inputs by construction. The paper is self-contained against external benchmarks in that its headline numbers are raw empirical maxima from the CrowdLimits experiment.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted model parameters or invented entities. The central results are direct measurements from video trajectories. The main load-bearing methodological choices are the percentile threshold for maxima, the speed smoothing interval, and the unverified assumption that one-minute intervals produce stationary states. These choices are not fitted to data, but they affect the reported capacity values and are not tested for robustness.

free parameters (2)

Extreme percentile threshold = 99th (Section 5.2) vs 95th (Table 4)
The authors chose a percentile to define the maxima, but the text and table disagree on which percentile was used. This choice directly determines the reported qmax, rhomax, and vmax values, and the inconsistency makes the reported numbers ambiguous.
Speed time interval delta-t = Five video frames
The speed in Equation 3 is computed over five consecutive video frames, a choice that smooths the trajectory. The sensitivity of the derived flow and capacity values to this interval is not reported.

assumptions (3)

domain assumption The Voronoi method yields accurate local density and speed at very high pedestrian densities up to 8.7 people per square meter.
The paper relies on Steffen & Seyfried (2009) and Zhang et al. (2011) for the validity of the Voronoi method, but those validations were not performed at the extreme densities reached here, so the accuracy at these densities is assumed.
domain assumption One-minute inflow increments are sufficient for the crowd to reach a quasi-stationary state.
The fundamental diagram interpretation requires that each measured density-flow pair represents a stable condition. The paper does not test stationarity or report convergence times, so the measured maxima may be transient.
domain assumption The tracked cap positions accurately represent pedestrian positions at high densities.
The paper states that trajectories were manually checked and corrected to be 100% accurate, but it does not quantify how often caps were occluded or how tracking errors during the densest phases were resolved.

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Cite this review

Pith. "Pith review of How many people can simultaneously move through a pedestrian space? The impact of complex flow situations on the shape of the fundamental diagram." pith.science (2026). https://pith.science/paper/OI5HSRBV

@misc{pith2026190807208,
  author       = {Pith},
  title        = {Pith review of: How many people can simultaneously move through a pedestrian space? The impact of complex flow situations on the shape of the fundamental diagram},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OI5HSRBV}},
  note         = {Machine review of arXiv:1908.07208}
}

read the original abstract

Pedestrian crowding occurs more frequent. As a result of the increasing pedestrian demand in public space, the limits of pedestrian spaces are of increasing interest. Some research on the maximum demand that can flow through a cross-section has been presented, which mainly features simple movement base cases, low-density situations and/or a homogeneous crowd. Consequently, it is currently unclear to what extent their findings apply to heterogeneous high-density crowds, which are often encountered during real-world scenarios. The CrowdLimits experiment attempted to reproduce crowd movement dynamics of heterogeneous crowds experiencing higher densities than have been recorded up to this moment. Here, the aim was to study the impact of the three important differences between the current laboratory studies and real-world crowd dynamics in crowded pedestrian spaces simultaneously, namely crowd heterogeneity, high densities movements, and (more) complex movement base cases. This study shows that there are substantial differences in the maximum sustainable flow rate, and the maximum local and global sustainable density for distinct movement base cases and flow ratios. Moreover, the results provide evidence of that continuation of flow under very high densities can be recreated under laboratory conditions using a heterogeneous population of pedestrians. Besides, the experimental results indicate that the maximum global flow rate decreases when the scenario becomes more difficult (i.e. bidirectional to intersecting) and the collision avoidance opportunities decrease (i.e. 80-20 to 50-50 flow ratio). Thus, this paper concludes that the maximum flow rate of pedestrian infrastructures decreases with increasing complexity.

Figures

Figures reproduced from arXiv: 1908.07208 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 9 canonical work pages

[1]

Crowding may occur at busy train stations, or large multi-modal hubs, during mass events in cities, or in busy city centers

INTRODUCTION Situations where pedestrian crowding occurs become more frequ ent. Crowding may occur at busy train stations, or large multi-modal hubs, during mass events in cities, or in busy city centers. As a result of the increasing demand of pedestrian movements in public space, the limits of pedestrian spaces are of increasin g interest. In particular...

work page 1971
[2]

BACKGROUND In recent years, the movement of crowds has been studied quite extensively. Two topics are at the core of these research efforts, namely, 1) the capacity of pedestrian infrastructures and 2) the shape of the fundamental relation between density and flow for pedestrian movements. This section will present a brief overview of the current state-of...

work page 1994
[3]

Moreover, one would not be able to record the characteristics of the individuals (e.g

EXPERIMENTAL SETUP CROWDLIMITS EXPERIMENT Studying this type of movements in real -life is difficult due to ethical restrictions. Moreover, one would not be able to record the characteristics of the individuals (e.g. age, gender, length, weight) in the crowd. Thus, Delft University of Technology has set out to find the answer to these two questions by des...

work page 2006
[4]

Intersecting 50-50 without assignment; 4) Intersecting 80-20 without assignment

work page
[5]

The center camera, an 8MP camera, was directed at the center of the infrastructure and captured the approach and exiting of all pa rticipants

DATA COLLECTION AND DATA EXTRACTION The movement dynamics of each individual in the crowd was ca ptured using a set of video cameras attached to the ceiling of an examination hall facing top-down at the height of 8 meters. The center camera, an 8MP camera, was directed at the center of the infrastructure and captured the approach and exiting of all pa rti...

work page 1957
[6]

These variables will be used in section 6 to analyze the data from the CrowdLimits experiment

QUANTIFYING THE IMPACT OF MOVEMENT BASE CASES This section presents a mathematical description of the variables that will be used to quantify the differences in the fundamental diagrams. These variables will be used in section 6 to analyze the data from the CrowdLimits experiment. First, the derivation of the pedestrian flow variables from the pedestrian ...

work page 2009
[7]

First, a visual comparison is made of both the local fundamental diagrams as well as the global fundamental diagrams

RESULTS In this section the fundamental diagrams resulting from the four distinct ‘basic’ scenarios of the CrowdLimits experiment and discusses the differences in the shape of the fundamental diagrams. First, a visual comparison is made of both the local fundamental diagrams as well as the global fundamental diagrams. 6.1 Local pedestrian fundamental diag...

work page 2015
[8]

Allegro: Unravelling slow mode travelling and traffic: with innovative data to a new transportation and traffic theory for pedestrians and bicycles

CONCLUSION & FUTURE WORK This research has studied the shape of the pedestrian fundamental diagram from a local and global perspective. Data from a large laboratory study coined CrowdLimits featuring a heterogeneous crowd were used to study the impact of differences in the movement base case and flow ratio. Trajectory data was derived from video recording...

work page 2007

Show all 10 references

[9]

REFERENCES Buchmüller, S., Wiedmann, U., Parameters of pedestrians, pedestrian traffic and walking facilities, ETH Zurich,
[2006]

(report) Cao, S., Seyfried, A., Zhang, J., Holl, S., and Song, W. (2017 ). Fundamental diagrams for multidirectional pedestrian flows. Journal of Statistical Mechanics: Theory and Experiment . https://doi.org/10.1088/1742 - 5468/aa620d Chattaraja, U., Seyfried, A., and Chakrob...

2009 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Crowding may occur at busy train stations, or large multi-modal hubs, during mass events in cities, or in busy city centers

INTRODUCTION Situations where pedestrian crowding occurs become more frequ ent. Crowding may occur at busy train stations, or large multi-modal hubs, during mass events in cities, or in busy city centers. As a result of the increasing demand of pedestrian movements in public space, the limits of pedestrian spaces are of increasin g interest. In particular...

work page 1971

[2] [2]

BACKGROUND In recent years, the movement of crowds has been studied quite extensively. Two topics are at the core of these research efforts, namely, 1) the capacity of pedestrian infrastructures and 2) the shape of the fundamental relation between density and flow for pedestrian movements. This section will present a brief overview of the current state-of...

work page 1994

[3] [3]

Moreover, one would not be able to record the characteristics of the individuals (e.g

EXPERIMENTAL SETUP CROWDLIMITS EXPERIMENT Studying this type of movements in real -life is difficult due to ethical restrictions. Moreover, one would not be able to record the characteristics of the individuals (e.g. age, gender, length, weight) in the crowd. Thus, Delft University of Technology has set out to find the answer to these two questions by des...

work page 2006

[4] [4]

Intersecting 50-50 without assignment; 4) Intersecting 80-20 without assignment

work page

[5] [5]

The center camera, an 8MP camera, was directed at the center of the infrastructure and captured the approach and exiting of all pa rticipants

DATA COLLECTION AND DATA EXTRACTION The movement dynamics of each individual in the crowd was ca ptured using a set of video cameras attached to the ceiling of an examination hall facing top-down at the height of 8 meters. The center camera, an 8MP camera, was directed at the center of the infrastructure and captured the approach and exiting of all pa rti...

work page 1957

[6] [6]

These variables will be used in section 6 to analyze the data from the CrowdLimits experiment

QUANTIFYING THE IMPACT OF MOVEMENT BASE CASES This section presents a mathematical description of the variables that will be used to quantify the differences in the fundamental diagrams. These variables will be used in section 6 to analyze the data from the CrowdLimits experiment. First, the derivation of the pedestrian flow variables from the pedestrian ...

work page 2009

[7] [7]

First, a visual comparison is made of both the local fundamental diagrams as well as the global fundamental diagrams

RESULTS In this section the fundamental diagrams resulting from the four distinct ‘basic’ scenarios of the CrowdLimits experiment and discusses the differences in the shape of the fundamental diagrams. First, a visual comparison is made of both the local fundamental diagrams as well as the global fundamental diagrams. 6.1 Local pedestrian fundamental diag...

work page 2015

[8] [8]

Allegro: Unravelling slow mode travelling and traffic: with innovative data to a new transportation and traffic theory for pedestrians and bicycles

CONCLUSION & FUTURE WORK This research has studied the shape of the pedestrian fundamental diagram from a local and global perspective. Data from a large laboratory study coined CrowdLimits featuring a heterogeneous crowd were used to study the impact of differences in the movement base case and flow ratio. Trajectory data was derived from video recording...

work page 2007