# Inflow process of pedestrians to a confined space

Takahiro Ezaki<EMAIL_ADDRESS>Japan Society for the Promotion of Science, 8
Ichibancho, Kojimachi, Chiyoda-ku, Tokyo 102-8472, Japan  Research Center for
Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, Tokyo 153-8904, Japan  Kazumichi Ohtsuka  Research Center for
Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, Tokyo 153-8904, Japan  Mohcine Chraibi  Jülich Supercomputing
Centre, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany  Maik Boltes
Jülich Supercomputing Centre, Forschungszentrum Jülich GmbH, 52425 Jülich,
Germany  Daichi Yanagisawa  Research Center for Advanced Science and
Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904,
Japan  Armin Seyfried  Jülich Supercomputing Centre, Forschungszentrum Jülich
GmbH, 52425 Jülich, Germany  Department of Computer Simulation for Fire Safety
and Pedestrian Traffic, Bergische Universität Wuppertal, 42285 Wuppertal,
Germany  Andreas Schadschneider  Institut für Theoretische Physik, Universität
zu Köln, 50937 Köln, Germany  Katsuhiro Nishinari  Research Center for
Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, Tokyo 153-8904, Japan

###### Abstract

To better design safe and comfortable urban spaces, understanding the nature
of human crowd movement is important. However, precise interactions among
pedestrians are difficult to measure in the presence of their complex
decision-making processes and many related factors. While extensive studies on
pedestrian flow through bottlenecks and corridors have been conducted, the
dominant mode of interaction in these scenarios may not be relevant in
different scenarios. Here, we attempt to decipher the factors that affect
human reactions to other individuals from a different perspective. We
conducted experiments employing the inflow process in which pedestrians
successively enter a confined area (like an elevator) and look for a temporary
position. In this process, pedestrians have a wider range of options regarding
their motion than in the classical scenarios; therefore, other factors might
become relevant. The preference of location is visualized by pedestrian
density profiles obtained from recorded pedestrian trajectories. Non-trivial
patterns of space acquisition, e.g., an apparent preference for positions near
corners, were observed. This indicates the relevance of psychological and
anticipative factors beyond the private sphere, which have not been deeply
discussed so far in the literature on pedestrian dynamics. From the results,
four major factors, which we call flow avoidance, distance cost, angle cost,
and boundary preference, were suggested. We confirmed that a description of
decision-making based on these factors can give a rise to realistic preference
patterns, using a simple mathematical model. Our findings provide new
perspectives and a baseline for considering the optimization of design and
safety in crowded public areas and public transport carriers.

pedestrian dynamics and inflow process and personal space

##  I Introduction

Human crowding in public areas is still a significant issue in social and
engineering science Altman1975 ; Hillier1993 . To design comfortable and safe
urban spaces, it is important to understand the nature of the interactions
among pedestrians and their consequences. However, due to the lack of such
knowledge, many inefficient or even disastrous situations are still found,
e.g., serious overcrowding in transportation systems in urban areas and
crowding during disasters, leading to a stampede. Meanwhile, relatively recent
attempts have partly succeeded in modeling the collective phenomena of
pedestrians Schadschneider2009 ; Helbing2001 ; Helbing2005b ;
Schadschneider2010 , including spontaneous lane formation in a bidirectional
flow Helbing1995 ; Burstedde2001 ; Kretz2006 ; Nowak2012 , crowd turbulence in
overcrowded areas Helbing2007 ; Yu2007 , and formation of stripe patterns in
intersecting flows Helbing2005b ; Ando1988 . In addition to experimental
Hoogendoorn2005 ; Kretz2006 ; Seyfried2009 ; Jelic2012 ; Yanagisawa2012a ;
Moussaid2012  and empirical Henderson1971 ; Weidmann1993 ; Moussaid2010
studies, several types of mathematical models that facilitate simulations and
analyses of central phenomena have been developed (see Schadschneider2009 ;
Helbing2001 ; Helbing2005b ; Schadschneider2010  for reviews). Such approaches
have also contributed to the understanding of how walking people recognize and
react to their environment.

In this paper we study the inflow process of pedestrians that was recently
proposed as a model process for understanding the passenger entrance behavior
to elevators, buses, trains, etc. Ezaki2012c ; Ezaki2013a . The inflow process
is defined by successive entry of pedestrians into a confined area and their
subsequent dwelling. While entering, each pedestrian performs a complex
decision-making process perceiving the current situation in the room (e.g.,
the distribution of persons), anticipating the behavior of subsequently
entering persons and planning to the exit from the room. These stimuli and
knowledge are evaluated for the decision, taking into account social
conventions. Among pedestrians, there is competition to optimize the final
location with respect to easy and fast exit under the limitation for avoiding
overcrowding to secure comfort. Classical studies of pedestrian dynamics
focused on motion in corridors and bottlenecks to investigate flow capacities
or jamming at high densities Schadschneider2009 ; Helbing2001 ; Helbing2005b ;
Schadschneider2010 . In such scenarios, the decision process is negligible,
because pedestrians are not given the freedom of choice (e.g., changing the
direction or their locations) because of predefined destinations in the setup
of the experiment or restrictions on the motion at high densities. In the
inflow process, no predefined destinations are given. Our goal is to reveal
the modes of interactions among pedestrians that become visible through the
decisions. The distribution of pedestrians’ positions could result from the
interplay among several factors. A dominant mechanism could be based on the
concept of personal space, the area individuals maintain around themselves
into which others cannot intrude without causing discomfort Hall1962 ;
Little1965 ; Sommer1959 . Note that the effect of personal space has been
integrated into computational models in various contexts in pedestrian
dynamics Was2006 ; Was2010 ; Ezaki2012c ; Manenti2012 .

To understand such human behavior, further experimental and empirical studies
are needed. Recent behavioral experiments on the inflow process Ezaki2013a ;
Liu2016  succeeded in revealing that pedestrians prefer the areas near the
boundaries, which had been predicted using a cellular automata model
Ezaki2012c . In addition, Liu et al. Liu2015 ; Liu2016  extended the
experiments described in Ref. Ezaki2013a  by varying the pedestrian number and
the motivation for future exiting.

The number of pedestrians, i.e., the final pedestrian density in the area is
considered to be a significant factor in pedestrian behavior because the main
mechanism of the inflow process is expected to be strongly related to the
avoidance reactions against other pedestrians. If the final density is unknown
or changes during the experimental trials, subjects might decide their final
positions based on an incorrect understanding of the future situations. To
easily control this factor, we restrict ourselves to considering a fixed
pedestrian number and area size in this study. Instead, we consider the
geometry of the room as a variable in our experiment. As minor changes in the
geometry of the area, we focus on (i) differences in entrance position and
(ii) presence of an obstacle. A previous study Ezaki2012c  suggested that the
entrance position strongly affects the pedestrian distribution pattern. In
fact, these experimental settings allowed us to obtain robust average behavior
and analyze the underlying decision-making. Using pedestrian trajectory
detection Boltes2013  and density estimation Zhang2011  techniques developed
in the context of pedestrian dynamics, we attempt to visualize the behavior
and the resulting decisions of pedestrians in such situations, thereby
providing an insight into the anticipation and evaluation of future situations
regarding their personal space.

In this study, we arranged a basic experiment of the inflow process (Fig. 1).
Test subjects were asked to enter a confined area, and temporarily stay there
as described in Sec. II. On the basis of the obtained results, we report their
interpretations (Sec. III). Details of techniques and analyses used are
summarized in Appendix. Finally we discuss the implications and limitations of
the results in Sec. IV.

##  II Experimental Design

Controlled experiments have been conducted in the Research Center for Advanced
Science and Technology (RCAST) at The University of Tokyo, Japan. A total of
25 male participants were recruited from students of The University of Tokyo,
who were paid for participation. The participants were strangers to each
other. The experiment was conducted for three different room structures (see
Fig. 1c): (i) With the entrance in the middle without obstacles (“normal
scenario”; N); (ii) With the entrance in the middle with an obstacle in front
(“obstacle scenario”; O); (iii) With the entrance near a corner but without an
obstacle (“corner scenario”; C). The obstacle scenario “O” is included because
it has been suggested for evacuation processes of pedestrians that an obstacle
facilitates faster exiting by impeding clogging Helbing2005b ; Yanagisawa2009
. However, its (potentially, adverse) effect on the inflow process is still
unknown. In addition, the obstacle models the effects of hand-rail poles in
buses or trains, which is important within the transportation engineering
context.

In each trial, 25 participants were asked to enter an area (3.6 m$\times$ 3.6
m) set up on the ground, which was enclosed within walls with a height of 2.0
m (Fig. 1a and Fig. 1b). A PVC pipe ($\phi$0.11 m; 2.0 m high) was used as an
obstacle. The initial positions of the pedestrians were marked on the ground
to regulate the inflow speed, and their entrance order was randomized after
each round. They were instructed not to hurry but behave as they would when
boarding an elevator. The area was covered with a blue sheet to cover the
support structure of the walls and the obstacle, which were sufficiently thin.
The width of the entrance was 0.6 m. This value was chosen not to let the
participants see inside the area before entrance in order to control the
information about the configuration of pedestrians inside.

The experimental area was recorded with a digital camera (SONY HDR-SR7;
recording format 1080i60) mounted at a window on the 4th floor (21 m high) of
a neighboring building. The participants were equipped with caps and black
shirts for video tracking, and their trajectories were collected using
tracking software (PeTrack Boltes2013 ). We asked the participants to remain
in the area for a while after the final participant stopped walking and then
asked them to exit the room through the entrance for the next trial. We
repeated nine trials for each scenario. Prior to the experiment, we conducted
a few test trials using the normal scenario to let the participants know the
size of the area, speed of walking, etc., to prevent them from being
bewildered in an unfamiliar environment. Owing to these test trials, we
observed no apparent difference in their behavioral patterns across
experimental trials.

Figure 1: (a) Snapshot of the experimental setup (obstacle scenario). The pole
indicated by a white arrow is the obstacle. (b) Initial conditions and
dimensions of the setup. (c) Three different scenarios (normal, N; obstacle,
O; corner, C). The obstacle is shown by a magenta cross. (d) Snapshot of the
experiment. Pedestrians wore on their heads a piece of cardboard with two
dots, by which head directions, as well as positions, were detected. (e)
Definitions of the $xy$-coordinate, the polar coordinate,
($r_{i},\theta_{i}$), and the distance to the boundary, $r_{\rm{max}}$.

##  III Results

###  III.1 Qualitative analysis of collective characteristics

Fig. 2 shows the pedestrian movement and final positions during one
experiment. Locations near the boundaries were occupied first (Fig. 1d, Fig.
2a and Fig. 2b). After stopping at their location, they remained standing
there except for small fluctuations (Fig. 2a; see also supplementary video).
Interestingly, in this occupation order, pedestrians were not required to and
did not choose to pass between two closely located pedestrians, minimizing the
invasion of personal space. The final distribution of pedestrians was uniform
to a certain extent, with slight deviations in Voronoi areas (Fig. 2b). For
larger distortions of the uniform state, pedestrians would fill up the gaps to
gain more space, which in turn reduced the inhomogeneity. Rather homogeneous
final state was thus achieved through a process of self-organization. The very
homogeneous final distribution was reached without significant changes to the
first choice of destination, which is a indication of good anticipation.

Just before reaching their final positions, pedestrians turn toward the
entrance (exit) where changes in the situation owing to successively entering
pedestrians are expected (Fig. 2c; see also supplementary video). This turning
explains the features near the end of the trajectories (Fig. 2a). The
resulting vector field of head directions is aligned toward the entrance (Fig.
2d). This coordinated alignment is universally observed in all three
experimental scenarios (N, O, and C).

This behavior could be interpreted as the process of reducing the discomfort
from not being able to see the next pedestrians entering the area. This
potentially allows avoidance motion in case of conflicts. As explained, the
personal space of pedestrians is fixed from the boundary of the area, and
therefore, the direction for the boundary contains less uncertainty, whereas
stimuli requiring a reaction are mainly expected from the direction of the
entrance. Furthermore, pedestrians who entered the space first and fixed their
positions are face-to-face with incoming pedestrians, potentially leading to
discomfort at eye contact. It is known that more personal space is necessary
when a person is confronted by others Argyle1965 . Therefore, the alignment of
head directions supports a more efficient use of space, with respective of the
level of comfort.

Note that across the three scenarios, we observed no significant difference in
the time required for 25 pedestrians to enter the area.

Figure 2: Detection of pedestrian motion in a single trial (obstacle
scenario). (a) Trajectories of pedestrians. (b) Final distribution of
pedestrians with Voronoi cells. Each number represents the entrance order. The
local density is defined as the reciprocal of each Voronoi area (see equation
(1) in Appendix). (c) Sample trajectories with head directions. Black arrows
correspond to the final state. (d) Head directions for a final state. For
visualization of the space usage, referential body areas (not obtained from
experimental data) are shown with ellipses of 0.4 m width and 0.2 m thickness.

###  III.2 Qualitative analysis of individual decisions

Next, we study pedestrians’ choice of location. By introducing polar
coordinates as shown in Fig. 1e, the location of pedestrian $i$ is expressed
as ($r_{i},\theta_{i}$). For the corner scenario, the origin of these
coordinates is in the corner where the entrance is located, i.e., at
$(x,y)=(1.8,0)$. We also define $r_{\rm{max}}(\theta_{i})$ for each direction
$\theta_{i}$, which denotes the distance between the entrance and the wall in
that direction. To evaluate the positions of pedestrians, we use the
normalized distance $r_{i}/r_{\rm{max}}\in[0,1]$ for a fair comparison between
different directions. Fig. 3a shows a decrease of this ratio as the number of
pedestrians in the room increases. This clearly illustrates that pedestrians
fill the area from the boundary walls to the entrance. This could be
interpreted as anticipative behavior because the positions near boundaries are
generally farther away from the entrance (exit). The normalized distance for N
and O cases approximately follow the same curve. Meanwhile, in scenario C, the
deviation of the normalized distance for the first four pedestrians is larger
than for the other two scenarios, which reflects the inhomogeneity in the
pedestrian distribution, which we will discuss next.

Figure 3: (a) Normalized distance from the entrance to the final position of
pedestrians. Each error bar represents the standard deviation ($N=9$ trials).
(b) Time and trial averaged density profiles using Voronoi diagrams. Between
scenarios N and O, no significant difference was observed, whereas scenario C
displayed a distinctive pattern. (c) Mean normalized walking distance with the
standard deviation for the normal scenario. The normalized walking distance
was obtained by dividing the walking distance by the shortest distance between
the entrance and the pedestrian’s final position (see Eq. (2)). As the turning
angle increased, the walking distance was more extended. The same feature was
also observed for the other two scenarios. Each asterisk represents
statistical significance assessed by the Welch’s $t$ test ($P<0.05$).

Fig. 3b displays the degree of space occupancy (the average pedestrian density
integrated with respect to time for each trial; see also Appendix B). High
occupancy values indicate positions that tend to be already occupied at an
early stage in each trial. In scenarios N and O with the entrance in the
middle, boundaries are uniformly preferred, except for the corners near the
entrance, which are more attractive. In contrast, in scenario C, an asymmetric
distribution with respect to a diagonal line starting at the entrance is
observed. In addition to the top-left and bottom-right corners, the top-middle
boundary attracts pedestrians, showing characteristics distinct from those of
the scenarios with a middle entrance. This preference for the top-middle
boundary contributes to the large deviations in the normalized distance (Fig.
3a). These qualitative differences in distributions due to the entrance
position indicate the motivation of pedestrians in the process, which could be
also viewed as a strong indicator for anticipation. Pedestrians entering in
later stages have fewer options for finding positions, usually limited to the
middle of the area. For a better understanding of the underlying mechanisms,
it is therefore important to focus on the choice of position of the
pedestrians entering in the early stages of the experiment.

###  III.3 Interpretation and theoretical description

As we have seen, the decision for which position to take is made based on
anticipation. We discuss the factors that could affect the location choice of
pedestrians to interpret the results based on the following assumptions.
First, the decision is made by considering an expected ideal configuration
based on the knowledge of the total number of persons. In this article,
pedestrians anticipate an ideal uniform distribution, upon which they base
their evaluation of each location. Second, a trade-off between finding a
desirable location and minimizing costs to reach there is considered, such
that an option that reconciles these factors is realized.

Each factor is assumed as follows. For comfortableness, pedestrians prefer
positions where less disturbance is expected. Here, we consider avoidance
factors against two types of interferences that are suggested by the
occupation order shown in Fig. 2 and Fig. 3b. The first is (i) flow avoidance,
i.e., the desire to avoid any dynamical interference into the private space
caused by succeeding pedestrians passing by to reach their desired positions.
This factor should be salient near the entrance, as schematically shown in
Fig. 4a.

The other factor is psychological pressure in the (expected) final state. The
degree of discomfort caused by the presence of other persons is known to
strongly depend on the distance between them (i.e. the local density)
Worchel1976 . In this measure, the boundaries, in particular corners, are
preferable because the level of discomfort is lower due to fewer neighbors
(Fig. 4b; see also Appendix D.2). Hence we refer to the second factor as (ii)
boundary preference. Similar behavior has been reported in other situations,
e.g., seat preference in trains Evans2007  and classrooms Kaya2007 .

The presence of flow avoidance is supported by the fact that in scenario C,
the area in front of the entrance is less preferred, even though it is near a
boundary (Fig. 3b). Furthermore, the boundary preference is clearly seen in
the same figure as the middle area, which is not directly affected by the
inflow, is less preferred than the boundaries.

Figure 4: Schematic representations of the factors explaining experimental
results. (a) Flow avoidance (Eq. (4)). (b) Boundary preference (Eq. (5)). (c)
Distance cost (Eq. (6)). (d) Angle cost (Eq. (7)). (e, f) Preference
distribution obtained by superposing these four factors (a–d). For both panels
(e, f), the identical parameter values were used, except for the entrance
position. Parameters were set as
$(f,f_{1}^{2},f_{2}^{2},b,d,a)=(0.6,0.1,0.3,3,0.4,0.05)$ for a dimensionless
area, $(x,y)\in[-0.5,0.5]\times[-1,0]$.

Additional contributions are (iii) distance cost and (iv) angle cost. The
distance cost takes into account the effort that has to be made to reach a
comfortable position. A position becomes more unattractive if it is farther
away from the entrance. It is natural to assume that this distance factor is
isotropic as shown in Fig. 4c. The dependence on the distance is less obvious
and not necessarily linear. Distance cost explains why the area that is most
distant from the entrance (bottom-left) is not preferred in scenario C,
although it should be most attractive from the perspective of flow avoidance
and boundary preference (Fig. 3b).

The angle cost takes into account the effort necessary when changing the
direction of motion. When a pedestrian selects a position corresponding to a
large angle $|\theta_{i}|$, he or she has to immediately change the walking
direction to directly reach there. This leads to additional energy consumption
and consequently discomfort. During the experiments, in many cases the
participants avoided immediate turning but instead chose a longer but more
comfortable path following an arc with a gradual angle variation. In addition,
the approach to such positions requires immediate decision-making. Any delay
might increase the walking distance. This can be clearly observed in the
experiment (see Fig. 3c). Therefore it is reasonable to believe that the angle
cost is especially salient for large angles, as shown in Fig. 4d. This factor
explains why the preference exhibits distinctive patterns in the areas near
the two corners, i.e., the top-left and bottom-right corners in scenario C
(Fig. 3b). These two areas have the same conditions of distance from the
entrance and boundary preference. Also, the flow avoidance factor should not
be effective in these two areas.

We suggest that these intuitively plausible factors (i)–(iv) predominantly
affect the pedestrians decision-making behavior. For a demonstration, we
simply superposed these factors for scenarios N and C (Fig. 4e and Fig. 4f),
choosing parameter values such that realistic patterns were qualitatively
reproduced. We want to emphasize that this interpretation by the four factors
is at least capable of reproducing realistic patterns and might be useful for
understanding the underlying human collective behavior. It is not designed to
quantitatively predict each pedestrian’s decision. Thus, the functions and
parameters used in Fig. 4a–Fig. 4d should be considered as rough qualitative
estimates. Furthermore it needs to be critically checked whether these four
factors are additive or not. However, this formulation contributes rich
insights to the understanding of the inflow process and serves as starting
point for further investigations of this problem.

##  IV Discussions

We reported our experimental findings on the inflow process for three
different scenarios, i.e., with two different entrance positions and with an
obstacle. We observed that the position of the entrance had a significant
impact on the pedestrians’ choice of location, whereas the obstacle did not
largely affect pedestrian movement. In the experiments, we intentionally
selected the number of participants and the area size (i.e., the average final
density) such that inter-pedestrian interactions were of intermediate
strength. Although its validity should be further investigated, our
description of the results using the four factors suggests expected preference
patterns in different scenarios, which could be considered as a base-line
reference for future studies. For example, when only a small number of
pedestrians enter the area (and they know that), expected pressure will be
small, which would result in approximately no preference for boundaries.
Moreover, flow avoidance would be less important, making positions near the
entrance more attractive. This becomes obvious in the limiting case of a
single passenger in an elevator (see Fig. 5 for example visualizations).
Meanwhile, when the area is very large, the preference for the boundary would
be outweighed by the distance cost, and the area close to the entrance would
be preferred (see Fig. 6 for example visualizations). Furthermore, for people
in cultures with high tolerance against interference of personal space, the
boundary preference factor might become weaker, which should be investigated
via cross-cultural experiments. Also, the presence of different types of
individuals (e.g., females and children) and social groups (e.g., families and
friends) Moussaid2010 ; Bode2015 ; Gorrini2015  may affect their boundary
preference. Therefore, such differences might be understood through the four
factors, although our experimental results are based on limited scenarios. In
reality, other attractions are often observed. For example, in trains and
buses in rush hours, exit doors are more attractive for passengers who attempt
to leave first, which has been experimentally confirmed Liu2015 ; Liu2016 .
Such behavioral differences may appear based on the destination of each
passenger. In future studies, for which our findings are useful for providing
a base-line case, such a trade-off should also be investigated. Furthermore,
the alignment of head direction to an entrance could be diverted by attractive
objects. When there is a common understanding of the next motion (e.g.,
leaving through a separate exit) passengers are attracted to that direction
and align their heads accordingly.

Detailed models of pedestrian motion inevitably require to take unobservable
internal states of humans into account, e.g., for decision processes and
interactions between pedestrians. Our findings suggest some principles in
evaluating the current and future states around each individual. Pedestrians
try to avoid interference in their personal space, considering the cost that
depends on physical constraints (distance, angle, etc.). Such future
anticipation cannot be captured by one of the current paradigms of pedestrian
motion that is described by a driving force to a predefined target and local
avoidance Helbing2000 ; Yu2005 ; Chraibi2010 , but requires models that
include decision processes, like most cellular automata Kirchner2002 . For
this reason, the inflow process has not received much attention in pedestrian
dynamics research until now.

A theory of space evaluation of pedestrians could provide a missing link for
better understanding pedestrian motion. In addition, it could potentially lead
to important applications for designing comfortable and safe facilities. In
our experiment, the shape of the area was suitable for attaining a uniform
pedestrian distribution. Meanwhile, space in mass transport systems (e.g.
trains and buses) is sometimes not open due to the presence of seats and hand
rails that could distort or blind pedestrians’ space recognition, and lead to
inefficient inhomogeneous distributions and local overcrowding. In our
experiments, an additional obstacle had only a small effect on pedestrian
trajectories, while leaving the system dynamics mostly unaffected. However,
the final distribution and the location preference could change if the
obstacle were placed at a more obstructive location (e.g. just in front of the
entrance), or if several obstacles were placed in the area. This is a topic of
future research.

Figure 5: Predicted location preference for various congestion levels,
$\tilde{\rho}$. By assuming that the flow avoidance is proportional to the
number of pedestrians, we set $f=\tilde{\rho}f_{0}$, where $f_{0}(=0.6)$ is
the strength of the flow avoidance corresponding to $\tilde{\rho}=1$ (25
pedestrians). The other parameters were left unchanged:
$(f_{1}^{2},f_{2}^{2},b,d,a)=(0.1,0.3,3,0.4,0.05)$. Figure 6: Predicted
location preference for various area sizes, $\tilde{L}\times\tilde{L}$, with
the same number of pedestrians. The condition $\tilde{L}=1$ corresponds to the
size of the original experimental area in the main text, 3.6 m $\times$ 3.6 m.
According to the extension of the area, the normalized density is reduced as
$\tilde{\rho}=\tilde{L}^{-2}.$ The other parameters were left unchanged:
$(f,f_{1}^{2},f_{2}^{2},b,d,a)=(0.6,0.1,0.3,3,0.4,0.05)$.

## Appendix: Materials and Methods

##  Appendix A Trajectory detection

To extract the trajectory of each individual person, the software PeTrack was
used Boltes2013 . After deinterlacing the image, the lens distortion was
removed. Then, the extrinsic calibration, i.e., the camera position and angle
of view according to the moving plane, was performed for measuring the real
position of each head. The perspective distortion had only a minor influence
on the calculated position because the viewing angle was small owing to the
high mounting position of the camera at a height of 21 m. Hence an average
body height of 1.73 m for all persons could be assumed. The black and red dots
on the white cardboard worn on each head were detected by searching for
directed isolines of the same brightness with a subsequent analysis of their
shape (size, relative position to each other, and aspect ratio). The isovalue
for the brightness threshold varied over a large interval to cope with the
different lighting conditions during the experiment. The two dots were applied
to determine the orientation of the head, where the red dot indicated the face
side of the head (Fig. 1d). The position of each person was defined to be in
the middle of the black and red dots. To track a person over time, similar
pixels related to the marker on the head were searched in successive frames
such that the detected positions could be concatenated correctly over time for
the resulting trajectory.

##  Appendix B Time-averaged density profile

Fig. 3b was produced by averaging the temporal density of pedestrians whose
walking-velocity values were smaller than 0.15 m/sec, over entrance time and
trials. In each frame, a set of pedestrian positions in the experimental area
defines the Voronoi diagram (Fig. 2b; see also Ref. Zhang2011 ). Let $A_{i}$
be the area of each Voronoi cell for pedestrian $i$ (who is in the
experimental area). The density at a point ($x,y$) is defined as

$\rho(x,y)=\frac{1}{A_{i}}\qquad{\rm if}\quad(x,y)\in A_{i}.$ (1)

First, we divided the experimental area into squares of side 0.2 m. For each
square $j$, the time-averaged density ($\bar{\rho_{j}}^{\prime}$) was
calculated using the Voronoi diagram for each trial. In this procedure, we
used the trajectory data of eight frames per second between the entrance times
of the first and final pedestrians. Then after normalizing the average values
to neutralize the difference in entrance time
($\bar{\rho_{j}}^{\prime}\rightarrow\bar{\rho_{j}}$ such that the sum over the
experimental area becomes a constant value
$\sum_{j}0.2^{2}\bar{\rho_{j}}=\frac{25}{2}$ for every trial , they were
averaged over trials. If the area is uniformly used and pedestrian inflow rate
is constant, the time averaged density is calculated as
$\langle\bar{\rho_{j}}\rangle=\frac{1}{2}\frac{25}{3.6^{2}}=0.965\;\rm{ped}/\rm{m}^{2}$.

##  Appendix C Deviation of walking trajectory from the shortest path

We define $s_{i}$ as the walking distance of pedestrian $i$ from the entrance
to the final position. If a pedestrian directly moves to his final position,
it coincides with the shortest distance $r_{i}=|{\bm{r}}_{i}(t_{\rm{f}})|.$
Here ${\bm{r}}_{i}(t)$ is the position vector of pedestrian $i$ at time $t$,
and $t_{\rm{f}}$ is the time when a pedestrian reached the final position,
which was collected using the velocity threshold condition,
$t_{\rm{f}}=\min\\{t>t_{\rm{e}}||\dot{{\bm{r}_{i}}}(t)|<0.15\;\rm{m}/\rm{sec}\\}$
in practice (Fig. 3c). Here, $t_{\rm{e}}$ represents the pedestrian’s entrance
time. Using these variables, the degree of the walking path deviation is
defined as

$\frac{s_{i}}{r_{i}}=\frac{\int_{t_{\rm{e}}}^{t_{\rm{f}}}|\dot{\bm{r}}_{i}(t)|dt}{|{\bm{r}}_{i}(t_{\rm{f}})|},$
(2)

which is, by definition, always larger than $1$.

##  Appendix D Schematic representations of location preference

To capture the elusive features of pedestrian interactions, we attempt to
illustrate the factors that determine the choice of locations, and thereby
reproduce Fig. 3b by considering a simple superposition

$P(x,y)=F+B+D+A,$ (3)

where $P,F,B,D$ and $A$ are abstract cost functions for the location
preference, the gain by the flow avoidance, the boundary preference, the
distance cost and the angle cost, respectively. These factors are defined to
increase as the location becomes more desirable for pedestrians. Of course,
these factors are not necessarily additive, but it is natural to consider that
$P$ is a monotonically increasing function of $F,W,D$ and $A$. As the simplest
form, we tentatively adopt this definition of $P$ in this article.

###  D.1 Flow avoidance

To express the flow avoidance we assume a Gaussian function (Fig. 4a)

$F(x,y)=-f\exp{\left(-\frac{(x-x_{0})^{2}}{f_{1}^{2}}-\frac{(y-y_{0})^{2}}{f_{2}^{2}}\right)},$
(4)

where $f$, $(x_{0},y_{0})$, and $(f_{1},f_{2})$ represent a positive constant,
the location of the entrance and the widths of flow avoidance, respectively.
Considering the direction of flow, it is natural to presume $f_{1}<f_{2}$.
Pedestrians might anticipate other pedestrians’ choice, which would affect
this flow avoidance. However, it is a formidable challenge to include such
feedback. Thus we here forbear to go into detail and instead define $F$ a
priori. Since this factor only reduces the preference near the entrance, the
conclusion in this study is not significantly affected by a specific choice of
this function.

###  D.2 Boundary preference

When describing the psychological pressure between pedestrians, it is natural
to consider a function $g$ that decays with the distance between two
pedestrians (Fig. 6). In the uniform pedestrian distribution preconceived by a
pedestrian, the expected number of pedestrians in an area $dA$ at position
$\bm{r^{\prime}}\in D$ ($D:$ experimental area) is denoted by
$\tilde{\rho}dA$, where $\tilde{\rho}$ is the normalized density (pedestrians
per area). This area provides pressure against position $\bm{r}$, whose
magnitude is $g(|{\bm{r}}-{\bm{r}^{\prime}}|)\tilde{\rho}dA.$ Thus, the total
(expected) psychological pressure on position $\bm{r}$ is obtained by taking
the sum of this factor for all the positions ($\bm{r^{\prime}}$) in the
experimental area. For simplicity we take the limit of $dA\rightarrow 0$,
which yields

$B({\bm{r}})=-\int\int_{\bm{r^{\prime}}\in
D}g(|{\bm{r}}-{\bm{r}^{\prime}}|)\tilde{\rho}dA.$ (5)

Because the area out of the boundaries does not contribute to this value, the
areas of boundaries become more preferable (see Fig. 4a). Thus, this factor
explains pedestrians’ preference to the boundaries. The pedestrian density is
normalized as $\tilde{\rho}=1$ in this study (for 25 pedestrians in a 3.6
m$\times$ 3.6 m area). Here, we tentatively defined the discomfort function as
$g(x)=b(1+x)^{-2}$ (with a positive constant $b$), avoiding a singularity at
$x=0$ because pedestrians do not normally fear being physically overlapped (at
the densities considered here). It could also be defined with the singularity,
for which case, the integral should be performed except for human’s body area
$|{\bm{r}}-{\bm{r}^{\prime}}|<r_{0}$ (body radius). Both definitions produce
qualitatively the same preference, and thus we adopted the simpler one because
our interest is not in finding an exact description of the factor.

Figure 7: Closeness and degree of discomfort. In this article we assumed the
following function: $g(x)=b(1+x)^{-2}$, where $x$ represents the distance
between the two individuals (see also Eq. (5)).

###  D.3 Distance cost

The distance cost should be a monotonically increasing function of $|\bm{r}|.$
For simplicity we assume the linear expression

$D({\bm{r}})=-d|{\bm{r}}|,$ (6)

with a positive constant $d$ (Fig. 4c).

###  D.4 Angle cost

For the angle cost we selected a function that nonlinearly decreases with the
absolute value of $\theta$: (Fig. 4d)

$A({\bm{r}})=-a|\theta|^{\alpha}.$ (7)

Here $a$ and $\alpha$ are positive constants. In this article, we tentatively
assume $\alpha=3$.

We emphasize that the absolute values of $F,B,D$, and $A$ are not essential,
but their relative strengths are important. As the exact forms of these
functions cannot be easily determined, we used simple functions that satisfy
the requirements discussed in Sec. III.3. In Fig. 4, we selected the parameter
values as $(f,f_{1}^{2},f_{2}^{2},b,d,a)=(0.6,0.1,0.3,3,0.4,0.05)$ for a
dimensionless area, $(x,y)\in[-0.5,0.5]\times[-1,0]$. These parameter values
were manually set such that the density patterns become consistent with the
experimental results, as a rough estimate. Examples with different parameter
values are shown in Fig. 5 and Fig. 6.

##  Appendix E Ethics statement

This experiment has been approved by the ethics committee of the office for
life science research ethics and safety, The University of Tokyo. All
participants provided written informed consent to participate in the
experiment.

###### Acknowledgements.

We acknowledge Haruki Ishikawa for assistance in preparatory works on
pedestrian detection. We would like to thank Ryosuke Nishi and Marina Dolfin
for their valuable comments on this manuscript. TE, DY, and KN acknowledge
financial support by JSPS Grant Number 13J05086, 15K17583 and 25287026,
respectively.

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