#  Social aspects of collision avoidance: A detailed analysis of two-person
groups and individual pedestrians

Adrien Gregorj  Okayama University, Okayama, Japan  adrien-
<EMAIL_ADDRESS>Zeynep Yücel  Okayama University, Okayama, Japan
ATR International, Kyoto, Japan  Francesco Zanlungo  Okayama University,
Okayama, Japan  Osaka International Professional University, Osaka, Japan  ATR
International, Kyoto, Japan  Claudio Feliciani  The University of Tokyo,
Tokyo, Japan  Takayuki Kanda  ATR International, Kyoto, Japan  Kyoto
University, Kyoto, Japan

###### Abstract

Pedestrian groups are commonly found in crowds but research on their social
aspects is comparatively lacking. To fill that void in literature, we study
the dynamics of collision avoidance between pedestrian groups (in particular
dyads) and individual pedestrians in an ecological environment, focusing in
particular on (i) how such avoidance depends on the group’s social relation
(e.g. colleagues, couples, friends or families) and (ii) its intensity of
social interaction (indicated by conversation, gaze exchange, gestures etc).
By analyzing relative collision avoidance in the “center of mass” frame, we
were able to quantify how much groups and individuals avoid each other with
respect to the aforementioned properties of the group. A mathematical
representation using a potential energy function is proposed to model
avoidance and it is shown to provide a fair approximation to the empirical
observations. We also studied the probability that the individuals disrupt the
group by “passing through it” (termed as intrusion). We analyzed the
dependence of the parameters of the avoidance model and of the probability of
intrusion on groups’ social relation and intensity of interaction. We
confirmed that the stronger social bonding or interaction intensity is, the
more prominent collision avoidance turns out. We also confirmed that the
probability of intrusion is a decreasing function of interaction intensity and
strength of social bonding. Our results suggest that such variability should
be accounted for in models and crowd management in general. Namely, public
spaces with strongly bonded groups (e.g. a family-oriented amusement park) may
require a different approach compared to public spaces with loosely bonded
groups (e.g. a business-oriented trade fair).

## Introduction

Groups represent an important component of pedestrian crowds [1, 2], and
lately they have been the subject of many studies, focusing on such themes as
their effect on crowd dynamics [3], the observation of their shape and
velocity [4, 1, 5, 6, 7, 8, 9, 10, 11], mathematical and computational
modeling of interaction dynamics [12, 13, 6, 14, 15, 16, 17], and dependence
on social structure and interaction level [18, 19, 20, 21, 22]. Many works are
based on ecological observations [23, 24, 23, 5, 1, 6, 14, 7, 19, 22, 20, 21,
25, 26, 27], which may be argued to be indispensable in dealing with social
aspects of human behavior [28, 29, 30, 31, 32, 33, 5].

To plan safe buildings [34, 35, 36, 37, 38] and manage crowds in busy public
places (i.e. transportation hubs) and events [39, 40, 41], it is important to
consider group behavior in crowd simulators and monitoring/predicting tools
[42, 43, 44, 45]. This includes taking into account factors such as the
structure of different types of groups, their social relations [46], and
cultural differences [47, 48] for more accurate simulations. It is also
crucial to model not only the internal dynamics of the group, but also their
reaction to the external environment, such as the presence of other
pedestrians, while keeping in mind the aforementioned characteristics of each
group. Furthermore, it is essential to consider the impact of the group on
other pedestrians and how their behavior may be influenced by the presence of
the group.

The emergence of autonomous navigating agents in industries like smart
vehicles, assistive robots, and drones has led to a significant increase in
research attention towards collision avoidance in recent years. Human beings
are naturally good at avoiding each other and the famous Shibuya crossing, in
Tokyo, is a good illustration of our ability to avoid collision, even in
crowded environments. Consequently, researchers tried to model this ability in
various ways [49, 50, 51]. Early on, in the Social Force Model [52], a
repulsive force between particles (representing pedestrians) was used to
account for collision avoidance. Such models have yielded good results when
used in tandem with path-finding algorithms to navigate autonomous agents
[53]. More recent research has delved into the specific scenario of pairwise
avoidance during face-to-face encounters among pedestrians, using real-life
trajectory data[27, 26, 54]. They measured the deviation of pedestrians from
their expected undisturbed trajectories when encoutering others (1vs1 and 1vsN
scenarios) and compared the results to a Langevin-like physics model. They
notably showed that interaction with multiple incoming pedestrians is better
discribed using non-linear superposition of short-ranged contact avoidance
forces.

However, the majority of the studies mentioned above concentrate solely on the
collective behavior of groups, i.e. the dynamics that drive the group to move
as one cohesive unit. Mathematically oriented works on group dynamics may
introduce the concept of a “group potential” and examine it by assuming that
interactions with pedestrians outside the group can be approximated as white
noise [6] or as an external “mean-field” potential [14] on average. Similarly,
observation-based works [7, 21, 19, 22] often describe group properties using
probability density functions defined by an average process that neglects the
specifics of the environment.

On the other hand, when introducing group behavior into a microscopic
simulator, it is essential to incorporate specific rules that describe
interactions between the group and the environment, particularly with
surrounding pedestrians. Introducing in-group dynamical rules is a logical
starting point, and these can simply be added to the collision-avoidance rules
used for individuals. For instance, if a two-person group encounters a single-
walking pedestrian in an acceleration-based or “Social Force” model [55, 52,
56], the acceleration terms of the group’s pedestrians can be obtained by
summing the individual-individual collision avoidance term and those resulting
from in-group interaction [57]. The behavior of the lone pedestrian can be
modeled by adding the two avoidance terms with respect to the pedestrians in
the group. In other words, collision avoidance is treated as a one-to-one
behavior, group dynamics are regarded as an exclusively in-group phenomenon,
and the overall dynamics is the sum of these distinct components.

This is the (classical) linear superposition principle [58], which is
typically assumed (and verified) in mechanics and simplifies physical models
and their dynamics to a great extent. However, this principle does not
necessarily apply to pedestrian dynamics. For instance, when a single walking
pedestrian encounters a two-person group walking side by side, particularly if
they are socially interacting, he or she may choose to avoid walking through
them, even if it seems like the best choice from a pure collision-avoidance
perspective. This intrusion decision would be likely, should the two
pedestrians be perceived as independent.

In this work, we investigate a relatively unexplored aspect of pedestrian
behavior and crowd dynamics: collision avoidance of (or against) groups, and
in particular, its dependence on the groups’ “social attributes”, which refer
specifically to social relation and intensity of interaction.

In doing that, we use two data sets of pedestrian trajectory including
annotations of groups’ social attributes to investigate the nature of
individual-group collision avoidance. Moreover, we focus particularly on
groups composed of 2 people (i.e. dyads), since they are much more common than
larger ones and they actually constitute their fundamental building block
together with triads (i.e. for an easy navigation and social interaction,
large groups break into sub-groups of 2 or 3 people) [5]. In addition, larger
groups (of 3 or more people) may require a categorization of pair-wise social
relations or interactions, which may be very complex to formulate or
generalize. In that respect, we use the word group to refer simply to dyads.
As groups’ counterpart in collision avoidance, we focus on individuals, which
is a term we use to refer to people not appertaining to a group.

## Methods

### Data sets

In this study, we used two data sets, namely the ATC data set and DIAMOR data
set, both of which are reviewed and approved for studies involving human
participants by the ATR ethics board [7, 6], are publicly available and
contain trajectories derived from range data [59, 60, 61]. From these
trajectories, we computed the normalized cumulative density maps of the
experiment environments shown in Figure 1.

Figure 1: The normalized cumulative density maps for (a) the ATC data set and
(b) DIAMOR data set. The environment is discretized as a 2D mesh with a grid
cell size of 10 cm by 10 cm, and the number of observations is counted in each
grid cell. Normalization refers to the scaling of this histogram with its
maximum value.

The data sets are annotated based on video footage for different social
attributes of groups. Specifically, the ATC data set is annotated from the
viewpoint of social relations, whereas the DIAMOR data set is annotated from
the viewpoint of the intensity of interaction.

For the ATC data set, possible options for social relation are couples,
colleagues, family and friends, which are determined through the domain-based
approach of Bugental [62] and correspond to the domains of mating,
coalitional, attachment and reciprocal, respectively. This annotation process
yields the values presented in Table 1-(a).

For the DIAMOR data set, the intensity of interaction is evaluated at 4
degrees, 0 representing no-interaction and 1, 2, and 3 representing weak, mild
and strong interaction, respectively. This annotation process yields the
values presented in Table 1-(b). Note that in order not to bias the coders’
assessment, we only defined the number of interaction levels as 4, but we did
not give any guidelines on what can be considered as weak, mild or strong
interaction [63]. Instead, we let the coders grasp a feeling about different
intensities of interaction through a free-viewing task (i.e. letting them
watch the videos for 3 hours before giving any labels, see Supplementary
Information Section 1 for further details.)

Table 1: Number of groups annotated with each (a) social relation (in ATC data set) and (b) intensity of interaction (in DIAMOR data set). (a) | (b)
---|---
| Social relation |  $\\#$ of annotations
---|---
Couples | 69
Colleagues | 314
Family | 180
Friends | 253
Total | 816
| Intensity of interaction |  $\\#$ of annotations
---|---
0 (no interaction) | 140
1 | 159
2 | 460
3 (strong interaction) | 100
Total | 859

### Approach

Provided that the group and the individual do not perform any collision
avoidance, we can expect their (relative) motion to be approximated by a
straight line. We are aware that this assumption requires implicitly the
environment to be sufficiently straight and wide (e.g. like DIAMOR, see Figure
1-(b) and the discussion in Supplementary Information Section 7) and is valid
up to a reasonable range (i.e. over a few meters). Namely, in environments
with complex geometries (curved or with many obstacles, intersections etc.),
the pedestrians need to deviate as part of their interaction with the
boundaries. Similarly, over long distances, they will eventually meet some
walls, or divert towards different goals, making their relative motion bent.
Nevertheless, in a sufficiently straight corridor and on a scale of few
meters, we can expect it to be a good approximation.

This trivial assumption can be considered to serve as a hypothesis, opposite
to what we actually anticipate. Based on such a hypothesis, the deviation of
(relative) motion from a straight line can be attributed to group-individual
collision avoidance. Specifically, by measuring this deviation with respect to
different social relations or intensities of interaction (of the group), we
may understand the reflections of such group attributes on collision
avoidance.

Figure 2: (a) Illustration of the scattering problem in physics. A mobile
particle (in blue) is projected toward a fixed particle (in green). The impact
parameter, $b$, is the straight-line distance between the particles, and
$r_{min}$ is the closest approach. The particle is deflected with an angle
$\theta$. (b) A typical pedestrian avoidance situation in the group-centered
reference frame. The individual enters the vicinity of the group (gray region)
at time $t^{\prime}$ with velocity $\mathbf{v}_{i}$. The straight-line
distance from the individual to the group is denoted by $r_{b}$. At time
$t_{c}$, the individual is closest to the group at a distance of $r_{0}$.

This formulation presents a striking resemblance to one of the fundamental
problems of Physics, namely, the “scattering problem”, where a “particle”
(blue ball in Figure 2-(a)) is shot on a “target” (green ball), and its
deviation from the straight line motion is used to study the interaction
potential. In the original scattering problem (see Figure 2), this deviation
is assessed by accounting for the straight-line distance $b$ (called the
impact parameter) and closest approach $r_{\text{min}}$, which is derived from
the scattering angle $\theta$, as an accurate measurement of particles’
location is very difficult. By repeating the experiment with different impact
parameters and estimating the corresponding closest approach $r_{\text{min}}$,
one can get an approximation for the potential acting on the particle.

In this study, we establish a simple duality relation between the above-
mentioned problem and our group-individual collision avoidance scenario.
Namely, the impact parameter $b$ is replaced with a straight-line distance
$r_{b}$ and the closest approach can simply be measured as the shortest
distance $r_{0}$ between pairs of trajectory data points of the group and the
individual. In Section Observables we elaborate in detail on how we define
these observable quantities.

Using this approach from physical sciences to describe human behavior
represents obviously a strong approximation, not only because human behavior
is too complex to be modeled through simple physical forces, but also because
it completely ignores the effect of the environment, which, in physical
parlance, is equivalent to a strong and non-uniform external force.
Nevertheless, as we will see, this approach still allows us to grasp the
fundamentals of the collision dynamics between groups and individuals and to
quantify the interaction. At this point, it is also worth stressing that the
proposed model in Section Modeling is aimed at assessing the effect of
different social attributes in a qualitative way, rather than reproducing
quantitatively human behavior.

### Observables

#### Data preparation and transformation

We first carry out a data preparation step by (i) removing atypical/non-
characterizing motion (waiting, running etc.), (ii) representing the group as
a single unit (its geometrical center) and (iii) focusing on frontal
encounters of groups and individuals, for which we expect the pedestrians to
be able to judge the social attributes of the incoming party.

Subsequently, we transform trajectories of the group $g$ and the individual
$i$ to a reference frame, which is co-moving with the group. Namely, at each
time instant (i) the positions of the group and the individual
$\mathbf{r}_{g,i}$ are translated such that the group (center of mass) is
positioned at the origin and (ii) their velocities $\mathbf{v}_{g,i}$ are
rotated such that the velocity of the group is directed towards $x^{+}$.
Finally, the velocities $\mathbf{v}_{g,i}$ are translated by
$-\mathbf{v}_{g}$, rendering the group immobile. The main purpose of this
transformation is to provide an easier visualization of relative position in
2D, which represents the position of the individual with respect to the group
center, while having the group motion as a preferential direction. On the
other hand, most of our analysis is based on the absolute value of the
relative distance between the group center and the individual, which is
rotationally invariant and independent of frame choice.

#### Relative distance r

Similar to Corbetta et al.[27], our analysis is based on $\mathbf{r}$, the
relative position between the group center and the individual,

$\mathbf{r}(t)=\mathbf{r}_{i}(t)-\mathbf{r}_{g}(t).$ (1)

Its time derivative is the relative velocity $\mathbf{v}$,

$\mathbf{v}(t)=\mathbf{v}_{i}(t)-\mathbf{v}_{g}(t).$ (2)

The absolute value (norm) of $\mathbf{r}$ is simply denoted as $r$.

#### Straight-line distance $r_{b}$

The straight-line distance $r_{b}$ is computed as the shortest distance from
the origin (i.e. translated position of the group) to the line, which passes
through the point at which the individual enters a pre-defined vicinity around
the group termed as window of observation. This refers to the area in the
group-centered reference frame from $-W$ to $W$ meters both along $x$ and $y$
axes (i.e. along the group’s motion direction and the direction orthogonal to
that). Empirically a $W$ of 4 m is seen to contain the most significant part
of the group-individual collision avoidance (see Figure 2-(b) for an
illustration. Refer to previous literature [64, 65] and Supplementary
Information Section 3 for details on the choice of $W$).

Let $t^{\prime}$ be the time instant at which the individual enters $W$ and
let $\mathbf{r}(t^{\prime})$ be its relative position at that instant.
According to the hypothesis mentioned in Section Methods, provided that there
is no collision avoidance the individual will follow a path starting at
$\mathbf{r}(t^{\prime})$ and move along its velocity vector at that instant
$\mathbf{v}(t^{\prime})$. In this case, the straight-line distance $r_{b}$ can
be computed as the shortest distance between this line and the origin (i.e.
translated position of the group),

$r_{b}=\frac{||\mathbf{r}(t^{\prime})\times\mathbf{v}(t^{\prime})||}{||\mathbf{v}(t^{\prime})||}.$
(3)

In the analysis, in order to alleviate the impact of orientation noise on the
velocity of the individual, we averaged its velocity vector over 4 time
instants (before $t^{\prime}$) and used this mean velocity in Equation 3
instead of $\mathbf{v}(t^{\prime})$.

#### Observed minimum distance $r_{0}$

The observed (i.e. actual) minimum distance $r_{0}$ between the group and the
individual is simply,

$r_{0}=\min_{t}\big{(}r(t)\big{)}=r(t_{c}),$ (4)

where $t_{c}$ is the time instant at which the individual is closest to the
group. The time steps, at which pedestrian positions are recorded, are
obviously discrete. Nevertheless, in order to have a more accurate estimation
of $r_{0}$, one can also interpolate $\mathbf{r}(t)$ between two consecutive
time steps $t_{k}$ and $t_{k+1}$ by using the velocity vector at time $t_{k}$,

$\mathbf{r}(t)\approx\mathbf{r}(t_{k})+(t-t_{k})\mathbf{v}(t_{k}),\;t\in[t_{k},t_{k+1}).$
(5)

This procedure allows detecting minimum distances not only exactly at sampling
instants, but also at intermediate time points between consecutive samples,
which yields a much more accurate estimation of $r_{0}$ (refer to
Supplementary Information Section 4 for details).

#### Scaled distances

Groups’ interpersonal distance is shown to depend on their social relation and
interaction intensity [21, 20]. Thus, we represent the distances defined above
in two ways: in a group-independent way (in meters) and in a group-dependent
way, in which the unit of distance is the average interpersonal distance of
dyads with the given social bonding [19]). In the text, we denote distances
measured in meters with the normal font (e.g. $r$) and scaled distances
measured in interpersonal distance units with a bar (e.g. $\bar{r}$). Since we
observed that results concerning scaled values are in general easier to
interpret, in the main text we mostly report those (for further details, refer
to Supplementary Information Section 5).

### Analysis of collision avoidance

As mentioned in Section Methods, our study of the collision avoidance dynamics
between groups and individuals is fundamentally based on examining the
relation between $r_{b}$ and $r_{0}$. The relation between these two
observables, although defined in a slightly different way, has been studied in
a similar way for 1-1 encounters by Corbetta [27]. In what follows we define
two different methods to analyze this relation and then propose a method to
model it.

#### Empirical relation between $r_{b}$ and $r_{0}$ and its statistical
analysis

To examine the distribution of $r_{b}$ versus $r_{0}$, the values of
$\bar{r}_{b}$ are quantized into bins of 0.5 unit and for each bin, the
average and standard error of the corresponding values of $\bar{r}_{0}$ are
computed. The choice of 0.5 as bin size was primarily driven by empirical
observations. For certain combinations of distance $r_{b}$ (or $\bar{r}_{b}$)
and bonding (social relation or interaction level), setting a smaller bin size
results in having bins with little or no data. Conversely, using a larger bin
size decreases the resolution. In that respect, we consider a bin size of 0.5
to strike a balance between these competing factors. The results will be
presented and discussed in Section Results on the relation between
$\bar{r}_{0}$ and $\bar{r}_{b}$.

#### Intrusions

Small groups, such as dyads, have been shown to usually prefer deviating to
avoid splitting [66]. Nonetheless, we found situations where the individual
passes through the group (i.e. between group members), and we refer to them as
“intrusion”. For simplicity’s sake, we define the probability of intrusion as
the probability of having $r_{0}$ smaller than the group interpersonal
distance (see Section Scaled distances). We perform a statistical analysis to
investigate the dependence of intrusion on the social attributes of the group.
The results will be shown and discussed in Section Intrusion, whereas the
details of the computational procedure can be found in Supplementary
Information Section 6.

#### Modeling

Many models of pedestrian collision avoidance are based on “Social Forces”
[55, 52, 56], which may be defined through a potential. It has been reported
that using position-dependent potentials in modeling of pedestrian collision
avoidance fails to reproduce detailed behavior. Even if we assume that a
“Social Force” approach may reproduce actual human behavior, the corresponding
potential should at least be velocity-dependent and based not on current
distance but on future distance at the moment of predicted closest approach
[56, 15]. Nevertheless, determining a potential that may, at least
qualitatively, describe the collision avoidance between groups and
individuals, represent an important first step towards a more realistic
quantitative modeling.

Let us first review how we can study the potential energy between two
interacting bodies in physics (note that while discussing the physical model,
we use the word “interaction” to refer to the effect that the bodies exert on
each other.). The study of such a “scattering” problem is obviously a
cornerstone of physics, and the non-Quantum formalism analyzed in this section
was used to study such important problems as the structure of atoms [67] and
gravitational lens effect due to space-time curvature [68] among others. In
general, the “bodies” in focus are very complex and composed of many particles
(e.g. planets, stars). Nevertheless, due to the scale of the problem, they may
be treated as point particles themselves (in our “pedestrian scenario”, the
group will be represented with a single point).

Their interaction is determined by a potential energy $U(\mathbf{r})$, which
is in general a function of only relative position (a result connected to
invariance under space translations and equivalent to Newton’s third law
[69]). Nevertheless, in many important applications, the potential is central,
i.e. rotationally invariant, and depends only on the magnitude of the
distance, $U(r)$.

In such a case, it is shown that the interesting (potential-dependent)
dynamics is studied in the $r$ variable [67]. Defining the reduced mass $\mu$
as

$\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}},$ (6)

where $m_{1}$ and $m_{2}$ are the masses of the two bodies, the angular
momentum $\mathbf{L}$ and energy $E$ result to be constants of motion,

$E=\frac{\mu}{2}\dot{r}^{2}+\frac{||\mathbf{L}||^{2}}{2\mu r^{2}}+U(r).$ (7)

In a scattering problem, the system is not bound and $r$ diverges for
$t\to\pm\infty$. In most physical applications, we can only measure the
scattering angle and the velocity far before/after the interaction. We can
then compare the measured angle with a theoretical result involving an
integral. However, if the full trajectory is known, a simpler way to study the
system is available.

We study the system far before interaction, i.e. for $t\to-\infty$ and
$r\to+\infty$, and call the corresponding asymptotic speed $v_{\infty}$. We
see that the absolute value of angular momentum can be written as

$L\equiv||\mathbf{L}||=\mu v_{\infty}b,$ (8)

where the impact parameter $b$ is the minimum value of $r$ assumed in case of
straight motion (i.e. no interaction). Assuming
$\lim\limits_{r\to+\infty}U(r)=0$, we obtain

$E_{\infty}=\frac{\mu}{2}v_{\infty}^{2}.$ (9)

On the other hand, since we actually have interaction, the minimum distance
$r_{\text{min}}$ in the observed trajectory turns out to be different than
$b$, i.e. $r_{\text{min}}\neq b$. At $r=r_{\text{min}}$, having a minimum, we
have $\dot{r}=0$ and the corresponding energy is

$E_{0}=\frac{\mu v_{\infty}^{2}b^{2}}{2r_{\text{min}}^{2}}+U(r_{\text{min}}).$
(10)

Conservation of energy implies $E_{\infty}=E_{0}$ and provides the following
relation for the value of $U(r)$ at $r=r_{\text{min}}$

$U(r=r_{\text{min}})=\frac{\mu
v_{\infty}^{2}}{2}\frac{r_{\text{min}}^{2}-b^{2}}{r_{\text{min}}^{2}}.$ (11)

This relation enables studying the potential $U(r)$, provided that
$v_{\infty}$, $b$ and $r_{\text{min}}$ are measured. In modeling collision
avoidance between pedestrians based on the above framework, we assume that

$\frac{\mathrm{d}U(r)}{\mathrm{d}r}<0\;\forall r\Rightarrow U(r)>0\;\forall
r.$ (12)

In other words, the force is assumed to be repulsive. Namely, denoting
$\mathbf{F}_{1}$ as the force acting on body 1, and recalling the usual
definition $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$, we have

$\mathbf{F}_{1}=-\boldsymbol{\nabla}U(r)=-\frac{\mathrm{d}U(r)}{\mathrm{d}r}\frac{\mathbf{r}}{r}.$
(13)

We apply these physical concepts in a pedestrian scenario to model the
“collision avoidance potential” between groups and individuals. As mentioned
in Section Methods, $r_{b}$ is inspired by the impact parameter $b$, whereas
$r_{0}$ corresponds to the closest approach $r_{\text{min}}$. Thus, the term
$v_{\infty}$ in Equation 11 should be approximated by using the relative
velocity when $r_{b}$ is computed. But since pedestrian velocities have a
small variation, we may consider it to be almost constant. In a similar way,
as usual when studying “forces” that determine the pedestrians’ cognitive
decisions, all masses are considered to be equal (to one) [55] and we may
remove $\mu$ from the equation. Finally, since the approach is completely of a
qualitative nature, we opt for ignoring the overall constant in Equation 11
and study the following simplified version,

$U^{\prime}(r=r_{0})=\frac{r_{0}^{2}-r_{b}^{2}}{r_{0}^{2}},$ (14)

to which we will refer to as the “collision avoidance potential” (defined as a
dimensionless pure number).

A comment on Equation 14 is probably needed. This equation does not represent
the functional form of the dependence of the potential on $r$. Instead, it
shows which is the value of $U$ attained at $r_{0}$ given that the straight-
line distance is $r_{b}$. Different values of $r_{b}$ allow us to probe
different values of $U$, where the smaller $b$ is, the higher $U^{\prime}$ is.
Nevertheless, Equation 14 clearly allows us only to probe values
$U^{\prime}<1$. This is due to the fact that in the computation of
$U^{\prime}$ the value of the initial kinetic energy

$\frac{\mu v_{\infty}^{2}}{2}$ (15)

is taken as fixed, and we are measuring the probed values of the collision
avoidance potential as multiples of such kinetic energy. Note that in particle
physics short distances are indeed probed by using very high kinetic energies.

The results are shown and discussed in Section Potential, whereas the details
of the computational procedure are described in Supplementary Information
Section 7. In addition, in Section Comparison to individual-individual
collision avoidance we also show the results concerning a similarly defined
potential describing individual-individual collision avoidance.

## Results and discussion

### Results on relative frame pdfs

The group-centered reference frame is particularly suitable to study the 2D
distribution of $\mathbf{r}$, i.e. of the position of the individual around
the group. Figures 3 and 4 show the 2D distributions in relation to different
social relations and interaction intensities of the group, respectively, using
as a distance unit the groups’ average interpersonal distance. Note that, in
order to highlight the specificities of each social attribute as compared to
the whole, we depict the difference between a given attribute and the overall
2D average, which is computed as an unweighted average of the distributions of
all relating cases. Therefore, positive values depict an increased likeliness
of presence for the individual, while, reciprocally, negative values depict a
decreased likeliness.

Figure 3: 2D probability distribution of individuals’ position
$\bar{\mathbf{r}}$ relative to overall average. Positions are shown in the
group-centered reference frame and the $x$ axis is aligned with the direction
of motion of the group. Each sub-figure depicts the difference between the
distribution relating to a certain social relation and an unweighted average
concerning all social relations. (a) Colleagues, (b) couples, (c) families,
(d) friends. The color scales are adjusted for highlighting the differences.

The effects of varying social relations are presented in Figure 3. Comparing
Figure 3-(a) with Figure 3-(b) and (d), one may notice that individuals do not
have a prominent preference to pass on the right or left side of colleagues,
whereas they prefer to pass more on the right for couples and on the left for
friends (as compared to the overall average). In addition, they pass with a
very small distance ($r\approx 0$) more often for families (see Figure 3-(c))
than for other kinds of social relations, which may be due to a more dispersed
configuration of family group members [21]. On the other hand, in Figure 3-(b)
we see very clearly two low probability horizontal stripes, roughly located
around $y=\pm 1$. As these stripes correspond more or less to group members’
positions, they suggest that the group’s abreast formation is rarely disturbed
in couples.

Figure 4: 2D distribution of individuals’ position $\bar{\mathbf{r}}$ relative
to overall average. Positions are shown in the group-centered reference frame
and the $x$ axis is aligned with the direction of motion of the group. Each
sub-figure depicts the difference between the distribution relating to a
certain intensity of interaction and an unweighted average of all intensities.
(a) 0, (b) 1, (c) 2, (d) 3. The color scales are adjusted for highlighting the
differences.

Concerning social interaction, the difference with respect to varying
intensities is much more noticeable, the most interesting one being between 0
and 3 (see Figures 4-(a) and (d)). Namely, concerning groups annotated as non-
interacting (i.e. with 0 intensity of interaction), the center stripe presents
positive values, while the lower and upper stripes $y\approx\pm 2.5$ present
negative values, indicating that individuals are more likely to maintain a
trajectory directly facing the group (possibly even intruding it) (see Figure
4-(a)). Reciprocally, from Figure 4-(d) we can see that individuals are less
likely to position themselves on a colliding trajectory with the group and
prefer to place themselves on its side, when it has a high intensity of
interaction. There are interesting left/right asymmetries in Figure 4, which
may be related to the tendency of Japanese pedestrians to move mainly on the
left, and overtake on the right [70]. This tendency may cause low-interaction
groups, when they are not intruded on, to have a relatively higher possibility
to be passed on their left than on their right, since they are expected to
have a higher speed than highly interacting ones. We do not have a clear
interpretation for the right/left asymmetry between the couples distribution
in Figure 3-(b) and the friends distribution in Figure 3-(d).

### Results on the relation between $\bar{r}_{0}$ and $\bar{r}_{b}$

Figure 5: Observed minimum distance $\bar{r}_{0}$ as a function of the
undisturbed straight-line distance $\bar{r}_{b}$ (a) for various social
relations and (b) intensities of interaction of the group. Error bars report
standard error intervals. The dashed line corresponds to the
$\bar{r}_{0}=\bar{r}_{b}$ linear dependence. $p$-values for the ANOVA of
$\bar{r}_{0}$ (c) for various social relations and (d) intensities of
interaction of the dyad. In (c), results for $\bar{r}_{b}<1$ are not displayed
as very low values were obtained ($p<10^{-6}$).

We divide the range of $\bar{r}_{b}$ into bins of 0.5 unit and compute the
mean and standard error of $\bar{r}_{0}$ corresponding to each bin. The
results are depicted in Figures 5-(a) and (b) and are similar to previous
results[27]. Smaller values of $\bar{r}_{b}$ indicate that the straight line
trajectory of the individuals would require them to pass very close to the
group. In addition, $\bar{r}_{b}<0.5$ signifies a distance smaller than half
of the group interpersonal distance, which means that the individual would
need to intrude on the group (if moving straight).

Concerning social relations, we observe that when $\bar{r}_{b}<0.5$, the
average value of $\bar{r}_{0}$ is considerably larger for couples and friends
than for colleagues and families (see Figure 5-(a)). In other words, there is
a strong resistance against intruding on groups with the former social
relations. Concerning intensity of interaction, we have a similar observation
for higher intensities of interaction (from 1 to 3, see Figure 5-(b)), while
for 0 intensity we have $\bar{r}_{0}\approx\bar{r}_{b}$ for all $\bar{r}_{b}$
values.

These observations make us believe that the social attributes of the group do
impact group-individual collision avoidance. Specifically, there is larger
avoidance, when there is a strongly-bonded group involved (i.e. couples,
friends or with high intensity of interaction).

The statistical significance of these results can be assessed through an ANOVA
(see Supplementary Information Section 8 for considerations regarding the
necessary assumptions). To that end, we compute the $p$ values concerning each
bin shown in Figures 5-(a) and (b) and demonstrate the results in Figures
5-(c) and (d), respectively. Regarding lower values of $\bar{r}_{b}$ (i.e.
$\bar{r}_{b}<1.5$), we observe statistical significance (i.e. $p<0.05$)
concerning both social relation and intensity of interaction. Regarding larger
values of $\bar{r}_{b}$ (i.e. $\bar{r}_{b}>2$), there is no statistically
significant difference, as it can be expected observing the overlapping curves
in the corresponding regions of Figures 5-(a) and (b).

Figure 6: Probability that the distance $\bar{r}_{0}$ is smaller than 1 for
(a) for various social relations and (b) intensities of interaction of the
dyad. Pearson’s $\chi^{2}$ p-values for the hypothesis of independence of the
frequencies of samples verifying $\bar{r}_{0}<1$ for (c) for various social
relations and (d) intensities of interaction (of the group).

Let us also notice that in Figure 5-(b) concerning the DIAMOR data set, for
$\bar{r}_{b}\gg 1$, we have $\bar{r}_{b}\approx\bar{r}_{0}$ regardless of
intensity of interaction, in agreement with the hypothesis that collision
avoidance can be ignored for such values (Equation 14). The fact that this is
not the case in ATC, where we actually observe $\bar{r}_{b}>\bar{r}_{0}$ for
$\bar{r}_{b}\gg 1$, is considered to be an effect of the ATC environment being
less straight and narrower (see Figure1-(a)).

To compensate for this effect in the computation of the potential, we perform
a linear correction in the computation of $\bar{r}_{b}$ in Section Potential.
The details of this correction are presented in Supplementary Information
Section 7. In addition, results concerning the relation between $r_{b}$ and
$r_{0}$, i.e. values measured in meters and not scaled with group
interpersonal distance, are shown in Section Comparison to individual-
individual collision avoidance.

### Intrusion

It is noticeable that the observed minimum distance $\bar{r}_{0}$ reaches
particularly low values in some encounters. For instance, the first bin in
Figure 5-(b) for intensity of interaction 0 presents an average value of
$\bar{r}_{0}$ smaller than 1. This means that the distance from the center of
mass of the group to the individual gets smaller than the group interpersonal
distance (see Supplementary Information Section 5). In such cases, it is
likely that the individual is actually intruding on the group instead of
deviating, essentially following the straight line trajectory.

To quantify the frequency of such intrusions, we computed the probability of
$\bar{r}_{0}$ being smaller than 1. Specifically, this is an empirical
probability computed as the ratio of the number of observations with
$\bar{r}_{0}<1$ to the total number of observations (for a given bin of
$\bar{r}_{b}$). The results are shown in Figures 6-(a) and (b). Here, we see
that there is indeed a correlation between the probability of intrusion and
the social bonding of the group being intruded on. Namely, individuals have a
higher probability to intrude on loosely-bonded groups (i.e. colleagues,
families and non- or slightly-interacting groups) than strongly-bonded groups
(couples, friends and strongly interacting groups).

The statistical significance of this observation is assessed through Pearson’s
$\chi^{2}$ test and the relating $p$-values are presented in Figures 6-(c) and
(d). The difference in probability of intrusion concerning different social
relations is significant ($p<0.05$), when $\bar{r}_{b}$ is smaller than 1.5.
On the other hand, for the intensity of interaction we have a significant
difference of intrusion for $\bar{r}_{b}<1$.

Actually, the average distance of a group member from the group center is
$\bar{r}_{0}=0.5$. The corresponding analysis for the probability of having
$\bar{r}_{0}<0.5$ is shown in Supplementary Information Section 6.

### Potential

Figure 7: Collision avoidance potential $U^{\prime}(\overline{r}_{0})$ (a) for
various social relations and (b) intensities of interaction of the group.
Dashed lines correspond to a power function fit of the quantized data.
Collision avoidance potential $U^{\prime}(r_{0})$ (c) for various social
relations and (d) intensities of interaction of the group. Dashed lines
correspond to an exponential fit of the quantized data. (c) and (d) report a
comparison to individual-individual (non-group) interaction using non-scaled
distances.

As described in Section Modeling, we study $U^{\prime}(r_{0})$ (see Equation
14) to model the “potential” representing the interaction between the group
and the individual. To that end, we again quantize the values of $\bar{r}_{b}$
and compute the corresponding mean values of $\bar{r}_{0}$ before calculating
the values of the potential $U^{\prime}(\bar{r}_{0})$ for each bin. Figures
7-(a) and (b) show the relating values. Additionally, to extrapolate outside
the range available, a function of the form ${k}/{r^{\beta}}$ is fitted to the
data using non-linear least squares, illustrated with dashed lines in Figures
7-(a) and (b).

Interestingly, the potential is shown to be affected by the nature of the
social bonding of the group. As a matter of fact, stronger bondings (e.g.
couples, high intensity of interaction) generate a “stronger potential” (i.e.
with a steeper negative derivative) which, as seen in Sections Results on the
relation between $\bar{r}_{0}$ and $\bar{r}_{b}$ and Intrusion “causes”
individuals to deviate more, and significantly decreases their probability to
intrude on the group. On the other hand, loosely-bonded groups (e.g.
colleagues and non- or slightly-interacting groups) generate a weaker
potential, resulting in a smaller deviation and a higher chance of intrusion.

The discussion above concerns results obtained using distances scaled with the
group interpersonal distance; results concerning computations performed using
distances measured in meters are shown in Section Comparison to individual-
individual collision avoidance.

### Comparison to individual-individual collision avoidance

Many practitioners simulate crowds on the basis of individuals. Thus, it is
interesting to compare the above-mentioned potentials with results obtained
for individual-individual interaction. The results (using distances measured
in meters, i.e. not scaled by group interpersonal distance) are shown in
Figures 7-(c) and (d).

Note that groups are larger (than individuals) and expected to exert a
stronger “social force”, but they are also susceptible to being disrupted and
intruded on (passed at $\approx 0$ distance to their geometrical center).
Also, while it is expected (statistically) that collision avoidance between
individuals is symmetric, it may be that groups interact less than individuals
by deviating very little. These effects seem to balance and potentials for
collision avoidance between individuals are quite similar to those with
groups.

Nevertheless, it may be seen that potentials describing low intensity social
interactions, colleagues and families have typically a less steep derivative
than the one for individual-individual encounters, while the opposite is
observed for high-intensity social interactions and (in particular) for
couples.

## Conclusion

In this work, we analyzed how group-individual pedestrian collision avoidance
depends on the group’s social relation and social interaction intensity.

In detail, we verified that when straight motion (i.e. absence of collision
avoidance) would lead to a possible collision, the actual minimum distance
$r_{0}$ between the individual and the group is a growing function of social
interaction intensity, and assumes a higher value for couples and friends.
Similarly, individuals have a stronger tendency to “intrude” or “disrupt” a
group by passing at a distance comparable to the group interpersonal distance
when they face groups with low interaction intensity and colleagues and
families, as can be verified both by studying 2D distance probability
distributions, and by performing a statistical analysis on the probability
that the minimum distance becomes smaller than the group interpersonal
distance.

We also introduced a “potential” to study the dependence of “intensity of
collision avoidance” on relative distance, by mimicking the theoretical
modeling of two-body scattering in classical mechanics. This approach, which
may be used as a guiding light in the development of a “social force model” of
individual-group interaction, shows again that the potential determining
collision avoidance tends to grow much faster with decreasing distance values
(i.e. it has a steeper negative derivative) for strongly interacting groups,
couples and families.

The latter result is particularly clear when studied using the group’s average
interpersonal distance as a length unit. A further comment on this result may
be necessary, since the tendency of individuals not to pass through “strongly
bonded dyads” (such as couples, friends and strongly interacting dyads) may be
due not only to some kind of “social rule”, but also to the fact that passing
through these groups is actually harder due to the narrower space between
them.

To this respect, we should finally comment also on the results concerning
families, which may be a little counter-intuitive by suggesting that families
are somehow perceived as weakly interacting and are often “intruded” [71]. It
should be stressed that, as reported by Zanlungo et al. [19], the families in
the ATC data set are mostly composed of parent-child pairs, that often do not
walk abreast, or at least have a weaker tendency to walk abreast. The authors
of the original study justify this tendency by referring to “the erratic
behavior of children”, but it may also be related to a stronger hierarchical
structure in a parent-child dyad with respect to couples, friends and
colleagues [19]. It may thus be argued that the tendency of individuals to
approach families at a shorter distance may depend on families being less
spatially structured, or correspondingly having a higher tendency to change
their spatial structure. Such role of group spatial structure in individual-
group interaction could be the subject of future studies, possibly when larger
data sets collected in more suitable environments will be available.

We believe that our results and inferences point out interesting variabilities
in pedestrian motion due to social aspects of human navigation [72]. A
valuable implication of our study is that infrastructure design could be
adapted to the nature of the social bonding of its users. We can speculate
that, for instance, if a particular environment is known to be frequented
mostly by strongly bonded groups, such as an amusement park, providing
additional space (e.g. by widening corridors or walkways) to allow for
collision avoidance may make it more comfortable. Nevertheless, these
qualitative considerations should ultimately be corroborated with quantitative
simulation models that include our findings. By taking into account the social
dynamics of the people using a particular space, designers and architects
could create environments that are more conducive to safe and efficient
movement. This could help to reduce the risk of accidents and improve the
overall user experience. We also hope that using models which account for the
expected social composition of the crowd may help in improving the performance
of tracking and simulation systems [73].

## Data availability

The data sets analyzed during the current study are available at
https://dil.atr.jp/ISL/sets/groups/.

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## Acknowledgements

This work was supported by the JSPS Grant-in-Aid for JSPS Fellows 22J20686.
This work was financially supported by the JST-Mirai Program Grant Number
JPMJMI20D1 and the JSPS KAKENHI Grant Number JP20K14992. This work was (in
part) supported by JST Moonshot R and D under Grant Number JPMJMS2011, Japan.

## Author contributions statement

A. G. took part in Data curation, Formal analysis, Investigation, Methodology,
Software, Validation, Writing – original draft, Writing – review & editing. Z.
Y. took part in Conceptualization, Formal analysis, Data curation, Formal
analysis, Methodology, Supervision, Writing – original draft, Writing – review
& editing. F. Z. took part in Conceptualization, Formal analysis, Methodology,
Writing – original draft, Writing – review & editing. C. F. took part in
Funding acquisition, Project administration, Writing – review & editing. T. K.
took part in Funding acquisition, Project administration, Writing – review &
editing. All authors reviewed the manuscript.

## Additional Information

The authors declare no competing interests.

##  Appendix A Specifics of experiments and data

In this section, we give details about experimental conditions and specifics
of data and annotations.

### ATC data set

The ATC data set is a large pedestrian data set, which is introduced by
Zanlungo et al. [7] and has since been used in several studies, especially
those focusing on social aspects of crowd dynamics [28, 29, 30, 31, 32, 33]
and it is freely available [59]. It has several properties, which make it
particularly fit for the investigation of such factors and we will point them
out as we explain the specifics of the data set below.

The recording location is the ground floor of a multi-purpose center, which
involves offices, shops and facilities for special events such as expositions.
In particular, the tracking area consisted of an atrium and a corridor which,
along with being bordered by shops, serve also as a passage between a train
station, a ferry terminal and the office area. Therefore, in addition to
accommodating the visitors of the multi-purpose center, it is also constantly
frequented by commuters. In that respect, the recorded pedestrians come from a
diverse social background (e.g. age group, occupation etc).

The recording area is quite large, i.e. approximately 900 m2. Therefore, it
allows continuous tracking of individuals for long distances (i.e. up to 50
m), which helps also in the annotation of social relations. Note that in the
analysis we discarded the trajectory segments collected in the atrium and used
the segments collected in the corridor illustrated in Figure 1-(a), which is
over 40 m long.

The recording time span is more than 800 hours within a one year time window.
In such, it involves workdays, weekends, national holidays, public festivals,
and special days of the business center such as exhibitions and trade shows.
These occasions lead to an increase in density and a diversity in the purpose
of visit.

The raw data are in specific depth and video information registered by an RGBD
sensor network. Based on the depth information and the algorithm of Brščić et
al. [60], the pedestrians are tracked automatically (on the 2D floor plane)
and the resulting trajectories can freely be downloaded [59]. The concerning
normalized cumulative density map of the environment is found as in Figure
1-(a). Note that the cumulative density map is basically a 2D histogram. We
consider the environment as a 2D mesh with a grid cell size of 10 cm by 10 cm
and count the number of observations in each grid cell. The normalization
refers to the scaling of this histogram with its maximum value.

Note that since the sampling frequency of the sensors is quite high (i.e. 20
Hz), tracked trajectories are influenced by gait dynamics. Furthermore, if
velocities were simply computed as discrete differences between the tracked
trajectory points, the relatively small error in position due to sensor noise
would be magnified. For this reason, Zanlungo et al. proposed re-sampling the
data by averaging pedestrian positions over 0.5 s time windows, in order to
reduce the influence of pedestrian gait and other sorts of measurement noise,
as well as to assure uniform sampling in time, and also in this work we are
going to use such re-sampled data [7].

Based on the video information, the pedestrians are annotated according to
group membership and several intrinsic group features [19]. Specifically,
human coders labeled which pedestrians constitute a social group and which
pedestrians move alone. In addition, for the groups they also annotated
intrinsic features such as age, purpose of visit etc. One specific feature,
which we focus on in this study, refers to the apparent social relation of the
group members. The possible options are couples, colleagues, family and
friends, which correspond to the domains of mating, coalitional, attachment
and reciprocal defined by Bugental [62], respectively. (See Table 1-(a) for
the outcome of the annotation process).

### DIAMOR data set

The DIAMOR data set was introduced by Zanlungo et al. [6] and used
particularly for the purposes of group recognition and motion modeling [74,
61, 6].

The recording location is an underground pedestrian street network in a
commercial district of Osaka, Japan. The entire underground network is
composed of a total of more than several kilometers of walking path, and it is
connected to the Osaka-Umeda railway and underground station complex, which is
considered to be one of the busiest in the world (the busiest outside Tokyo)
and visited daily by millions of pedestrians. For example, according to the
Osaka municipal transportation bureau, the three metro stations located in the
underground area had a daily number of passengers of more than 700K in the
fiscal year 2019. The DIAMOR data set includes recordings from a junction of
two straight corridors in a relatively peripheral portion of this street
network and we focus on one of these. Similar to ATC data set, several train
stations, business centers, shopping malls etc. are accessible from the
recording location leading to diversity in pedestrian profile.

The recording area is roughly 200 m2 and allows continuous tracking along
approximately 50 m. The recording time span is two weekdays and a total of
eight hours of recordings are available, which, although shorter than the ATC
data set, we consider to be enough for the purposes of this study.

Similar to the ATC data set, it is composed of depth and video information.
The depth information is used to derive the trajectories of the pedestrians
based on the method reported by Glas et al. [75] and the trajectories can be
freely downloaded [59]. As a result of this tracking process, the normalized
cumulative density map shown in Figure 1-(b) is obtained.

Based on the video information, human coders were asked to annotate groups and
individuals (people who do not belong to a group). Of course, it is not wrong
to say that each group member is an “individual”, but within the context of
this study we refer to them explicitly as “group member”, and use the word
“individual” specifically to refer to people who do not belong a group. Coders
also annotated whether or not members of dyads were engaged in interaction
(oral communication, possibly accompanied with non-verbal elements such as
gestures or gaze exchange, as defined by Knapp et al. [63]), and the
corresponding intensity of interaction (evaluated at 4 degrees from 0
representing no-interaction to 3 representing strong-interaction). Note that
the annotations of ATC data set are inherently disjoint, i.e. the annotation
labels can be considered to be mutually exclusive nominal variables. On the
other hand, the annotation labels of the DIAMOR data set can be viewed as
ordinal variables (i.e. with a gradual relation). The outcome of the
annotation process (i.e. the number of observations for each intensity of
interaction) is summarized in Table 1-(b).

##  Appendix B Data preparation

Both the ATC data set and the DIAMOR data set are collected in an ecological
environment. Namely, they contain trajectories collected from uninstructed
people moving freely. Although they were not recorded secretly (i.e. there
were signboards informing the pedestrians that a data collection campaign was
being carried out), the pedestrians’ awareness of being recorded is
anticipated to have a negligible effect on how they move, in particular as
compared to participant experiments performed in artificial (laboratory)
environment. In that respect, since the data are collected under uncontrolled
settings, the tracked trajectories may contain behaviors like waiting, running
etc. From the point of view of this study, such cases are not of interest. In
order to eliminate atypical/non-characterizing observations, each trajectory
is treated as explained below.

Let a group (dyad) be described as an unordered pair composed of (two members)
$p$ and $q$, i.e. $g=(p,q)$ and let $i$ denote an individual. For the sake of
simplicity, we reduce a group to a single mobile agent in the data preparation
phase. Namely, the location of the group is represented by the group center of
mass $\mathbf{r}_{g}$ and its velocity is represented with group velocity
$\mathbf{v}_{g}$. Specifically, concerning a group $g=(p,q)$, $\mathbf{r}_{g}$
and $\mathbf{v}_{g}$ are represented as the average positions and velocities
of $p$ and $q$ at each time step, respectively.

$\begin{split}&\mathbf{r}_{g}=\frac{\mathbf{r}_{p}+\mathbf{r}_{q}}{2},\\\
&\mathbf{v}_{g}=\frac{\mathbf{v}_{p}+\mathbf{v}_{q}}{2}.\\\ \end{split}$ (16)

Thereby, we treat a group $g$ and an individual $i$ in the same manner and
first check the sufficiency of the number of trajectory data points. Provided
that $\mid\left\\{\mathbf{r}_{g,i}\right\\}\mid\geq 16$, the trajectories are
considered to have enough data points for characterizing locomotion. Note that
this corresponds to a minimum of 8 seconds of observation, since the sampling
time step is 0.5 s. Any trajectory which includes fewer samples is discarded.

Next, we check instantaneous speeds ${{v}}_{g,i}$ and remove the trajectories
which are associated with too low or too high speeds (i.e. out of walking
range). For judging the typical speed range of pedestrians, we referred to the
literature on human locomotion. Based on the results reported by Zanlungo et
al. [6], $g$ and $i$ are considered to depict typical walking motion, if their
instantaneous speed lies within the range $0.5\leq v_{g,i}\leq 3$ (in m/sec).
Otherwise, they are assumed to be “not walking” and discarded.

As mentioned in Section Introduction of the main track, we focus on the effect
of social attributes of (the group) on collision avoidance. To have an
understanding of those, the peers need to have sufficient visual information
about each other. Therefore, we start with conditioning on having a frontal
view of each other, which implies moving in opposite directions. In other
words, if $g$ and $i$ move in the same direction, one party will be leading
and the other following, such that the leading party will not see and thus not
be aware of the other, and the following party will have limited information
about the leading one (e.g. on social relation, age, interactions, etc.). In
theory, the group and individual might also approach each other at an angle,
but in the studied environments such cases are rare and we do not consider
them in this work. However, if they move in opposite directions, they will
have the opportunity to watch the incoming party and get a sense of its social
features.

For ensuring a frontal view, we detected the relative motion direction of $g$
and $i$ and considered only those $g$ and $i$, which move in opposite
directions. Let $\phi$ represent the angle between the velocity vectors
$\mathbf{v}_{g}$ and $\mathbf{v}_{i}$ at a given time instant,

$\phi=\arccos(\mathbf{v}_{g}\cdot\mathbf{v}_{i})/(||\mathbf{v}_{g}||\,||\mathbf{v}_{i}||).$
(17)

Then, $g$ and $i$ are considered to be moving in opposite directions, if
$3\pi/4\leq\phi<\pi$. Note that, in addition to boasting a bigger potential
from the viewpoint of our purposes, in the studied “bi-directional”
environments, considering opposite relative motion direction has also the
advantage that the number of observations associated with it is significantly
higher than those with other directions (namely, $68\%$ of all observations),
which is preferable for the statistical analysis performed in our study.

##  Appendix C Window of observation

Figure 8: Illustration of the impact of the choice of the window size $W$ on
the computation of the straight-line distance $r_{b}$. In red, a small window
violating completeness, in blue, a large window violating atomicity and in
green, an "optimal" window which satisfies both constraints. Note that since
we study frontal encounters in the group reference frame, individuals will
always enter the window from the right.

As measures at infinity are obviously not feasible, in order to measure the
straight-line distance $r_{b}$ we need to define a window of observation
(centered on the group). In particular, this window should satisfy two
competing properties, completeness and atomicity. On the one hand, the window
needs to be large enough to verify that, when entering and exiting from the
side of the window the individual is not yet (significantly) influenced by the
group (i.e. he/she is not yet engaged in an avoidance maneuver). This
guarantees that the deviation is entirely contained in the window of
observation, hence the notion of completeness. On the other hand, the window
should not be too large, so that the assumption that the individual would walk
on a straight line in the absence of the group stays valid. Indeed, although
pedestrians certainly do not walk on straight lines at all times, at a
relatively small scale and in the absence of perturbations (resulting from
either external sources such as the environment or other pedestrians, or
internal sources such as a change of planned destination) it can be expected
that one’s trajectory will be close to a straight line (atomicity). Figure 8
illustrates the impact of the size of the window.

Regarding completeness, we referred to literature on collision avoidance and
searched for a reasonable threshold value. Cinelli and Patla found that, the
“safety zone”, i.e. the area in which individuals allow a moving object to
approach before initiating an avoidance behavior, is on average 3.73 m [64].
Furthermore, Kitazawa et al. showed that pedestrians gaze most at other
approaching individuals, when they are on average 3.97 m away, and that they
seldom look at pedestrians at longer distances than this [65]. Therefore, we
deliberated that 4 m is a reasonable lower bound for considering that their
mutual influence is still null.

Regarding atomicity, the environment needs to be taken into consideration. In
a straight and wide corridor, like in DIAMOR, the straight line assumption is
more founded than for a more complex environment, like the bent and narrow
corridor in ATC where pedestrian have to follow the curve and will have
naturally less straight trajectory. Nonetheless, the discussion provided in
Section G suggests that the disparity due to environment geometry can be
accounted for with a linear correction. Therefore, we argue that the value of
4 m derived from the completeness condition is adequate also to satisfy the
atomicity condition.

##  Appendix D Improving accuracy of estimation for the observed minimum
distance $r_{0}$

The trajectories provided at the ATR pedestrian group data set [59] are
derived by the algorithm of Brščić et al. [60], whose output rate depends on
the rate of sensor readings, which may be non-uniform. In order to make the
rate of trajectory samples uniform and also to eliminate the effect of gait
and sensor noise, we re-sampled the trajectories at 2 Hz. However, the new
time resolution can be too sparse for the purpose of computing the observed
minimum distance. As a remedy to this issue, we propose interpolating the
position of an individual $i$ between two consecutive time steps $t_{k}$ and
$t_{k+1}$ with its own velocity vector at time $t_{k}$,
$\mathbf{v}^{\prime}_{i}(t_{k})$ (see Figure 9).

Figure 9: Illustration of computation of the observed minimum distance
$r_{0}$.

For each time step $t_{k}$, the smallest distance between $g$ and $i$ can be
computed by measuring the distance from the origin to the line passing through
$p^{\prime}_{i}(t_{k})$, the position of $i$ at $t_{k}$ and directed by
$\mathbf{v}^{\prime}_{i}(t_{k})$, the velocity of $i$ at $t_{k}$. We can
consider this distance as the actual minimal distance to the group $g$ in the
interval $\left[t_{k},t_{k+1}\right]$, only if $i$ can reach the position
where this distance is observed within $t_{k+1}-t_{k}$ seconds (0.5 s in this
study). We denote the signed distance from $p^{\prime}_{i}(t_{k})$ to that
position with $\lambda$, which can be computed by taking the projection of the
vector from $p^{\prime}_{i}(t_{k})$ to the origin (i.e.
$-\mathbf{v}^{\prime}_{i}(t_{k})$) onto the unit vector
$\frac{\mathbf{v}^{\prime}_{i}(t_{k})}{||\mathbf{v}^{\prime}_{i}(t_{k})||}$,

In explicit terms,

$\lambda=\frac{-\mathbf{v}^{\prime}_{i}(t_{k})\cdot\mathbf{p}^{\prime}_{i}(t_{k})}{||\mathbf{v}^{\prime}_{i}(t_{k})||},$
(18)

The distance $\lambda$ is traveled by $i$ in $t_{min}$,

$t_{min}=\frac{\lambda}{||\mathbf{v}^{\prime}_{i}(t_{k})||}.$

Note that if $t_{min}<t_{k+1}-t_{k}$, a smaller distance is achieved within
the time interval $\left[t_{k},t_{k+1}\right]$ than at its initial and final
instants ($t_{k}$ and $t_{k+1}$). Thus, this value is registered as the
minimum distance concerning this time interval. This implies that the minimum
distance concerning that time interval is achieved at an intermediate instant.
Otherwise the lower one of $\left\|\mathbf{p}^{\prime}_{i}(t_{k})\right\|$ and
$\left\|\mathbf{p}^{\prime}_{i}(t_{k+1})\right\|$ is registered. In this case,
if $||\mathbf{p}^{\prime}_{i}(t_{k})||<||\mathbf{p}^{\prime}_{i}(t_{k+1})||$,
it means that $g$ and $i$ are getting further away. If
$||\mathbf{p}^{\prime}_{i}(t_{k})||>||\mathbf{p}^{\prime}_{i}(t_{k+1})||$, it
means that $g$ and $i$ are getting closer and that it is highly likely that in
the next time interval, they will be even closer. The same operation is
carried out for all time intervals and the minimum of all the registered
values is used as $r_{0}$.

##  Appendix E Scaling by interpersonal distance

Yücel et al. [21] showed that interpersonal distance between members of a dyad
strongly depends on their social relation. For instance, couples were shown to
walk with an interpersonal distance significantly smaller than for other
social relations (values of interpersonal distance for various social
relations and intensities of interaction are show in Table 2).

This variability may affect the behavior of the individual approaching the
group. Namely, an individual $i$ may choose not to intrude a group $g$ due to
insufficient space (between its members) or may prefer to intrude $g$ due to
ample space. So the presence or lack of intrusion may simply be due to
geometric circumstances and not stemming from social factors (e.g. strength of
social bonding). This effect is actually better studied by measuring distances
using a common unit (i.e. meters).

On the other hand, when $i$ does not intrude on $g$, the effect of the social
bonding may be better expressed, if distances are measured with respect to the
physical size of the group (interpersonal distance between the members), since
avoiding a small group at a distance of, e.g. 2 meters, may involve a stronger
avoidance behavior, if the group size is smaller.

Table 2: Average interpersonal distance of groups annotated with each (a) social relation (in ATC data set) and (b) intensity of interaction (in DIAMOR data set).  (a) | (b)
---|---
| Social relation | Average interpersonal distance (mm)
---|---
Couples | $780$
Colleagues | $891$
Family | $935$
Friends | $835$
| Intensity of interaction | Average interpersonal distance (mm)
---|---
0 (no interaction) | $1115$
1 | $951$
2 | $842$
3 (strong interaction) | $805$

##  Appendix F Probability of intrusion

In the main track, to study intrusion, we examined the probability of
$\bar{r}_{0}<1$, i.e. that the individual reaches a distance from the group
center smaller than the average group interpersonal distance. One could argue
that an alternative, if not better, definition of intrusion, can be based on
studying the probability of having $\bar{r}_{0}<0.5$, i.e. a distance smaller
than the average distance of a group member from the group’s center. For this
reason, in this section we perform a similar analysis to the one presented in
the main track, but using the value of $0.5$. Namely, Figure 10 shows the
probability $P(\bar{r}_{0}<0.5)$ and Figure 11 shows the corresponding
$p$-values.

(a)

(b)

Figure 10: Probability that the distance $\bar{r}_{0}$ is smaller than $0.5$
for (a) for various social relations and (b) intensities of interaction of the
group.

(a)

(b)

Figure 11: Pearson’s $\chi^{2}$ $p$-values for the hypothesis of independence
of the frequencies of samples verifying $\bar{r}_{0}<0.5$ and $\bar{r}_{0}\geq
0.5$ for (a) for various social relations and (b) intensities of interaction
of the group.

Nicely, the probabilities show a similar trend to those given in the main
track and confirm our inference that loosely-bonded groups are more likely to
be intruded on than strongly-bonded ones.

##  Appendix G Linear correction to $r_{b}$

As mentioned in Section C, we tried to calibrate the window of observation in
such a way that the trajectory of the individual should be close to a straight
line for large values of ${r}_{b}$. If the straight-line distance ${r}_{b}$ is
large, it means that the individual should have enough space to pass
comfortably without deviating and we would expect the minimum distance
${r}_{0}$ to be somewhat similar to ${r}_{b}$.

Nevertheless, it seems clear that the curved and narrow nature of the ATC
environment puts a limit on the applicability of the straight line hypothesis.
In particular, as the width of the ATC corridor is comparable to the size of
the chosen window of observation (see Figure 1), we can expect that the
environment will pose some constraint on the motion of the pedestrians in
particular for large values of $r_{b}$.

This is indeed confirmed by the data. Namely, by looking at Figure 12-(a)
relating to ATC dataset, we observe that the curves are all noticeably offset
from the $x=y$ (dashed) line for high values of $\bar{r}_{b}$. As
$\bar{r}_{0}$ is smaller than $\bar{r}_{b}$, it seems as if the individual
steers towards the group.

|
---|---
(a) | (b)
Figure 12: Observed minimum distance $\bar{r}_{0}$ as a function of the
undisturbed straight-line distance $\bar{r}_{b}$ (a) for various social
relations and (b) intensities of interaction of the group. The dashed line
corresponds to the $\bar{r}_{0}=\bar{r}_{b}$ linear dependence.

It is not trivial to propose a geometric model for such deviation, due to the
relatively complex nature of the ATC environment, but for simplicity’s sake we
may thus assume the correction on $r_{b}$ to be linear. Following this
hypothesis, we evaluate the impact of environment geometry by computing an
average value of the observed distance $r_{0}$ (resp. $\bar{r}_{0}$) for large
values of $r_{b}$ (resp. $\bar{r}_{b}$) (corresponding to the highest bin in
Figure 12), for all groups and individuals. We then compute a correction
coefficient $c$ defined as the ratio between the observed average value and
the expected value $r_{0}=r_{b}$ (as stated above, when $r_{b}$ is large, we
expect no deviation during encounters). Such coefficients are found to be
$0.82$ for scaled values and $0.77$ for unscaled values. These coefficients
can then be used to multiply the values and alleviate the effect of the
curvature of the environment. Specifically, when computing the potential
$U^{\prime}$, we replace the values of $r_{b}$ (resp. $\bar{r}_{b}$) by the
corrected values $r^{\prime}_{b}=cr_{b}$ (resp.
$\bar{r}^{\prime}_{b}=c\bar{r}_{b}$).

In Figure 13, we show the scaled distances for groups with various social
relations, along with the line corresponding to the correction coefficient. By
definition, the correction fits the various curves for larger values of
$r_{b}$. For further reference, Figure 14 shows the correction coefficient for
the unscaled values concerning all groups and individual pedestrians used for
the potential of Figure 7 in the main track. The qualitative agreement between
the scaled ATC plots and the unscaled DIAMOR ones suggests that the linear
correction used to obtain $r^{\prime}_{b}$ and $\bar{r}^{\prime}_{b}$ is
reasonable.

Figure 13: Observed minimum distance against straight-line distance for various social relations (scaled). The dashed black line corresponds to the linear correction applied.  |
---|---
Figure 14: Observed minimum distance $r_{0}$ against straight-line distance
$r_{b}$ for (a) various social relations (unscaled) and (b) groups and
intensities of interaction (unscaled). The dashed black line corresponds to
(a) the linear correction applied for the ATC dataset and (b) to the line
$y=x$ for the DIAMOR dataset.

##  Appendix H Justification of using ANOVA in assessment of statistical
significance

To validate the statistical significance of the differences observed in the
relation between $\bar{r}_{0}$ and $\bar{r}_{b}$ with regard to the social
bonding of the group, we performed analysis of variance tests. The standard
one-way ANOVA requires the data to confirm three conditions, namely (i)
independence of observations, (ii) normality of residuals and (iii) equality
of variances. In what follows, we verify and discuss these assumptions.

For (i) independence of observations, we argue that the independence is
naturally verified, since the distances are computed for different pairs of
groups and individuals, with each group being classified with only one social
bonding characteristic.

Regarding (ii) normality of residuals, we carried out D’Agostino’s $K^{2}$
test on the residuals in each bin (i.e. the distance from which the average
values for that bin was subtracted). We found that in 31% of the comparisons
(i.e. 5 out of the 16 bins where the test is performed), the assumption of
normality was verified ($p$-value $>0.05$).

As for (iii) equality of variances, we used Levene’s test to verify that the
various samples have equal variance in each bin. We found that the requirement
is satisfied in 75% of the cases ($p$-value for Levene’s test $>0.05$).

From the above, we inferred that normality of the residuals is the requirement
which is not upheld most often, shedding a doubt on the validity of our
conclusions. In that respect, we employed an additional statistical test,
namely Kruskal-Wallis H-test, which is a non-parametric alternative for the
one-way ANOVA. The reason for choosing this test is that it does not require
normality of residuals unlike the standard one-way ANOVA. The comparison of
the results of Kruskal-Wallis H-test to those of ANOVA is presented in Figure
15. From this figure, one can easily notice that the judgment of statistical
significance is not affected by the choice of the test (significance is
verified for identical bins). Note that the only bin for which there is a
strong difference is $\bar{r}_{b}\approx 2$ (see Figure 15-(b)), which is
nevertheless (in both Figure 15-(a) and (b)) the transition between
significant and insignificant.

|
---|---
Figure 15: $p$-values for the ANOVA and Kruskal-Wallis H test of $\bar{r}_{0}$
(a) for various social relations and (b) intensities of interaction of the
dyad. Very low values are not displayed.

##  Appendix I Goodness of fit of the model

Concerning both the scaled and not-scaled distances, the goodness of fit of
the proposed models was assessed based on the Akaike information criterion
(AIC) and the Kolmogorov–Smirnov (KS) goodness of fit test (see Table 3).

It is hard to interpret the outcomes of AIC concerning a single model, since
it is not bounded (it will tend to $-\infty$ when the residual sum of squares
gets smaller). Nonetheless, AIC can be used to compare several models, and
thus we use it as such. The fit of the model is seen to be relatively poor for
non-interacting dyads (see Table 3-(b)) compared to other intensities of
interaction and social relations. We also note that the fit is generally
better, for scaled distances as compared to the unscaled distances (5 models
out of 8).

Concerning the Kolmogorov–Smirnov goodness of fit test, the null hypothesis is
that the underlying distributions of the two samples are identical. Our
results show that we cannot reject that hypothesis in most cases (note the
high $p$ values in Table 3).

For the intensity of interaction of 0 with a $p$ value of 0.02, it is not
surprising that the fit is not good, since there is virtually no effect of
groups social interaction at that level, as it can be seen in Figure 5-(b) of
the main track.

Table 3: AIC and Kolmogorov-Smirnov $p$-values for the goodness of fit of
models describing the collision avoidance potential as a function of the
distance $r$ of the form ${k}/{r^{\beta}}$ for (a) for various social
relations and (b) intensities of interaction of the group. (a)
---
| Social relation | AIC | KS $p$-value
---|---|---
| Not-Scaled | Scaled | Not-Scaled | Scaled
Couples | -8.51 | -29.10 | 0.66 | 1.00
Colleagues | -27.10 | -33.98 | 0.28 | 0.98
Family | -50.76 | -40.92 | 0.98 | 0.98
Friends | -42.72 | -39.34 | 0.98 | 0.98
Individuals | -44.73 | — | 0.98 | —
(b)
| Intensity of interaction | AIC | KS $p$-value
---|---|---
| Not-Scaled | Scaled | Not-Scaled | Scaled
0 | 54.37 | 57.90 | 0.02 | 0.02
1 | -27.78 | -28.82 | 0.28 | 0.28
2 | -42.55 | -37.55 | 0.09 | 0.28
3 | -18.22 | -29.45 | 0.28 | 0.66
Individuals | -48.41 | — | 0.28 | —