# Microscopic dynamics of escaping groups
through an exit and a corridor

E.A. Rozan  Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires,
Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.  G.A. Frank
Unidad de Investigación y Desarrollo de las Ingenierías, Universidad
Tecnológica Nacional, Facultad Regional Buenos Aires, Av. Medrano 951, 1179
Buenos Aires, Argentina.  F.E. Cornes  I.M. Sticco  C.O. Dorso  Instituto de
Física de Buenos Aires,
Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.

###### Abstract

This research explores the dynamics of emergency evacuations in the presence
of social groups. The investigation was carried out in the context of the
basic Social Force Model (SFM). We included attractive feelings between people
belonging to the same social group, as proposed in Ref. [1]. We focused on the
escaping dynamics through an emergency exit and through a corridor. We
confirmed the results appearing in Ref. [1] for a desired velocity of
$4\,$m/s, but further extended the analysis on the emergency exit to the range
1-8 m/s. We noticed that the presence of groups worsens the evacuation
performance. However, very strong feelings can improve the escaping time with
respect to moderate feelings. We call this phenomenon “Closer-Is-Faster”, in
analogy to the “Faster-Is-Slower” effect. The presence of social groups also
affects the dynamic within a corridor by introducing an additional slow-down
in the moving crowd.

###### keywords:

Evacuation , Social Force Model (SFM) , Social Groups , Clogging Delays

###### PACS:

45.70.Vn , 89.65.Lm

hyperrefToken not allowed

##  1 Introduction

The basic Social Force Model (SFM) introduced by Hebling et al. [2] has been
used to describe many emergency scenarios [1, 3, 4, 5, 6]. The model first
succeeded to explain why the crowd dynamics slows down as pedestrians try
harder to escape through a narrow exit [2, 7, 8]. This is known as the
“Faster-Is-Slower” phenomenon, and it has been reported to occur in very
crowded environments [9, 10, 11]

Recent research shows that the “Faster-Is-Slower” phenomenon is not the only
one occurring at high pressure levels [12]. As pressure undergoes dangerous
levels, the crowd is supposed to behave as a compact human cluster because of
the friction among the pedestrians. This is known as the “Faster-Is-Faster”
regime [7, 13, 14, 9, 15].

The basic SFM deals with the collective behavior of pedestrians under panic,
but lacks for “social cohesion”. That is, the behavioral effects at the “group
level” [16]. The absence of these effects in the SFM can lead to the wrong
estimation of the time required for an emergency evacuation, according to
experimental research. For instance, Refs. [17, 18] report that the egress of
previously assigned groups reduces the evacuation time. However, Refs. [19,
20] report that the evacuation is slightly slowed down by the presence of
groups. The matter seems to be under discussion, but it is clear that a better
understanding of the consequences of social cohesion is still necessary.

Analyzes from video cameras confirm the fact that large groups have the
tendency to move in spatial patterns in order to facilitate the voice
communication between members [21, 22]. These patterns occur in the context of
normal walking conditions, while there is not enough evidence for this
behavior under disturbing conditions (to our knowledge). This suggests,
however, that group behavior fits differently into crowd models than
individualistic ones. Refs. [21, 22, 23, 24] propose intra-group forces within
the SFM context in order to mimic the group behavioral patterns.

Ref. [1] studied the dynamics of dyads (groups of two members) in the context
of an emergency evacuation. They introduced an attractive force between
partners, additional to the usual socio-psycological forces appearing in the
basic SFM (see Ref. [25] for details on this matter). The force intensity
attained the emotional intimacy in the dyad. The investigation reports that
the feelings of closeness play a relevant role in the evacuation time, and can
yield to a significant loss in the time performance.

Our main objective is to extend the above investigation to two quite different
scenarios: escaping through a bottleneck, and passing along a straight
corridor in a unidirectional flow. Both scenarios attain highly dense
situations. We will limit the corridor investigation, though, to non-panicking
situations for a better comparison with experimental data. More specifically,
we will compute the fundamental diagram (i.e. the relationship between the
density and the pedestrian flow), and qualitatively compare our results with
empirical measurements from Ref. [26].

The investigation is organized as follows. In Section 2 we present the
highlights of the SFM and its extension to the group model, as defined in Ref.
[1]. We also include definitions for clustered structures and the fundamental
diagram. In Section 3 we compare the extended SFM with the (scarce) empirical
data on social groups. Section 4 details the simulation procedures for
studying the bottleneck and the corridor scenarios. In Sections 5.1, 5.2 and
5.3 we present the results corresponding to the bottleneck scenario, while
Section 5.4 shows the corresponding results for the corridor. A discussion on
the results is shown in Section 6. Our main conclusions can be found in
Section 7.

##  2 Theoretical Background

###  2.1 Social Force Model (SFM)

Our research was carried out in the context of the Social Force Model (SFM)
proposed by Helbing et al. [2]. This model states that human motion is
motivated by the desire of people to reach a certain destination, and can be
further affected by environmental factors. The basic SFM considers the
following equation of motion for any pedestrian $i$:

$m_{i}\,\displaystyle\frac{d\mathbf{v}_{i}}{dt}(t)=\mathbf{f}_{d}^{(i)}(t)+\displaystyle\sum_{j}\displaystyle\mathbf{f}_{s}^{(ij)}(t)+\displaystyle\sum_{j}\mathbf{f}_{g}^{(ij)}(t)$
(1)

where $j$ represents any other pedestrian or a wall.

Three kind of forces are included in the above equation: the desire force, the
social force and the sliding friction force. An additional force will be
introduced in Section 2.2.

The “desire force” $\mathbf{f}_{d}$ corresponds to the pedestrians own
willings to move at the desired velocity $v_{d}$. It is associated with the
required acceleration (or deceleration) from their current velocity
$\mathbf{v}_{i}(t)$ to the desired one. The corresponding expression is as
follows

$\mathbf{f}_{d}^{(i)}(t)=m_{i}\,\displaystyle\frac{v_{d}^{(i)}(t)\
\hat{\mathbf{e}}_{d}^{(i)}(t)-\mathbf{v}_{i}(t)}{\tau}$ (2)

where $m_{i}$ is the mass of the pedestrian $i$ and $\tau$ represents the
relaxation time needed to reach his (her) desired velocity.
$\hat{\mathbf{e}}_{d}$ is the unit vector pointing to the target position,
while $v_{d}$ attains the anxiety level of the pedestrian. For simplicity, we
will assume that $v_{d}$ remains unchanged during the entire process and is
the same for all of the individuals, but $\hat{\mathbf{e}}_{d}$ changes
according to the current position of the pedestrian.

In the context of an evacuation process, if no acquittance, friendship or
family engagements exist, the most common tendency is to keep some space
between each other, or from the walls. The “social force”
$\mathbf{f}_{s}^{(ij)}$ represents this socio-psychological tendency between
any two pedestrians, say $i$ and $j$, in order to preserve their private
space. It is assumed to be

$\mathbf{f}_{s}^{(ij)}=A_{i}\,e^{(r_{ij}-d_{ij})/B_{i}}\mathbf{n}_{ij}$ (3)

where $A_{i}$ and $B_{i}$ are two fixed parameters. $r_{ij}$ = $r_{i}+r_{j}$
is the sum of the pedestrians radius (say, the shoulder-neck distance), and
$d_{ij}$ is the distance between their centers of mass. $\mathbf{n}_{ij}$
stands for the unit vector in the $\vec{ji}$ direction. If $j$ represents a
wall, then $d_{ij}$ corresponds to the shortest distance between the
pedestrian and the wall, and $r_{j}$ is set to zero.

The emotional reactions due to friendship or family engagements may also be
handled as a “socio-psychological force”. We will discuss this matter in
Section 2.2 below.

The sliding friction is present whenever two individuals (or an individual and
the wall) get in contact. It is known as the “granular force” within the SFM,
and is considered to be a linear function of the relative tangential
velocities of the contacting individuals. Its mathematical expression is as
follows

$\mathbf{f}_{g}^{(ij)}=\kappa\,g(r_{ij}-d_{ij})\,\left(\Delta\mathbf{v}_{ij}\cdot\>\hat{\mathbf{t}}_{ij}\right)\>\hat{\mathbf{t}}_{ij}$
(4)

where $\kappa$ is the friction coefficient. The function $g(r_{ij}-d_{ij})$ is
equal to its argument when this is positive (that is, when $d_{ij}<r_{ij}$,
meaning that pedestrian $i$ and $j$ are closer than the contact distance) and
equals zero for any other case.
$\Delta\mathbf{v}_{ij}\cdot\>\hat{\mathbf{t}}_{ij}$ represents the difference
between the tangential velocities of the sliding bodies (if $j$ represents a
wall, then $\Delta\mathbf{v}_{ij}=\mathbf{v}_{i}$). The whole set of
parameters’ values can be found in Section 4.1.

###  2.2 The Attractive Force

Ref. [1] introduced an attractive force into the equation of motion in order
to mimic the behavioral patterns of groups. For instance, when one of the
members of the group is pushed aside, he (she) will try to move back to the
the space in common within the group. This behavior is considered in Ref. [1]
to be associated with attractive feelings among the members of the group, and
are supposed to be balanced with the “private sphere” preservation.

The attractive force in Ref. [1] is assumed to derive from a Fermi-like
potential. Its mathematical expression reads as follows

$\mathbf{f}_{a}^{(ij)}=-\frac{\epsilon}{4D_{i}}\cosh^{-2}\left(\frac{C_{i}-d_{ij}}{2D_{i}}\right)\mathbf{n}_{ij}$
(5)

where $\epsilon$ represents the intensity of the attraction and will vary
depending on the the “closeness” to the group members. $C_{i}$ and $D_{i}$ are
fixed parameters, set to $D_{i}=0.5B_{i}$ and $C_{i}=r_{ij}+7B_{i}$ in Ref.
[1]. We will hold these values throughout our research.

Fig. 1 shows the attractive potential and the force for the set of parameters
used in the present investigation (see caption for details).

Figure 1: Attractive potential ($U$) and force
($F=-\frac{\mathrm{dU}}{\mathrm{d}r}$). The parameters values are $C$ = $1.02$
m and $D=0.04$ m. Note that the curves are normalized and therefore do not
depend on the value of $\epsilon$.

The feeling intensity can be as strong as $\epsilon=10^{9}\,$N$\cdot$m, as
reported in Ref. [1]. Thus, we will express our results in the more convenient
variable defined as follows:

$\varepsilon=\log_{10}\left(\frac{\epsilon}{\text{N}\cdot\text{m}}\right)$ (6)

###  2.3 Clustering structures

Human clusters arise when pedestrians get in contact between each other and
are responsible for the time delays during the evacuation process [7, 8, 13,
14, 3]. Clusters of pedestrians can be defined as the set of individuals that
for any member of the group (say, $i$) there exists at least another member
belonging to the same group ($j$) in contact with the former. Thus, we define
a spatial cluster following the mathematical formula [27]

$C_{g}:P_{i}\in C_{g}\Leftrightarrow\exists\ P_{j}\in
C_{g}\>/\>d_{ij}<r_{i}+r_{j}$ (7)

where $p_{i}$ represents the $i$th pedestrian, $r_{i}$ their radius (shoulder-
neck).

From all granular clusters, the blocking clusters are the minimum set of
contacting pedestrians (belonging to a cluster) that connects the walls on
both sides of the exit. Roughly speaking, it refers to the shortest chain of
contacting pedestrians that links both sides of the exit door. An example of a
blocking cluster is shown in Fig. 2.

Figure 2: Example of a blocking cluster formed during an evacuation process.
Individuals are represented as red circles, and those with black borders are
in contact, thus they are part of a cluster. As one of them is in contact with
the upper wall and another one is in contact with the bottom one, this set of
pedestrians is a blocking cluster.

###  2.4 Fundamental Diagram

Many researches on pedestrian dynamics focus their attention on the relation
between the flow and the density of a moving crowd. This relation is
represented by the “fundamental diagram” and it has become one of the most
common ways to characterize the pedestrians’ dynamics along a corridor [28,
26, 29, 30]. The flow is defined as

$\mathbf{J}=\rho\mathbf{V}$ (8)

where $\rho$ represents the local density (say, the number of pedestrians
within a region divided by its area) and $\mathbf{V}$ corresponds to the local
velocity (the averaged velocity of the pedestrians in that region).

Details on empirical and simulated fundamental diagrams and measurement
conditions can be found in Refs. [26, 28].

##  3 Experimental background

In this section, we fit our numerical simulations into the available
experimental data, in order to obtain estimates of $\varepsilon$ (the
intensity of the attractive force) for each category of social group.

Ref. [31] captured, by means of video cameras, the walking characteristics of
group members. The authors classified this information into group categories,
and determined the distance distribution function between members of dyads
(that is, groups of two individuals). Their results can be seen in Fig. 3.

Figure 3: (Color online only) Probability distribution function for the
distance ${\Delta r=|\mathbf{r}_{1}-\mathbf{r}_{2}|}$ within a group. Each
curve corresponds to a different group category. These recordings were taken
in “The Asia and Pacific Trade Center” in Osaka (see Ref. [31] for details).

We are now able to link the expected distances for couples or colleagues with
the intensity of the attraction feelings. We proceed as follows: first, we
compute the equilibrium distances for a wide range of intensity values
$\varepsilon$ (see A for details). Secondly, we compare these distances with
the corresponding values for couples or colleagues. Finally, we associate an
intensity range for $\varepsilon$ within each category.

While all the experimental distributions in Fig. 3 are quite similar, the less
involving categories display wider distributions. For instance, the most
probable distances for the couples category ranges from 0.5 m to 0.65 m ($\pm$
one standard deviation). The most probable distances for the colleagues,
instead, ranges from 0.6 m to 0.9 m ($\pm$ one standard deviation). This kind
of dispersion makes it difficult to classify groups when the walking distance
is around $\Delta r\sim 0.6\,$m. Thus, we will not try to fit the data for the
intermediate categories (family and friends), in order to avoid ambiguities.

Figure 4: (Color online only) Equilibrium distance between both members of a
walking dyad in a low density corridor, as a function of the attractive
intensity $\varepsilon$. Details on the parameters used for the SFM can be
found in Section 4 below. The red and green shaded regions correspond to the
most probable distances for colleagues and couples, respectively, while the
blue shaded region corresponds to the presumed intimate couples category (see
text for details).

Fig. 4 shows the equilibrium distance of a walking dyad as a function of the
attractive force intensity. Notice that a linear relation exists between
$\Delta r$ and $\varepsilon$. The shaded regions in Fig. 4 indicate the
mapping from $\varepsilon$ to the couple or colleagues category (see caption
for details). We see that the range of $\varepsilon$ corresponding to
colleagues is $1<\varepsilon<6$, and the one corresponding to couples is
$5<\varepsilon<7.5$.

Notice that if the attraction feelings $\varepsilon$ lie between 5 and 6, we
are not able to tell precisely if they are colleagues or a couple. This
requires additional information that is out of the scope of our investigation.

Besides, note from Fig. 3 that all the distribution functions vanish for
distances shorter than 0.5 m. We presume that this is a consequence of
cultural habits (say, japanese culture in the case of Fig. 3). We leave open
the possibility that other cultures might allow distances shorter than 0.5 m
(hugging, holding hands, etc.). We will refer to dyads attaining distance
intervals between 0.4 and 0.5 m as intimate couples. This distance range maps
to $\varepsilon$ between 7.5 and 9 in Fig 4.

##  4 Numerical Simulations

We first present the general simulation conditions. We then go over the
specific conditions for the bottleneck and the straight corridor situations
separately.

###  4.1 General Simulation Conditions

Pedestrians were modeled as soft spheres. Initially, the individuals were
randomly distributed along the simulation box with random initial velocities,
following a Gaussian distribution with null mean value. At the beginning of
the simulation, group members were separated by a random distance between 0.4
and 0.7 m and placed in a random orientation. The desired velocity $v_{d}$ was
the same for all the individuals, and the attractive feelings intensity
$\varepsilon$ was the same for all of the groups in each simulation process.

We set the SFM parameters to the usual literature values, as shown in Table 1
(see details in Refs. [1, 2, 3, 32]). Notice that two friction coefficients
appear, which correspond to the commonly accepted values in the literature.
However, researchers did not arrive to a unique value for the friction at
bottlenecks or corridors [28, 11]. We will accept two possible values, without
introducing a discussion on this matter here.

Parameter | Value | Meaning
---|---|---
$m$ | 70 Kg | Pedestrian’s mass
$r$ | 0.23 m | Pedestrians’ radius (shoulder-neck)
$\tau$ | 0.5 s | Relaxation time
$A$ | 2000 N | Intensity of $\mathbf{f}_{s}$
$B$ | 0.08 m | Characteristic length of $\mathbf{f}_{s}$
$C$ | 1.02 m | Maximum of $\mathbf{f}_{a}$
$D$ | 0.04 m | Halved characteristic length of $\mathbf{f}_{a}$
$\kappa$ |  $2.4\cdot 10^{5}\,$Kg m${}^{-1}\,$s-1 | Friction coefficient in bottlenecks
$\kappa_{\text{augmented}}$ |  $1.2\cdot 10^{6}\,$Kg m${}^{-1}\,$s-1 | Friction coefficient in corridors
Table 1: Parameters used for the SFM simulations. Notice that there are two
different values for $\kappa$ (see text for details).

Each simulation process had two kinds of individuals: single pedestrians and
group members. Single individuals interact with others through the social and
friction forces ($\mathbf{f}_{s}$ and $\mathbf{f}_{g}$, respectively). Group
members, additionally, are mutually attracted by the attractive force
$\mathbf{f}_{a}$ (see Section 2.1 and 2.2 for details).

The simulations were performed using Lammps molecular dynamics simulator with
parallel computing capabilities [33]. The time integration algorithm followed
the velocity Verlet scheme with a time step of $10^{-4}\,$s. We implemented
special modules in C++ for upgrading the Lammps capabilities to perform the
SFM simulations. Data recordings were done at time intervals of $0.05\,$s,
that is, at intervals as short as 10% of the pedestrian’s relaxation time
$\tau$.

###  4.2 Evacuation in the bottleneck scenario

We simulated evacuation processes from a 20 m $\times$ 20 m room with 225
pedestrians. The room had a single exit placed symmetrically, as shown in Fig.
5. The door width was 0.92 m (equal to the diameter of two individuals), as in
Refs. [1, 8, 7, 13]. All the individuals had the willing to go to the exit
door. That is, at each time-step the desired direction $\hat{\mathbf{e}}_{d}$
was updated in order to point to the exit, as depicted in Fig. 5. The
simulation process lasted until 160 individuals ($\sim$70%) left the room. If
this condition could not be fulfilled in the first 3000 s, the process was
stopped. We ran 100 processes for each set of $v_{d}$ and $\varepsilon$
values, in order to get enough data for statistical analysis.

Figure 5: (Color online only) Initial configuration for an evacuation process
through a bottleneck. 225 pedestrians are located in a 20 m $\times$ 20 m room
with a single door. Single individuals are represented as red circles, while
each dyad is represented by a other colors (online version only).
$\mathbf{e}_{d}^{(i)}$ points directly to the exit.

We focused on two cases: 25% of the pedestrians belong to dyads (say, 56
individuals grouped in 28 dyads), or 100% of the pedestrians belong to dyads
(224 individuals grouped in 112 dyads). The latter is supposed to be an
extreme situation, but is intended as a bounding case for our results.

We did not include bigger groups in order to avoid the effects of group
walking patterns.

###  4.3 Straight corridor scenario

This scenario explored the flow of pedestrians along a straight corridor of
width $w~{}=22$ m (say, similar to the width of the entrance at the Jamaraat
Bridge; see Ref. [26] for details) and length $L=28\,$m with periodic boundary
conditions (see Fig 6). The global density ranged from $\rho=0.5$ to 9 P/m2.

The desired velocity for every pedestrian was set to $v_{d}=1\,$m/s, which
correspond to low anxiety levels. The desired direction $\hat{\mathbf{e}}_{d}$
always pointed from left to right (see Fig. 6).

The sampling area was located in the middle of the corridor as shown in Fig.
6. The velocity $\mathbf{V}$ of the individuals inside the circle was recorded
every 0.5 s for 80 s, after the first 20 s of the simulation. The flow
$\mathbf{J}$ was further computed according to Eq. (8).

Figure 6: Schematic diagram for individuals in the corridor. White circles
represent pedestrians moving towards the positive values of $x$. The red
circle in the middle corresponds to the sampling region, and its radius was
set to 1 m.

Recall from Table 1 that the accepted friction coefficient for the corridor
scenario is five times higher than the one for the bottleneck. This value
yields to a fundamental diagram that agrees with the experimental data
appearing in Ref. [26] (see Ref.[28]). Therefore, we will consider this
augmented value for simulations in the corridor scenario without entering into
further discussions.

In all cases, 70% of the individuals belonged to groups. We studied situations
of groups of up to 5 members in Section 5.4. However, we will not analyze the
groups pattern formation or the decision-making processes.

##  5 Results

###  5.1 Evacuation time with social groups

In this section we present the results corresponding to the bottleneck
scenario. Fig. 7 shows the mean evacuation time $\langle t\rangle$ of a crowd
for a wide range of anxiety levels and attractive intensities (see caption for
details).

(a) 25%

(b) 100%

Figure 7: (Color online) Mean evacuation time for the egress of 160
individuals as a function of $v_{d}$ and the attractive feelings intensity
level $\varepsilon$. The door width was 0.92 m (equal to two pedestrians’
diameter). The black curve, $\epsilon=0$, corresponds to no dyads within the
crowd. (a) 25% of the pedestrians were grouped in dyads. (b) 100% of the
pedestrians were grouped in dyads.

Fig. 7 exhibits the usual “Faster-Is-Slower” and “Faster-Is-Faster” behaviors,
as explained in the literature (see Refs. [8, 14, 13]). However, the presence
of dyads worsens the evacuation performance. This phenomenon becomes more
significant as the fraction of dyads increases (see Fig. 7(b)).

Our concern is with the intensity of the feelings within the dyads,
$\varepsilon$. Thus, we explored the evacuation performance for four
representative values of $v_{d}$. Fig. 8 shows the corresponding mean egress
time as a function of $\varepsilon$.

(a) 25%

(b) 100%

Figure 8: (Color online) Mean evacuation time for 160 individuals as a
function of $\varepsilon$ for fixed values of $v_{d}$. (a) 25% of the
pedestrians were grouped in dyads. (b) 100% of the pedestrians were grouped in
dyads. The simulation conditions were the same as in Fig. 7

The very first inspection of the plots shows that the slopes are almost flat,
except at intensities $\varepsilon$ between 2 and 3, and above 8. Fig. 8(b)
exhibits more noticeable changes at these intensities than in Fig. 8(a).

The intensities $\varepsilon$ between 2 and 3 correspond to feeling forces in
the range of 625 to 6250 N. Recall that the social force approaches 2000 N
near the contact distance. Thus, the attractive feelings appear quite weak
with respect to the social repulsion below $\varepsilon=2$, but quite strong
beyond $\varepsilon=3$. The range $(2<\varepsilon<3)$ produces a significant
change in $\langle t\rangle$, according to Fig. 8(b). Notice, however, that
the egress time settles to a well established level above $\varepsilon=3$,
depending on the degree of anxiety (say, the desired velocity $v_{d}$).

The egress time performs somewhat better beyond $\varepsilon=7$ if all the
individuals are grouped in dyads and $v_{d}>1.25\,$m/s (see Fig. 8(b)). This
corresponds to intimate couples evacuating in a crowded environment. Ref.[1]
reports this phenomenon for $v_{d}=4\,$m/s only. The authors explain that
dyads are so tightly linked that the whole dyad mimics a single big person,
improving the evacuation time.

The time improvement observed for intimate couples ($\varepsilon>7$) does not
apply to the low anxiety curve in Fig. 8(b) (say, $v_{d}=1.25\,$m/s). The
reason for this is explained in B. We will not make further comments on this
phenomena since it belongs to a non-panic scenario.

The behavior of $\langle t\rangle$ as a function of $\varepsilon$ is
qualitatively similar to its counterpart as a function of $v_{d}$ (see Fig.
7). We will call “Closer-Is-Slower” (CIS) and “Closer-Is-Faster” (CIF) the
regimes in Fig. 8 that mimic the “Faster-Is-Slower” and “Faster-Is-Faster”
regimes in Fig. 7, respectively. That is, CIS stands for the interval
attaining a positive slope Fig. 8(b) (say, $2<\varepsilon<5$) and CIF
corresponds to the interval attaining a negative one ($\varepsilon>5$). Recall
that the former is associated with the expected feelings among colleagues
while the latter is associated with intimate couples.

The contour-map for the evacuation time as a function of both $v_{d}$ and
$\varepsilon$ is shown in Fig. 9. This map combines Figs. 7 and 8 and resumes
the expected perfomance levels for $\varepsilon$ or $v_{d}$ simultaneously.

(a) 25% dyads

(b) 100% dyads

Figure 9: (Color online only) Mean egress time for 160 individuals as a
function of both the attractive feeling intensity $\varepsilon$ and the
desired velocity $v_{d}$, averaged over 100 simulations. (a) 25% of the
pedestrians were grouped in dyads. (b) 100% of the pedestrians were grouped in
dyads.

Fig. 9 presents a wide view on how dyads affect the evacuation dynamics. The
worst situation occurs among very close colleagues attaining desired
velocities of $v_{d}=4\,$m/s. This is in agreement with Ref. [1], where the
authors showed that colleagues persist moving together among the crowd, and
thus, are responsible for slowing down the evacuation. This phenomenon appears
somehow blurred in Fig. 9 for extremely high stressing situations (say,
${v_{d}>5\,}$m/s).

###  5.2 Delay analysis

In order to analyze the microscopic dynamics behind the CIS and CIF regimes,
we measured the time intervals between successive individuals that leave the
room (regardless if they belong to the same dyad or not). We will refer to
these time intervals as “delays”. Fig. 10 shows the corresponding histograms
for different values of $\varepsilon$ and two representative $v_{d}$’s (see
caption for details). Only the results for 100% dyads are shown for clarity
reasons.

(a) $v_{d}=3$ m/s

(b) $v_{d}=5$ m/s

Figure 10: (Color online) Occurrence distribution of delays between successive
individuals. The bin size is 0.25 s. Note that the delays of all 100
simulations included. The simulation conditions were the same as in Fig. 7 for
100% dyads. (a) $v_{d}=3$ m/s. (b) $v_{d}=5$ m/s.

We can distinguish two frequency patterns in Fig. 10. The “short delays” (say,
$\Delta t<1\,$s) are quite considerable. The “intermediate delays”
($1\,$s${\,\leq\Delta t\leq 4\,}$s), instead, extend as a long tail. However,
the situation for $v_{d}=5\,$m/s exhibits a heavier tail than the one for
$v_{d}=3\,$m/s. As a first instance, this partially explains the egress time
patterns shown in Figs. 7, 8 and 9, but a more meaningful comparison should be
made according to the weighted distribution

$\displaystyle F(C)=\sum_{\Delta t_{i}\,\in\,C}\Delta t_{i}\cdot f(\Delta
t_{i})$ (9)

where $f(\Delta t_{i})$ is the frequency of the delays with duration $\Delta
t_{i}$ and $C$ means any category set. For simplicity we will analyze only
three categories of delays: short ($\Delta t<1$ s), intermediate (1
s$\leq\Delta t\leq 4$ s) and long ($\Delta t>4$ s). Fig. 11 shows each
distribution as a function of $\varepsilon$.

(a) $v_{d}=1.25$ m/s

(b) $v_{d}=3$ m/s

(c) $v_{d}=5$ m/s

(d) $v_{d}=8$ m/s

Figure 11: (Color online only) Weighted sum of short, intermediate and long
delays (see text for details). The total evacuation time is also included for
comparison. (a) $v_{d}=1.25$ m/s. (b) $v_{d}=3$ m/s. (c) $v_{d}=5$ m/s. (d)
$v_{d}=8$ m/s.

Recall that $v_{d}=1.25\,$m/s is out of the panic scenario, and thus, we will
not attempt to analyze this case. The reader can find some notes on this issue
in B.

Notice that short delays remain unchanged for every value of $\varepsilon$ and
any given $v_{d}$. Thus, we infer that the CIS or CIF regimes should be
related to the presence of intermediate and long delays. Fig. 11 also shows
that the intermediate delays become relevant in context of the panic. These
contribute along both the CIS and CIF regimes. The long delays contribute in
less degree and diminish along the CIF regime (say, for $\varepsilon>6$).

We further examined the animations of the simulations, and observed that
individuals get released from the blocking clusters more frequently in the CIF
regime, and consequently, the time that the exit remains blocked diminishes.
This explains the decrease of the long delays and the overall decrease of the
egress time. The observed behaviors relating the time lapses and the blocking
clusters were also reported in Ref. [14].

Fig. 12 summarizes the above results as a function of $v_{d}$ and
$\varepsilon$. We can see that the intermediate delays attain a maximum at the
top of the heat-map ($8<\varepsilon<9$, that is, for intimate couples), while
this maximum moves to the middle of the map ($5<\varepsilon<6$) for long
delays. We conclude that there is a “displacement” from long to intermediate
ones, which ultimately implies an improvement for the evacuation time.
Finally, for $\varepsilon>9$, both categories of delays decrease.

(a) Sum of intermediate delays

(b) Sum of long delays

Figure 12: (Color online only) Heat-maps for the weighted the sum of (a)
intermediate delays (1 s $\leq\\!\Delta t\\!\leq$ 4 s) and (b) long delays
($\Delta t\\!>$ 4 s). The simulation conditions are the same as in Fig. 7.

###  5.3 Ordered vs. disordered dynamics in the bottleneck

We now open the question on how many of the individuals that egress
consecutively are actually dyad-partners. This question concerns the
“ordering” of the evacuation process, where by “order” we mean that dyad
members egress together. For instance, in an “ideally ordered” evacuation
process where 100% of the individuals are grouped, exactly 50% of the delays
should correspond to partners — the first one egresses, the second one is his
(her) dyad-partner, then the third one is from another dyad, the fourth one is
his (her) partner and so on.

Notice that as the door is wide enough for two pedestrians to exit
simultaneously, there is a chance that an individual might sneak in between a
dyad, without loosing the attractive feelings between the partners (say,
without missing each other). An example of such a situation is shown in Fig.
13 (see caption for details). We will, therefore, consider that both dyad
members egress together if not more than three individuals evacuated between
them, as sketched in Fig. 13(c).

(a) $t=68.10$ s

(b) $t=69.95$ s

(c) Order of egress

Figure 13: (Color online only) Snapshots of an evacuation process for
$v_{d}=5$ m/s and $\varepsilon=5$. Each dyad is represented by a different
color (online version only). The purple individual egresses from the bottom
half of the door at $t=68.10$ s. Then the light blue and dark blue individuals
egress from the top half of the door at $t=68.15$ s and $t=68.55$ s,
respectively. However, as the second member of the purple dyad egresses from
the bottom half at $t=69.95$ s, we consider that the dyad partners do not miss
each other, and both egress (seemingly) together.

Fig. 14 shows the fraction of individuals that egress together with their
dyad-partner. As $\varepsilon$ increases for any fixed value of $v_{d}$, the
number of partners that egress together increases, which was to be expected,
since stronger attractive feelings will motivate partners to remain coupled.
In particular, when $\varepsilon=10$ this fraction surpasses 40% for all the
explored anxiety levels, almost reaching an “ideally ordered” scenario.

Figure 14: (Color online) Percentage of individuals that egress together with
their partners. The “ideally” ordered scenario corresponds to 50%. The
simulation conditions are the same as in Fig. 7.

On the other hand, for any fixed value of $\varepsilon$, as $v_{d}$ increases
the ordering fraction decreases. This means that the dyads miss the egress
“order” as their anxiety level increases (say, $v_{d}$ increases). This is
quite noticeable within the range of colleagues and couples
($4<\varepsilon<6$). Intimate couples, however, are scarcely affected by panic
(within the explored range of $v_{d}$).

We further linked the degree of “order” in the evacuation process with the
corresponding egress delays. Fig. 15 shows the distribution of delays within
individuals belonging to the same dyad (see caption for details). We call
these “intra-group delays”.

Figure 15: (Color online only) Intragroup delay distribution for three
representative values of $\varepsilon$ for $v_{d}=5\,$m/s. All the individuals
where grouped in dyads. The bin size of the histograms is 0.25 s.

It is clear from Fig. 15 that the more intense attractive feelings, the
shorter the intra-group delays. For instance, the intra-group delays that are
longer than 15 s represent 22% for $\varepsilon=3$, 12% for $\varepsilon=6$
and 6% for $\varepsilon=9$. These occur often on the sides of the door, as
illustrated in Fig. 16 (see caption for details).

(a) $t=21.15$ s

(b) $t=22.60$ s

(c) $t=59.75$ s

(d) $t=61.65$ s

Figure 16: (Color online only) Snapshots of an evacuation process for 100%
dyads and $v_{d}=5$ m/s. The door is centered at $x=20$ m, $y=10$ m and has a
0.92 m width. All dyads have a very strong attractive feelings
($\varepsilon=9$). Each dyad is represented by a different color, and those
with a black border have a fairly long intragroup delay (online version only).
The orange partners egress at snapshots (a) and (d), while the blue partners
egress at snapshots (b) and (c), respectively. The intra-group delay for the
orange dyad is 40.5 s, while the corresponding delay for the blue one is 37 s.

We conclude from this Section that “ordering” is a major issue whenever social
groups are present. Strong feelings within the couples ensures that both will
evacuate together. On the contrary, weak feelings will allow individuals to
sneak in between the dyads, making partners miss each other, and therefore
slowing down the evacuation.

###  5.4 Ordered vs. disordered dynamics in the corridor

We report in this Section the dynamical consequences of introducing social
groups in a corridor (within the context of the SFM). We focus on the
fundamental diagram (i.e. flux vs. density, see Section 2.4) for crowds where
70% of the individuals belong to groups of up to 5 individuals. The sampling
procedure is as shown in Fig. 6.

Fig. 17 captures how individuals are grouped in our model while the corridor
is in a non-congested regime (see caption for details).

(a) Groups of 3 members

(b) Groups of 5 members

Figure 17: (Color online only) Snapshots of simulated processes for corridors
with groups of 3 and 5 members, respectively, attaining strong attractive
feelings ($\varepsilon=8$). Each group is represented by a different color,
and individuals that are not members of a group are represented in red. In
both cases, the total number of individuals is $N=1848$, which divided by the
area of the corridor gives the global density $\rho=3$ P/m2 (this corresponds
to the the free-flow regime). The desired velocity of all individuals was
$v_{d}=1\,$m/s. Both situations are stationary, right after the pedestrians
are accommodated within the corridor.

Recall that “ordering” in Section 5.3 means that groups egress together, while
no one can sneak in between. The situation in the corridor is somehow
different, although some individuals might sneak in between the members of a
group when the attractive feelings are those associated to colleagues and
couples (say, $\varepsilon<7$) and the corridor is congested. This situation
is shown in Fig. 18. We will therefore associate these inhomogeneities to a
somewhat “disordered” crowd.

Figure 18: (Color online only) Snapshot at $t=32.2\,$s of a corridor with
dyads attaining $\varepsilon=5$. Each group is represented by a different
color, and individuals that are not members of a group are represented in red.
The global density is $\rho=5$ P/m2 (which corresponds to the congested
regime). The desired velocity for all the individuals was $v_{d}=1\,$m/s. The
dyads with a black border are a few examples of how other individuals can
sneak in between groups at high densities.

Fig. 19 shows the fundamental diagram for 70% of the pedestrians grouped in
dyads. A slight slowing down of the flux can be observed in the congested
regime (say, $\rho>4.5$ P/m2) as $\varepsilon$ increases. Notice that varying
$\varepsilon$ does not significantly change the behavior of the flow (within
the explored range). We present in C a more detailed analysis regarding the
slight differences between the curves in the congested regime, by computing
the velocity profiles among the corridor.

Figure 19: (Color online only) Mean flow as a function of the global density
($\rho=N/\mathrm{Area}$) for the corridor of 22 m wide and 28 m long (with
periodic boundary conditions). The flow was sampled within a circle of
$R=1\,$m in the middle of the corridor. The data points are mean values over
80 s, sampled at a frequency of 2 times per second. 70% of the individuals
were grouped in dyads, and every dyad has the same attractive intensity
$\varepsilon$ in each simulation. The desired velocity was $v_{d}=1\,$m/s for
all the individuals. The empirical diagram from Ref. [26] is shown in the
inset. Figure 20: (Color online only) Fundamental diagram for a crowd moving
along a corridor, where 70% of the pedestrians belong to groups of up to 5
individuals. The attractive intensity was $\varepsilon=8$. GS stands for the
group size (all groups were of the same size in each simulated process). The
black curve for $\epsilon=0$ corresponds to the no grouping situation. The
desired velocity was $v_{d}=1\,$m/s for all individuals.

We hypothesize that the slight slowing down could become more relevant when
groups are enlarged (say, of up to 5 members). Fig. 20 shows the fundamental
diagram obtained for groups with 3 to 5 members with an attractive intensity
of $\varepsilon=8$. We can see that the free-flow regime (say, the non-
congested regime) extends to higher densities as the number of group members
increases. This means that groups move in a free-flow environment at densities
up to $4-5\,$P/m2.

A re-examination of Fig. 17 shows that there is more space between groups as
they become larger. Thus, we presume that the “effective density” of the crowd
is somewhat lower than the global density (say, $N$ divided by the total
corridor area). We will discuss this point in Section 6.

Fig. 20 also shows that no matter the size of the groups (within the explored
range), the flow slows down to a common value for extremely high global
densities. This corresponds to a very compact scene, as shown in Fig. 21 where
all the pedestrians are in contact.

Figure 21: (Color online only) Snapshot at $t=33.5\,$s for a corridor with
groups of 3 members attaining $\varepsilon=8$. The global density is
$\rho=7\,$P/m2 (which belongs to the congested regime). The simulation
conditions are otherwise the same as in Fig. 17.

The flow in Fig. 19 drops to 2 P(ms)-1 for dyads at very high densities (and
$\varepsilon=8$). The corresponding flow in Fig. 20 drops to a somewhat lower
value, no matter the group size (within the explored range). This means that
the group size is not a relevant magnitude at very high densities, except for
dyads. Fig. 21 shows that more than two members per group form quite closely
shaped structures. As a drawback, these groups can not move among the crowd as
smoothly as dyads or single individuals do. Moreover, as groups attain
stronger attractive feelings (such as $\varepsilon=8$), this drawback becomes
more relevant. Thus, it seems reasonable to obtain a more significant slowing
down with respect to the dyads situation (at very high densities).

We summarize the above results as follows. Social groups introduce a somewhat
inhomogeneous (disordered) scene along the corridor. This yields to a
fundamental diagram that differs from the one reported for single individuals.
The differences, however, become more significant if the groups’ size exceeds
two members. The first difference occurs at the intermediate density interval,
where the free-flow regime extends into higher densities (say,
$\rho=4-5\,$P/m2). The second difference, though, concerns the slowing down.
Social groups slow the flow down more pronouncedly than single individuals at
the extremely congested regime. However, the size of the groups seems to not
play a role in this phenomenon (within the explored group sizes).

##  6 Discussion

We analyzed in Section 5 the evacuation dynamics of social groups in the
bottleneck and the corridor scenarios. Although quite different situations,
both show that the presence of attractive feelings introduce additional delays
(bottlenecks) or slowing downs (corridor) in the movement dynamics. We now
discuss how these occur according to the intensity of the attraction. We focus
on dyads, but the discussion can be extended to larger groups.

We present two schemes in Fig. 22. The scheme on the left displays the
situation of “weakly” attracted partners, where mutual feelings are not strong
enough to get into body contact. The scheme on the right displays the
situation expected for intimate partners attaining a “strong” attractive
feeling, and thus, getting into the contact distance.

(a) Weak $\varepsilon$

(b) Strong $\varepsilon$

Figure 22: (Color online only) Schemes comparing crowds with different types
of dyads. Each color represents a dyad while gray individuals are not grouped.
The desired and attractive forces are shown in green (fd) and red (fa),
respectively. All the desire forces point to the right for simplicity, but it
is also possible to point to a nearby target. (a) Crowd with dyads attaining
weak attractive feelings intensity $\varepsilon$ (colleagues, couples). (b)
Crowds with dyads attaining very strong $\varepsilon$ (intimate couples).

As shown in Fig. 22(a), when the attractive feelings are not strong enough to
guarantee that all dyads will be at contact distance, some individuals are
able to squeeze among partners. This brings out negative consequences to the
evacuation. For instance, the pink individual on the rightmost of Fig. 22(a)
squeezed through the green dyad. But the green dyad is still blocking the way
to the pink individual on the left. At this point, the whole pink dyad gets
delayed since none of the pink individuals will leave behind his (her)
partner. Similar situations are represented in Fig. 22(a). See, for example,
the orange dyad and the gray individual.

Besides, if $\varepsilon$ is strong enough to ensure that dyads will be at the
contact distance, the scene will look like the one in Fig. 22(b). No squeezing
is now possible among the members of a dyad. These move together as a single
bigger structure (the dyad) towards the target. The delays occur due to
congestion (and friction) between dyads during the way out. Notice that this
scenario corresponds to the “Closer-Is-Faster” (CIF) regime since higher
$\varepsilon$ leads to more packed dyads, and therefore, is somehow more
favorable to the evacuation process than the one in Fig. 22(a).

Recall our observation in Section 5.4 that strong attraction feelings reduce
the “local effective density”. This becomes quite clear from Fig. 22(b). The
global density can stay moderate, say at 4-5 P/m2, although the social groups
form closely packed structures. As a consequence, there is more space left
between the groups, allowing a free-moving regime.

The schemes represented in Fig. 22 resume the CIS and CIF phenomena. However,
these may occur in different contexts. For instance, Fig. 13(a) shows the
situation where a member of the purple dyad is trapped among the blocking
cluster while the other is trying to cross the door. Figs. 18 and 21 are
corresponding examples of weak and strong attractive feelings in a crowed
corridor.

We call the attention on the fact these phenomena can be obscured by other
circumstances. We mentioned the groups’ shape as one of these circumstances in
Section 5.4. But the limiting walls (of the corridor or door sides) can be
thought as another one (examine Figs. 13, 16 and 17 as examples). Further
investigation will focus on these ones.

##  7 Conclusions

Our investigation focused on the evacuation of social groups in the context of
the SFM. We upgraded the basic SFM according to the attractive feelings
suggested in Ref. [1]. We calibrated the model parameters with novel empirical
results appearing in Ref. [31]. We then focused on two scenarios: the
bottleneck and the corridor situations.

Our major result concerning the bottleneck scenario is that dyads (social
groups of two members) slow down the emergency evacuations, although the final
performance depends on the emotional involvement of the partners: moderately
involved partners (colleagues) yield the worst evacuation performance, while
highly involved partners (intimate couples) develop a better perfomance. We
call both situations as the “Closer-Is-Slower” regime (for colleagues) and the
“Closer-Is-Faster” regime (for intimate couples), respectively.

The investigation on the nature of the delays in each regime showed that long
delays are associated to some kind of disorder among the congested crowd. That
is, moderately involved partners allow others to squeeze through them, and
therefore, slow the evacuation down. Intimate couples, instead, remain always
together, keeping some kind of “order” during the evacuation.

We emphasize that the “Closer-Is-Slower” and “Closer-Is-Faster” regimes
express some kind of parallelism with the known “Faster-Is-Slower” and
“Faster-Is-Faster” regimes. We correlated all of them along the attractive
intensities vs. anxiety levels in Section 5.2.

The corridor scenario experiences two significant phenomena whenever social
groups are present, which can be summarized as follows:

(a) The free-flow regime extends towards more densely crowded situations than
in the case of single individuals (to around 4-5 P/m2). This means that social
groups yield to a more easily moving situation, as discussed in Section 6.

(b) In the highly congested regime, the flow slows down more pronouncedly than
in the case of single individuals. This means that social groups do not move
as smoothly as the single individuals in a highly dense environment. This was
studied in Section 5.4.

These phenomena become more noticeable when the groups consist of at least
three members, and if these are highly emotionally involved.

In summary, our results show that socials groups tend to slow down the crowd
dynamics (in the context of the SFM). However, the bottleneck and corridor
situations also show that the final performance depends strongly on other
considerations, say, the specific geometry where the situation takes place.

## Acknowledgments

This work was supported by the Agencia Nacional de Promoción Científica y
Tecnológica, Argentina) grant FONCYT 2019 Number PICT-2019-2019-01994. G.A
Frank thanks Universidad Tecnológica Nacional (UTN) for partial support
through Grant PID Number SIUTNBA0006595.

##  Appendix A Balance between social and attractive forces

This appendix analyses the dynamic of an isolated dyad. It may be considered
as an approach for low density crowds, where the dyad partners keep a
respectful distance with others.

We will neglect the sliding friction between partners since we do not expect
any relevant tangential velocity difference between them. Thus, partners will
only interact via the social and attractive forces. Fig. 1 shows the sum of
these forces as a function of the distance between them, $d_{ij}$, for two
values of $\varepsilon$.

(a) $\varepsilon=3$

(b) $\varepsilon=5$

Figure 1: Net force between partners (sum of the social force and the
attractive force). The blue line corresponds to $d_{ij}$ for which
$|\mathbf{f}_{s}(d_{ij})|=|\mathbf{f}_{a}(d_{ij})|$. (a) $\varepsilon=3$. (b)
$\varepsilon=5$.

We call $d_{eq}$ as the equilibrium distance between partners. If
$d_{ij}<d_{eq}$ the net force will be positive (repulsive force), and if
$d_{ij}>d_{eq}$ it will be negative (attractive force). The balance equation
may be approximated to the following first order expression

$f_{s}(d_{12})+f_{a}(d_{12})\simeq-k(d_{12}-d_{eq})\qquad\text{ if
}|d_{12}-d_{eq}|\ll 1$ (10)

where $k$ depends on the parameters of the forces, like $A$ and $\varepsilon$.
Notice that this is actually similar to an harmonic force equation.

In addition to the interaction forces, each individual is motivated to move to
a target position through the “desired velocity” (see Section 2.1). If the
desired velocity is the same for both partners, say $\mathbf{v}_{d}=v_{d}\
\mathbf{\hat{e}}_{d}$, the equations of motion are as follows

$m\ddot{\mathbf{r}}_{1}=\mathbf{f}_{a}^{12}+\mathbf{f}_{s}^{12}+\mathbf{f}_{d}^{1}\simeq-k\left(|\mathbf{r}_{1}-\mathbf{r}_{2}|-d_{eq}\right)\mathbf{\hat{r}}_{12}+m\frac{\mathbf{v}_{d}-\dot{\mathbf{r}}_{1}}{\tau}$
$m\ddot{\mathbf{r}}_{2}=\mathbf{f}_{a}^{21}+\mathbf{f}_{s}^{21}+\mathbf{f}_{d}^{2}\simeq{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-}k\left(|\mathbf{r}_{1}-\mathbf{r}_{2}|-d_{eq}\right)\mathbf{\hat{r}}_{12}+m\frac{\mathbf{v}_{d}-\dot{\mathbf{r}}_{2}}{\tau}$

Subtracting the bottom one to the top one, and defining $\mathbf{r\equiv
r_{1}-r_{2}}$, and $\omega^{2}\equiv 2k/m$, leads to

$\ddot{\mathbf{r}}=-\omega^{2}(|\mathbf{r}|-d_{eq})\mathbf{\hat{r}}-\frac{1}{\tau}\
\dot{\mathbf{r}}\ \Rightarrow\ \boxed{\ \ddot{\mathbf{r}}+\frac{1}{\tau}\
\dot{\mathbf{r}}+\omega^{2}\mathbf{r}=\omega^{2}d_{eq}\mathbf{\hat{r}\ }}$
(11)

This is the equation of a damped harmonic oscillator in two dimensions, with
frequency $\omega$, decay time $2\tau$ and equilibrium distance $d_{eq}$. So,
in a negligible amount of time compared to an egress time, each dyad reaches
its equilibrium distance. It is interesting to note that this damping is
caused by the desire force but does not depend on $v_{d}$, as the decay time
is given by $\tau$.

Fig. 2 shows the distance between the members of a dyad as a function of time.
Varying $A$, $\varepsilon$ and $v_{d}$, showed no significant differences in
the decay time, and it only changed when varying $\tau$, as expected.

Figure 2: Distance between the members of a dyad in a simulation. The
attractive intensity is $\varepsilon=5$. The initial separation is 0.84 m.

Notice that for $\varepsilon=5$, the equilibrium distance is the same as the
one depicted in Fig. 1(b). Similarly, for any other $\varepsilon$, the
equilibrium distance found from the balance equation
$|\mathbf{f}_{s}(d_{eq})|=|\mathbf{f}_{a}(d_{eq})|$ and from low density
simulations are the same.

##  Appendix B Evacuation time from low to moderate anxiety levels and very
strong attractive feelings

We notice in Figs. 9 and 10 (see Sections 5.1 and 5.2), that emergency
evacuations with low anxiety levels ($v_{d}<1.5\,$m/s) slow down as
$\varepsilon$ gets to $\varepsilon\simeq 8$. Video animations of the
simulation processes show that dyads are permanently in contact during the
evacuation. This means that partners behave as a single body. Fig. 1 captures
an example of the animations.

Figure 1: (Color online) Snapshots of an evacuation simulation process with
$v_{d}=1.25\,$m/s and $\varepsilon=9$. Each dyad is represented by a different
color. The purple dyad with a green dot is highlighted as an example of how
dyads advance towards the exit (see text for details).

For moderate and high $v_{d}$, we see no empty spaces surrounding the exit, as
can be seen in Figs. 13 and 16. Fig. 1 shows, however that the individuals
prefer to stay with their partners instead of moving to the available spots
near the exit. In other words, the attractive feelings surpass the desire to
get out of the room, delaying both partners even when they have the option to
advance.

The above behavior occurs because panic is not really present, and thus, other
emotions dominate the scene. We do not focus on these situations since they
are out of the emergency context.

##  Appendix C The velocity profiles of dyads in the corridor

To further examine the ordering and disordering in the corridor scenario, Fig.
1 shows the velocity profiles (the velocity in the direction of the corridor
as a function of the transverse coordinate, say, $v_{x}$ vs. $y$).

Figure 1: Velocity profile ($v_{x}$ vs. $y$) for various densities $\rho$
(shown in P/m2 units) and $\varepsilon$. The simulated corridor is 28 m long
with periodic boundary condition along the walking direction, $x$. The
corridor is delimited by walls at $y=0$ and $y=22\,$m.

For densities within the free-flow regime, all pedestrians move at the desired
velocity ($v_{x}\\!=\\!v_{d}\\!=\\!1\,$m/s) for all $\varepsilon$, and thus
the flow grows linearly as a function of the density (see Eq. (8)). As the
density increases within the congested regime, the friction with the walls and
between individuals slows the pedestrians down and generates a seemingly
parabolic velocity profile. This result is similar to the velocity profiles
for laminar flows in viscous fluids, where the velocity attains a maximum at
the center and decreases towards the walls. This behavior has been studied
before in Ref. [28] and has also been observed in real crowds (see Ref. [29]).

From Fig. 1, we can notice two different behaviors:

  * 1.

At $\rho=5\,$P/m2, increasing $\varepsilon$ from 3 to 5 decreases the
velocity, while increasing $\varepsilon$ to 8 increases it. This is consistent
with the CIS and CIF regimes.

  * 2.

At $\rho=8\,$P/m2, increasing $\varepsilon$ decreases the measured velocity,
meaning that only the CIS regime is present.

These behaviors can also be seen in the fundamental diagrams in Fig. 19 at the
mentioned densities.

Fig. 2 shows the percentage of grouped pedestrians that have their dyad-
partner as the closest individual to them. As the crowd gets more congested,
it becomes harder for dyads to remain together. Notice that for higher
$\varepsilon$ at a fixed density amount of partners at contact distance
increases. For instance, only 10% of the dyads are connected at $\rho=5\,$P/m2
and $\varepsilon=3$ and 5. This scales to 50% at $\varepsilon=8$.

Figure 2: Fraction of dyads in contact as a function of the global density.
The simulation conditions are the same as in Section 5.4, and measurements
were taken after 50 s of simulation.

The above result supports our discussion in Section 6, say, that when dyad
members are not physically in contact but are close enough to attract each
other, the attractive forces will slow down the pedestrians’ evacuation (see
Fig. 22).

In summary, for each global density there is a range of $\varepsilon$ in which
the dyads members tend to separate, and a different range in which the
majority of them stay at contact distance. Within the first one, increasing
$\varepsilon$ is not enough to significantly increase the amount of dyads that
stay together, but it does increase the attractive forces that slow the flow
down, thus this range coincides with the “Closer-Is-Slower” regime (see Fig.
22(a)). Similarly, in the second range increasing $\varepsilon$ increases the
amount of dyads that stay together, reducing the slowing down of the flow (see
Fig. 22(b)), therefore it corresponds to the “Closer-Is-Faster” range.

For very high densities, only the first regime is found. For example, for
$\rho\geq 6.5\,$P/m2 in Fig. 2, even when $\varepsilon=8$ the percentage of
connected dyads is below 20%. Thus, only the “Closer-Is-Slower” regime is
present.

Figure 3: Velocity profiles for $\rho=4.5\,$P/m2 in a corridor where 70% of
the pedestrians belong to groups of up to 5 individuals. The attractive
intensity was $\varepsilon=8$. GS stands for the group size, and all groups
were of the same size in each simulated process. The respective fundamental
diagrams for densities near $4.5\,$P/m2 are also shown for reference.

The behavior is quite different for larger groups. Recall from Section 5.4
that at $\rho$ = 4.5 P/m2, the congestion degree varies depending on the group
size. We further inspect the velocity profiles at this density value in Fig. 3
corresponding to each case. We can distinguish three patterns, as follows:

  * 1.

The fundamental diagram corresponding to crowds with no groups and with groups
of 3 individuals is in the congested regime (black and blue curves,
respectively), the velocity profile have a parabolic shape.

  * 2.

With groups of 5 individuals (green curve), the free-flow regime is still
present, and all the pedestrians move at their desired velocity
$v_{x}~{}=~{}v_{d}~{}=~{}1$ m/s.

  * 3.

With groups of 4 individuals (orange curve), the corridor is barely congested
and the flow is transitioning between the two regimes. The velocity profile is
thus in an intermediate stage, morphing from a straight line to a parable
shape.

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