=Paper=
{{Paper
|id=None
|storemode=property
|title=Crystals of Crowd: Modelling Pedestrian Groups Using MAS-based Approach
|pdfUrl=https://ceur-ws.org/Vol-741/ID12_ManentiManzoni.pdf
|volume=Vol-741
|dblpUrl=https://dblp.org/rec/conf/woa/ManentiM11
}}
==Crystals of Crowd: Modelling Pedestrian Groups Using MAS-based Approach==
https://ceur-ws.org/Vol-741/ID12_ManentiManzoni.pdf
Crystals of Crowd: Modelling Pedestrian Groups
Using MAS-based Approach
Lorenza Manenti1,2 , Sara Manzoni1,2
1
Complex Systems and Artificial Intelligence Research Center
Dipartimento di Informatica, Sistemistica e Comunicazione
Universita’ di Milano–Bicocca
viale Sarca 336/14, 20126 Milano
2
Centre of Research Excellence in Hajj and Omrah,
Umm Al-Qura University
Makkah, Saudi Arabia
{manenti,manzoni}@disco.unimib.it
Abstract—The paper presents an agent-based model for the application in the pedestrian dynamics area are presented in
explicit representation of groups of pedestrians in a crowd. The the literature [7]–[9].
model is the result of a multidisciplinary research (CRYSTALS Despite simulators can be found on the market and they are
project) where multicultural dynamics and spatial and socio-
cultural relationships among individuals are considered as first commonly employed by end-user and consultancy companies
class elements for the simulation of crowd of pilgrims taking to to provide suggestions to crowd managers and public events
the annual pilgrimage towards Makkah. After an introduction organizers about questions regarding space management (e.g.
of advantages of Multi-Agent System approach for pedestrian positioning signals, emergency exits, mobile structures), some
dynamics modelling, a formal description of the model is pro- main open issues in Pedestrian Dynamics community are
posed. The scenario in which the model was developed and some
examples about modelling heterogeneous groups of pedestrians highlighted as specific modelling requirements. For instance,
are described. theoretical studies and empirical evidences demonstrated that
the presence of groups strongly modifies the overall dynamics
I. I NTRODUCTION of a crowd of pedestrians [10], [11].
Models for the simulation of pedestrian dynamics and In this paper, we propose an agent-based model for the
crowds of pedestrians have been proposed and successfully explicit representation and modelling of groups of pedestrians,
applied to several scenarios and case studies: these models starting from some fundamental elements we derived from
are based on physical approach, Cellular Automata approach theories and empirical studies from sociology [12], anthropol-
and Multi-Agent System approach (see [1] for a state of ogy [13] and direct observations gathered during experiments
the art). In this work, we refer to the Multi-Agent System in collective environments [14]. This work is the result of
(MAS) approach according to which crowds are studied as CRYSTALS project, a multidisciplinary research project where
complex systems whose dynamics results from local behaviour multicultural dynamics and spatial and social relationships
of individuals and the interactions with their surrounding among individuals are considered as first class elements for
environment. A MAS is a system composed of a set of the simulation of crowd of pilgrims taking to the annual Hajj
autonomous and heterogeneous entities distributed in an en- (the annual pilgrimage towards Makkah).
vironment, able to cooperate and coordinate with each other In modelling groups, considering the differences in the
[2], [3]. Many research areas contribute to the development agent-based tools before mentioned, our goal was to provide
of tools and techniques based on MAS for the modelling a general platform-independent model, without an explicit de-
and simulation of complex systems, as crowds of pedestrians scription of space, time, perception functions and behavioural
are. In particular, Artificial Intelligence (AI) has contributed functions which are usually strictly related to the development
in different ways [4]. At the very beginning, AI researchers of the tool. On the contrary, we focus on the organization
mainly worked towards encapsulating intelligence in agent of pedestrians and on the study of relationships among indi-
behaviours. Other main aspects which AI researchers recently viduals and the relative group structure, both as static feature
investigated concern modeling and computational tools to and dynamic evolution. The main contribution of the approach
deal with interactions [5], [6]. The result of this line of we are presented concerns the expressiveness of modelling.
research is that we currently can exploit sounding tools that are Considering the explicit representation of relationships among
flexible, adaptable, verifiable, situated and distributed. Due to pedestrians, it is moreover possible to apply methods of
the suitability of agents and of MAS-approach to deal with network analysis, in particular regarding the identification of
heterogeneity of complex systems, several examples of its relevant structures (i.e. borders and spatially located groups
[15], [16]). An analysis focused on the presence of groups according
Differently, other proposals about group modelling pre- to cultural relationships highlighted that four main types of
sented in the pedestrian dynamics literature do not explicitly groups can be identified within Hajj pilgrims crowds:
investigate the whole concept of group (both from static 1) primary groups, the basic units social communities are
and dynamic way) and do not consider elements derived built on consisting in small units whose members have
from anthropological and sociological studies: in [17] e.g. a daily direct relationships (e.g. families);
proposal in which the concept of group is related to the idea 2) residential groups, characterized by homogeneous spa-
of attraction force applied among pedestrians is presented as tial localization and geographical origin;
an extension of social force field model [18]; [19] proposes 3) kinship groups, based on descent;
a model of pedestrian group dynamics using an agent-based 4) functional groups,“artificial” groups which exist only
approach, based on utility theory, social comparison theory to perform a specific functions (i.e. executive, control,
and leader-follower model; in [20] a MAS-based analysis in expressive function). Relationships among members are
which social group structures is presented, exploiting inter and only based on the fulfilment of a goal.
intra relationships in groups by means of the creation of static To model groups during Hajj, four kinds of static rela-
influence weighted matrices not depending on the evolution of tionships have to be considered: primary, residential, kinship,
the system. functional. Moreover, every group can be characterized by a
The paper is organized as follows: we focus on the descrip- set of features like the country of origin, the language, the
tion of basic elements of the model and on the description of social rank. Differently, every pedestrian can be characterized
agent behavioural rules, directly connected with the analysis of by personal features like the gender, the age, the marital status,
internal states of agents. First, in section II, the scenario of the the impaired status. In Fig. 2 and 3 some examples on the
CRYSTALS project in which the presence of heterogeneous previously presented groups are shown.
groups is particularly evident is explained. At last, an idea of
application of the model to the case study, some conclusions III. C ROWD C RYSTALS : A F ORMAL M ODEL
and future directions are presented. In this model, we refer to some considerations about orga-
nizing structures related to particular patterns of pedestrians
II. T HE SCENARIO OF A RAFAT I S TATION ON M ASHAER such as crystals of crowd. This concept is directly derived by
L INE the theory of Elias Canetti [12]:
In this section we describe a case of study in which model Crowd Crystals are the small, rigid groups of men,
requirements have been developed with the study of affluence strictly delimited and of great constancy, which serve
and entrance on Arafat I station of new Mashaer train line (Fig. to precipitate crowds. Their structure is such that
1) during Hajj 2010, the annual Pilgrimage towards Makkah. they can be comprehended and taken in at a glance.
Hajj is a phenomenon in which millions of pilgrims organized Their unity is more important than their size. The
in groups come from all the continents and stay and live crowd crystal is a constant: it never changes its size.
together for a limited period of time. In this situation, a lot of Starting from this definition, a crowd can be seen as a set of
groups with different cultural characteristics live together and crystals (i.e. groups of agents); a crowd of crystals is a system
create the whole crowd of the Pilgrimage. formally described as:
S = hA, G, R, O, Ci
where:
• A = {a1 , . . . , an } is the population of agents;
• G = {G1 , . . . , Gm } is a finite set of groups;
• R = {r1 , . . . , rl } is a finite set of static binary relation-
ships defined on the system;
• O = {o1 , . . . , ok } is a finite set of goals presented in the
system;
• C = {C1 , C2 , . . . , Cs } is a family of features defined on
the system regarding the groups where each Ci is a set
of possible values that the ith feature can assume.
In the next sections we formally define groups and agents.
A. Crystals
We define the concept of group in a crowd starting from
the previously presented definition of crystals of crowd. Every
group is defined by a set of agents and by a relationship that
Fig. 1. A representation of the scenario of Mashaer train station in Arafat I defines the membership of agents to the group. We derive the
Fig. 2. Figure on the left shows a group of people following a domestic flag: this group is a residential group, in which people are characterized by the same
geographical origin. Figure on the right shows some primary and kinship groups, composed of few people interconnected by means of descent relationships.
Fig. 3. These figures show the situation in a waiting box in which a lot of people are waiting to enter the station. Considering the whole group of people
who are waiting, we can identify it as a functional group: they are interconnected by a functional relationship, based on the goal of the group (i.e. enter the
station).
importance and the connection between the notion of group attributes of the group and the goal. The latter idea is not a
and the notion of relationship by multidisciplinary studies: restriction: following multidisciplinary studies, people involve
informally, a group is a whole of individuals in a relationship in a group share the same objective or project. The problem
with a common goal and/or a common perceived identity. to mediate the goal associated to the group and the “local”
Every group is defined a priori by a set of agents: this set has goal associated to agent as single entity is not dealt with in
a size (i.e. the cardinality of the group) and the composition of this first proposal.
members can not change. Moreover, among group members, We define a group Gi as a 4-tuple:
a static relationship already exists: the kind of relationship
determines the type of group, e.g. a family, a group of friends, Gi = hAi , zi , ri , oi i
a working group and so on.
In order to characterize pedestrian groups, it is possible to where:
identify a set of features, shared among all groups in a system: • Ai ⊆ A is a finite set of agents belonging to Gi ;
these features allow to analyse and describe more in detail • zi ∈ C1 × C2 × . . . × Cs is a vector with the values of
different aspects which is necessary to take into account in features related to Gi group;
the modelling of the system. On the basis of this assumption, • ri ∈ R is a static irreflexive, symmetric relationship
a vector with the values of features as associated to every among agents which belong to the group Gi and such
group. These values are shared and homogeneous on agents that for all a, b ∈ Ai with a 6= b, the pair (a, b) is in the
belonging to the same group. In the same way, every group transitive closure of ri . This means that the graph given
has a goal that is shared among all the group members. In fact, by ri is undirected and connected without self-loops;
every agent belonging to a group inherits from it the global • oi ∈ O is the goal associated to the group Gi .
In this first proposal, we assume that agents can not belong p is symmetric) and p(a, a) = 0D , where D is a domain
to two different groups at the same time: of distances, described as a totally ordered set with 0D as
\ a minimal element. We introduce D with the scope to not
Ai Aj = ∅ ∀i, j = 1, . . . , m and i 6= j restrict the definition of the environment in a spatial domain:
different simulation tools describe space both in a continuous
This constraint is certainly a restriction for the generaliza- and discrete way. In order to be platform-independent, in this
tion of the model. Future works are related to the extension of work, we do not explicitly define the environment and, i.e.,
the model to lead with this aspect. We can also describe the distances, in a spatial domain.
population of agents A as the union the populations of every From p we derive, for any specific agent a ∈ A, a function
group: pa : A 7→ D that associates to a its distance from any other
m
[ agents in A. Given two agents a, b ∈ A, pa (b) = pb (a).
A= Ai Moreover, for every group Gi we introduce another pseudo-
i=1 semi-metric:
Visually, we can represent each group as a graph GAi =
(Ai , Ei ) where Ai is the set of agents belonging to Gi and Ei vi : Ai × Ai 7→ D
is the set of edges given by the relationship ri . We require that
GAi is a non-oriented and connected graph (i.e. every pair of that denotes the distance between two different agents
distinct nodes in the graph is connected). belonging to the same group Gi . Given two agents a, b ∈ Gi ,
vi (a, b) = vi (b, a) (i.e., vi is symmetric) and vi (a, a) = 0D .
B. Agents
From vi we derive, for any specific agent a ∈ Gi a function
Another fundamental element besides groups is the agent via : Ai 7→ D that associates to the agent its distance
population A in which every agent represents a pedestrian from any other agents in Gi . Given two agents a, b ∈ Gi ,
in a crowd. In order to introduce characteristics related the via (b) = vib (a).
pedestrians, we introduce L = {L1 , . . . , Lq } as a family of In fact, we introduce two different functions p and vi due to
agent features where every Li is a set of possible values that a potential difference in their semantic from a theoretical point
the ith feature can assume. Every agent can have different of view. Actually, considering scenarios of crowd simulations,
values related to a set of characteristics L: this distinction is not necessary: in this sense, we assume that p
and vi functions have the same semantic ∀Gi . A simplification
a = hwa i is possible:
where wa ∈ L1 × L2 × . . . × Lq is a vector with the values
of features related to agent a. ∀Gi , vi (a, b) = p(a, b) ∀a, b ∈ A
C. Agent Behavioural rules In the next section we will use p in order to calculate
After the characterisation of the main elements of the the distance among agents and to guide the behaviours of
system, we now focus on behavioural rules of pedestrians agents inside and outside groups. As previously written, we
belonging to a group in a crowd. have introduced the distance domain D in order to allow
We deeply focus on two behavioural rules: the fact that us to not restrict the definition of distance to a spatial
pedestrians tend to maintain a minimum distance from pedes- domain. Obviously, all crowd simulations are situated in a
trians belonging the other groups (i) and the fact that pedes- particular environment in which distances can be measured in
trians in a group tend to keep a maximum distance from other R+ : thinking about a spatially located or binary (true/false)
agents belonging to the same group (ii). systems simulating pedestrians, only positive real values are
These rules are directly derived by Proxemics a theory first admissible. For this reason, we can reduce the complexity of
introduced by E.T. Hall [13] and related to the study of the set D and admit that D ⊂ R+ , in which also binary values are
of measurable distances between people as they interact. The included (i.e. false=0 and true=1).
core of this theory is the fact that different persons perceive 1) Safe Proxemic Rule: The first rule we want to intro-
the same distance in different way, due to personal attitude. In duce is related to the behaviour during interaction between a
order to develop these rules, it is necessary to introduce a set pedestrian and other pedestrians belonging to a different group.
of functions to measure distances among agents in the case From this point of view, in order to introduce the importance of
of a pedestrian inside and outside a group, depending on the personal differences derived, for instance, by cultural attitude
semantic of space. and social context, in the pedestrian simulating context, we
On A we define a pseudo-semi-metric: associate to every agent a ∈ A belonging to a group Gi a
personal distance da ∈ D.
p : A × A 7→ D,
We introduce a function da that, considering the feature
that is a function that measures distances between agents, values associate to the agent and to its group, derives da as
such that, given two agents a, b ∈ A, p(a, b) = p(b, a) (i.e., follows:
! ! defines the presence or absence of the safe group condition.
Y Y In other words S (hrj | j ≤ ti) ∈ {0, 1}. The fact that S
da : C × L 7→ D
is dependent on the whole history of the graph structure is
C∈C L∈L
motivated by the necessity to take care of particular conditions
Given an agent a ∈ Gi , with a = hwa i and its group Gi that can temporary change the graph structure but that can be
with features zi , its personal distance is da(zi , wa ) = da . This quickly recovered. By using the whole history we can avoid
distance derives both from the global characteristics of group to consider unsafe (respect to safe) a group that is, in fact, in
(i.e. zi ) and from the local characteristics of agent (i.e. wa ) a safe (respect to unsafe) condition. For instance, considering
we are considering. a simulation placed into two rooms separated by a turnstile.
Considering the distance among a and the other agents not The passage of a group through the turnstile can divide the
belonging to its group, we require that a ∈ Gi is in a safe group: in fact the group is not in an unsafe condition if we
proxemic condition if the distance pa (b) is above da for all can detect that the passage through the turnstile is a temporary
b ∈ A \ Ai . condition.
Formally, we define that an agent a ∈ Gi is in a safe The safe group rule represents the fact that pedestrians in
proxemic condition iff: a group tend to keep a maximum distance from other agents
@b ∈ A \ Ai : pa (b) ≤ da belonging to the same group: if the safe group condition is
violated, agents tend to restore the condition of group safeness.
This first rule represents the fact that pedestrians tend to
maintain a minimum distance from pedestrians belonging the
D. Agent Internal State
other groups; if the safe proxemic condition is violated, agents
tend to restore the condition of proxemic safeness. On the basis of behavioural rules before introduced, it is
2) Safe Group Rule: Every group Gi is characterized by a possible to introduce an analysis about conditions (i.e. internal
private defined distance δGi ∈ D that depends on the values of states) of agents. This analysis can be useful in order to study
group features zi . We introduce a function dg that calculates how agents change their internal conditions considering the
δGi as follows: application of the model in a simulation.
Y
dg : C 7→ D
C∈C
1
Given a group Gi , dg(zi ) = δGi . SPG SG
Previously, the introduced relationships R were called static 1
relationships. The introduction of time into the model gives the
possibility to define relationships that are time dependent: due 6 3
to the fact that time can be modelled in a continuous or discrete 2 2 5 5
way, the proposed model is defined in a way applicable to both 3 6
continuous and discrete modelling. Considering a particular
4
time t ∈ T ⊆ R and t0 as the starting time, the evolution of
SP U
the system is given by a map ϕ : S × T 7→ S, where S is
4
the space of possible systems. The state of the system at time
t is ϕ(S0 , t), where S0 is the state of the system at time t0 .
We use the definition of time in order to introduce a new kind
Fig. 4. Finite state automata representing states and transitions in an agent
of relationship time-dependent (differently from the previous
one). We call dynamic relationship a function r such that rt
is a dynamic irreflexive, symmetric relationship among agents Considering the two behavioural rules before introduced,
which belong to the group Gi . rt represents the relation at time an agent can be in a safe proxemic condition or in unsafe
t that is dependent on the whole evolution of the system from proxemic condition; moreover, it can be in a safe group
time t0 to time t. For each group Gi at time t it is possible condition or in unsafe group condition. Four states, depending
to consider the graph given by the relation rt . In particular, to on the verification of the behavioural rules are admitted:
model the proximity relationship between agents, a possible 1) Safe Proxemic and Group state (SPG): an agent is in this
definition of rt is the following: state if both the safe proxemic and safe group condition
are verified;
∀a, b ∈ Gi , (a, b) ∈ rt iff p(a, b) ≤ δGi
2) Safe Proxemic state (SP): an agent is in this state if only
recalling that vi is potentially different for each ϕ(S0 , t) since the safe proxemic condition is verified;
it is defined into the system. 3) Safe Group state (SG): an agent is in this state if only
It is possible to define a group as having the safe group the safe group condition is verified;
condition at time t on the basis of the history of the evolution 4) Unsafe state (U): an agent is in this safe if neither the
of the graph structure given by rt . Let S be the function that safe proxemic nor safe group conditions are verified.
An overview about internal states and the transitions among • oi = C1.
them is presented in the Fig. 4. Note that all the transitions Regarding the definition of characteristics of agents,
among states are admissible. a plausible family of characteristics can be L =
Now, let us consider two particular configurations on the {gender, age, marital status, impaired status}. From this
population of agents A = {a1 , a2 , . . . , an } with G1 , . . . , Gm point of view, an agent a ∈ A can be defined for instance as
groups: if m = 1 there is only one group coinciding with the ai = {male, adult, married, no}.
whole A; if m = n, |Ai | = 1 ∀i = 1, . . . , m, all the groups
have a size equal to 1 and every agent in the population is a V. C ONCLUSION AND F UTURE D IRECTIONS
singleton. In these cases we note that the previous finite state
automata can be simplified as follows (Fig. 5 and Fig. 6): In this paper, we proposed an agent-based model for the
explicit representation and modelling of groups of pedestrian
• if there is only one group coinciding with A, only two
in a crowd, focusing on the organization of pedestrians and on
states are admissible: SPG and SP. In fact, SG and U
the study of relationships among individuals and the relative
states are not possible because all agents of the population
group structure.
belong to the same Gi group;
Future directions are related to the development of simula-
tion in the presented scenario in order to test and validate the
model, and in the application of methods for network analysis
2 on the group structures, in order to identify and study, for
SPG SP example, the presence of recursive patterns.
2
ACKNOWLEDGMENT
Fig. 5. Simplification of the finite state automata referring to an agent (I)
This research was fully supported by the Center of Research
• if every agent represents a singleton, only two states are Excellence in Hajj and Omrah, Umm Al-Qura University,
admissible: SPG and SG. In fact, SP and U states are not Makkah, Saudi Arabia, grant title “Crystal Proxemic Dynam-
possible because every agent is always in a safe group ics of Crowd & Groups”.
condition.
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