Simulation of complex systems: the case of crowds (Phd course - lesson 1/7)

Simulation of complex systems:
the case of crowds
Giuseppe Vizzari

Complex Systems and Artiﬁcial Intelligence Research Center (CSAI)

University of Milano-Bicocca, Italy

Outline
• Complex Systems?

• Pedestrian and crowd simulation, a brief introduction and
motivations

• Levels of analysis of pedestrian/crowd behaviour

• Approaches to pedestrian modelling (macro vs. micro)

• Macroscopic modelling rationale

• Microscopic modelling alternatives and styles

• A reﬂection on the speciﬁcities of crowd simulation simulation
projects

• Conclusions and discussion

Outline
motivations





projects


Complex Systems?
• Several (more ore less formal) definitions:

• A complex system is a highly structured
system, which shows structure with variations

• A complex system is one whose evolution is
very sensitive to initial conditions or to small
perturbations, one in which the number of
independent interacting components is large,
or one in which there are multiple pathways by
which the system can evolve

• A complex system is one that by design or
function or both is difficult to understand and
verify

• A complex system is one in which there are
multiple interactions between many different
components

• Complex systems are systems in process that
constantly evolve and unfold over time

• … 
• Features of complex systems:

• Composed by several interacting elements

• Nonlinearity
• Networked or hierarchical structure

• Positive and negative feedbacks
• Possibility to evolve and adapt

• Robustness and plasticity
• …

• Complex systems research is a hot topic
for scientists… but also for engineers!

• One of their main features is their internal
structure and the interaction among their
composing parts… that very often is
studied by means of simulations

Simple vs Complicated vs Complex
• Simple system

• Cause and effect relationships known, stable,
repeatable and predictable

• Known system

• Example: bicycle

• Complicated system

• No fundamental difference with respect to cause and
effect relationships, but on a much larger scale and with
increased requirements around coordination or
specialised expertise

• “Knowable" system

• Examples: car, bus, airplane, rocket

• Complex system

• Cause and effect relationships often understandable in
retrospect, but not necessarily easily reproduced and
predictable

• Systems that can (to a certain extent) be analysed by
means of simulation

• Examples: living organism (even simple as yeast),
human organisation (not just an organisational chart, a
real organisation, e.g. a Department)… a CA model (the
Game of Life)!

Outline

motivations




projects


Motivations of 
crowd modeling and simulation
• Designer’s decision support

• Normal and evacuation situations

• Positioning of signs

• Malls and shopping centres

• Support crowd management by means
of the elaboration of what-if scenarios

• Situations where large crowds are
frequent (sport events, festivals,
religious events)

• Public transport systems, in
particular stations

• Support the study of pedestrian
behaviour

• Envisioning of diﬀerent behavioural
models in realistic environments

• Possibility to perform ‘in-machina’
experiments

Crowds of pedestrians as 
complex systems
• Overall system behaviour depends on
individuals’ decisions and actions…

• … that are generally inﬂuenced by a
large number of factors

• … intertwined in an often
unpredictable way

• Mixed and conﬂicting mechanisms

• Competition for the shared space…

• … but also cooperation (non written
social norms) to prevent stall
situations

• Imitation...

• ... but also natural tendency to stay
at a distance (proxemics)

• Emergent phenomena
• …but also rules, norms
• …

Considerations About
Elder Pedestrians
• Elder pedestrians are frequent

• Their impact on the transportation
infrastructures can be signiﬁcant

• Structures should be ready to host
them properly

• Service points should be organised
and managed taking them into
consideration

• Simulation can support the
evaluation of designs and highlight
potential issues

• New models for supporting the should
be able to deal with

• Heterogeneity in pedestrian speed

• Systematic diﬀerences in reaction
times

• Presence of groups, potentially
structured, some of which with
strong cohesion

Impact of groups in pedestrian
and crowd dynamics
• Current approaches generally
consider every pedestrian as a
individual with no relationships

• Considering only his/her own
goals

• Considering other pedestrians
as moving obstacles

• Nonetheless, in several situations
pedestrians are bound by
relationships inﬂuencing their
movement

• Generally speaking, a crowd is
made up of groups of
pedestrians...

• What do we miss by neglecting this
aspect of pedestrian behaviour?

Not just “physical” crowds

Simulation: a deﬁnition and
motivations

motivations
• (Computer) Simulation represents a
way to exploit a computational
model

motivations
model
– to evaluate designs and plans
without actually bringing them into
existence in the real world

motivations
model
– to evaluate theories and models of
complex systems by envisioning the
eﬀect of the modelling choices, with
the aim of gaining insight of their
functioning

motivations
model
functioning
• The use of “synthetic environments”
is sometimes necessary, because the
simulated system cannot actually be
observed

motivations
model
functioning
observed
– Because it is actually being designed

motivations
model
functioning
observed
– Because it is actually being designed
– For ethical or practical reasons

Simulation life-cycle
• From the target system to its computational model and a simulator
Modeling 
and design 
of a simulator
Target System
Model and simulator
5

• Execution of a simulation campaign
Simulation execution
Modeling 
and design 
of a simulator
Target System
Model and simulator Data generated by the 
simulation(s)
5

• Evaluation/validation of the model (and simulator) against collected
data
Dynamics of Target System
Modeling 
and design 
of a simulator
Analysis of results +
interpretation 
(model evaluation
leading to explanation
or 
prediction)
Target System
simulation(s)
Collected Data
5

• Evaluation/validation of the model (and simulator) against collected
data
• Possible usage for explanation and/or prediction
Dynamics of Target System
Modeling 
and design 
of a simulator
Analysis of results +
interpretation 
(model evaluation
leading to explanation
or 
prediction)
Target System
simulation(s)
Collected Data
5

A comprehensive framework for integrated
synthesis and analysis

Outline

motivations




projects


Pedestrians and crowds: levels of analysis
Chapter 5. Identification of processes and elements in a pedestrian flow model 105
The operational level pertains to immediate decisions of the pedestrian concerning his
behaviour. Interactions with other pedestrians play an important role at this level.
An overview of the mentioned decision levels, related processes at and interactions be-
tween different decision levels are given in figure 5.1. Also, inputs and outputs on each
level are indicated in this figure.
STRATEGIC
TACTICAL
OPERATIONAL
Activity set choice
Derived
activity set
Network topology
Timetable
Route
Activity schedule
Activity areas
Route choice
Activity area choice
Activity scheduling
Dynamic network
characteristics
Geometry
Obstacles
Vehicle characteristics
Walking
Waiting
Performing an activity
Trajectory choice
Figure 5.1: Levels in pedestrian behaviour based on Hoogendoorn et al. (2001) and figure 3.1[Hoogendoorn et al., 2001]

Outline

motivations




projects


Modeling approaches: what to model?
Pedestrian/Crowd
models
Macroscopic
(continuous ﬂuids)
Microscopic
(individual pedestrians)
Overall equations (mass conservation,
ﬂow velocity) determining velocity and
density in the analysed area
Individual state, position, decision are
modelled; aggregate measures derive
from these decisions
+ lower computational costs
for large crowd
- limited applicability
+ generally greater applicability
- higher computational costs
for large crowd

Macroscopic crowd models rationaleMathematical models Numerical tests Conclusion
Macroscopic models
Pedestrians as "thinking fluid"1
Averaged quantities:
⇢(t, x) pedestrians density
~v(t, x) mean velocity
Mass conservation
(
@t⇢ + divx(⇢~v) = 0
⇢(0, x) = ⇢0(x)
for x 2 ⌦ ⇢ R2
, t > 0
Two classes
1st order models: velocity given by a phenomenological
speed-density relation ~v = V (⇢)~⌫
2nd order models: velocity given by a momentum balance equation
1
R.L. Hughes, Transp. Res. B, 2002
P. Goatin (INRIA) Mathematical modeling of crowds January 12, 2014 5 / 22
Mathematical models Numerical tests
Eikonal equation: level set curves for |rx | = 1
In an empty space: potential is proportional to distance to destination
P. Goatin (INRIA) Mathematical modeling of crowds January 12, 20
[Goatin, 2014]
• Global constraints are specified, e.g.

• Mass conservation

• Velocity function over the analysed area

• Different types models according to the
velocity function definition

• 1st order: goal orientation and speed
density relation

• 2nd order: momentum balance equation

• Considerations

• All pedestrians share the same goals
and attitude

• Difficult to consider dynamic aspects in
the environment and situation

• Not able to capture all aspects of crowd
dynamics (e.g. some emerging
phenomena)

• Suitable for optimisation problems in
specific contexts

Example application and results of a recent
macroscopic approach
x−axis (m)
y−axis(m)
Densities at t = 50.1 s
5 10 15 20 25 30 35 40 45 50
5
10
15
20
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30
35
40
45
50
0
0.5
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1.5
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x−axis (m)
y−axis(m)
5 10 15 20 25 30 35 40 45 50
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25
30
35
40
45
50
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 2: Evacuee assignment using optimal free ﬂow routing.
y−axis(m)
10
15
20
25
30
35
40
45
50
1
1.5
2
2.5
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3.5
4
4.5
5
y−axis(m)
10
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x−axis (m)
y−axis(m)
5 10 15 20 25 30 35 40 45 50
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50
0
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x−axis (m)
y−axis(m)
5 10 15 20 25 30 35 40 45 50
5
10
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25
30
35
40
45
50
0
0.5
1
1.5
2
2.5
3
3.5
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Figure 2: Evacuee assignment using optimal free ﬂow routing.
x−axis (m)
y−axis(m)
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
0
0.5
1
1.5
2
2.5
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x−axis (m)
y−axis(m)
5 10 15 20 25 30 35 40 45 50
5
10
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30
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45
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0
0.5
1
1.5
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4.5
5
Figure 3: Evacuee assignment (iteration 30) for two time stamps. [Hoogendoorn et al., 2014]

Agent Based Models for simulation:
peculiarities, advantages, risks
7

• The analytical unit is the
individual agent, not aggregate
variables
7

variables
• This means, on one hand, that
additional insight on the
modelled system is required
7

variables
• On the other hand such a model
should be able to
7

variables
should be able to
• Generate the same
aggregate dynamics as
traditional ones
7

variables
should be able to
• Generate the same
aggregate dynamics as
traditional ones
• Be able to represent, manage,
analyse additional aspects,
such as for instance spatial
ones
7

Alternative microscopic
pedestrian and crowd models
• Particle based
• Mostly, but not exclusively, social force
model and derivatives

• Continuous space and time
• Cellular Automata
• Ad-hoc rules (e.g. Blue & Adler,
intersections) or ﬂoor ﬁeld approach
(e.g. Nishinari, Schadschneider, ...)

• Discrete in time and space
• Multi-Agent Systems
• Several approaches from computer
graphics (e.g. Thalmann, Terzopoulos,
Donikian, Manocha), some approaches
are extensions of CA, ...

• Behavioural models generally more
complex that in CA approaches

Particle based approach
• Pedestrians à particles
subject to forces

subject to forces
• Goals: forces of attraction
generated by points/
reference point in the space
Lane formation

subject to forces
Lane formation
‘Freezing by heating’

subject to forces
• Interaction among
pedestrians: forces
generated by particles
Lane formation

subject to forces
pedestrians: forces
• Social forces
Lane formation

subject to forces
pedestrians: forces
• Social forces
• Repulsive à tendency to stay
at a distance
Lane formation

subject to forces
pedestrians: forces
• Social forces
• Repulsive à tendency to stay
at a distance
• Attractive à imitative
mechanisms
Lane formation

Cellular Automata and crowd modelling

• Environment à bi-dimensional lattice of
cells

cells
• Pedestrian à speciﬁc state of a cell
(e.g. occupied, empty)

cells
• Movement à generated thanks to the
transition rule

cells
transition rule
– an occupied cell becomes empty and an
adjacent one, which was previously
vacant, becomes occupied

cells
transition rule
• Choice of destination cell in a transition
generally includes information which is
not provided by basic CAs

cells
transition rule
– Beneﬁt-Cost/Gradient: predeﬁned
information related to cell desirability

cells
transition rule
– Benefit-Cost/Gradient: predefined
information related to cell desirability
– Magnetic Force/Social Force: model
the effect of presence of other agents
in the environment (attraction/
repulsion of crowds)

From CA to Situated MAS
– Entities are reiﬁed, separated from the
environment

environment
– Agents, not just cell states

environment
– They may have different behaviours
– Possibility to grant agents different
information about the environment
– Possibility to integrate several
different action deliberation models

environment
– Possibly heterogeneous system

environment
– Entities interact by means of
mechanisms not necessarily related to
underlying cell’s adjacency

environment
– Entities interact by means of
mechanisms not necessarily related to
underlying cell’s adjacency
– Action at a distance is allowed

Situated MAS: 
action and interaction

Situated MAS: 
– Agents are situated

Situated MAS: 
– they perceive their context and situation

Situated MAS: 
– their behaviour is based on their local
point of view

Situated MAS: 
point of view
– their possibility to act (move) and
interact is inﬂuenced by the
environment

Situated MAS: 
point of view
environment
– Situated Agents Interaction models

Situated MAS: 
point of view
environment
– Often inspired by biological systems
(e.g. pheromones, computational ﬁelds)

Situated MAS: 
point of view
environment
– Generally provide a modiﬁcation of the
environment, which can be perceived
by other entities

Situated MAS: 
point of view
environment
– Generally provide a modiﬁcation of the
environment, which can be perceived
by other entities
– But may also provide a direct
communication (as for CAs interaction
among neighbouring cells)

Groups in the literature -
Modeling and Simulation
• Extensions to the social force model

• Helbing, Theraulaz et al. 2009, 2010

• Small groups (2,3,4), unstructured

• Low to moderate densities
• Validation based on actual observations

• Xu and Duh, 2010

• Only couples (groups of 2 pedestrians)

• Low to moderate densities
• Shallow validation based on literature
(Daamen, 2004)

• CA models

• Sarmady, Haron, Zawawi Hj, 2009

• Leaders and followers
• Groups of 2 to 6 members experimented

• Not validated

• Agent-based models

• Qiu and Hu 2010

• Structured groups (intra and inter group
matrices)

• Large groups experimented (60 pedestrians)

• Not validated

• Group members tend to stay close to other group
members (additional behavioural component)

Meso level approaches: the best of both worlds?

Outline

motivations





• A reﬂection on the speciﬁcities of crowd simulation
simulation projects

Why not use simple queues?
• One could model such a system
with one (or more) simple queues...
Pedestrian
in the lecture
hall
Exit
(one pedestrian
every t ms)

Pedestrian
in the lecture
hall
Exit
(one pedestrian
every t ms)
Exit
(one pedestrian
every t ms)

• ... but the model would not be able
to ‘answer’ some of the questions
we can pose to the previous models
• What if t is unknown?
• Only aggregate quantities are
managed
• No heterogeneity
Pedestrian
in the lecture
hall
Exit
(one pedestrian
every t ms)
Exit
(one pedestrian
every t ms)

managed
• Moreover, it would be very diﬃcult to
manage more complex situations,
in terms of:
Pedestrian
in the lecture
hall
Exit
(one pedestrian
every t ms)
Exit
(one pedestrian
every t ms)

managed
in terms of:
• Environmental structure
Pedestrian
in the lecture
hall
Exit
(one pedestrian
every t ms)
Exit
(one pedestrian
every t ms)

managed
in terms of:
• Environmental structure
• Possible behaviours for the
pedestrians
Pedestrian
in the lecture
hall
Exit
(one pedestrian
every t ms)
Exit
(one pedestrian
every t ms)

So, no “best” modeling approach?

• The choice of the abstract/
computational model depends on
several factors:

several factors:
• Available knowledge on the
simulated phenomenon/
situation/reality

several factors:
situation/reality
• Available data on actual
scenarios, for sake of
calibration, veriﬁcation and
validation

several factors:
situation/reality
validation
• Goals of the simulation activity

several factors:
situation/reality
validation
• Goals of the simulation activity
• Possible tension between these
elements!

What about Agent Based Modeling and
Simulation methodologies?

• It is very diﬃcult to deﬁne a methodology
that is both general and actually useful

• General methodologies tend to deﬁne
well understood macro phases, but no
useful suggestion on how actually carry
them out in a speciﬁc context

• For instance, in diﬀerent domains it is
not even clear what is the most proper
modelling granularity level

– In crowd modelling, a pedestrian is
modelled as an agent (at least in
microscopic models)…

– But in biological systems, what should be
represented as an agent? An organism?
An organ? A cell? A molecule?

– But in biological systems, what should be
represented as an agent? An organism?
An organ? A cell? A molecule?
• Speciﬁc useful (more or less) formalised
modelling approaches and
methodologies only in speciﬁc contexts
?

A reflection: from reality, to models, to a simulation
system
• The overall simulation project involves
several phase, roles, types of
knowledge and competences
• The frequent passages (translation,
encoding, decoding, interpretation...)
between different levels of abstraction
can lead to several problems

• Non documented assumptions

• Unrealistic/unfeasible simplifications
• Simulation projects are difficult
• R. Shannon, “Introduction to the Art and
Science of Simulation” (1998)

• ... But some even talk of “dark arts” [J.P.
Marney and H. Tarbert, “Why do
simulation? Towards a working
epistemology for practitioners of the dark
arts” (2000)]
Reality
Subsystem
Abstract model
Computational model
Simulator

Conclusions and
discussion
• The study of pedestrians and crowd behaviour
fruitfully comprises activity of analysis and
synthesis

• The behaviour of pedestrians and crowds can
be analysed at different levels

• Both macro and microscopic approaches to the
modelling of these phenomena are possible…

• … both have merits and limits in their
applicability

• Microscopic approaches take a very different
perspective on pedestrians and their
environment

• The contribution of different disciplines, experts
is crucial but the overall simulation project
requires particular attention

• Of course, these model, studies and application
domain can be a source for different research
area…

• … a source of new problems and potential
opportunities

• … a source of suggestions, potentially
interesting approaches/solutions

ありがとうございます。 Giuseppe Vizzari

Minimal bibliography
• Stefania Bandini, Andrea Gorrini, Giuseppe Vizzari: Towards an integrated approach to crowd analysis and crowd synthesis: A
case study and first results. Pattern Recognition Letters 44: 16-29 (2014).

• Hoogendoorn, S.P., P.H.L. Bovy & W. Daamen (2001), Microscopic pedestrian wayfinding and dynamics modelling, In: M.
Schreckenberg & S. Sharma, (eds.), Pedestrian and Evacuation Dynamics, Springer, Berlin, 123–154.

• Andreas Schadschneider, Wolfram Klingsch, Hubert Klüpfel, Tobias Kretz, Christian Rogsch, Armin Seyfried: Evacuation
Dynamics: Empirical Results, Modeling and Applications. Encyclopedia of Complexity and Systems Science 2009: 3142-3176.

• Paola Goatin (2014), Mathematical modeling of crowds. Presentation at TRB 2014 Workshop “Crowd Flow Dynamics,
Modeling and Management” - available in “TRB Subcommittee on Crowd Flow Dynamics, Modeling, and Management”
Facebook group

• Serge Hoogendoorn, Winnie Daamen, Dorine Duives, and Femke van Wageningen-Kessels: Optimal Crowd Evacuation. TRB
2014

• Helbing, D., A. Johansson, and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study. Phys. Rev. E, Vol. 75, 2007,
p. 046109.

• Colombo, Rinaldo M., Paola Goatin, Giulio Maternini, and Massimiliano D. Rosini. "Macroscopic Models for Pedestrian Flows."
In Big events and transport: the transportation requirements for the management of large scale events. 2010.

• Daamen, Winnie, Serge P. Hoogendoorn, and Piet HL Bovy. "First-order pedestrian traffic flow theory." Transportation
Research Record: Journal of the Transportation Research Board 1934, no. 1 (2005): 43-52.

• Helbing, Dirk, and Peter Molnar. "Social force model for pedestrian dynamics." Physical review E 51, no. 5 (1995): 4282.

• Klüpfel, H., A Cellular Automaton Model for Crowd Movement and Egress Simulation. Ph.D. thesis, University Duisburg-Essen,
2003.

• Burstedde, C., K. Klauck, A. Schadschneider, and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional
cellular automaton. Physica A: Statistical Mechanics and its Applications, Vol. 295, No. 3 - 4, 2001, pp. 507 – 525.

• Vizzari, G., L. Manenti, and L. Crociani, Adaptive Pedestrian Behaviour for the Preservation of Group Cohesion. Complex
Adaptive Systems Modeling, Vol. 1, No. 7, 2013.

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