“Un modelo basado en agentes para el estudio de la actividad en redes sociales online”

A Majority Rule model to simulate twitter-like activity
in complex networks
Arezky H. Rodríguez
Academia de Matemáticas,
Colegio de Ciencia y Tecnología
Universidad Autónoma de la Ciudad de México (UACM)

Yamir Moreno
BIFI, Univ de Zaragoza

Sandro Meloni
BIFI, Univ de Zaragoza

Instituto de
BioComputación y Física
de Sistemas Complejos.

Supported by Grant “Programa de Fomento a la Movilidad de Investigadores del
Gobierno de Aragón 2011”

Outline:

• Motivation.
• The Model.
• Views of the simulation using Netlogo.
• First results.
19/08/13

Seminario de Complejidad y Economía
CEIICH-UNAM

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Protest in the connected society
From New York to Istanbul, and Rio to
Tunis, waves of social unrest have been
sweeping across the world. Whatever they
are called – Occupy Wall Street in New
York (2011), 15M Movement in Spain
(2011), the Jasmine Revolution in Tunisia
(2010) or the Arab Spring (2010), and the
Salad Uprising in Brazil (2013) – the mass
mobilisations share several common
features.
Espousing public discontent over a range
of sometimes unrelated, even conflicting
issues, they were driven largely by new
communication technologies coupled with
an abiding distrust of government policies.
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Protest in the connected society
Unlike the formal, planned protests of earlier times, the latest ones are, for the
most part, informal and relatively spontaneous. As such, scientists say, they reflect a
shift away from conventional social hierarchies towards what some call leaderless
networks.
Similar to demonstrations leading to the Arab Spring, the protests across 100
Brazilian cities were facilitated largely by social media such as Facebook and
Twitter.

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A Majority Rule model to simulate twitter-like
activity in complex networks
Purpose:
Characterize the dynamics of people activation on a
network like Twitter as a function of different internal and
external parameters. A person is considerer active when is
broadcasting a message to her link neighbours. The
influence of an external factor which initially influences
the people on the network is considered.

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A Majority Rule model
The Ambient:
The formalism of agent-based models is used.
People and their contacts are modeled as a weighted
network of N nodes, where a node represents a person and
the links of the network represent the real people
connections with other persons. The parameter ωij account
for the link weight between agent i and j.
As a fist approximation it is considered the network with
undirectional links.
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The external stimulus:
There is an external stimuls (mass media, broadcasting
radio station, problematic topic, etc) of intensity Wo which
is percived by all the agents of the network.
The intensity of the external stimulus exponentialy
decreases on time with a factor ε. It pretends to simulate
the damping on time of an initial stimulus due to memory
lose.
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The Agents design:
Each node is characterized by a vector of two parameters:
(feeling-awkward?, active?)
Parameter feeling-awkward? ∈{true, false}
Parameter active? ∈{true, false}
The parameters feeling-awkward? and active? of the nodes
are coupled between them and also coupled with the
feeling-awkward? and active? of their link neighbours in a way
that will be explained later.
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The Agents design:
Each agent possesses also a set of possitive real parameters
βie, βiup, and βidown
which account for the inner disposition of the agent i to
become active following the external stimulus or following its
link-neighbors, respectively.

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Initial conditions:
All nodes have
feeling-awkward? = false
active? = false
It is selected a number of nodes no and it is set
active? = true
It resembles certain amount of agents which react to the
external stimuls Wo at t = 0.
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Initial conditions:
Three ways of select the initially active nodes no are
implemented:

• Random
•Target with higher connectivity first (hcf).
•Target with lower connectivity first (lcf).
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Temporal evolution:
Each time-step agent updates its values of feeling-awkward? and
active?, in this order.

•Agent feeling-awkward?(t) =

ƒ(own active?(t -1), neighbours active?(t -1))

•Agent active?(t) = ƒ(own active?(t - 1), own feeling-awkward?(t))
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Temporal updating:
It is implemented two secuential updating:

•Syncronous (in Parallel):

all network nodes update its feeling-awkward? first and then all
network nodes update its active?

•Asyncronous – Random (resembling continuous updating):

all network nodes are ordering in a list in random order (each time)
and following this order each node updates its feeling-awkward? and
inmediatelly updates its active? value.
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Details:
Mathematical expresions:
up
i

Q =1−e

up
i

up
i

−(βie Wo f (t )+β W )

up
i

W =∑ j neighbours ωij s j (t )
Si = 1 for agent i with active? = true
Si = 0 for agent i with active? = false

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Q

down
i

W

−βidown W down
i

=1−e

down
i

=∑ j neighbours ωij (1−s j (t))
f (t)=e

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−ϵ t

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Details:
Agents update its feeling-awkward? according to:
feeling-awkward?(t) =ƒ(own active?(t-1), neighbours active?(t-1))

Agents update its active? according to:
active?(t) = ƒ(own active?(t-1), own feeling-awkward?(t))

if feeling-awkward? = false → active?(t) =
active?(t – 1)
if feeling-awkward? = true → active?(t) = not active?(t - 1)
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Parameters used:
βie= βiup= βidown = 1
Wo =1
ωij = 1
asyncronous-random updating

The simulation ends when all agents have
feeling-awkward? = false
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Results:

Erdös-Rényi network of 10 000 nodes.
Each point is an average over 1000 runs.
Asyncronous-random updating.
Absorving states and trapped states
Tipping points!!
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Results:

Erdös-Rényi network of 10 000 nodes.
Tipping points!!
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Caracterizing the tipping points:
Diversity Index

DI =

Entropy Information

1

∑i P

S =−∑i P i log 2 ( P i )

2
i

Amount of posible states you
have in your system.

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Amount of information your
system containts.

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Results:
Erdös-Rényi network
of 10 000 nodes.
Each point is an
average over 1000 runs.
Asyncronous-random
updating.
Absorving states and
trapped states
Tipping points!!

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Results:
Erdös-Rényi network
of 10 000 nodes.
Each point is an
Asyncronous-random
updating.
trapped states
Tipping points!!

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Results:

Scale-Free network of 10 000 nodes.
Tipping points!!
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Results:
Scale Free network of
10 000 nodes.
Each point is an
Asyncronous-random
updating.
trapped states
Tipping points!!

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Results:
Scale Free network of
10 000 nodes.
Each point is an
Asyncronous-random
updating.
trapped states
Tipping points!!

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Further explorations:

• Report the activation distribution: amount of neighbors which are active
when the agent becomes active

• Tipping points as a function of the initial stimulus intensity W

o

• Tipping points as a function of the network size
• Trapped states characterization
• β distributions dependences
• Syncronous vs Asyncronous-random updating
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Further explorations:

• Social agents have roles
• Social agents are autonomous
• Social agents interact locally with a few number of neighbors
Necessity of other models!!
Explanation rather than prediction

Gracias....
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CEIICH-UNAM

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“Un modelo basado en agentes para el estudio de la actividad en redes sociales online”

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