A Majority Rule model to simulate twitter-like activity in complex networks

Arezky H. RodríguezAcademia de Matemáticas,Colegio de Ciencia y TecnologíaUniversidad Autónoma de la Ciudad de México (UACM)

Yamir MorenoBIFI, Univ de Zaragoza

Sandro MeloniBIFI, Univ de Zaragoza

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

Instituto de BioComputación y Física de Sistemas Complejos.

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Outline:

• Motivation.

• The Model.

• Views of the simulation using Netlogo.

• First results.

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Protest in the connected societyFrom 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 societyUnlike 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 ω

ij 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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A Majority Rule model

The external stimulus:

There is an external stimuls (mass media, broadcasting radio station, problematic topic, etc) of intensity W

o 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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A Majority Rule model

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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A Majority Rule model

The Agents design:

Each agent possesses also a set of possitive real parameters β

ie, β

i

up, and βi

down

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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A Majority Rule model

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 W

o at t = 0.

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A Majority Rule model

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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A Majority Rule model

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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A Majority Rule model

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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A Majority Rule model

Details:Mathematical expresions:

Qiup=1−e−(βie Wo f (t )+βi

up W iup )

f ( t)=e−ϵ t

W iup=∑ j neighbours

ωij s j (t )

Si = 1 for agent i with active? = true

Si = 0 for agent i with active? = false

W idown

=∑ j neighboursωij(1−s j ( t))

Qidown=1−e−βi

down W idown

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A Majority Rule model

Details:Agents update its feeling-awkward? according to:

Agents update its active? according to:

if feeling-awkward? = false → active?(t) = active?(t – 1)if feeling-awkward? = true → active?(t) = not active?(t - 1)

feeling-awkward?(t) =ƒ(own active?(t-1), neighbours active?(t-1))

active?(t) = ƒ(own active?(t-1), own feeling-awkward?(t))

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Parameters used:

βie= β

i

up= βi

down = 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 statesTipping points!!

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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 statesTipping points!!

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Caracterizing the tipping points:

Diversity Index Entropy Information

DI =1

∑iP i

2S=−∑i

Pi log2(P i)

Amount of posible states you have in your system.

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 average over 1000 runs.

Asyncronous-random updating.

Absorving states and trapped states

Tipping points!!

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Results:

Scale-Free network of 10 000 nodes.Each point is an average over 1000 runs.Asyncronous-random updating.

Absorving states and trapped statesTipping points!!

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Results:

Scale Free 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:

Scale Free 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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• 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 Wo

• Tipping points as a function of the network size

• Trapped states characterization

• β distributions dependences

• Syncronous vs Asyncronous-random updating

Further explorations:

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Gracias....

• 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

Further explorations: