http://ifisc.uib-csic.es - Mallorca - Spain

Master Thesis

Updating rules and the voter modelImplementing heterogeneous activity patterns in

Agent Based Models

JUAN FERNÁNDEZ GRACIASUPERVISORS: Maxi San Miguel and Víctor M. Eguíluz

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OUTLINE

Motivation

The Voter Model

Standard update rules

New update rule: exogenous vs. endogenous

Conclusions

Updating rules and the voter model

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Updating rules and the voter model

MOTIVATION: HUMAN ACTIVITY PATTERNSMany human activities display a bursty interevent time distribution (power law like or at least heavy-tailed), i.e. bursts of rapidly ocurring events are separated by long periods of inactivity.

Examples: e-mail communication, short messages communication, phone calls, timing of finantial trades ...

Candia et al J. Phys. A: Math. Theor. 41 224015 (2008)Wu et al PNAS 107 18803 (2010)

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Updating rules and the voter model

The activity displayed in Agent Based Models (ABMs) is much homogeneous, having narrow distributions for interevent times (times between changes of state of single agents).

Thesis: The update rule, i.e. the sequence and timing in which the agents' states are updated are responsible for these patterns of activity.- Propose a new update rule that is able to account for heavy tails in the interevent time distributions of such models.- Focus on a simple model for consensus formation, like the voter model, and analize what is the effect on the macroscopic outcome of the dynamics.

QUESTIONS

Origin of power-law interevent time distributions

Effects of this timing on social dynamics

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Updating rules and the voter model

THE VOTER MODEL- Model for opinion formation.- N agents on a network, each one with a binary variable x

i=+1 or -1(opinion,

state...).- Imitation mechanism. If agent i's state is updated, a random neighbour j is chosen and i copies j's state x

i=x

j.

- Two absorbing configurations, consisting of all agents having the same state (consensus).

What is the final state of the system? - Consensus or coexistence of states?How is the transient to that state?

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Updating rules and the voter model

SYNCHRONOUS UPDATE (SU)

All agents updated simultaneously.

RANDOM ASYNCHRONOUS UPDATE (RAU)

One random agent updated per simulation step.

SEQUENTIAL ASYNCHRONOUS UPDATE (SAU)

One agent updated per simulation step. Agents always updated in same order.

STANDARD UPDATE RULES

On regular lattices periodic or chaotic solutions

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Updating rules and the voter model

Voter model and standard update rulesPrevious studies focused on RAU dynamics. Here we see that the same results hold for the three standard update rules on the interaction networks we used.

No ordering in the thermodynamic limit!

Homogeneous activity patterns with a well defined characteristic

interevent time.

Complete graph

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Updating rules and the voter model

NEW UPDATE RULEEach agent is characterized by two variables: state x and 'persistence time' .Simulation step:

1. with probability each agent i becomes active, otherwise it stays inactive;2. active agents update their state according to the dynamical rules of the model (in this case we focus on the voter model);3. all agents increase their persistence time in one unit.

Limits: → SU; → RAU

EXOGENOUS UPDATEActive agents reset after step 2.

ENDOGENOUS UPDATEOnly active agents that change state in step 2 reset .

We are interested in activation probabilities that decay with . The longer an agent stays innactive, the harder is to activate. We will consider .

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Updating rules and the voter model

Voter model with exogenous update rule

decays slower for bigger N → constant in thermodynamic limit No ordering in the thermodynamic limit!

Complete graph

For b=1 we recover a power law tail of exponent -1 in the

cumulative persistence C(t)

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Updating rules and the voter model

Voter model with endogenous update ruleComplete graph

Ordering in the thermodynamic limit!Plateau as 1/N

For all system sizes.

For b=1 we recover a power law tail of exponent -1 in the

cumulative persistence C(t)

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Updating rules and the voter modelCONCLUSIONS- The voter model with standard update rules : No order is reached in the thermodynamic limit. For finite size systems rho and S(t) decays exponentially with a characteristic time that scales as N for complete graph and random networks and as N/log(N) for scale-free of exponent 3. Persistence M(t) is homogeneous with well defined characteristic time scaling as N1/2.

- Exogenous update and voter model: no order is reached in the thermodynamic limit. The activity patterns can be heterogeneous in the form of power-law tails in the persistence M(t), but for a small range of b (not shown).

- Endogenous update and voter model: coarsening process in the thermodynamic limit, with algebraic decay of rho and S in such a way that the mean time to reach consensus is not defined. The power law tail in persistence M(t) is found for a wider range of b values.

- The way in which heterogeneous activity patterns are implemented can give rise to qualitatively different macroscopic outcomes (exogenous vs. endogenous).

- Provide theoretical framework that bridges the efforts devoted to uncover properties of human dynamics with modelling efforts in opinion dynamics.

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Updating rules and the voter model

THANKS FOR THE ATTENTION

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Updating rules and the voter model

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Updating rules and the voter model

Approximating the exponent of the cumulative persistence

Taking that every time an agent's state is updated, it changes we can relate the persistence M(t) and the activation probability p(t).

M(t): probability of changing state t timesteps after the last change of state.p(t): probability of being updated t timesteps after last change of state.

It follows then

Going to the continuum time and remembering the definition of the cumulative persistence:

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Updating rules and the voter model

Varying parameter b (varying persistence exponent)

EXOGENOUS UPDATE ENDOGENOUS UPDATE

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Updating rules and the voter model

Meanfield approach

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Updating rules and the voter model

Voter model with standard update rulesPrevious studies focused on RAU dynamics. Here we see that the same results hold for the three standard update rules on the interaction networks we used.

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Updating rules and the voter model

Voter model with standard update rules

reaches a plateau independent of system size → surviving runs stay disordered.

No ordering in the thermodynamic limit!

Homogeneous activity patterns with a well defined characteristic interevent time.

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Updating rules and the voter model

Voter model with exogenous update rule

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Updating rules and the voter model

Voter model with endogenous update rule

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Updating rules and the voter model

INTERACTION NETWORKS

Periodic solution of the voter model with SU on a square lattice.

To avoid traps in the dynamics of the voter model with any of the update rules we choose the following three interaction networks.

Complete graph.Every agent can interact with every other agent

Random graph.The interaction between any two pair of agents is present with probability p.

Scale-free graph.The degree distribution of the graph is a power-law. Extremely highly connected nodes are to be expected.

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Updating rules and the voter model

MACROSCOPIC DESCRIPTIONMagnetization: not good as order parameter, since it is conserved under certain situations.

Density of interfaces: good order parameter, not generally conserved

Ensemble avarages: averages over many realizations of the system. Denoted as .

Survival probability: probability that a realization has not reached one of the absorbing configurations. Denoted as S(t). The mean time to reach consensus is

Density of interfaces averaged over surviving runs: reflects level of order of realizations which have not reached one of the absorving states.

Persistence M(t) and cumulative persistence C(t): persistence or interevent time distribution M(t) measures the probability of an agent to change state t timesteps after her last change of state. C(t) is its cumulative distribution defined as the probability of not having changed state t timesteps after the last change of state.

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Updating rules and the voter model

STANDARD UPDATE RULES

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Updating rules and the voter model

RANDOM ASYNCHRONOUS UPDATE (RAU)

At each simulation step one randomly chosen agent is updated. Then the number of simulation steps is divided by the number of agents N to have the time in Montecarlo steps.

Standard update rules

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Updating rules and the voter model

SEQUENTIAL ASYNCHRONOUS UPDATE (SAU)

At each simulation step one agent is updated. The agents are always updated in the same order. Then the number of simulation steps is divided by the number of agents N to have the time in Montecarlo steps.

Standard update rules

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Updating rules and the voter model

SYNCHRONOUS UPDATE (SU)

At each simulation all agents are updated simultaneously. Time is measured directly in simulation steps.

Standard update rules

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