Temporal activity patterns in social dynamics

Oriol Artime (Advisors: Maxi San Miguel & Jose Javier Ramasco)

PhD Thesis (2020)

PhD Thesis (2020)

A system is considered complex when it is formed by a set of elements that

interact in a simple way, and by virtue of these interactions, an emergent

behavior appears. This means that the global behavior of a complex system

cannot be neither explained nor deduced if the individual constituents are isolated

and studied separately: the reductionism and the superposition principle, broadly

used in physics, are no longer valid. Classical examples of this emergence are the

origination of consciousness from neural interactions, the collapses in a financial

market, or the synchronized motion of a flock of birds. The definition of a complex

system is flexible, therefore, seemingly different systems can be studied under the

same set of concepts, methods and techniques. This is the reason why their study

has become a multidisciplinar and interdisciplinar challenge.

Human societies are a paradigmatic example of a complex system, and it will

be the one that we explore in this thesis. The constituents of this system are

the individuals, or agents, that interact among them through different channels.

We will focus on the collective phenomena of opinion dynamics and emergence of

consensus, which are problems that have been traditionally tackled in sociology

and psychology, but in the last years there have been valuable contributions

coming from other scientific disciplines, such as physics, mathematics or computer

science. These new viewpoints look for simple models that attempt to find the

minimal conditions and the mechanisms that generate the phenomena that one

wants to understand. The study of these minimal models usually analyzes either

the type of mechanism that sustains the dynamics through which the individuals

interact (opinion changes by imitation, social pressure of a majority, etc.) or

the effects that the topology of the interactions (who interacts with whom) has

on the model. However, a fundamental element that tends to be ignored is the

temporal dimension. To offer a more detailed description of opinion dynamics,

non-trivial statistical properties in the activity patterns of the agents, such as

long-tailed interevent time distributions or memory effects, need to be included.

By doing so, the description not only becomes more realistic, but, as we will see,

the dynamical behavior of the models is radically different.

The common element among all chapters is precisely the study of the origin

and the impact of these temporal properties in the context of opinion dynam-

ics. The first two chapters serve as introduction: the first one offers a general

perspective of the type of problems that we will deal with and the tools to solve

them, while the second one is a detailed description of the models that we will

use and their properties.

The first main block of the thesis contains three chapters, each of them cor-

responding to a publication. In the third and fourth chapter we investigate the

mechanism of aging, understood as the influence that the time that an agent has

been without changing state, the age, has on the next state change. We analyze

the effects of this type of aging in the Kirman model, also known as the noisy

voter model. By adding aging the system passes from a discontinuous finite-size

phase transition to a continuous one, robust in the thermodynamic limit. We

describe in detail, analytically and numerically, some properties of this new tran-

sition: critical point, universality class, stationary properties of the age of the

agents. The role of aging in the standard voter model on multilayer networks

will be also considered. We give a description of the model’s asymptotic states,

such as a non-absorbing partially ordered one, in terms of the fraction of nodes

participating simultaneously in the different layers, the topology of the networks

and the time scales of the evolution of the models. In the last chapter of the first

block, the fifth, we study how the distribution of arrival times to the absorbing

state in the standard voter model and in the Susceptible-Infected (SI) model of

epidemics is affected by different types of correlations. In this case, the correla-

tions do not come from a modified dynamics, such as aging, but they are added

manually so their strength is easily tunable. We show that positive temporal

correlations in the interevent time activation speed up the dynamics, while topo-

logical correlations in the form of communities slow it down. When both types of

correlations are considered together, the models show a high sensitivity to their

relative strength, speeding up or slowing down the dynamics (with respect to the

case without correlations) depending on their combination.

The second main part of the thesis is formed by only one chapter, the sixth,

where we study the first-passage time distributions for different opinion dynamics

models. These probability distributions give the time that is needed to achieve

for first time a state (for instance, the consensus) from a predetermined initial

state (for instance, the state of exact coexistence of opinions). The main con-

tribution of the chapter is the proof that the functional dependence of these

distributions for the family of models described by the Fokker-Planck equation

is determined uniquely by the initial and final states, and the eventual presence

of absorbing states in the dynamics. We explain under which conditions we find

either scale-free first-passage distributions, with their corresponding exponent, or

single-peaked fast-decaying distributions. In this way, we overcome the limitation of the mean first-passage time, which is the most used quantity in this kind of problems, since it is unable to distinguish between these two very different behaviors. To complete the analysis, we verify the analytical predictions for models of different nature.

To summarize, in this thesis we investigate opinion dynamics models through

the lens of the statistical physics of collective phenomena. We put special em-

phasis on the temporal dimension. On the one hand, we study the behaviors that

emerge from the modification of temporal interaction patterns among agents. On

the other hand we analyze the first-passage time properties of different models,

being able to classify their functionality in terms of few elements. Our results

evince the need to include realistic temporal interaction statistics in the modeling,

since their absence can lead to wrong or misleading conclusions.

interact in a simple way, and by virtue of these interactions, an emergent

behavior appears. This means that the global behavior of a complex system

cannot be neither explained nor deduced if the individual constituents are isolated

and studied separately: the reductionism and the superposition principle, broadly

used in physics, are no longer valid. Classical examples of this emergence are the

origination of consciousness from neural interactions, the collapses in a financial

market, or the synchronized motion of a flock of birds. The definition of a complex

system is flexible, therefore, seemingly different systems can be studied under the

same set of concepts, methods and techniques. This is the reason why their study

has become a multidisciplinar and interdisciplinar challenge.

Human societies are a paradigmatic example of a complex system, and it will

be the one that we explore in this thesis. The constituents of this system are

the individuals, or agents, that interact among them through different channels.

We will focus on the collective phenomena of opinion dynamics and emergence of

consensus, which are problems that have been traditionally tackled in sociology

and psychology, but in the last years there have been valuable contributions

coming from other scientific disciplines, such as physics, mathematics or computer

science. These new viewpoints look for simple models that attempt to find the

minimal conditions and the mechanisms that generate the phenomena that one

wants to understand. The study of these minimal models usually analyzes either

the type of mechanism that sustains the dynamics through which the individuals

interact (opinion changes by imitation, social pressure of a majority, etc.) or

the effects that the topology of the interactions (who interacts with whom) has

on the model. However, a fundamental element that tends to be ignored is the

temporal dimension. To offer a more detailed description of opinion dynamics,

non-trivial statistical properties in the activity patterns of the agents, such as

long-tailed interevent time distributions or memory effects, need to be included.

By doing so, the description not only becomes more realistic, but, as we will see,

the dynamical behavior of the models is radically different.

The common element among all chapters is precisely the study of the origin

and the impact of these temporal properties in the context of opinion dynam-

ics. The first two chapters serve as introduction: the first one offers a general

perspective of the type of problems that we will deal with and the tools to solve

them, while the second one is a detailed description of the models that we will

use and their properties.

The first main block of the thesis contains three chapters, each of them cor-

responding to a publication. In the third and fourth chapter we investigate the

mechanism of aging, understood as the influence that the time that an agent has

been without changing state, the age, has on the next state change. We analyze

the effects of this type of aging in the Kirman model, also known as the noisy

voter model. By adding aging the system passes from a discontinuous finite-size

phase transition to a continuous one, robust in the thermodynamic limit. We

describe in detail, analytically and numerically, some properties of this new tran-

sition: critical point, universality class, stationary properties of the age of the

agents. The role of aging in the standard voter model on multilayer networks

will be also considered. We give a description of the model’s asymptotic states,

such as a non-absorbing partially ordered one, in terms of the fraction of nodes

participating simultaneously in the different layers, the topology of the networks

and the time scales of the evolution of the models. In the last chapter of the first

block, the fifth, we study how the distribution of arrival times to the absorbing

state in the standard voter model and in the Susceptible-Infected (SI) model of

epidemics is affected by different types of correlations. In this case, the correla-

tions do not come from a modified dynamics, such as aging, but they are added

manually so their strength is easily tunable. We show that positive temporal

correlations in the interevent time activation speed up the dynamics, while topo-

logical correlations in the form of communities slow it down. When both types of

correlations are considered together, the models show a high sensitivity to their

relative strength, speeding up or slowing down the dynamics (with respect to the

case without correlations) depending on their combination.

The second main part of the thesis is formed by only one chapter, the sixth,

where we study the first-passage time distributions for different opinion dynamics

models. These probability distributions give the time that is needed to achieve

for first time a state (for instance, the consensus) from a predetermined initial

state (for instance, the state of exact coexistence of opinions). The main con-

tribution of the chapter is the proof that the functional dependence of these

distributions for the family of models described by the Fokker-Planck equation

is determined uniquely by the initial and final states, and the eventual presence

of absorbing states in the dynamics. We explain under which conditions we find

either scale-free first-passage distributions, with their corresponding exponent, or

single-peaked fast-decaying distributions. In this way, we overcome the limitation of the mean first-passage time, which is the most used quantity in this kind of problems, since it is unable to distinguish between these two very different behaviors. To complete the analysis, we verify the analytical predictions for models of different nature.

To summarize, in this thesis we investigate opinion dynamics models through

the lens of the statistical physics of collective phenomena. We put special em-

phasis on the temporal dimension. On the one hand, we study the behaviors that

emerge from the modification of temporal interaction patterns among agents. On

the other hand we analyze the first-passage time properties of different models,

being able to classify their functionality in terms of few elements. Our results

evince the need to include realistic temporal interaction statistics in the modeling,

since their absence can lead to wrong or misleading conclusions.