Temporal activity patterns in social dynamics

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used in physics, are no longer valid.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.