# Aging in some opinion formation models: a comparative study

Jaume Llabres  Instituto de Física Interdisciplinar y Sistemas Complejos IFISC
(CSIC-UIB), Campus Universitat de les Illes Balears, 07122 Palma de Mallorca,
Spain  Sara Oliver-Bonafoux  Instituto de Física Interdisciplinar y Sistemas
Complejos IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, 07122
Palma de Mallorca, Spain  Celia Anteneodo  Department of Physics, Pontifical
Catholic University of Rio de Janeiro, PUC-Rio, Rua Marquês de São Vicente,
225, 22451-900 Rio de Janeiro, Brazil  National Institute of Science and
Technology for Complex Systems, INCT-SC, Rio de Janeiro, Brazil  Raúl Toral
Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB),
Campus Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain

###### Abstract

Changes of mind can become less likely the longer an agent has adopted a given
opinion state. This resilience or inertia to change has been called “aging”.
We perform a comparative study of the effects of aging on the critical
behavior of two standard opinion models with pairwise interactions. One of
them is the voter model, which is a two-state model with a dynamic that
proceeds by simple social contagion, another is the so-called kinetic exchange
model, which allows a third (neutral) state and the formed opinion depends on
the previous opinions of both interacting agents. Furthermore, in the noisy
version of both models, random opinion changes are also allowed, regardless of
the interactions. Due to aging, the probability of change diminishes with the
age, and to take this into account we consider algebraic and exponential
kernels. We investigate the situation where aging acts only on pairwise
interactions. Analytical predictions for the critical curves of the order
parameters are obtained for the opinion dynamics on a complete graph, in good
agreement with agent-based simulations. For both models considered, consensus
is optimized by an intermediate value of the parameter that rules the rate of
decrease of the aging factor.

Aging, opinion formation model, voter model, kinetic-exchange model, social
dynamics

##  I Introduction

As well known, the ideas and concepts coming from the Statistical and
Nonlinear Physics toolbox have been successful to understand social phenomena
[1, 2]. The main goal is to explain a “macroscopic”, emergent, behavior in
terms of the “microscopic”, individual’s, components in the same way that one
derives the laws of macroscopic physical systems, e.g. the equation of state,
from microscopic information such as the forces amongst particles. One lesson
we have learnt from the modern development of Statistical Physics is that the
emergent behavior, manifested through macroscopic phase transitions, is
largely independent on some details of the microscopic interactions, and that
there are some universal features that only depend on a few ingredients, such
as dimensionality, symmetries, etc. It is in this spirit that the modeling of
social systems has relied on the development of very simple agent-based models
that take as starting points some stylized facts about the interactions
amongst agents. Consider, for example, the subject of the evolution of
cultures. In a seminal paper, Axelrod asked the question “If people tend to
become more alike in their beliefs, attitudes, and behavior when they
interact, why do not all such differences eventually disappear?” [3]. He then
introduced a model whose basic assumption is that interactions amongst
individuals lead to an augmentation in the similarity between them. A detailed
analysis [4] shows that depending on the initial diversity there is a phase
transition between an ordered monocultural state and a disordered
multicultural state, demonstrating the apparent paradox of the emergence of
global polarization through the mechanism of local convergence. It was then
established that the dimensionality of the space of interactions was a
relevant variable of the model [5]. Similar ideas appear in other areas. In
the context of opinion formation, for example, one expects, on the one hand,
that social influences reduce differences between individuals by social
contagion but, on the other, interactions can also produce differentiation by
repulsive forces, associated to anticonformist, ”contrarian”, agents that tend
to deviate from the behavior adopted by the neighbors in the network of
connections [6]. Another example is Galam’s model for minority opinion spread
[7] that, although based on a local majority rule, can lead to an advantage of
the hostile minority views.

Diverse other mechanisms can compete with the tendency to uniformity promoted
by social contagion. One is the ’noisy’ behavior, ruled by a ’social
temperature’ parameter, associated to imperfect imitations due to random
factors or independent choices, such that random changes of opinion
interpreted as idiosyncratic behavior can occur. Standard models that study
the effect of independence are the noisy voter model [8, 9, 10, 11] and the
noisy kinetic-exchange model [12, 13]. In the former, agents adopt opinions
$\pm 1$, copying the opinion of a random contact, in the latter, the states
$\pm 1$ and 0 can be adopted by a rule that takes into account the current
opinion of both interacting contacts. In both cases the rule of social
influence acts with probability $1-a$, while otherwise, with probability $a$,
a random change can occur. That is, these models rely on the assumption that
opinion dynamics is mainly governed by mechanisms of imitation or social
contagion that mold opinion formation through the interaction between
individuals, but random factors are always present. In both cases, noise
opposes the system ability to reach a self-organized state with a winning
opinion. Furthermore, both present a critical value $a_{\mathrm{c}}$ below
which order can be achieved, a main difference being that $a_{\mathrm{c}}\to
0$ in the thermodynamic limit, when the number of agents tends to infinity,
for the noisy voter, while $a_{\mathrm{c}}$ remains finite in the kinetic-
exchange model in the same limit.

Another ingredient that has been proven to be relevant in the modeling of
social systems is that of “aging”, or the larger resistance an agent offers to
change its state of opinion the longer it has been holding the current state.
Initially termed as “inertia” [14], it was shown that the slowing down of the
microscopic dynamics induced by aging actually decreased the time needed to
reach the macroscopically ordered state state of consensus. In this respect,
and aligned with previous work [14, 15, 16, 17, 18, 19], we will consider that
aging acts on the mechanism associated to social contagion.

It is the aim of this paper to compare the effects of aging on the two above-
mentioned paradigmatic representatives of the class of noisy opinion models,
the voter model and the kinetic-exchange model, highlighting the similarities
and differences amongst them. To this end we will consider an all-to-all
interaction scheme in which every agent is connected to every other agent, and
will develop an adiabatic approximation able to provide us with the phase
diagram, including the location of the critical points separating regions of
consensus with disordered regions. Afterwards, we compare the location of the
critical points for both models.

The rest of the paper is organized as follows: In Section II, we introduce a
general setup to study models of opinion dynamics under the presence of
idiosyncratic, random, changes of opinion, as well as the interaction between
the agents modulated by an aging mechanism. In Section III we derive a mean-
field type approach for the noisy voter model under an adiabatic
approximation, leaving the more technical details for Appendix A.1. In the
same section we also compare with the results of numerical simulations for two
different functional forms for the aging probability, algebraic and
exponential, focusing mainly on the magnetization, measuring the amount of
order, or amount of consensus, in the system. In Section IV, after reviewing
briefly the main results for the noisy kinetic-exchange model, we compare the
results for the magnetization and the dependence of the critical value of the
noise intensity as a function of a parameter measuring the rate of decrease of
the aging probability, for the voter and the kinetic-exchange model. Finally,
in Section V we end with some conclusions and outlook for future work.

##  II Models of noisy opinion dynamics with aging

In this work we analyze two models that have been widely used as prototypical
examples in the study of the dynamics of opinion formation in social systems:
the noisy voter model and the noisy kinetic-exchange model. Both models
consider a set of $N$ agents endowed with an internal state variable
$s_{i},\,i=1,\cdots,N$, representing the possible positions of agent $i$
concerning a given topic. For the voter model the internal variable can take
two possible values $s_{i}\in\\{-1,+1\\}$, representing a position against or
in favor of the topic. The kinetic-exchange model introduces a third neutral
state such that $s_{i}\in\\{-1,0,+1\\}$. This internal variable evolves due to
two different generic mechanisms that act stochastically: idiosyncratic
changes and social influence. The difference between them is that the former
changes randomly the state variable $s_{i}$ independently on the states of
other agents, while the latter changes $s_{i}$ following a rule that depends
on the state variables of another agent $s_{j}$. In the voter model the social
rule is that $s_{i}\to S(s_{i},s_{j})=s_{j}$, modeling the mechanism of
imitation. In the kinetic-exchange model the social rule is $s_{i}\to
S(s_{i},s_{j})={\rm sgn}[s_{i}+s_{j}]$, where ${\rm sgn}[s]$ is the sign
function. The rule is such that neutral agents can not modify the opinion of
another agent, and agents with a well defined opinion (either positive or
negative) can convince a neutral agent or turn into the neutral state an agent
with the opposite opinion.

Previously, both models have been considered under the influence of aging in
the social mechanism [9, 20]. It is the goal of this paper to generalize some
of the previous work and to compare the effects that aging induces in both
models.

To be precise, let us spell out in detail the evolution rules for these
models. Initially we assign a random value to each state variable $s_{i}$ and
set all internal times $\tau_{i}$ to zero. Then,

  1. 1.

At each iteration, an agent $i$ is randomly selected.

  2. 2.

With probability $(1-a)$ the social rule is chosen: With probability
$q(\tau_{i})$, modeling the persistence or reaction of the individual $i$ to
change as a function of its age $\tau_{i}$, a neighbor $j$ is randomly
selected and the opinion of agent $i$ is modified according to the rule
$s_{i}\to S(s_{i},s_{j})$.

  3. 3.

Otherwise, with probability $a$, the idiosyncratic rule is chosen: the state
$s_{i}$ is replaced by a randomly selected value among all possible opinion
states.

  4. 4.

Irrespective of the update mechanism actually used by agent $i$ (random change
or pairwise interaction, with aging or not) its age $\tau_{i}$ is updated in
the following way: If the state $s_{i}$ has changed, then age is reset, i.e.,
$\tau_{i}\to 0$, otherwise, age is incremented in one unit, i.e.,
$\tau_{i}\to\tau_{i}+1$.

We will restrict ourselves in this work to the all-to-all connectivity, where
each agent is linked to every other agent. Hence, the set of neighbors of an
agent $i$ is the whole set of agents (excluding itself). Time is measured in
Monte Carlo steps, such that one unit of time corresponds to $N$ agent
selections for updating.

Although other forms are possible, in the present work, we adopt the following
two general functional forms for the probabilities (we use $q(\tau)\equiv
q_{\tau}$, for brevity in the notation),

$\displaystyle\text{algebraic decay}:\hskip 20.0pt$ $\displaystyle
q_{\tau}=\frac{q_{\infty}\tau+q_{0}\tau^{*}}{\tau+\tau^{*}},$ (1)
$\displaystyle\text{exponential decay}:\hskip 20.0pt$ $\displaystyle
q_{\tau}=q_{\infty}+(q_{0}-q_{\infty})e^{-\tau/\tau^{*}},$ (2)

where $q_{0}$ and $q_{\infty}$, satisfying $0\leq q_{\infty}<q_{0}\leq 1$,
denote, respectively, the initial ($\tau=0$) and asymptotic ($\tau\to\infty$)
values of $q_{\tau}$, and $\tau^{*}$ is a parameter dictating the rate of
decrease of the aging-probability, i.e. a larger value of $\tau^{*}$ implies a
slower decay. Both forms have been considered before in the context of the
study of the way that the voter model (without idiosyncratic changes)
approaches the asymptotic consensus state [21, 22]. A previous study of the
noisy voter model [17] was limited to the algebraic case with
$q_{0}=1/2,\,\tau^{*}=2$. The aging-less case is recovered formally taking the
limit $\tau^{*}\to\infty$, implying $q_{\tau}=q_{0}$ for all values of the age
$\tau$. Our theoretical treatment is rather general, but we adopt the values
$q_{0}=1$ and $q_{\infty}=0$ in the numerical simulations.

##  III Noisy voter model with aging

In this section we present the results obtained for the noisy voter model with
aging. We build over the treatment of Ref. [17] but present a more general
one, valid for arbitrary forms of the aging-update probability.

In this model each agent $i$ is characterised by its binary state variable,
$s_{i}\in\\{-1,+1\\}$, and its age or residence time in state $s_{i}$,
$\tau_{i}$. We denote by $x^{\pm}_{\tau}(t)$ the fractions of agents in states
$\pm 1$ and with age $\tau$, such that
$x(t)=\sum_{\tau=0}^{\infty}x^{+}_{\tau}(t)$ is the total fraction of agents
in state $+1$ at time $t$, and $1-x(t)=\sum_{\tau=0}^{\infty}x^{-}_{\tau}(t)$
is the total fraction of agents in state $-1$ at time $t$. As detailed in
Appendix A.1, it is possible to derive a closed evolution equation for $x(t)$.
The procedure starts by writing down rate equations for the densities
$x^{\pm}_{\tau}(t)$. One then performs an adiabatic approximation that assumes
that the evolution of $x^{\pm}_{\tau}(t)$ is slaved to that of $x(t)$ which
evolves at a longer time scale. Under this approximation, the equation for the
dynamical evolution of the fraction $x(t)$ of agents in state $+1$ at time $t$
is

$\begin{split}\frac{dx}{dt}&=G^{\text{\tiny{V}}}(x),\\\
G^{\text{\tiny{V}}}(x)&=\frac{a}{2}(1-2x)+(1-a)x(1-x)[\Phi^{\text{\tiny{V}}}(x)-\Phi^{\text{\tiny{V}}}(1-x)],\end{split}$
(3)

where the function $\Phi^{\text{\tiny{V}}}(x)$ is defined as

$\Phi^{\text{\tiny{V}}}(x)=\frac{\sum_{\tau=0}^{\infty}q_{\tau}F^{\text{\tiny{V}}}_{\tau}(x)}{\sum_{\tau=0}^{\infty}F^{\text{\tiny{V}}}_{\tau}(x)},$
(4)

with

$F^{\text{\tiny{V}}}_{0}(x)=1,\quad
F^{\text{\tiny{V}}}_{\tau}(x)=\prod_{k=0}^{\tau-1}\gamma_{2}(q_{k}\,x,a),\quad\tau\geq
1,$ (5)

and

$\gamma_{n}(z,a)=\frac{a}{n}+(1-a)(1-z).$ (6)

The steady state-solutions $x_{\text{\tiny{st}}}$ are obtained by setting the
time derivative of the density $x(t)$ equal to zero, or
$G^{\text{\tiny{V}}}(x_{\text{\tiny{st}}})=0$, according to Eq. (3). Note that
due to its very nature, the adiabatic approximation does provide the correct
values of these steady-state solutions. This is because the exact calculation
of the steady-state solutions requires that the rates of change of all
densities are set to zero. This leads to the condition
$G^{\text{\tiny{V}}}(x_{\text{\tiny{st}}})=0$ which is also a result of the
adiabatic approximation. What the approximate dynamical equation (3) provides
is the stability of the different steady-state solutions as determined by the
sign of the derivative $\frac{dG^{\text{\tiny{V}}}(x)}{dx}$ evaluated at
$x_{\text{\tiny{st}}}$.

In the aging-less case, $q_{\tau}=q_{0},\,\forall\tau$, it is
$\Phi^{\text{\tiny{V}}}(x)=q_{0}$ and the only fixed point of Eq. (3) is
$x_{\text{\tiny{st}}}=1/2$, which is stable. This indicates that the
disordered phase is the only asymptotic solution. It is clear from the
structure of Eq. (3) that $x_{\text{st}}=1/2$ is, trivially, also a steady-
state solution for the aging situation, for an arbitrary function
$\Phi^{\text{\tiny{V}}}(x)$. The stability of this trivial solution and the
existence of other steady-state solutions depend solely on the function
$G^{\text{\tiny{V}}}(x)$.

For the algebraic functional form of the aging-update probabilities given by
Eq. (1), it is possible, as explained in Appendix B, to obtain an analytical
form for the function $\Phi^{\text{\tiny{V}}}(x)$ in terms of hypergeometric
functions. However, even in this case, the non-trivial steady state values and
their stability have to be obtained numerically. For the exponential form of
the aging-update probabilities given by Eq. (2), it does not seem to be
possible to express $\Phi^{\text{\tiny{V}}}(x)$ in terms of other known
functions, but we outline in Appendix B.2 a convenient numerical algorithm for
its evaluation.

Both for the algebraic and the exponential forms for the aging-update
probabilities, the numerical analysis concludes that, in the case
$q_{0}=1,\,q_{\infty}=0$, the solution $x_{\text{\tiny st}}=1/2$ is stable for
$a>a_{\mathrm{c}}(\tau^{*})$ and unstable for $a<a_{\mathrm{c}}(\tau^{*})$ and
that a new pair $x_{\text{\tiny{st}}},1-x_{\text{\tiny{st}}}$ of symmetric
stable solutions emerge for $a\leq a_{\mathrm{c}}(\tau^{*})$. These two
solutions share the same stability as it can be easily proved that
$\left.\frac{dG^{\text{\tiny{V}}}(x)}{dx}\right|_{x=x_{\text{\tiny{st}}}}=\left.\frac{dG^{\text{\tiny{V}}}(x)}{dx}\right|_{x=1-x_{\text{\tiny{st}}}}$.
This change of stability of the symmetric solution $x_{\text{\tiny st}}=1/2$
is understood as a phase transition between a disordered phase at
$a>a_{\mathrm{c}}(\tau^{*})$ where both opinions $\pm 1$ coexist in equal
proportion, to an ordered, consensus, phase for $a<a_{\mathrm{c}}(\tau^{*})$
where one of the opinions is majoritarian. The existence of this phase
transition is one of the main results presented in Ref. [17] for a particular
form of the aging probability (corresponding to the algebraic case with
$q_{0}=1/2,\,\tau^{*}=2$). The same transition was understood as a mapping of
the problem with aging to another one with suitable nonlinear rates in the
social mechanism [23]. We now present a general study for both the algebraic
and exponential dependence of the aging probability and different values of
the parameter $\tau^{*}$.

In Fig. 1 we plot the magnetization,
$m_{\text{\tiny{st}}}=|2x_{\text{\tiny{st}}}-1|$, as a function of the noise
intensity, $a$, both for the algebraic and the exponential functional forms of
the aging update probability, and for several values of the parameter
$\tau^{*}$. In the same graph we plot the results of numerical simulations of
the stochastic rules for the agent-based model detailed in Section II. The
simulations agree remarkably well with the analytical results, validating the
adiabatic approximation introduced in the analysis. There are some deviations
with respect to the analytical calculation for values of the probability
parameter $a$ which are close to the critical value $a_{\mathrm{c}}$. This is
attributed to the necessary consideration of a finite number of agents
$N=10^{4}$ in the numerical simulations, while the theoretical treatment
considers the thermodynamic limit $N\to\infty$.

Figure 1: Noisy voter model with aging. Magnetization,
$m_{\mathrm{st}}=|2x_{\mathrm{st}}-1|$, as a function of the noise intensity
$a$ for different values of the $\tau^{*}$ parameter, for the algebraic and
exponential kernels. Solid lines correspond to the theoretical results, while
symbols come from numerical simulations with $N=10^{4}$ agents, averaged over
$5\times 10^{6}$ MCS after a transient of $5\times 10^{6}$ MCS. Notice the
non-monotonic dependency of the critical point with the parameter $\tau^{*}$.

For a given value of $\tau^{*}$, the critical value $a_{\mathrm{c}}$ can be
found by calculating the point at which the symmetric solution
$x_{\text{\tiny{st}}}=1/2$ changes from being stable to being unstable, i.e.,
by solving the equation
$\left.\frac{dG^{\text{\tiny{V}}}(x)}{dx}\right|_{x=1/2}=0$. A simple
calculation shows that this condition is equivalent to

$(1-a_{\mathrm{c}})\left.\frac{d\Phi^{\text{\tiny{V}}}(x)}{dx}\right|_{x=1/2}=2a_{\mathrm{c}}.$
(7)

Note that $a_{\mathrm{c}}$ also appears in the function
$\Phi^{\text{\tiny{V}}}(x)$. The dependence of the critical value
$a_{\mathrm{c}}$ on the parameter $\tau^{*}$, obtained by solving numerically
the previous equation, is shown in Fig. 2. The larger the value of
$a_{\mathrm{c}}$, the larger the region in parameter space in which a
consensus-like phase is present. Remarkably, both for the algebraic and the
exponential cases, the critical value tends to zero for $\tau^{*}\to 0$ and
$\tau^{*}\to\infty$, indicating that in those limits the only asymptotic
solution is the disordered one. The limit $\tau^{*}\to\infty$ leads to
$q_{\tau}=1,\,\forall\tau$, which corresponds to the standard version (without
aging) of the noisy-voter model. It is known that this model displays a
finite-size noise-induced phase transition between disorder and consensus at a
critical value of the parameter $a_{\mathrm{c}}=2/N$. In the thermodynamic
limit, it is $a_{\mathrm{c}}\to 0$, in accordance with our result. The reason
of the disappearance of the consensus in the limit $\tau^{*}\to 0$ stems from
the fact that the social mechanism is activated with a probability $q(\tau)$
that tends to $0$ for all $\tau$, except $\tau=0$, for which $q(0)=q_{0}>0$.
Hence, the only effective updating mechanism acting at all times is random
updates, that necessarily lead to disorder. In between these two limits there
is an optimal value $\tau^{*}_{\mathrm{c}}$ for which $a_{\mathrm{c}}$ is
maximum. For the algebraic case, it is $\tau^{*}_{\mathrm{c}}=2.01$ and the
corresponding value of the noise intensity is $a_{\mathrm{c}}=0.224$, while
for the exponential case we find $\tau^{*}_{\mathrm{c}}=3.77$ and
$a_{\mathrm{c}}=0.242$. The critical lines $a_{\mathrm{c}}(\tau^{*})$ of the
algebraic and exponential cases cross at the point $\tau^{*}_{0}=2.33$, such
that for $\tau^{*}<\tau^{*}_{0}$, the algebraic aging shows a larger consensus
region (larger value of $a_{\mathrm{c}}$) and the opposite for
$\tau^{*}>\tau^{*}_{0}$.

Figure 2: Theoretical curves of the critical noise value $a_{\mathrm{c}}$
versus the $\tau^{*}$ parameter in the algebraic and exponential cases, for
the noisy voter (V) and kinetic-exchange (K) models. The horizontal lines
correspond to the critical value in the aging-less case in the noisy voter
(solid line) and in the kinetic-exchange (short-dashed line) models.

##  IV Comparison with the noisy kinetic-exchange model with aging

We now compare the main similarities and differences between the noisy voter
model and the kinetic-exchange both under the presence of aging.

The kinetic exchange model introduces a third, neutral, value of the state
variable, $s_{i}\in\\{-1,0,+1\\}$. At variance with the noisy voter model, a
macroscopic characterization uses both densities $x^{\pm}(t)$ of agents in
states $\pm 1$ at time $t$. By normalization, the density of agents in state
$0$ is given by $x^{0}(t)=1-x^{+}(t)-x^{-}(t)$.

Along the lines used for the voter model, and developed in more detail in Ref.
[20], it is possible to write down a closed system of equations for the
densities $x^{\pm}(t)$, namely,

$\begin{split}\frac{dx^{+}}{dt}&=G^{\text{\tiny{K}}}(x^{+},x^{-}),\\\
\frac{dx^{-}}{dt}&=G^{\text{\tiny{K}}}(x^{-},x^{+}),\\\
G^{\text{\tiny{K}}}(z,w)&=(1-a)\left[z(1-z-w)\Phi^{\text{\tiny{K}}}(z+w)-z\,w\,\Phi^{\text{\tiny{K}}}(w)\right]+\frac{a}{3}\left(1-3z\right).\end{split}$
(8)

where

$\Phi^{\text{\tiny{K}}}(x)\equiv\frac{\sum_{\tau=0}^{\infty}q_{\tau}F^{\text{\tiny{K}}}_{\tau}(x)}{\sum_{\tau=0}^{\infty}F^{\text{\tiny{K}}}_{\tau}(x)}.$
(9) $F^{\text{\tiny{K}}}_{0}(x)=1,\quad
F^{\text{\tiny{K}}}_{\tau}(x)=\prod_{k=0}^{\tau-1}\gamma_{3}(q_{k}\,x,a),\quad\tau\geq
1,$ (10)

where $\gamma_{n}(z,a)$ is given by Eq. (6). Although the dynamical system for
the kinetic model, Eqs. (8), is very different from that of the voter model,
Eqs. (3), a very similar structure is found for the functions
$\Phi^{\text{\tiny{V}}}(x)$ and $\Phi^{\text{\tiny{K}}}(x)$. As explained in
Appendix B, both functions are expressible in terms of hypergeometric
functions for the algebraic dependence of the aging probabilities and can be
computed using a very efficient numerical algorithm in the case of the
exponential dependence.

Let us summarize now the main results of the analysis of the dynamical system
Eqs. (8). In the aging-less case, $\Phi^{\text{\tiny{K}}}(x)=q_{0}$, the
dynamical equations always admit the steady-state solution
$x_{\text{\tiny{st}}}^{+}=x_{\text{\tiny{st}}}^{-}=1/3$. This solution becomes
unstable for values of the noise intensity $a$ less than the critical value
$a_{\mathrm{c}}=\frac{q_{0}}{3+q_{0}}$, where a pair of non-symmetric
solutions $x_{\text{\tiny{st}}}^{+}\neq x_{\text{\tiny{st}}}^{-}$ emerge. The
behavior of the magnetization
$m_{\mathrm{st}}=|x_{\text{\tiny{st}}}^{+}-x_{\text{\tiny{st}}}^{-}|$ is given
by

$m_{\mathrm{st}}=\begin{cases}{\displaystyle\frac{\sqrt{(q_{0}-(3+q_{0})a)(3q_{0}-(1+3q_{0})a)}}{\sqrt{3}\,q_{0}\,(1-a)}},&a\leq
a_{\text{c}}\\\ 0,&a\geq a_{\text{c}}\end{cases},$ (11)

a result already obtained by Crokidakis [12] in the case $q_{0}=1$.

When aging is included, the basic structure of the solution remains [20]:
there is always a symmetric phase
$x_{\text{\tiny{st}}}^{+}=x_{\text{\tiny{st}}}^{-}$ steady-state solution,
which is stable for $a~{}>~{}a_{\mathrm{c}}(\tau^{*})$. For
$a~{}<~{}a_{\mathrm{c}}(\tau^{*})$ this solution becomes unstable and a pair
of non-symmetric stable solutions emerge. It is not possible to give explicit
expressions for the solutions $x_{\text{\tiny{st}}}^{\pm}$ for an arbitrary
function $\Phi^{\text{\tiny{K}}}(x)$ and they have to be determined
numerically. The stability of these solutions is determined also numerically
from the sign of the eigenvalues of the Jacobian matrix

$\left.\begin{pmatrix}\dfrac{\partial\dot{x}^{+}}{\partial
x^{+}}&\dfrac{\partial\dot{x}^{-}}{\partial x^{+}}\\\\[15.0pt]
\dfrac{\partial\dot{x}^{+}}{\partial
x^{-}}&\dfrac{\partial\dot{x}^{-}}{\partial
x^{-}}\end{pmatrix}\right|_{x^{\pm}=x_{\text{\tiny{st}}}^{\pm}}$ (12)

evaluated at the fixed points.

In Fig. 3 we show the results of the magnetization $m_{\mathrm{st}}$ as a
function of the noise intensity $a$ for two values of the $\tau^{*}$ parameter
for the noisy voter ($m_{\mathrm{st}}=|2x_{\mathrm{st}}-1|$) and for the
kinetic-exchange ($m_{\mathrm{st}}=|x_{\mathrm{st}}^{+}-x_{\mathrm{st}}^{-}|$)
models with aging. In both cases we find excellent agreement between
theoretical results and numerical simulations using the stochastic rules of
the processes.

Figure 3: Comparison of the noisy voter model (V) and the noisy kinetic-
exchange model (K), both under the influence of aging. Magnetization
$m_{\mathrm{st}}$ as a function of the noise intensity $a$ in the algebraic
and in the exponential cases for two values of the $\tau^{*}$ parameter:
$\tau^{*}=2$ (left panel) and $\tau^{*}=10$ (right panel). Results of the
mean-field theory are plotted with solid lines, while those of numerical
simulations are displayed with symbols. The simulations have been performed
with $10^{4}$ agents and averaged over $5\times 10^{6}$ MCS after $5\times
10^{6}$ thermalization MCS.

As can be seen in Fig. 2, for sufficiently small values of $\tau^{*}$ (strong
or moderate aging, e.g., $\tau^{*}=2$ in Fig. 3), the most determining factor
for the existence of a given consensus region is not the type of model
considered (either voter or kinetic-exchange), but the type of aging kernel.
In particular, algebraic aging, which decays most slowly with $\tau$, is the
one that most enhances this effect. On the other hand, for large values of
$\tau^{*}$ (weak aging, e.g., $\tau^{*}=10$ in Fig. 3), the kinetic-exchange
model gives rise to considerably larger consensus regions than the voter
model, regardless of the type of aging considered (be algebraic or
exponential), consistently with the fact that in the aging-less limit
($\tau^{*}\to\infty$) the kinetic model has a non-null critical point in the
thermodynamic limit, in contrast to the voter model.

The most interesting result for the kinetic model is the non-monotonic
behavior of the magnetization: the critical value $a_{\mathrm{c}}(\tau^{*})$
exceeds the value corresponding to the aging-less case $a_{\mathrm{c}}=1/4$
and it then approaches asymptotically to $1/4$. As can be seen in Fig. 2, this
phenomenon occurs for both profiles of aging. Moreover, as in the noisy voter
model with aging, also in the kinetic model there is a value $\tau_{0}^{*}$
such that for $\tau^{*}<\tau_{0}^{*}$ the algebraic aging gives rise to a
larger consensus region than the exponential aging, while for
$\tau^{*}>\tau_{0}^{*}$ the situation is reversed. For the noisy kinetic-
exchange model it is $\tau_{0}^{*}=5.40$.

##  V Final remarks

We have considered two models of opinion formation, namely, the noisy versions
of the 2-state voter and 3-state kinetic-exchange models, and introduced the
effect of aging through the probability of change $q_{\tau}$ acting on the
interaction between agents. Moreover, we considered two different families of
kernels for $q_{\tau}$: either algebraically or exponentially decaying with
$\tau$.

A mean-field description has been given, for generic form of $q(\tau)$,
unifying previous results for the two models [17, 20] and extending part of
them to embrace a wider family of models with $n$ opinion states. This
description relies on obtaining the evolution equations for the density of
agents in each opinion state $s$ and with given age $\tau$. Then, the steady
state solutions of this system of (infinite) equations and the analysis of its
stability (via an adiabatic approximation valid when separation of timescales
holds) allowed to obtain the plots of the order parameter $m_{st}$ vs. the
noise probability $a$, in agreement with agent-based simulations in a complete
graph.

We have considered particularly the algebraic form
$q_{\tau}=(q_{\infty}\tau+q_{0}\tau^{*})/(\tau+\tau^{*})$, and the exponential
form $q_{\tau}=q_{\infty}+(q_{0}-q_{\infty})\exp(-\tau/\tau^{*})$, with $0\leq
q_{\infty}<q_{0}\leq 1$. In numerical examples, we focused on the case
$(q_{0},q_{\infty})=(1,0)$. Our results indicate that qualitatively similar
tendencies emerge for both kernels in both models as follows. For very large
$\tau^{*}$ (weak aging), there is a good agreement with the aging-less
situation, as expected. For all values of $\tau^{*}$, the magnetization
$m_{\mathrm{st}}$ as a function of the noise intensity $a$ displays a
continuous transition from order to disorder at a critical value
$a_{\text{c}}$. In all the analyzed cases we observe an optimal value of
$\tau^{*}$ to obtain consensus (maximizing this critical value
$a_{\mathrm{c}}$). It is also noteworthy that, while both models produce
different results under weak aging, when aging is strong or moderate, the
critical value $a_{c}$ becomes nearly insensitive to the particular social
interaction rule $(s_{i},s_{j})$ and number of opinion states $n$.

An interesting continuation would be to go beyond all-to-all interactions and
to study aging effects on the opinion dynamics in random networks. Moreover,
other forms of the aging kernel might also bring new features.

###### Acknowledgements.

Partial financial support has been received from the Agencia Estatal de
Investigación (AEI, MCI, Spain)
MCIN/AEI/10.13039/501100011033 and Fondo Europeo de Desarrollo Regional
(FEDER, UE) under Project APASOS (PID2021-122256NB-C21) and the María de
Maeztu Program for units of Excellence in R&D, grant CEX2021-001164-M. C.A.
also acknowledges partial support received from Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq)-Brazil (311435/2020-3) and
Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ)-Brazil (CNE
E-26/201.109/2021).

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##  Appendix A Rate equations

In the general case of $n$ possible state values $s$, the rate equations for
the density $x^{s}_{\tau}$ of agents in state $s$ and age $\tau$ are

$\frac{dx_{\tau}^{s}}{dt}=\Omega_{ss}(\tau-1)-\sum_{s^{\prime}}{\Omega_{ss^{\prime}}(\tau)},\quad\tau\geq
1,$ (13)

where $\Omega_{ss^{\prime}}(\tau)$ is the transition rate from state $s$ to
state $s^{\prime}$. The first term accounts for those updates that did not
result in a change of state and hence increase their internal time from
$\tau-1$ to $\tau$, while the second accounts for those that resulted in a
change of state and therefore resulted in a change $\tau\to 0$ of the internal
time. For $\tau=0$ the rate equation includes all those changes that result in
a reset of the internal time,

$\frac{dx_{0}^{s}}{dt}=\sum_{\tau=0}^{\infty}\sum_{s^{\prime}\neq
s}{\Omega_{s^{\prime}s}(\tau)}-\sum_{s^{\prime}}{\Omega_{ss^{\prime}}(0)}.$
(14)

Summing Eqs. (13, 14) over all values of $\tau$, one obtains the rate
equations for the density of agents in each state $s$ as
$x^{s}=\sum_{\tau=0}^{\infty}x^{s}_{\tau}$. We derive now the rate expressions
for the noisy voter. A similar treatment for the kinetic-exchange model can be
found in Ref. [20].

###  A.1 Noisy voter model with aging

The state can take two possible values $s\in\\{-1,+1\\}$. For the all-to-all
connectivity adopted in this paper the set of transition rates
$\Omega_{ss^{\prime}}$ are

$\begin{split}\Omega_{--}(\tau)&=x_{\tau}^{-}\left(\frac{a}{2}+(1-a)(1-x^{+}q_{\tau})\right),\\\
\Omega_{-+}(\tau)&=x_{\tau}^{-}\left(\frac{a}{2}+(1-a)x^{+}q_{\tau}\right),\\\
\Omega_{+-}(\tau)&=x_{\tau}^{+}\left(\frac{a}{2}+(1-a)x^{-}q_{\tau}\right),\\\
\Omega_{++}(\tau)&=x_{\tau}^{+}\left(\frac{a}{2}+(1-a)(1-x^{-}q_{\tau})\right).\end{split}$
(15)

In the main text we have used the notation $x$ for $x^{+}$ and $1-x$ for
$x^{-}$. We show now the derivation of the expression for the first of these
rates $\Omega_{--}(\tau)$: The probability that an agent with internal time
$\tau$ and state value $s=-1$ remains $-1$ requires first to select an agent
in that state (probability $x^{-}_{\tau}$), then if the social rule is chosen
(probability $1-a$), the copying mechanism activates with probability
$q_{\tau}$ and the selected neighbor must be in state $-1$ (probability
$x^{-}$) but, if the copying mechanism is not activated (probability
$1-q_{\tau}$) the state remains $-1$. If, on the other hand, the idiosyncratic
rule is activated (probability $a$), and the new state is chosen equal to $-1$
with probability $1/2$. This leads to

$\Omega_{--}(\tau)=x^{-}_{\tau}\left((1-a)(x^{-}q_{\tau}+(1-q_{\tau}))+a\,\frac{1}{2}\right),$
(16)

which, after replacement of $x^{-}=1-x^{+}$ leads to the first of Eqs. (15).
Other transition rates in these equations are obtained similarly.

From these rates and Eqs. (13, 14) one can derive the rate equations for
$x_{\tau}^{s}$ as

$\begin{split}\frac{dx_{\tau}^{-}}{dt}&=\Omega_{--}(\tau-1)-x_{\tau}^{-},\\\
\frac{dx_{\tau}^{+}}{dt}&=\Omega_{++}(\tau-1)-x_{\tau}^{+},\end{split}$ (17)

for $\tau\geq 1$, and

$\begin{split}\frac{dx^{-}_{0}}{dt}&=\frac{a}{2}x^{+}+(1-a)x^{-}y^{+}-x_{0}^{-},\\\
\frac{dx^{+}_{0}}{dt}&=\frac{a}{2}x^{-}+(1-a)x^{+}y^{-}-x_{0}^{+},\end{split}$
(18)

for $\tau=0$, where we have defined

$y^{s}\equiv\sum_{\tau=0}^{\infty}q_{\tau}x^{s}_{\tau}.$ (19)

To obtain a closed equation for the time evolution of the densities $x^{s}$ we
use an adiabatic approximation whereby we assume that the time derivative Eq.
(17) of the age-dependent density $x^{s}_{\tau}$ can be set to zero, allowing
to express $x^{\pm}_{\tau}$ in terms of $x^{\pm}_{\tau-1}$. The solution of
this recursive relation is

$\begin{split}x_{\tau}^{-}&=x_{0}^{-}F_{\tau}(x^{+}),\\\
x_{\tau}^{+}&=x_{0}^{+}F_{\tau}(x^{-}),\end{split}$ (20)

with

$F_{0}(x)=1,\hskip
28.45274ptF_{\tau}(x)\equiv\prod_{k=0}^{\tau-1}\gamma_{2}(q_{k}\,x,a),\quad\tau\geq
1,$ (21)

and the function $\gamma_{2}$ is defined in Eq. (6). Summing Eq. (20) over all
$\tau\geq 0$, we obtain

$\begin{split}x^{-}&=x_{0}^{-}\sum_{\tau=0}^{\infty}F_{\tau}(x^{+}),\\\
x^{+}&=x_{0}^{+}\sum_{\tau=0}^{\infty}F_{\tau}(x^{-}),\end{split}$ (22)

which substituted in Eqs. (20) leads to

$\begin{split}x_{\tau}^{-}&=x^{-}\frac{F_{\tau}(x^{+})}{\sum_{\tau}F_{\tau}(x^{+})},\\\
x_{\tau}^{+}&=x^{+}\frac{F_{\tau}(x^{-})}{\sum_{\tau}F_{\tau}(x^{-})},\end{split}$
(23)

which are now expressed in terms of the global variables $x^{\pm}$.

Adding Eqs. (17) and (18) over all values of $\tau$, we obtain the
corresponding equations for the density $x^{s}$ of each state $s$.

$\begin{split}\frac{dx^{-}}{dt}&=\frac{a}{2}(x^{+}-x^{-})+(1-a)(x^{-}y^{+}-x^{+}y^{-}),\\\
\frac{dx^{+}}{dt}&=\frac{a}{2}(x^{-}-x^{+})+(1-a)(x^{+}y^{-}-x^{-}y^{+}).\end{split}$
(24)

Moreover note that one of the two equations can be eliminated since
$x^{+}+x^{-}=1$. Then, in order to obtain a closed evolution equation for the
global variable $x^{+}$, we need to express the variables $y^{s}$ appearing in
Eqs. (24) in terms of $x^{+}$. This can be done by using Eq. (23) into Eqs.
(19), yielding

$y^{-}=(1-x^{+})\Phi(x^{+}),\;\;\;\;\;y^{+}=x^{+}\Phi(1-x^{+}),$ (25)

where we have introduced the function

$\Phi(x)\equiv\frac{\sum_{\tau=0}^{\infty}q_{\tau}F_{\tau}(x)}{\sum_{\tau=0}^{\infty}F_{\tau}(x)}.$
(26)

Let us note here, for consistency, that in the aging-less case, $q_{\tau}=1$,
it is $\Phi(x)=1$ and, hence, $y^{s}=x^{s}$, for $s=-1,+1$.

Replacement of Eqs. (25) in Eqs. (24) leads to the rate equation for $x\equiv
x^{+}$, Eq. (3). Note that the function $\Phi(x)$ depends on the aging profile
$q_{\tau}$ both through the explicit dependence of Eq. (26) and in the
expression of $F_{\tau}(x)$ of Eq. (21). As shown in Appendix B the function
$\Phi(x)$ can be related to hypergeometric functions in the case of an
algebraic dependence of the aging probability.

##  Appendix B Calculation of sums involving $F_{\tau}(x)$

###  B.1 Algebraic aging

In the case of a rational function of the age of the general form

$q_{\tau}=\frac{q_{\infty}\tau+q_{0}\tau^{*}}{\tau+\tau^{*}},$ (27)

where $\tau^{*}>0$ and $0\leq q_{\infty}<q_{0}$, the function $F_{\tau}(x)$
defined is given by

$F_{\tau}(x)\equiv\prod_{k=0}^{\tau-1}\gamma_{n}(q_{k}\,x,a)=\gamma_{n}(q_{\infty}x,a)^{\tau}\frac{\left(\tau^{*}\xi_{n}(x,a)\right)_{\tau}}{(\tau^{*})_{\tau}},\quad\tau\geq
1,$ (28)

where

$\xi_{n}(x,a)\equiv\frac{\gamma_{n}(q_{0}x,a)}{\gamma_{n}(q_{\infty}x,a)},$
(29)

with the function $\gamma_{n}(z,a)$ defined in Eq. (6), and
$(z)_{\tau}\equiv\Gamma(z+\tau)/\Gamma(z)$ is the Pochhammer symbol.

In order to compute the function
$\Phi(x)=\displaystyle\frac{\sum_{\tau=0}^{\infty}q_{\tau}F_{\tau}(x)}{\sum_{\tau=0}^{\infty}F_{\tau}(x)}$,
we need the following sums:

$\displaystyle\sum_{\tau=0}^{\infty}F_{\tau}(x)=$
$\,{}_{2}F_{1}\left(1,\tau^{*}\xi_{n}(x,a);\tau^{*};\gamma_{n}(q_{\infty}x,a)\right),$
(30) $\displaystyle\sum_{\tau=0}^{\infty}q_{\tau}F_{\tau}(x)=$ $\displaystyle
q_{0}\,{}_{2}F_{1}\left(1,\tau^{*}\xi_{n}(x,a);1+\tau^{*};\gamma_{n}(q_{\infty}x,a)\right)+$
$\displaystyle\frac{q_{\infty}}{1+\tau^{*}}\gamma_{n}(q_{0}x,a)\,_{2}F_{1}\left(2,1+\tau^{*}\xi_{n}(x,a);2+\tau^{*};\gamma_{n}(q_{\infty}x,a)\right).$
(31)

In this work we cover the algebraic case corresponding to $q_{0}=1$ and
$q_{\infty}=0$.

###  B.2 Exponential aging

In the definition of $F_{\tau}(x)$, Eq. (28), where $\gamma_{n}(z,a)$ is given
by Eq. (6), the substitution of $q_{\tau}=e^{-\tau/\tau^{*}}$ yields

$F_{\tau}(x)=\biggl{(}\underbrace{1-\frac{n-1}{n}a}_{\alpha}\biggr{)}^{\tau}\biggl{(}\underbrace{nx\frac{1-a}{n-(n-1)a}}_{b};e^{-1/\tau^{*}}\biggr{)}_{\tau},$
(32)

where $(r;s)_{0}=1$ and $(r;s)_{k}=\prod_{i=0}^{k-1}(1-rs^{i})$ for $k\geq 1$,
is the $q$-Pochhammer symbol, and where $\alpha\in(1/n,1)$ and $b\in(0,1)$.
Therefore, the function $\Phi(x)$ is given by

$\Phi(x)=\frac{\sum_{\tau=0}^{\infty}q_{\tau}F_{\tau}(x)}{\sum_{\tau=0}^{\infty}F_{\tau}(x)}.$
(33)

A way to compute numerically the infinite series in the numerator and
denominator of Eq. (33) is to cut-off them by finite sums up to an upper index
$\tau=M$. However, the convergence seems to be slow and large values of $M$
are needed for a precise calculation, specially for small values of $\tau^{*}$
or $a$. To overcome this difficulty, an efficient procedure has been described
in Ref. [20]. It starts by introducing the function

$\displaystyle\phi(z,b,s)=\sum_{\tau=0}^{\infty}(b;s)_{\tau}\,z^{\tau}\,,$
(34)

from where Eq. (33) reads

$\Phi(x)=\frac{\phi(\alpha s,b,s)}{\phi(\alpha,b,s)},\quad
s=e^{-1/\tau^{*}}\in(0,1).$ (35)

Then, we use the iteration relation that follows from its definition, Eq.
(34), namely,

$\phi(\alpha,bs^{k},s)=1+\alpha(1-bs^{k})\,\phi(\alpha,bs^{k+1},s),\quad
k=L,L-1,\dots,0$ (36)

with the initial condition

$\phi(\alpha,bs^{L+1},s)=\frac{1}{1-\alpha},$ (37)

that results from the approximation $bs^{L+1}\approx 0$ and the use of
$(0;s)_{k}=1$ in the definition (34). We have taken $L$ such that
$bs^{L+1}<\epsilon=10^{-12}$, or
$L\sim\log(\epsilon/b)/\log{s}=\tau^{*}\log(b/\epsilon)$ although other
smaller values of $\epsilon$ produced the same results.