# Chain Reaction of Ideas: Can Radioactive Decay Predict Technological
Innovation?

G. S. Y. Giardini  C. R. da Cunha<EMAIL_ADDRESS>
###### Abstract

This work demonstrates the application of a birth-death Markov process,
inspired by radioactive decay, to capture the dynamics of innovation
processes. Leveraging the Bass diffusion model, we derive a Gompertz-like
function explaining the long-term innovation trends. The validity of our model
is confirmed using citation data, Google trends, and a recurrent neural
network, which also reveals short-term fluctuations. Further analysis through
an automaton model suggests these fluctuations can arise from the inherent
stochastic nature of the underlying physics.

###### keywords:

Markov Chain, Cellular Automata, Neural Networks

††journal: Physica A

organization= School of Informatics, Computing, and Cyber-Systems, Northern
Arizona University, addressline=1295 S. Knoles Dr., city=Flagstaff, state=AZ,
postcode=86011, country=USA

##  1 Introduction

The radioactive decay of a specified isotope, such as ${}^{238}_{92}$U
undergoing a decay into ${}^{234}_{90}$Th $+$ ${}^{4}_{2}$He, can be
effectively modeled as a stochastic Markov process [1]. Unstable nuclei
spontaneously decay and emit radiation (e.g., $\alpha$-particles in this
process) at a rate characterized by the death transition rate $\mu$.
Simultaneously, other stable nuclei in the vicinity may capture these emitted
particles and become unstable, initiating a new decay chain. This capture
process is modeled by the birth transition rate $\lambda$. Both $\lambda$ and
$\mu$ depend solely on the current number of remaining nuclei (denoted by $n$)
and exhibit the memoryless property characteristic of Markov chains, meaning
the probability of a decay event depends only on the current state $n$, not on
the history of previous decays.

Building upon the Bass diffusion model [2, 3, 4], this work presents a novel
approach to modeling the innovation process inspired by the dynamics of
radioactive decay. While the Bass model identifies innovators (early adopters)
and imitators (influenced by existing adopters), we propose a birth-death
Markovian framework focusing on the agents who either adopt or abandon a
specific technology. Analogous to radioactive decay coefficients, our model
utilizes birth and death transition rates governed by the current number of
adherents. This framework transcends the Bass model’s emphasis on imitation by
capturing the inherent stochasticity of adoption and abandonment decisions,
enabling a more nuanced understanding of the innovation process dynamics.

Several existing approaches attempt to model innovation dynamics. Some
strategies are based on cognitive processes as random walks on complex
networks, where nodes symbolize ideas [5]. Catastrophe theory has also been
employed to understand innovation through the lens of cusp geometry [6].
Population dynamics, especially those utilizing models influenced by social
regularities and information flows, offer another avenue for understanding the
growth patterns elucidated in this research paper [7]. Additionally, various
models leverage frameworks such as epidemics, probit regressions, information
cascades, competition, and population dynamics [8, 9, 10].

Our proposed model distinguishes itself from these methods by establishing a
clear analogy to radioactive decay processes through the formal framework of
Markov chains. This direct connection with well-understood physical phenomena
offers unique advantages in terms of interpretability and potential predictive
power.

In the next section, we discuss our model in detail and present simple
ansatzes for modeling the birth and death, or attachment and detachment rates.
We will show that our estimates lead to a Gompertz-like function [11] that
models the number of active species. We test our model in the following
section with both the number of scientific papers and cultural trends captured
by Google searches. The estimates for the birth and death curves are then
tested with a recursive neural network, which also predicts fluctuations in
the model. The last section examines the origins of these fluctuations using
an automaton to model both the radioactive and the innovation processes.

##  2 The Model

Our model considers a population of individuals that either attach to or
detach from a technology. The state corresponding to a population of $n$
agents adhering to a new technology is mapped to the stochastic process
$X_{n}$. The attachment rate per single cell for the new technology is given
by $\lambda_{n}(t)$, whereas the detachment rate is given by $\mu_{n}(t)$.
Moreover, due to social conformity [12, 13], the chance an individual attaches
to or detaches from a technology depends only on the number of individuals
already attached to it. Therefore, we choose $\lambda_{n}(t)=n\lambda_{0}(t)$
and $\mu_{n}(t)=n\mu_{0}(t).$

We make the assumption that the stochastic process $X_{n}$ adheres to the
Markov property, which asserts its independence from its entire historical
trajectory. Consequently, given the information available up to time $n$,
denoted by a set of observations $x_{n}$ and encapsulated within the sigma-
algebra
$\mathcal{F}_{n}=\sigma(X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n-1}=x_{n-1})$, the
Markov property asserts that the conditional probability of $X_{n+1}$ at time
$n+1$ given $\mathcal{F}_{n}$ is equivalent to the conditional probability of
$X_{n+1}$ given the immediate preceding state $X_{n}$. Mathematically, this is
expressed as
$P(X_{n+1}=x_{n+1}|\mathcal{F}_{n})=P(X_{n+1}=x_{n+1}|X_{n}=x_{n})$.

The process $X_{n}$ induces a Markov chain, reflecting its dependence on its
current state. This dependence is captured by three components:

  1. 1.

Persistence: The initial term,
$(1-\lambda_{n}\Delta-\mu_{n}\Delta_{t})X_{n}(t)$, captures the likelihood of
remaining in the current state ($X_{n}$). This term encompasses scenarios
where there is no radioactive decay/particle transfer, reflecting the absence
of individuals either adopting or abandoning the technology. Here,
$\lambda_{n}$ represents the birth rate at instant $n$, and $\mu_{n}$ denotes
the death rate at the same instant.

  2. 2.

Adoption: The second term, $\lambda_{n-1}(t)\Delta_{t}X_{n-1}(t)$, describes
the increase in state $X_{n}$ resulting from transitions originating in
preceding state $X_{n-1}$. This term analogously reflects the receipt of
radioactive particles or the adoption of new technology by individuals, where
the influence is quantified by the birth rate associated with the prior state
and the duration of the time interval $\Delta_{t}$.

  3. 3.

Abandonment: The third term, $\mu_{n+1}(t)\Delta_{t}X_{n+1}(t)$, represents
the reduction in state $X_{n}$ attributed to transitions towards the
subsequent state $X_{n+1}$. Analogous to unstable nuclei emitting particles or
individuals disengaging from the technology, this reduction is contingent upon
the death rate associated with the succeeding state.

Mathematically, these concepts are combined in the following equation:

$\displaystyle X_{n}(t+\Delta_{t})=$
$\displaystyle(1-\lambda_{n}\Delta-\mu_{n}\Delta_{t})X_{n}(t)+\lambda_{n-1}(t)\Delta_{t}X_{n-1}(t)$
(1) $\displaystyle+\mu_{n+1}(t)\Delta_{t}X_{n+1}(t).$

At the limit of very short intervals, $\Delta_{t}\rightarrow 0$, this equation
becomes:

$\displaystyle\frac{dX_{n}(t)}{dt}=$
$\displaystyle-(\lambda_{n}(t)+\mu_{n}(t))X_{n}(t)+\mu_{n+1}(t)X_{n+1}(t)$ (2)
$\displaystyle+\lambda_{n-1}(t)X_{n-1}(t).$

While this equation focuses on individual transitions, one important question
arising by this model pertains to the average number of individuals engaged
with the technology at any given time. This average engagement is captured by
the following expression:

$M(t)=\sum_{n=1}^{\infty}nX_{n}(t).$

The rate of change of this average engagement is determined by:

$\displaystyle\frac{dM(t)}{dt}$
$\displaystyle=-\left(\lambda_{0}(t)+\mu_{0}(t)\right)\sum_{n=1}^{\infty}n^{2}X_{n}(t)$
(3) $\displaystyle+\mu_{0}(t)\sum_{n=1}^{\infty}n(n+1)X_{n+1}(t)+$
$\displaystyle+\lambda_{0}(t)\sum_{n=1}^{\infty}n(n-1)X_{n-1}(t)$
$\displaystyle=\left(\lambda_{0}(t)-\mu_{0}(t)\right)M(t).$

Building upon the understanding of the average adoption level from the
transition probabilities, let’s now delve deeper into the temporal dynamics of
technology engagement. To capture the evolving landscape of opinions and
preferences, we propose that the attachment rate ($\lambda_{0}$) representing
new users joining may decrease over time due to factors like competing
technologies, changing trends, and fading novelty. We suggest modeling this
decrease using a simple hyperbolic function given by
$\lambda_{0}(t)=\alpha/t$, where $\alpha$ is a characteristic time constant.
This simplified model employs a functional form akin to pulsed neutron-induced
activation [14, 15, 16], where the decay rate may undergo transient changes
over time [17]. While the actual birth rate dynamics in the system can be
intricate, this model effectively captures the initial trend and facilitates
the derivation of analytical solutions.

Conversely, the detachment rate $\mu_{0}$ represents users abandoning the
technology, influenced by various factors. We expect this rate to follow a
diffusion pattern, with rapid initial adoption followed by a gradual slowdown
and stabilization at a long-term average detachment rate. This characteristic
resembles the behavior of certain radioactive processes, like pulsed neutron-
induced activation, where temporary changes in activation rates might occur
due to external influences but the underlying dynamics remain governed by a
constant decay constant. To capture this mean-reverting behavior, we propose
employing an Ornstein-Uhlenbeck (OU) process [13], described by
$d\mu_{0}(t)=\frac{\beta-\mu_{0}(t)}{\sigma}dt+\kappa dW$, where $\sigma$ is
the mean-reversion time, $\beta$ is the equilibrium death rate, $\kappa$ is a
volatility parameter, and $dW$ is a Wiener process.

The death rate may initially be low but gradually increases as the initial
excitement wanes and limitations become apparent. Given our interest in major
trends, we assume $\kappa\sigma<<1$ and investigate the quasi-deterministic
regime of the rate. This simplification facilitates the analysis, allowing us
to focus on the general trend of the death rate. Furthermore, we assert that
the death rate at the beginning of the process is zero, as there has been no
adherence to the technology yet. Under these conditions, the OU process
predicts a death rate $\mu_{0}(t)=\beta\left(1-e^{-t/\sigma}\right)$.

Given these considerations about the birth and death rates, the time variation
of the expected number of individuals attached to the technology becomes:

$\frac{dM(t)}{dt}=\left(\frac{\alpha}{t}-\beta\left[1-exp\left(-\frac{t}{\sigma}\right)\right]\right)M(t).$
(4)

The solution of this equation is:

$M(t)=M_{0}t^{\alpha}e^{-\beta(\sigma e^{-t/\sigma}+t)}.$ (5)

It is interesting to note that the exponential component of the equation is a
Gompertz function [11], that describes, for example, the population dynamics
in confined spaces [18].

Parameters $M_{0}$, $\alpha$, $\beta$, and $\sigma$ can be found using a
standard Levenberg - Marquardt fitting [19]. These parameters allow us to
compare the temporal dynamics of different technologies.

###  2.1 Results

To validate our theoretical model’s ability to capture real-world dynamics, we
analyzed two contrasting datasets: scientific impact measured through citation
counts for three physics papers published in $1980$ (A–C [20, 21, 22]) and
popular cultural trends represented by Google Trends interest data for
“facebook” and “snapchat” spanning 20 years (2004–2023). We retrieved the
citation data using the Web of Science citation report tool over a $44$-year
period. Google defines interest as the worldwide number of searches relative
to the highest point over the time series on a base $100$. To ensure robust
comparisons, both datasets were normalized and included only non-zero data
points.

Figure 1: Annual citation counts for three scientific papers (A–C [20, 21, 22]) are shown. Square, circle, and triangle symbols denote experimental data for each paper, respectively. Model predictions based on Eq. 5 are presented as dashed and dotted curves. Table 1: Fitting parameters and the root mean square (RMS) error of the residuals for three scientific papers (A–C [20, 21, 22]) # of Citations | RMS of Res. | $\sigma$ | $\beta$ | $\alpha$ | $M_{0}$
---|---|---|---|---|---
A | 8.2545 | 1.27918 | 0.0622717 | 0.644351 | 23.1076
B | 4.06217 | 17.924 | 0.0632812 | 0.40297 | 23.6886
C | 1.33186 | 37.5037 | 0.302128 | 4.67661 | 0.58463

Figure 1 demonstrates the model’s ability to capture diverse citation
patterns, revealing both rapid (Paper A) and gradual (Paper C) initial
citation growth. These contrasting patterns align with two distinct innovation
models: the Socratic and Schumpeterian models [23]. The Schumpeterian model is
characterized by a slower, more deliberative initial growth in citations,
while the Socratic model is marked by a faster, disruptive initial increase.
Notably, our model accurately captures the citation dynamics of both
innovation types, as evident in the figure and the estimated parameters
presented in Table 1.

Figure 2 expands our analysis to popular cultural trends, as reflected in
Google Trends data for the keywords “Facebook” and “Snapchat”. The
corresponding estimated parameters are presented in Table 2. While the overall
fitting results are satisfactory, noteworthy divergences appear after month
$100$ for “snapchat” and after month $190$ for “Facebook”. Remarkably, these
points in time coincide with the latter half of $2019$, aligning with the
onset of the impact of COVID-$19$. This divergence suggests that the pandemic
can be considered a perturbation, disrupting the natural trajectory of
innovation adoption and detachment. In the context of our radioactive analogy,
this scenario resembles the introduction of a new neutron-induced reaction
while an existing one is already in place.

Figure 2: Temporal evolution of interest in the keywords ‘Facebook’ and
‘Snapchat,’ featuring experimental data represented by square and circle
symbols. Model predictions using Eq. 5 are depicted by the dashed curves.

It is worth noting that our model demonstrates the capability to detect
external perturbations, such as those caused by significant events like the
COVID-$19$ pandemic. This capacity makes our approach valuable not only for
understanding historical trends but also for proactively identifying and
characterizing disruptions in the adoption and detachment patterns of cultural
phenomena.

Table 2: Fitting parameters and the root mean square (RMS) error of the residuals for the Google Trends data for keywords “Facebook” and “Snapchat” Social Media | RMS of Res. | $\sigma$ | $\beta$ | $\alpha$ | $M_{0}$
---|---|---|---|---|---
Facebook | 5.87717 | 235.071 | 0.119982 | 4.50933 | 1.35927e+06
Snapchat | 8.3044 | 44.292 | 0.096952 | 4.39675 | 0.00122246

Moreover, our model allows us to make predictions about the business cycles.
The time for an innovation to peak and vanish can be readily calculated from
Eq. 5. These results are shown in Tab. 3.

Table 3: Expected time to peak and extinction for the influence of scientific papers A–C and social media “Facebook” and “Snapchat” Innovation | A | B | C | ‘Snapchat’ | ‘Facebook’
---|---|---|---|---|---
Peak [years] | 11 | 13 | 29 | 5 | 9
Extinction [years] | 99 | 79 | 49 | 11 | 20

##  3 Estimating $\lambda_{0}$ and $\mu_{0}$

A good way to test our ansatz for birth and death functions is to use a neural
network to estimate them and compare the results with our theoretical
predictions.

In our analytical model, the birth and death rates can be discretized to:

$\displaystyle\lambda_{0}^{(m)}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{m},$ $\displaystyle\mu_{0}^{(m)}$ $\displaystyle=$
$\displaystyle\beta\left(1-e^{-m/\sigma}\right).$ (6)

With these equations, we can obtain the main trend of adoption and detachment
from innovations in a way that mimics pulsed neutron-induced radioactive
activation. These equations, however, only capture general trends and are
unable to capture fine details of the innovation process. Moreover, they were
constructed based on a set of assumptions that may not happen in real
situations.

To investigate the birth and death rates in real situations, we employ a
simple Elman recurrent neural network (RNN) [19, 24] with one input $x^{(n)}$,
one hidden layer $h_{1,2}^{(n)}$ with hyperbolic activations, and two linear
outputs $y_{1,2}^{(n)}$, as shown in Fig. 3. The layers are mathematically
given as:

$\displaystyle h_{1}^{(n)}$ $\displaystyle=$
$\displaystyle\tanh\left(w_{1}x+s_{11}h_{1}^{(n-1)}+s_{21}h_{2}^{(n-1)}+b_{1}\right)$
$\displaystyle h_{2}^{(n)}$ $\displaystyle=$
$\displaystyle\tanh\left(w_{2}x+s_{12}h_{1}^{(n-1)}+s_{22}h_{2}^{(n-1)}+b_{1}\right)$
$\displaystyle z_{1}^{(n)}$ $\displaystyle=$ $\displaystyle
r_{11}h_{1}^{(n)}+r_{21}h_{2}^{(n)}+c_{1}$ $\displaystyle z_{2}^{(n)}$
$\displaystyle=$ $\displaystyle r_{12}h_{1}^{(n)}+r_{22}h_{2}^{(n)}+c_{2},$
(7)

where parameters $\mathbf{s},\mathbf{r},\mathbf{b}$ and $\mathbf{c}$ are
obtained after training the network.

Figure 3: Recurrent neural network used to estimate $\lambda^{(n)}$ and
$\mu^{(n)}$ as outputs $y_{1,2}$ given inputs $x^{n}$. The dotted circle
indicates an input, the partially filled circles indicate a hyperbolic
activation and the solid circle represents an identity activation.

The realistic birth and death rates are estimated in a two-step approach.
First, the network is pre-conditioned to predict theoretical rates in Eq. 6
from synthetic data produced by 5. This ensures that all simulations with real
data depart from the same conditions and they reflect the theoretical
expectation. Then, the network is allowed to relax by adjusting the predicted
values of $\lambda_{0}$ and $\mu_{0}$ to values that satisfy Eq. 3 with
experimental values for $M$.

In the pre-conditioning stage, values for $\alpha,\beta$, and $\sigma$ are
randomly selected. Then values of $M$ are obtained using Eq. 5, and the
network is trained to produce estimates for the corresponding $\lambda_{0}$
and $\mu_{0}$. For this purpose, the network is trained with the loss:

$\mathcal{L}_{pc}=\frac{1}{N}\sum_{i=1}^{N}\left[\left(y_{1}^{(i)}-\lambda_{0}^{(i)}\right)^{2}+\left(y_{2}^{(i)}-\mu_{0}^{(i)}\right)^{2}\right],$
(8)

where $N$ is the length of the time series.

With the network pre-conditioned with the theoretical values, we let it relax
to new estimates given experimental data for $M$ and a loss function given by:

$\mathcal{L}_{ex}=\frac{1}{N}\sum_{i=1}^{N}\left[\left(\frac{dM^{(i)}/dt}{M^{(i)}}-y_{1}^{(i)}+y_{2}^{(i)}\right)^{2}+\delta\left(\left|y_{1}^{(i)}\right|+\left|y_{2}^{(i)}\right|\right)\right].$
(9)

The loss function is comprised of two terms. The first penalizes deviations
from Eq. 3, while the second discourages large output values through
regularization. In all simulations, we added noise to $M$ at a magnitude one
order smaller than its minimum value. The derivative of $M$ was estimated
using the Savitzky-Golay procedure [25, 26] with $15$ kernel nodes. Finally,
all simulations employed $\delta=10^{-5}$ and the ADAM optimizer [19] with
parameters $\eta=3\times 10^{-3}$, $\beta_{1}=0.9$, and $\beta_{2}=0.999$.
Training was conducted until the Gaussian curvature of the loss function,
computed with $500$ data points, reached the value of $\rho=10^{10}$. The
Gaussian curvature for a dataset $\\{x\\}$ is defined as:

$\rho_{x}(t)=\left|\frac{\left(1+dx/dt\right)^{3/2}}{d^{2}x/dt^{2}}\right|.$
(10)

###  3.1 Results

Figure 4 shows the estimated birth and death rates for scientific papers A–C
using both Eq. 6 and the RNN. We quantified the relative difference between
them as:

$\Delta_{n}=\frac{\left|\sum_{i=1}^{n}X^{(i)}-\sum_{i=1}^{n}y_{1,2}^{(i)}\right|}{\sum_{i=1}^{n}X^{(i)}},$
(11)

where $X$ is either $\lambda_{0}$ or $\mu_{0}$.

This approach of calculating the relative difference with the accumulated time
series attenuates short-term fluctuations and emphasizes underlying trends
through temporal integration. By mitigating the influence of transient
variations, it facilitates the detection of long-term divergences and subtle
shifts in trend dynamics.

Figure 4: Estimated $\lambda_{0}$ and $\mu_{0}$ for the citations of papers A
(a) and (d), B (b) and (e), and C (c) and (f). Blue solid lines correspond to
the theoretical curves, while the red solid curves correspond to the estimated
curves using RNN. The dashed gray lines are the relative errors.

The same analysis was made for the interest in keywords “Snapchat” and
“Facebook”. These results are shown in Fig. 5.

Figure 5: Estimated $\lambda_{0}$ and $\mu_{0}$ for the temporal evolution of
interest in the keywords ‘Snapchat’ (a) and (c), and ‘Facebook’ (b) and (d).
Blue solid lines correspond to the theoretical curves, while the red solid
curves correspond to the estimated curves using RNN. The dashed gray lines are
the relative errors.

In a neutron-induced radioactive process, bursts may induce transitions
between different decay modes or interact with various isotopes or nuclear
states, resulting in shifts in the decay-active population [27]. Additionally,
resonances in nuclear cross-sections can impact neutron capture, leading to
oscillations in the decay rate [28, 29]. The decay products can exert
influence on the environment, initiating feedback loops that, in turn, affect
the birth and death rates [30, 31]. Finally, fission events within the system
can generate additional neutrons, contributing to fluctuations in the decay
rates.

These phenomena find a metaphorical parallel in human behavior [13]. Social
interactions are susceptible to external shocks such as wars, pandemics, and
crises, capable of triggering rapid behavioral changes. Career transitions and
migration can further contribute to population shifts and fluctuations.
Cultural trends may introduce biases that shape our perception of information
[32, 33, 34]. Lastly, individual actions exhibit a feedback loop effect,
resulting in complex dynamics and oscillations in behavior.

Our model demonstrates the ability to capture the outcomes of these intricate
phenomena and estimate their impact on birth and death rates. To investigate
the origin of these features in more depth, we propose an automaton that
mimics both the behaviors of radioactive and innovation processes.

##  4 Capturing the Observed Behavior with an Automaton

As discussed in the previous section, the dynamics of innovation can be a
complex process often involving multiple interacting factors. In this last
section, we propose an automaton that effectively captures the main trends of
the observed behaviors through a set of simplified rules. This approach allows
us to verify our findings, explore different scenarios, identify key elements
of the system, and represent other complex systems that might have potential
similarities.

Our automaton is described by a tuple $A=(Z,S,N,f)$ where $Z$ is a $d\times
d\in\mathbb{N}^{2}$ lattice composed of discrete cells that can hold states
$S=\\{1,2,3\\}$. The transition from one cell value to another is given by the
transition function $f:S\rightarrow S$. The transition of states for a single
cell is considered within a von Neumann neighborhood, described by the set
$N(\mathbf{n}_{0})=\\{\mathbf{n}_{k}:\sum_{i}^{d}|(\mathbf{n}_{k})_{i}-(\mathbf{n}_{0})_{i}\leq
1\\}$. The configuration of the automaton is described by a function
$c:Z\rightarrow S$ that assigns a state to each cell. The next state of a cell
is given by $c(\mathbf{n}_{0})^{t+1}=f(\\{\mathbf{n}_{m}|\mathbf{n}_{m}\in
N(\mathbf{n}_{0})^{t}\\}$. The next state for a cell is determined through a
random selection from the values of its neighboring cells, with the selection
probabilities being proportional to the frequencies of those values.
Therefore, each agent tends to align its state to the state of the majority of
its neighbors. To avoid boundary effects, we used periodic boundary
conditions.

In our simulations, the grid is randomly initiated with $S_{0}=\\{1,2\\}$. The
automaton runs for $100$ steps to reach thermalization and a new state is
introduced to $n$ elements of the grid to create $S=S_{0}\cup\\{3\\}$ at
random locations on the lattice. Snapshots taken immediately after introducing
a new state, as well as after $10$, $30$, and $50$ steps, are shown in Fig. 6.

Figure 6: Output grid of the automaton: a) immediately after introducing a new
state, b) after $10$ steps, c) after $30$ steps, and d) after $50$ steps.

To verify the relationship of the automata with our model, we ran a series of
simulations on a $50\times 50$ lattice, and a fraction $\frac{n}{N}=\\{0.005$,
$0.01$, $0.02$, $0.04\\}$ of $n$ agents initialized with the state $S=\\{3\\}$
after the thermalization process. We then count the number of agents in state
$S=\\{3\\}$ after each simulation step. The temporal evolution of cells in
state $3$ is shown in Fig. 7.

Figure 7: Number of agents in state $S=\\{3\\}$ as a function of Monte Carlo
iteration steps on a $50\times 50$ lattice. $n/N$ represents the ratio between
cells starting in state $3$ after thermalization and the total number of
cells. Dashed lines represent the analytical solution of our innovation model
given by a fitting of Eq. 5 with data generated by the automaton.

Fitting the number of cells in state $S=\\{3\\}$ using our model, given by
equation 5, using Levenberg-Marquadt’s procedure, gives the dashed lines in
Fig. 7. The results visually match the model, and are confirmed by parameters
and errors shown in Table 4.

Table 4: Fitting parameters and the root mean square (RMS) error of the residuals for the data generated by the automaton n/N | RMS of Res. | $\sigma$ | $\beta$ | $\alpha$ | $M_{0}$
---|---|---|---|---|---
0.005 | 1.47763 | 3.04674 | 0.247942 | 0.975034 | 10.9288
0.01 | 3.1669 | 8.28948 | 0.314912 | 1.43169 | 53.8677
0.02 | 5.0286 | 15.1689 | 0.21077 | 0.754141 | 439.929
0.04 | 12.6278 | 1.63855 | 0.0610844 | 0.374988 | 62.7025

We used the results shown in Fig. 7 as inputs for the RNN and obtained
estimates for $\mu_{0}(t)$ and $\lambda_{0}(t)$ which are shown in Fig. 8.

Figure 8: Estimated $\lambda_{0}$ and $\mu_{0}$ for the evolution of three-
state automata with different fractions of cells initialized in a new state
$\\{3\\}$: $0.5$ % a) and d), $2$ % b) and e), and $4$ % c) and f). Blue solid
curves correspond to theoretical values, while the red curves correspond to
the estimated curves using RNN. The dashed gray curves are the relative
errors.

The fact that our automaton model can successfully capture major trends of
both radioactive and innovation processes primarily suggests that they share
similar underlying physics related to the birth and death of species or to the
attachment and detachment of agents to new states. Also, the good fitting with
long-term trends suggests that our estimates for the temporal evolution of
birth and death functions are reasonable. Finally, the oscillations and
deviances we see from the long-term trends are likely natural features caused
by the stochastic nature of the processes involved. Although we can estimate
their impact on the birth and death curves, they are the result of
uncertainties related to the degrees of liberties intrinsic to the processes.
Nonetheless, large deviations from the long-term trends, as depicted in Fig. 2
can indicate the application of external perturbations or the onset of new
competing processes.

##  5 Conclusions

In this work, we have shown how a Markovian innovation model can capture the
behaviors of both radioactive decay processes and innovation processes. We
built upon the Bass diffusion model and used simple ansatzes to create a
birth-death Markov process that explains the long-term trend of innovation
processes. To find the birth and death functions we used two simple ansatzes.
First, we argue that the birth function must be high at the beginning of the
process, but as the material is consumed it must vanish. Second, we argue that
the death function must be akin to a mean-reverting process. Given these birth
and death curves obtained under these assumptions, we found a Gompertz-like
function that models the number of active species.

To test our model, we applied it to two different innovation processes, namely
the number of citations for three physics papers, and cultural trends captured
by Google searches for trend words “facebook” and “snapchat”. Our model
correctly captures the long-term trends for all these cases.

To check the validity of our ansatzes for the birth and death curves, we used
a recurrent neural network model that was previously trained on the
theoretical model but is allowed to relax to fit real data. The neural network
confirmed the long-term trends of our model and showed short-term oscillations
and deviations for the birth and death curves.

The oscillations in the model were probed deeper with an automaton created to
model both radioactive and innovation processes. The results obtained with the
automaton show the same oscillation features observed with real data. The good
fitting obtained with the automaton suggests that both processes share similar
underlying physics, and the oscillations are caused by the intrinsic
stochastic behavior of the processes involved.

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