Noisy voter model with poll-induced delays

Here we share a Python implementation of a noisy voter model with poll-induced delays. The model itself was introduced in [1]. Here we just briefly discuss the differences between the "usual" noisy voter model and how to use our implementation.

In the noisy voter model only one agent (of $N$) is allowed to change their state (opinion) at any single time. Therefore the model is completely described by two rates: one corresponding to an agent switching from state "0" to state "1" (letting $X$ be the number of agents in state "1"):

$$ \lambda (X \rightarrow X + 1)=\lambda^+=(N-X)\cdot\left[\varepsilon_1 + X\right], $$

and the other corresponding to agent switching from state "1" to state "0":

$$ \lambda (X \rightarrow X - 1 )=\lambda^-=X\cdot\left[\varepsilon_0 + (N-X)\right]. $$

These rates correspond to the "non-extensive" parametrization of the noisy voter model. In this parametrization the model converges to a steady state distribution even in the thermodynamic limit ($N \rightarrow \infty$), instead of converging to a fixed point as is common in the existing literature.

We introduce polls by rewriting the peer-pressure (or imitation) terms in the rates to be time-delayed:

$$ \lambda^+(t)=\left(N-X(t)\right)\cdot\left[\varepsilon_1 + X\left(\left\lfloor\frac{t}{\tau} - 1\right\rfloor\tau\right)\right], $$

$$ \lambda^-(t)=X(t)\cdot\left[\varepsilon_0 + \left(N-X\left(\left\lfloor\frac{t}{\tau} - 1\right\rfloor\tau\right)\right)\right]. $$

In the above $\tau$ stands for both polling period, and the delay to the announcement of the poll outcome. So, effectively every $t=k\tau$ (with $k=0,1,2,\ldots$) an outcome of a previous poll $A_{k-1}$ (conducted at $t=(k-1)\tau$) is announced to the agents, and a new poll $A_k$ is conducted (this outcome will be announced at $t=(k+1)\tau$).

It is trivial to see that for small $\tau$ the time-delayed model is equivalent to the original model, but for larger $\tau$ interesting phenomena emerge (see [1] for details).

Requirements

• numpy (tested with 1.26.4)
• scipy (tested with 1.12)
• numba (tested with 0.59)

Table of contents

• Usage of the adapted Gillespie method
• Usage of the macroscopic simulation method
• Usage of the semi-analytical (Markov chain) method
• Obtaining the scaling law
• License
• References

Usage of the adapted Gillespie method

This method follows the original Gillespie method, but modifies it to take into account delays specific to the poll-delayed noisy voter model. We have found it to be somewhat faster than the more general next reaction method.

The code below simulates and plots a sample trajectory of the poll-delayed model. Simulation is conducted up to the fifth poll (i.e., until $t=5\tau$), in between each poll $100$ equidistant sampled values are retained.

import matplotlib.pyplot as plt
import numpy as np

from gillespie_sim import generate_series

# set simulation parameters
n_polls = 5
n_inter = 100

tau = 1e-2
epsi = 1.0
n_agents = 100

initial_state = 70
initial_poll = 30

seed = 10459

# generate trajectory
series = generate_series(
    n_polls,
    tau=tau,
    n_inter=n_inter,
    epsi_0=epsi,
    epsi_1=epsi,
    n_agents=n_agents,
    initial_state=initial_state,
    initial_poll=initial_poll,
    seed=seed,
)
poll_indices = np.linspace(0, n_polls, n_polls * n_inter + 1)

# plot the trajectory
plt.figure()
plt.xlabel(r"$t / \tau$")
plt.ylabel(r"$X(t)$")
plt.plot(poll_indices, series)
plt.show()

The code above should produce the figure below.

Usage of the macroscopic simulation method

This method was developed specifically for the poll-delayed model, it is a more efficient way to simulate the model when $\tau$ is large, because it does not simulate individual transitions, but only the outcomes of the polls.

Below you can see a sample code for a typical model run. This code simulates $10^3$ trajectories (which we collectively refer to as ensemble) with the same initial conditions and parameter sets. Every trajectory in the ensemble is simulated up to $k=3500$. Then the program takes ensemble mean, ensemble variance, and calculates probability mass function (both over ensemble, and trajectories). All numerical simulations are hidden behind generate_series function, which takes simulation parameters and initial conditions as its inputs. For those interested in analytical predictions for the moments, there are also get_mean and get_var functions which allow users to obtain predicted values for the moments.

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import betabinom  # type: ignore

from macro_sim import generate_series, get_mean, get_var

# setup simulation parameters
ensemble_size = 1000

n_polls = 3500
n_agents = 1000
epsi_0 = 2
epsi_1 = 2
tau = 0.03
initial_poll = int(np.ceil(0.3 * n_agents))
initial_state = int(np.ceil(0.7 * n_agents))

seed = 43061
seed_inc = 273

# generate ensemble of trajectories
ensemble = np.array(
    [
        generate_series(
            n_polls,
            tau=tau,
            epsi_0=epsi_0,
            epsi_1=epsi_1,
            n_agents=n_agents,
            initial_state=initial_state,
            initial_poll=initial_poll,
            seed=seed + i * seed_inc,
        )
        for i in range(ensemble_size)
    ]
)

# calculate mean and variance over ensemble
poll_ids = np.arange(0, n_polls + 1, 1)
mean_series = np.mean(ensemble, axis=0)
var_series = np.var(ensemble, axis=0)

# determine simulated PMF
#
# PMF is taken both over ensemble, and over trajectories to get a better
# quality picture
bin_edges = np.arange(-0.5, n_agents + 0.5, 1)
bin_counts = np.histogram(ensemble[:, -1000:], bins=bin_edges)[0]
bin_centers = (bin_edges[1:] + bin_edges[:-1]) / 2
del bin_edges

# calculate analytical results
theory_mean = get_mean(
    poll_ids, epsi_0=epsi_0, epsi_1=epsi_1, initial_state=initial_state,
    initial_poll=initial_poll
)
theory_var = get_var(
    poll_ids, epsi_0=epsi_0, epsi_1=epsi_1, initial_state=initial_state,
    initial_poll=initial_poll
)
theory_pmf = betabinom.pmf(bin_centers, n_agents, 2 * epsi_1, 2 * epsi_0)

# plot the figure
plt.figure(figsize=(9, 2))
plt.subplot(131)
plt.xlabel(r"$k$")
plt.ylabel(r"$\langle A_k\rangle$")
plt.plot(poll_ids[::2][::10], mean_series[::2][::10])
plt.plot(poll_ids[::2][::10], theory_mean[::2][::10], "k--")
plt.plot(poll_ids[1::2][::10], mean_series[1::2][::10])
plt.plot(poll_ids[1::2][::10], theory_mean[1::2][::10], "k--")
plt.subplot(132)
plt.xlabel(r"$k$")
plt.ylabel(r"$\text{Var}[A_k]$")
plt.plot(poll_ids[::2][::10], var_series[::2][::10])
plt.plot(poll_ids[::2][::10], theory_var[::2][::10], "k--")
plt.plot(poll_ids[1::2][::10], var_series[1::2][::10])
plt.plot(poll_ids[1::2][::10], theory_var[1::2][::10], "k--")
plt.subplot(133)
plt.xlabel(r"$A_k$")
plt.ylabel(r"$p(A_k)$")
plt.plot(bin_centers, bin_counts / np.sum(bin_counts))
plt.plot(bin_centers, theory_pmf, "k--")
plt.tight_layout()
plt.show()

The code above should produce the figure below.

Usage of the semi-analytical (Markov chain) method

This method relies on building the transition matrix (extremely time consuming) and then either "simulating" it for set number of steps (using simulate_pmfs function), or solving the eigenproblem to get stationary PMF (using get_stationary_pmf function). If you need, you can also obtain transition matrix and use it for your other purposes (use make_transition_matrix function).

Code below calculates poll outcome PMFs over $1000$ steps (single step corresponds to single polling period). Figure of PMF evolution is then obtained and plotted.

import matplotlib.pyplot as plt
import numpy as np
from matplotlib import colormaps
from mc_sim import simulate_pmfs
from scipy.stats import betabinom

# set simulation parameters
n_steps = 1000

tau = 1
epsi_0 = 2
epsi_1 = 2
n_agents = 100

# obtain PMF history
history = simulate_pmfs(
    n_steps, tau=tau, epsi_0=epsi_0, epsi_1=epsi_1, n_agents=n_agents
)

# plot the figure
plt.figure(figsize=(3, 2))
plt.semilogy()
plt.xlabel(r"$A_k$")
plt.ylabel(r"$p(A_k)$")
plt.ylim([1e-7, 2e1])
for idx, h in enumerate(history):
    plt.plot(
        h[:, 0], h[:, 1], c=colormaps["coolwarm"](1 - (1 - idx / len(history)) ** 20)
    )
X = np.arange(0, n_agents + 1)
plt.plot(X, betabinom.pmf(X, n_agents, 2 * epsi_1, 2 * epsi_0), "k--")
plt.show()

The code above should produce the figure below.

Obtaining the scaling law

In [1] we have observed that for any polling period the stationary distribution of the model is well approximated by the Beta-binomial distribution. With the shape parameters of the distribution scaling according to a non-trivial law. ar2 module contains get_scaling_law function, which calculates the law $L(\tau)$ for the given model parameters. Recall that $\hat{\alpha}(\tau) = \varepsilon_1 L(\tau)$ and $\hat{\beta}(\tau) = \varepsilon_0 L(\tau)$ [1], so you'll have to multiply by corresponding parameter value to get the expected shape parameter values.

The code below generates two plots by relying on analytical results obtained with AR(2) approach. The left subplot shows the scaling law of the shape parameters, and the right subplot shows how the stationary variance scales with the polling period. Calculation of the stationary variance relies on a hidden function _get_variance.

import matplotlib.pyplot as plt
import numpy as np

from ar2 import get_scaling_law
from ar2.theory import _get_variance

taus = np.logspace(-6, -1, 301)

epsi_0 = 0.5
epsi_1 = 2.0
n_agents = 1000

law = get_scaling_law(taus, epsi_0=epsi_0, epsi_1=epsi_1, n_agents=n_agents)
var = _get_variance(taus, epsi_0=epsi_0, epsi_1=epsi_1, n_agents=n_agents)

plt.figure(figsize=(6, 2))
plt.subplot(121)
plt.xlabel(r"$\tau$")
plt.ylabel(r"$L(\tau)$")
plt.semilogx()
plt.minorticks_off()
plt.plot(taus, law)
plt.subplot(122)
plt.xlabel(r"$\tau$")
plt.ylabel(r"$\mathrm{Var}[A_\infty]$")
plt.semilogx()
plt.minorticks_off()
plt.plot(taus, var)
plt.tight_layout()
plt.show()

The code above should produce the figure below.

License

If you like the code presented here, you may use it for your purposes. Though, if you find the model interesting, referencing the paper would be highly appreciated.

References

1. A. Kononovicius, R. Astrauskas, M. Radavičius, F. Ivanauskas. Delayed interactions in the noisy voter model through the periodic polling mechanism. (under review). arXiv:2403.10277 [physics.soc-ph].