Continuous-time noisy threshold model (CNTM)

This page explains what the CNTM is and how it is implemented in this package. For the code reference see here.

About the CNTM

On a network (undirected simple graph) of \(N\) nodes, each node \(i\) has one of two opinions \(x_i \in \{0, 1\}\). At the rate \(r \geq 0\), each node evaluates to change their opinion from its current opinion \(m\in \{0, 1\}\) to the other opinion \(n=1-m\). It changes the opinion if the percentage of neighbors of opinion n exceeds the threshold \(b_{m,n}\). Additionally, each node changes its state randomly at rate \(\tilde{r} \geq 0\) (noise). Hence, the rate at which node \(i\) switches from opinion \(m\) to opinion \(n\) is

\[ r \ \delta_{\left( \frac{d_{i,n}(x)}{d_{i}} \geq b_{m,n} \right)} + \tilde{r} \]

where \(d_{i,n}(x)\) denotes the number of neighbors of node \(i\) with opinion \(n\) and \(d_i\) is the degree of node \(i\). Thus, in contrast to the CNVM, the CNTM assumes that a switch to a different opinion only occurs if that opinion is already sufficiently established in the neighborhood.

Basic Usage

First define the model parameters:

from sponet import CNTMParameters
import networkx as nx

num_nodes = 100
r = 1
r_tilde = 0.1
threshold_01 = 0.5
threshold_10 = 0.3
network = nx.erdos_renyi_graph(n=num_nodes, p=0.1)

params = CNTMParameters(
    network=network,
    r=r,
    r_tilde=r_tilde,
    threshold_01=threshold_01,
    threshold_10=threshold_10,
)

Then simulate the model, starting in state x_init:

from sponet import CNTM
import numpy as np

x_init = np.random.randint(0, 2, num_nodes)
model = CNTM(params)
t, x = model.simulate(t_max=50, x_init=x_init)

The output t contains the time points of state jumps and x the system states after each jump.