Model

Initial conditions

Network generation

The initial social network, network, \(\mathcal{G}\), is constructed according to standard network construction algorithms the user can select from.

Available network constructors

• complete: A graph in which every node is connected to every other node.
• erdos_renyi: An Erdős-Rényi random graph.
• watts_strogatz: A Watts–Strogatz small-world graph.
• newman_watts_strogatz: A Newman–Watts–Strogatz small-world graph.
• barabasi_albert: A random graph generated by Barabási–Albert preferential attachment.

Attitude specification

Each node in \(\mathcal{G}\) is assigned an initial attitude, \(\theta_i\), sampled from a specified distribution and are constrained to the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Available attitude distributions

• normal: A Gaussian normal distribution.
• uniform: A uniform distribution.
• laplace: A Laplace distribution.
• vonmises: A von Mises distribution.

Dynamics

The time dynamics of the network's edges and node attitudes are governed by:

\[
A_{ij}'(t) = T_{ij}(t) \sin(\Theta_{ij}(t))
\]

\[
\theta_i'(t) = \beta \sum_j T_{ij}(t) A_{ij}^\alpha(t) \sin(\Theta_{ij}(t))
\]

where \(\alpha\) and \(\beta\) are scaling parameters and \(\mathbf{T}(t)\) is an interaction matrix constructing by generating a random matrix, \(\mathbf{M}(t)\), with elements in \([0, 1)\) and comparing it to the network's adjacency matrix, \(\mathbf{A}(t)\). Where \(M_{ij}(t) < A_{ij}(t)\), \(T_{ij}(t) = 1\) and nodes \(i\) and \(j\) interact. Otherwise \(T_{ij} = 0\) and nodes \(i\) and \(j\) don't interact at that time. where \(\Theta_{ij}(t)\) are the elements of the matrix of differences in node attitudes:

\[
\mathbf{\Theta}_{ij}(t) = \mathbf{\theta_i}(t) - \mathbf{\theta_j}(t)
\]

This produces an adaptive complex network where edge strengths and node attitudes are reinforced according to each other over time.

When \(\mathbf{A}(t)\) is symmetric, this is a modified Kuramoto model (Kuramoto, 1975), with the modifications:

1. The mean over all phases is replaced by the simple sum.
2. The coupling constant is the adjacency matrix of the network, which may be weighted.
3. The coupling constant has an exponent.

When \(\mathbf{A}(t)\) isn't symmetric, this is a circle map (Arnol'd, 1961; Ott, 2002).