Testing Symmetry of Two Networks with the Density Test

The “inference” module of graspologic contains functions that enable quantitative comparison of two networks to assess whether they are statistically similar. This “similarity” can be assessed in a few different ways, depending on the details of the networks to be compared and the preferences of the user.

The simplest test that can be performed is the density test, which is based upon the Erdos-Renyi model. Under this model, it is assumed that the probability of an edge between any two nodes of the network is some constant, p. To compare two networks, then, the question is whether the edge probability for the first network is different from the edge probability for the second network. This test can be performed easily with the inference module, and the procedure is described in greater detail below.

The Erdos-Renyi (ER) model

The Erdos-Renyi (ER) model is one of the simplest network models. This model treats the probability of each potential edge in the network occuring to be the same. In other words, all edges between any two nodes are equally likely.

```{admonition} Math Let \(n\) be the number of nodes. We say that for all \((i, j), i \neq j\), with \(i\) and \(j\) both running from \(1 ... n\), the probability of the edge \((i, j)\) occuring is:

\[P[A_{ij} = 1] = p_{ij} = p\]

Where \(p\) is the the global connection probability.

Each element of the adjacency matrix \(A\) is then sampled independently according to a Bernoulli distribution:

\[A_{ij} \sim Bernoulli(p)\]

For a network modeled as described above, we say it is distributed

\[A \sim ER(n, p)\]

```

Thus, for this model, the only parameter of interest is the global connection probability, \(p\). This is sometimes also referred to as the network density.

Testing under the ER model

In order to compare two networks \(A^{(L)}\) and \(A^{(R)}\) under this model, we simply need to compute these network densities (\(p^{(L)}\) and \(p^{(R)}\)), and then run a statistical test to see if these densities are significantly different.

```{admonition} Math Under this model, the total number of edges \(m\) comes from a \(Binomial(n(n-1), p)\) distribution, where \(n\) is the number of nodes. This is because the number of edges is the sum of independent Bernoulli trials with the same probability. If \(m^{(L)}\) is the number of edges on the left hemisphere, and \(m^{(R)}\) is the number of edges on the right, then we have:

\[m^{(L)} \sim Binomial(n^{(L)}(n^{(L)} - 1), p^{(L)})\]

and independently,

\[m^{(R)} \sim Binomial(n^{(R)}(n^{(R)} - 1), p^{(R)})\]

To compare the two networks, we are just interested in a comparison of \(p^{(L)}\) vs. \(p^{(R)}\). Formally, we are testing:

\[H_0: p^{(L)} = p^{(R)}, \quad H_a: p^{(L)} \neq p^{(R)}\]

Fortunately, the problem of testing for equal proportions is well studied. Using graspologic.inference, we can conduct this comparison using either Fisher’s exact test or the chi-squared test by using method=”fisher” or method = “chi2”, respectively. In this example, we use Fisher’s exact test. ```

import numpy as np
import matplotlib.pyplot as plt

from graspologic.inference.density_test import density_test
from graspologic.simulations import er_np
from graspologic.plot import heatmap

np.random.seed(8888)

%matplotlib inline

/opt/hostedtoolcache/Python/3.8.18/x64/lib/python3.8/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
  from .autonotebook import tqdm as notebook_tqdm
/home/runner/work/graspologic/graspologic/graspologic/models/edge_swaps.py:215: NumbaDeprecationWarning: The keyword argument 'nopython=False' was supplied. From Numba 0.59.0 the default is being changed to True and use of 'nopython=False' will raise a warning as the argument will have no effect. See https://numba.readthedocs.io/en/stable/reference/deprecation.html#deprecation-of-object-mode-fall-back-behaviour-when-using-jit for details.
  _edge_swap_numba = nb.jit(_edge_swap, nopython=False)

Performing the Density Test

To illustrate the density test, we will first randomly generate two networks of known density to compare using the test.

A1 = er_np(500, 0.6)
A2 = er_np(400, 0.8)
heatmap(A1, title='Adjacency Matrix for Network 1')
heatmap(A2, title='Adjacency Matrix for Network 2')

<Axes: title={'center': 'Adjacency Matrix for Network 2'}>

Visibly, these networks have very different densities. We can statistically confirm this difference by conducting a density test.

stat, pvalue, er_misc = density_test(A1,A2)
print(pvalue)

0.0

/home/runner/work/graspologic/graspologic/graspologic/inference/group_connection_test.py:362: UserWarning: This test assumes that the networks are directed, but one or both adjacency matrices are symmetric.
  warnings.warn(msg)