cross-entropy-for-combinatorics

Code accompanying the manuscript "Constructions in combinatorics via neural networks" by A Z Wagner.

The following code allows you to optimize any score function on the space of graphs with a fixed number of vertices.

Demo

The algorithm learns to generate graphs with low proximity + distance eigenvalue, as in Conjecture 2.3. in the paper.

Installation instructions

Clone this repository as follows:

git clone https://github.com/dpaleka/cross-entropy-for-combinatorics
cd cross-entropy-for-combinatorics

Install conda on your system, and run

conda env create --file environment.yml 
conda activate cross-entropy-for-combinatorics

How to use this for a custom score function?

Replace the score_graph method in score.py with the function you want to maximize.

For example, to minimize the absolute ratio of the first and the last eigenvalue of the adjacency matrix of a connected graph, put this in your score.py file:

def score_graph(adjMatG, edgeListG, Gdeg):
    """
    Reward function. The arguments adjMatG, edgeListG, Gdeg are numpy arrays.
    """
    N = Gdeg.size
    INF = 100000

    _, conn = bfs(Gdeg,edgeListG)
    if not conn:
        return -INF
        
    lambdas = np.flip(np.sort(np.linalg.eigvals(adjMatG)))
    return -abs(lambdas[0]/lambdas[-1])

Check the parameters at the top of optimize.py, in particular the number of vertices N. Then run

python optimize.py

Of course, this will just generate connected bipartite graphs after a few iterations.

Improving performance

If the generated graphs are not improving with regards to your score function, there are some straightforward directions to try:

1. Modify the parameters at the top of optimize.py. In particular, increasing the super_percentile parameter forces more exploration, and decreasing the LEARNING_RATE parameter can get you out of local optima.

2. Use a more powerful model to parametrize the graph generation. See You et al., 2018 (it has code for the Graph RNN) or the permutation-equivariant Li et al., 2018.

3. Change the loss function. There is a variety of policy gradient algorithms to choose from.