AI-based library to work with extremely large graphs.
Supporting: Cayley graphs, Schreier coset graphs, more to be added.

Extremely large graphs (e.g. googol size, ~10¹⁰⁰ nodes) cannot be approached in a usual way, it is impossible neither to create, neither to store them by standard methods.

Typically such graphs arise as state-transition graphs. For chess, Go or any other games - nodes of the graphs are positions, edges correspond to moves between them. For Rubik's cube - nodes are configurations, edges corresponds to configurations different by single moves.

The most simple and clear examples of such graphs - are Cayley graphs in mathematics. (and Schreier coset graphs ). Initial developments will focus on these graphs, supporting other types later.

We plan to support:

• ML/RL methods for pathfinding
• Estimation of diameters and growths
• Embeddings
• Efficient BFS for small subgraphs
• Efficient random walks generation
• Efficient Beam Search
• Hamiltonian paths finding
• Efficient computing on CPU, GPU, TPU (with JAX), usable on Kaggle.
• Etc.

Mathematical applications:

• Estimation of diameters and growths
• Approximation of the word metrics and diffusion distance
• Estimation of the mixing time for random walks of different types
• BFS from given state (growth function, adjacency matrix, last layers).
• Library of graphs and generators (LRX, TopSpin, Rubik Cubes, wreath, globe etc., see here).
• Library of datasets with solutions to some problems (e.g. growth functions like here).

See the following Kaggle notebooks for examples of library usage:

• Basic usage - defining Cayley graphs for permutation and matrix groups, running BFS, getting explicit Networkx graphs.
• Computing spectra.
• Library of puzzles in GAP format in CayleyPy.
• Path finding in Cayley Graphs:
  • Beam search with CayleyPy - simple example of finding paths for LRX (n=12) using beam search and neural network.
  • Finding shortest paths for LRX (n=8) using BFS.
  • Finding shortest paths for LRX cosets (n=16 and n=32) using BFS.
  • Beam search with neural network for LRX cosets (n=32).
  • Beam search for LRX, n=16.
  • Beam search for LRX, n=32
• Growth function computations:
  • For LX.
  • For TopSpin cosets.
• Benchmarks:
  • Benchmarks versions of BFS in CayleyPy.
  • Benchmark BFS on GPU.

We recommend installing the latest version from GitHub:

pip install git+https://github.com/cayleypy/cayleypy

You may also install using pip, although this might be missing recently added features:

pip install cayleypy

Documentation (API reference) for the latest version of the library is available here.

To start development, run:

git clone https://github.com/cayleypy/cayleypy.git
cd cayleypy
pip install -e .[torch,lint,test,dev,docs]

To run all tests, including some slow running tests:

RUN_SLOW_TESTS=1 pytest

Before committing, run these checks:

./lint.sh
pytest 

To check coverage, run:

coverage run -m pytest && coverage html

To rebuild documentation locally, run:

./docs/build_docs.sh 

This repository uses the Black formatter. If you are getting error saying that some files "would be reformatted", you need to format your code using Black. There are few convenient ways to do that:

• From command line: run black .
• In PyCharm: go to Setting>Tools>Black, and check "Use Black formatter": "On code reformat" (then it will run on Ctrl+Alt+L), or "On save", or both.
• In Visual Studio code: install the Black Formatter extension, then use Ctrl+Shift+I to format code. If you are asked to configure default formatter, pick the Black formatter.

• In general, this repository follows Google Python Style Guide. All contributors should read it.
• When writing comments, use punctuation. In particular, always put a period (".") in the end of sentences.
• We have pylint checks to enforce some style rules. You should fix pylint warnings instead of disabling the check.

Cayley graphs must be defined by a function that returns CayleyGraphDef. First, you need to decide where in the library to put it:

• If it's a graph generated by permutations, the function should be added to
  PermutationGroups in cayleypy/graphs_lib.py, annotated as @staticmethod.
• If it's a graph generated by matrices, the function should be added to
  MatrixGroups in cayleypy/graphs_lib.py.
• If it's a graph for a physical puzzle, the function should be added to Puzzles in caylepy/puzzles/puzzles.py. If it requires non-trivial construction, move that to separate function(s) and put them in separate file in cayleypy/puzzles. If the puzzle is defined by hardcoded permutations, put them in cayleypy/puzzles/moves.py.
• If it's a graph for a puzzle, and you have definition in GAP format, put the .gap file in puzzles/gap_files/default. It will become available via cayleypy.GapPuzzles.
• If it's a new type of graph, check with @fedimser where to put it.

Do not add new graphs to prepare_graph! We want new graphs to be added in different places to avoid merge conflicts.

Then, you need to define your graph. Definition consists of the following:

• Generators.
• Generator names (optional).
• Central state (optional, defaults to neutral element in the group, e.g. identity permutation).

When you are ready do the following:

1. Create a fork of this repository.
2. Clone your fork and make a new bracnch in your fork.
3. Add your function to an approritate place. See how other graphs are defined and follow that as an example.
4. Write a docstring for your function, describing your graph. If possible, include reference (e.g. to Wikipedia article, Arxiv paper or a book) where the graph is defined.
5. Add a test that creates an instance of your graph for small size and checks something about it (at least check number of generators).
6. Create a pull request from your fork to the this repository.
7. Make sure that all automated checks (like unittests and style checks) are passed.
8. Get two approvals from the reviewers team. They might make comments or request changes before provding an approval -- implement those. If needed, discuss how to proceed. If a commit is pushed after the approval, the approval has to be obtained again.
9. Once approvals are granted -- merge the PR.

For more details on how to make PRs with forks refer to this guide.

CayleyPy contains a library of machine learning models to be used as predictors in the beam search algorithm for finding paths in Cayley graph. These models can be easily accessed using Predictor.pretrained (example).

Each such model is a PyTorch neural network which consists of 3 parts:

• Model architecture description (a subclass of nn.Models) - defined in cayleypy/models.py.
• Model architecture hyperparameters (such as input size or sizes of hidden layers) - defined by models.ModelConfig.
• Model weights - these are stored on Kaggle.

List of currently available models is here.

1. Train your model.
2. Verify that when used with beam search, it reliably finds the paths.
3. Export weights to a file (using torch.save(model.state_dict(), path).
4. Upload weights as model on Kaggle, make it public and use open source license (MIT license is recommended).
5. Make sure the graph for which your model should be used has unique name (that is, CayleyGraphDef.name). For example, PermutationGroups.lrx(16) has name "lrx-16". Also prepare_graph given this name should return this graph (this is needed for tests).
6. Define ModelConfig for your model:
   • weights_kaggle_id is identifier of your saved model on Kaggle. This is what you would pass to kagglehub.model_download.
   • weights_path is the name of file with weights.
   • If your can be exactly described by one of available model types in models/models.py, use that model type with appropriate hyperparameters. If needed, add new hyperparameters to ModelConfig.
   • If your model architecture is very different from we already have in library, define new model type for it.
   • For example, we already have model type "MLP" (multi-layer perceptron) defined by MlpModel with the following parameters: input_size, num_classes_for_one_hot, layers_sizes.
7. Verify that when you define your model config, call load on it and then use that as predictor in beam search, it works.
8. Add your model to PREDICTOR_MODELS in models_lib. Use graph name as a key.
9. Run pytest cayleypy/models/models_lib_test.py. This will check that your model can be loaded from Kaggle and used for inference (i.e. has correct input and output shape), but it doesn't check quality of your model.
10. Optionally, add a test that beam search with your model successfully finds a path.

Our community has recently launched several Kaggle competitions to develop and benchmark our methods and make it easier for the larger audience to get involved. Most of those competitions don't require the usage of this library, however it might be handy as those tasks are the ones

The idea of the project - Alexander Chervov - see https://arxiv.org/abs/2502.18663, https://arxiv.org/abs/2502.13266, discussion group https://t.me/sberlogasci/1, Early ideas and prototypes appeared during Kaggle challenge Santa 2023: Prototype: https://www.kaggle.com/code/alexandervc/santa23-globe26-modeling5, Description: https://www.kaggle.com/competitions/santa-2023/discussion/466399, https://www.kaggle.com/competitions/santa-2023/discussion/472594.

The initial code developments can be found at Kaggle dataset: https://www.kaggle.com/datasets/alexandervc/growth-in-finite-groups (see paper https://arxiv.org/abs/2502.13266 ) Other developments can be found at: https://www.kaggle.com/competitions/lrx-oeis-a-186783-brainstorm-math-conjecture/code, https://www.kaggle.com/datasets/alexandervc/cayleypy-development-3-growth-computations, see also beam-search part: Cayleypy (Ivan Koltsov) , Rubik's cube part: Piligrim (Kirill Khoruzhii).

Also, code from the following Kaggle notebooks was used:

• https://www.kaggle.com/code/ivankolt/generation-of-incidence-mtx-pancake (advanced BFS).
• https://www.kaggle.com/code/avm888/cayleypy-growth-function.
• https://www.kaggle.com/code/avm888/jax-version-cayleypy (how to use JAX).
• https://www.kaggle.com/code/fedimser/bfs-for-binary-string-permutations (bit operations).
• https://www.kaggle.com/code/ivankolt/lrx-4bit-uint64?scriptVersionId=221435319 (fast BFS)