Compute discrete Ricci curvatures and Ricci flow on NetworkX graphs.

# GraphRicciCurvature

Compute Discrete Ricci curvature and Ricci flow on NetworkX graph.

This work computes the Ollivier-Ricci Curvature[Ni], Ollivier-Ricci Flow[Ni2,Ni3] and Forman-Ricci Curvature(or Forman curvature)[Sreejith]. Curvature is a geometric property to describe the local shape of an object. If we draw two parallel paths on a surface with positive curvature like a sphere, these two paths move closer to each other while for a negative curved surface like saddle, these two paths tend to be apart.

In [Ni], we observe that the edge Ricci curvature play an important role in graph structure. An edge with positive curvature represents an edge within a cluster, while a negatively curved edge tents to be a bridge within clusters. Also, negatively curved edges are highly related to graph connectivity, with negatively curved edges removed from a connected graph, the graph soon become disconnected.

Ricci flow is a process to uniformized the edge Ricci curvature of the graph. For a given graph, the Ricci flow gives a "Ricci flow metric" on each edge as edge weights, such that under these edge weights, the Ricci curvature of the graph is mostly equal everywhere. In [Ni3], this "Ricci flow metric" is shown to be able to detect communities.

Both Ricci curvature and Ricci flow metric can be act as a graph fingerprint. Different graph gives different edge Ricci curvature distributions and different Ricci flow metric.

Video demonstration of Ricci flow for community detection:

## Package Requirement

• NetworkX (Based Graph library)

• CVXPY (LP solver for Optimal transportation)

• NumPy (CVXPY support)

• POT (For approximate Optimal transportation distance)

• NetworKit (Optional: for faster parallel shortest path computation)

## Installation

### Installing via pip

```pip3 install [--user] GraphRicciCurvature
```

### Installing via pip (with NetworKit)

```pip3 install [--user] "GraphRicciCurvature [faster_apsp]"
```
• Notice that the NetworKit is not required. It is optional for faster all pair shortest path computation for larger graphs that NetworkX performs poorly. If the installation failed, please refer to NetworKit' Installation instructions. In most of the cast build this package from source is recommended.

## Getting Started

• See the jupyter notebook tutorial on nbviewer or github for a walk through for the basic usage of Ricci curvature, Ricci flow, and Ricci flow for community detection.
• Or you can run it in directly on binder (no account required) or Google colab (Faster but Google account required).

## Simple Example

```import networkx as nx
from GraphRicciCurvature.OllivierRicci import OllivierRicci
from GraphRicciCurvature.FormanRicci import FormanRicci

# import an example NetworkX karate club graph
G = nx.karate_club_graph()

# compute the Ollivier-Ricci curvature of the given graph G
orc = OllivierRicci(G, alpha=0.5, verbose="INFO")
orc.compute_ricci_curvature()
print("Karate Club Graph: The Ollivier-Ricci curvature of edge (0,1) is %f" % orc.G["ricciCurvature"])

# compute the Forman-Ricci curvature of the given graph G
frc = FormanRicci(G)
frc.compute_ricci_curvature()
print("Karate Club Graph: The Forman-Ricci curvature of edge (0,1) is %f" % frc.G["formanCurvature"])

# -----------------------------------
# Compute Ricci flow metric - Optimal Transportation Distance
G = nx.karate_club_graph()
orc_OTD = OllivierRicci(G, alpha=0.5, method="OTD", verbose="INFO")
orc_OTD.compute_ricci_flow(iterations=10)
```

More example in example.py.

## Reference

[Ni]: Ni, C.-C., Lin, Y.-Y., Gao, J., Gu, X., and Saucan, E. 2015. "Ricci curvature of the Internet topology" (Vol. 26, pp. 2758–2766). Presented at the 2015 IEEE Conference on Computer Communications (INFOCOM), IEEE. arXiv

[Ni2]: Ni, C.-C., Lin, Y.-Y., Gao, J., and Gu, X. 2018. "Network Alignment by Discrete Ollivier-Ricci Flow", Graph Drawing 2018, arXiv

[Ni3]: Ni, C.-C., Lin, Y.-Y., Luo, F. and Gao, J. 2019. "Community Detection on Networks with Ricci Flow", Scientific Reports, arXiv

[Sreejith]: Sreejith, R. P., Karthikeyan Mohanraj, Jürgen Jost, Emil Saucan, and Areejit Samal. 2016. “Forman Curvature for Complex Networks.” Journal of Statistical Mechanics: Theory and Experiment 2016 (6). IOP Publishing: 063206. arxiv

## Project details

This version 0.3.1 0.3 0.2.1.1 0.2.1 0.2 0.1