Library for detecting community structure in graphs

## Project description

# communities

`communities`

is a Python library for detecting community structure in graphs. It implements the following algorithms:

- Louvain method
- Girvan-Newman algorithm
- Hierarchical clustering
- Spectral clustering
- Bron-Kerbosch algorithm

You can also use `communities`

to visualize these algorithms. For example, here's a visualization of the Louvain method applied to the karate club graph:

## Installation

`communities`

can be installed with `pip`

:

```
$ pip install communities
```

## Getting Started

Each algorithm expects an adjacency matrix representing an undirected graph, which can be weighted or unweighted. This matrix should be a 2D `numpy`

array. Once you have this, simply import the algorithm you want to use from `communities.algorithms`

and plug in the matrix, like so:

```
import numpy as np
from communities.algorithms import louvain_method
adj_matrix = np.array([[0, 1, 1, 0, 0, 0],
[1, 0, 1, 0, 0, 0],
[1, 1, 0, 1, 0, 0],
[0, 0, 1, 0, 1, 1],
[0, 0, 0, 1, 0, 1],
[0, 0, 0, 1, 1, 0]])
communities, _ = louvain_method(adj_matrix)
# >>> [{0, 1, 2}, {3, 4, 5}]
```

The output of each algorithm is a list of communities, where each community is a set of nodes. Each node is referred to by the index of its row in the adjacency matrix.

Some algorithms, like `louvain_method`

and `girvan_newman`

, will return two values: the list of communities and data to plug into a visualization algorithm. More on this in the *Visualization* section.

## Algorithms

### Louvain's Method

`louvain_method(adj_matrix : numpy.ndarray, n : int = None) -> list`

Implementation of the Louvain method, from *Fast unfolding of communities in large networks*. This algorithm does a greedy search for the communities that maximize the modularity of the graph. A graph is said to be modular if it has a high density of intra-community edges and a low density of inter-community edges.

Louvain's method runs in *O(nᆞlog ^{2}n)* time, where

*n*is the number of nodes in the graph.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph`n`

*(int or None, optional (default=None))*: Terminates the search once this number of communities is detected; if`None`

, then the algorithm will behave normally and terminate once modularity is maximized

**Example Usage:**

```
from communities.algorithms import louvain_method
adj_matrix = [...]
communities, _ = louvain_method(adj_matrix)
```

### Girvan-Newman algorithm

`girvan_newman(adj_matrix : numpy.ndarray, n : int = None) -> list`

Implementation of the Girvan-Newman algorithm, from *Community structure in social and biological networks*. This algorithm iteratively removes edges to create more connected components. Each component is considered a community, and the algorithm stops removing edges when no more gains in modularity can be made. Edges with the highest betweenness centralities (i.e. those that lie between many pairs of nodes) are removed. Formally, edge betweenness centrality is defined as:

where

*σ(i,j)*is the number of shortest paths from node*i*to*j**σ(i,j|e)*is the number of shortest paths that pass through edge*e*

The Girvan-Newman algorithm runs in *O(m ^{2}n)* time, where

*m*is the number of edges in the graph and

*n*is the number of nodes.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph- If your graph is weighted, then the weights need to be transformed into distances, since that's how they'll be interpreted when searching for shortest paths. One way to do this is to simply take the inverse of each weight.

`n`

*(int or None, optional (default=None))*: Terminates the search once this number of communities is detected; if`None`

, then the algorithm will behave normally and terminate once modularity is maximized

**Example Usage:**

```
from communities.algorithms import girvan_newman
adj_matrix = [...]
communities, _ = girvan_newman(adj_matrix)
```

### Hierarchical clustering

`hierarchical_clustering(adj_matrix : numpy.ndarray, metric : str = "cosine", linkage : str = "single", n : int = None) -> list`

Implementation of a bottom-up, hierarchical clustering algorithm. Each node starts in its own community. Then, the most similar pairs of communities are merged as the hierarchy is built up. Communities are merged until no further gains in modularity can be made.

There are multiple schemes for measuring the similarity between two communities, *C _{1}* and

*C*:

_{2}**Single-linkage:**min({sim(i, j) | i ∊ C_{1}, j ∊ C_{2}})**Complete-linkage:**max({sim(i, j) | i ∊ C_{1}, j ∊ C_{2}})**Mean-linkage:**mean({sim(i, j) | i ∊ C_{1}, j ∊ C_{2}})

where *sim(i, j)* is the similarity between nodes *i* and *j*, defined as either the cosine similarity or inverse Euclidean distance between their row vectors in the adjacency matrix, *A _{i}* and

*A*.

_{j}This algorithm runs in *O(n ^{3})* time, where

*n*is the number of nodes in the graph.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph`metric`

*(str, optional (default="cosine"))*: Scheme for measuring node similarity; options are "cosine", for cosine similarity, or "euclidean", for inverse Euclidean distance`linkage`

*(str, optional (default="single"))*: Scheme for measuring community similarity; options are "single", "complete", and "mean"`n`

*(int or None, optional (default=None))*: Terminates the search once this number of communities is detected; if`None`

, then the algorithm will behave normally and terminate once modularity is maximized

**Example Usage:**

```
from communities.algorithms import hierarchical_clustering
adj_matrix = [...]
communities = hierarchical_clustering(adj_matrix, metric="euclidean", linkage="complete")
```

### Spectral clustering

`spectral_clustering(adj_matrix : numpy.ndarray, k : int) -> list`

Implementation of a spectral clustering algorithm. This type of algorithm assumes the eigenvalues of the adjacency matrix hold information about community structure. Here's how it works:

- Compute the Laplacian matrix,
*L = D - A*, where*A*is the adjacency matrix and*D*is the diagonal matrix - Compute the
*k*smallest eigenvectors of*L*, skipping the first eigenvector - Create a matrix
*V*containing eigenvectors*v*,_{1}*v*, ..._{2}*v*as columns_{n} - Cluster the rows in
*V*using k-means into*k*communities

This algorithm is NP-hard.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph`k`

*(int)*: Number of communities to cluster nodes into

**Example Usage:**

```
from communities.algorithms import spectral_clustering
adj_matrix = [...]
communities = spectral_clustering(adj_matrix, k=5)
```

### Bron-Kerbosch algorithm

`bron_kerbosch(adj_matrix : numpy.ndarray, pivot : bool = False) -> list`

Implementation of the Bron-Kerbosch algorithm for maximal clique detection. A maximal clique in a graph is a subset of nodes that forms a complete graph and would no longer be complete if any other node was added to the subset. Treating maximal cliques as communities is reasonable, as cliques are the most densely connected groups of nodes in a graph. Because a node can be a member of more than one clique, this algorithm will sometimes identify overlapping communities.

If your input graph has less than *3 ^{n/3}* maximal cliques, then this algorithm runs in

*O(3*time (assuming

^{n/3})`pivot=True`

).**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph- Note that this algorithm treats the graph as unweighted

`pivot`

*(bool, optional (default=False))*: If`True`

, the pivot variant of the algorithm (described here) will be used- This will make the algorithm more efficient if your graph has several non-maximal cliques

**Example Usage:**

```
from communities.algorithms import bron_kerbosch
adj_matrix = [...]
communities = bron_kerbosch(adj_matrix, pivot=True)
```

## Visualization

### Plot communities

`draw_communities(adj_matrix : numpy.ndarray, communities : list, dark : bool = False, filename : str = None, seed : int = 1)`

Visualize your graph such that nodes are grouped into their communities and color-coded.

Returns a `matplotlib.axes.Axes`

representing the plot.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph`dark`

*(bool, optional (default=False))*: If`True`

, the plot will have a dark background and color scheme, else it will have a light color scheme`filename`

*(str or None, optional (default=None))*: If you want to save the plot as a PNG,`filename`

is the path of the file to save it as; set to`None`

to display the plot interactively`dpi`

*(int or None, optional (default=None))*: Dots per inch (controls the resolution of the image)`seed`

*(int, optional (default=2))*: Random seed

**Example Usage:**

```
from communities.algorithms import louvain_method
from communities.visualization import draw_communities
adj_matrix = [...]
communities, frames = louvain_method(adj_matrix)
draw_communities(adj_matrix, communities)
```

### Animate the Louvain method

`louvain_animation(adj_matrix : numpy.ndarray, frames : list, dark : bool = False, duration : int = 15, filename : str = None, dpi : int = None, seed : int = 2)`

Use this to animate the application of the Louvain method to your graph. In this animation, the color of each node represents the community it's assigned to, and nodes in the same community are clustered together. Each step of the animation will show a node changing color (i.e. being assigned to a different community) and being moved to a new cluster, and the corresponding update to the graph's modularity.

This function returns a `matplotlib.animation.FuncAnimation`

object representing the animation.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph`frames`

*(list)*: List of dictionaries representing each iteration of the algorithm- Each dictionary has two keys:
`"C"`

, which holds a node-to-community lookup table, and`"Q"`

, the modularity value of the graph - This list of dictionaries is the second return value of the
`louvain_method`

- Each dictionary has two keys:
`dark`

*(bool, optional (default=False))*: If`True`

, the animation will have a dark background and color scheme, else it will have a light color scheme`duration`

*(int, optional (default=15))*: The desired duration of the animation in seconds`filename`

*(str or None, optional (default=None))*: If you want to save the animation as a GIF,`filename`

is the path of the file to save it as; set to`None`

to display the animation as an interactive plot`dpi`

*(int or None, optional (default=None))*: Dots per inch (controls the resolution of the animation)`seed`

*(int, optional (default=2))*: Random seed

**Example Usage:**

```
from communities.algorithms import louvain_method
from communities.visualization import louvain_animation
adj_matrix = [...]
communities, frames = louvain_method(adj_matrix)
louvain_animation(adj_matrix, frames)
```

## Utilities

### Inter-community adjacency matrix

`intercommunity_matrix(adj_matrix : numpy.ndarray, communities : list, aggr : Callable = sum) -> numpy.ndarray`

Creates an inter-community adjacency matrix. Each node in this matrix represents a community in `communities`

, and each edge between nodes *i* and *j* is created by aggregating (e.g. summing) the weights of edges between nodes in `communities[i]`

and nodes in `communities[j]`

.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of the graph from which communities were extracted`communities`

*(list)*: List of communities`aggr`

*(Callable, optional (default=sum))*: Function that takes a list of inter-community edge weights and combines them into a single edge weight

**Example Usage:**

```
from statistics import mean
from communities.algorithms import louvain_method
from communities.utilities import intercommunity_matrix
adj_matrix = [...]
communities = louvain_method(adj_matrix)
intercomm_adj_matrix = intercommunity_matrix(adj_matrix, communities, mean)
```

### Graph Laplacian

`laplacian_matrix(adj_matrix : numpy.ndarray) -> numpy.ndarray`

Computes the graph Laplacian. This matrix is used in the `spectral_clustering`

algorithm, and is generally useful for revealing properties of a graph. It is defined as *L = D - A*, where *A* is the adjacency matrix of the graph, and *D* is the degree matrix, defined as:

where *w _{ik}* is the edge weight between a node

*i*and its neighbor

*k*.

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph

**Example Usage:**

```
from communities.utilities import laplacian_matrix
adj_matrix = [...]
L = laplacian_matrix(adj_matrix)
```

### Modularity matrix

`modularity_matrix(adj_matrix : numpy.ndarray) -> numpy.ndarray`

Computes the modularity matrix for a graph. The modularity matrix is defined as:

where

*A*is the weight of the edge between nodes_{ij}*i*and*j**k*and_{i}*k*are the sum of the weights of the edges attached to nodes_{j}*i*and*j*, respectively*m*is the sum of all of the edge weights in the graph

**Parameters:**

`adj_matrix`

*(numpy.ndarray)*: Adjacency matrix representation of your graph

### Modularity

`modularity(mod_matrix : numpy.ndarray, communities : list) -> float`

Computes modularity of a partitioned graph. Modularity is defined as:

where

*A*is the weight of the edge between nodes_{ij}*i*and*j**k*and_{i}*k*are the sum of the weights of the edges attached to nodes_{j}*i*and*j*, respectively*m*is the sum of all of the edge weights in the graph*c*and_{i}*c*are the communities of the nodes_{j}*δ*is the Kronecker delta function (*δ(x, y) = 1*if*x = y*,*0*otherwise)

**Parameters:**

`mod_matrix`

*(numpy.ndarray)*: Modularity matrix computed from the adjacency matrix representation of your graph`communities`

*(list)*: List of (non-overlapping) communities identified in the graph

**Example Usage:**

```
from communities.algorithms import louvain_method
from communities.utilities import modularity_matrix, modularity
adj_matrix = [...]
communities = louvain_method(adj_matrix)
mod_matrix = modularity_matrix(adj_matrix)
Q = modularity(mod_matrix, communities)
```

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