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)
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(nᆞlog2n) time, where n is the number of nodes in the graph.
Parameters:
adj_matrix
(numpy.ndarray): Adjacency matrix representation of your graphn
(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(m2n) 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, C1 and C2:
- Single-linkage: min({sim(i, j) | i ∊ C1, j ∊ C2})
- Complete-linkage: max({sim(i, j) | i ∊ C1, j ∊ C2})
- Mean-linkage: mean({sim(i, j) | i ∊ C1, j ∊ C2})
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, Ai and Aj.
This algorithm runs in O(n3) time, where n is the number of nodes in the graph.
Parameters:
adj_matrix
(numpy.ndarray): Adjacency matrix representation of your graphmetric
(str, optional (default="cosine")): Scheme for measuring node similarity; options are "cosine", for cosine similarity, or "euclidean", for inverse Euclidean distancelinkage
(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 v1, v2, ... vn as columns
- 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 graphk
(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 3n/3 maximal cliques, then this algorithm runs in O(3n/3) time (assuming 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 graphdark
(bool, optional (default=False)): If True
, the plot will have a dark background and color scheme, else it will have a light color schemefilename
(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 interactivelydpi
(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 graphframes
(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
dark
(bool, optional (default=False)): If True
, the animation will have a dark background and color scheme, else it will have a light color schemeduration
(int, optional (default=15)): The desired duration of the animation in secondsfilename
(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 plotdpi
(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
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 extractedcommunities
(list): List of communitiesaggr
(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 wik 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
- Aij is the weight of the edge between nodes i and j
- ki and kj are the sum of the weights of the edges attached to nodes 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
- Aij is the weight of the edge between nodes i and j
- ki and kj are the sum of the weights of the edges attached to nodes i and j, respectively
- m is the sum of all of the edge weights in the graph
- ci and cj are the communities of the nodes
- 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 graphcommunities
(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)