# Louvain Community

Louvain community detection seeks to maximize modularity which measures, in quantitative terms, how effectively the graph has been partitioned. It actually compares the paritioning to a random partitioning. Modularity is caculated as follows.

See [Louvain community on Wikipedia](https://en.wikipedia.org/wiki/Louvain_method) for more details on the algorithm.

$$
Q = \frac{1}{2m} \sum_{i,j} \left[ A_{ij} - \frac{k_i k_j}{2m} \right] \delta(c_i, c_j)
$$

$A_{ij}$ - Does an edge exist between nodes i and j. if the edge exists, $A_{ij} = 1$ if not, it is 0.

$\frac{k_i k_j}{2m}$ - the number of edges in the case of a random graph

$delta(c_i, c_j)$ - Are the two nodes in the same community? (0 for No. 1 for Yes)

To summarize, Louvain starts by putting each node in its own community, then iteratively, putting nodes into the same community and recalculating the modularity.

## When to use Louvain Community

* Louvain works well when communities are tightly connected internally but loosely connected between each other (community to community).
* Undirected graphs
* Unweighted graphs

Louvain works well in cases where there are natural communities. For example, social networks, product co-purchasing and biological networks.


## When not to use Louvain Community
In addition to the inverse of the helpful characteristics listed above, several other graph-level attributes can hinder Louvain.

* Graphs with many small communities which are often merged
* Overlapping communities, since nodes will only exist in a single cluster
* Sparse graphs since low modularity can make the algorithm results useless.

## How computationally expensive is it?
While cuGraph's parallelism mitigates run time, [Big O notation](https://en.wikipedia.org/wiki/Big_O_notation) is still the standard to compare algorithm costs.


Where:

* $m$ - number of edges

The cost is $O(m)$

Louvain is especially efficient with GPU-acceleration because it lends itself to parallelization.

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