# cluster_optimal: Optimal community structure In igraph: Network Analysis and Visualization

## Description

This function calculates the optimal community structure of a graph, by maximizing the modularity measure over all possible partitions.

## Usage

 `1` ```cluster_optimal(graph, weights = NULL) ```

## Arguments

 `graph` The input graph. Edge directions are ignored for directed graphs. `weights` Optional positive weight vector for optimizing weighted modularity. If the graph has a `weight` edge attribute, then this is used by default. Supply `NA` to ignore the weights of a weighted graph. Larger edge weights correspond to stronger connections.

## Details

This function calculates the optimal community structure for a graph, in terms of maximal modularity score.

The calculation is done by transforming the modularity maximization into an integer programming problem, and then calling the GLPK library to solve that. Please the reference below for details.

Note that modularity optimization is an NP-complete problem, and all known algorithms for it have exponential time complexity. This means that you probably don't want to run this function on larger graphs. Graphs with up to fifty vertices should be fine, graphs with a couple of hundred vertices might be possible.

## Value

`cluster_optimal` returns a `communities` object, please see the `communities` manual page for details.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## Zachary's karate club g <- make_graph("Zachary") ## We put everything into a big 'try' block, in case ## igraph was compiled without GLPK support ## The calculation only takes a couple of seconds oc <- cluster_optimal(g) ## Double check the result print(modularity(oc)) print(modularity(g, membership(oc))) ## Compare to the greedy optimizer fc <- cluster_fast_greedy(g) print(modularity(fc)) ```

## Author(s)

Gabor Csardi csardi.gabor@gmail.com

## References

Ulrik Brandes, Daniel Delling, Marco Gaertler, Robert Gorke, Martin Hoefer, Zoran Nikoloski, Dorothea Wagner: On Modularity Clustering, IEEE Transactions on Knowledge and Data Engineering 20(2):172-188, 2008.

`communities` for the documentation of the result, `modularity`. See also `cluster_fast_greedy` for a fast greedy optimizer.