edge.betweenness.community {igraph} R Documentation

## Community structure detection based on edge betweenness

### Description

Many networks consist of modules which are densely connected themselves but sparsely connected to other modules.

### Usage

```edge.betweenness.community (graph, directed = TRUE,
edge.betweenness = TRUE, merges = TRUE, bridges = TRUE,
labels = TRUE)
edge.betweenness.community.merges (graph, edges)
```

### Arguments

 `graph` The graph to analyze. `directed` Logical constant, whether to calculate directed edge betweenness for directed graphs. It is ignored for undirected graphs. `edge.betweenness` Logical constant, whether to return the edge betweenness of the edges at the time of their removal. `merges` Logical constant, whether to return the merge matrix representing the hierarchical community structure of the network. This argument is called `merges`, even if the community structure algorithm itself is divisive and not agglomerative: it builds the tree from top to bottom. There is one line for each merge (ie. split) in matrix, the first line is the first merge (last split). The communities are identified by integer number starting from zero. Community ids smaller than ‘`N`’, the number of vertices in the graph, belong to singleton communities, ie. individual vertices. Before the first merge we have ‘`N`’ communities numbered from zero to ‘`N-1`’. The first merge, the first line of the matrix creates community ‘`N`’, the second merge creates community ‘`N+1`’, etc. `bridges` Logical constant, whether to return a list the edge removals which actually splitted a component of the graph. `labels` Logical constant, whether to contain the labels of the vertices in the result. More precisely, if the graph has a vertex attribute valled ‘name’, it will be part of the result object. `edges` Numeric vector, the ids of the edges to be removed from a graph, all edges should be present in the vector, their order specifies the order of removal.

### Details

The edge betweenness score of an edge measures the number of shortest paths through it, see `edge.betweenness` for details. The idea of the edge betweenness based community structure detection is that it is likely that edges connecting separate modules have high edge betweenness as all the shortest paths from one module to another must traverse through them. So if we gradually remove the edge with the highest edge betweenness score we will get a hierarchical map, a rooted tree, called a dendrogram of the graph. The leafs of the tree are the individual vertices and the root of the tree represents the whole graph.

`edge.betweenness.community` performs this algorithm by calculating the edge betweenness of the graph, removing the edge with the highest edge betweenness score, then recalculating edge betweenness of the edges and again removing the one with the highest score, etc.

`edge.betweeness.community` returns various information collected throught the run of the algorithm. See the return value down here.

`edge.betweenness.community.merges` gets a list of edges and by gradually removes them from the graph it creates a merge matrix similar to the one returned by `edge.betweenness.community`.

### Value

A named list is returned by `edge.betweenness.community`, with the following components:

 `removed.edges` Numeric vector, the edges of the graph, in the order of their removal. `edge.betweenness` Numeric vector, the edge betweenness value of the removed edges, the order is the same as in `removed.edges`. `merges` Matrix containing the merges (ie. divisions) the algorithm performed, see the `merges` argument for the format. `bridges` Numeric vector, the steps (ie. edge removals) which resulted a split of a component in the graph. `labels` The `name` argument of the vertices.

normal-bracket57bracket-normal Note that some components may be missing or `NULL` if you do not request them, see the parameters.
A numeric matrix is returned by `edge.betweenness.community.merges`. The matrix has two column and its format is the same as the `merges` slot of the result of `edge.betweenness.community`.

### Author(s)

Gabor Csardi csardi@rmki.kfki.hu

### References

M Newman and M Girvan: Finding and evaluating community structure in networks, Physical Review E 69, 026113 (2004)

`edge.betweenness` for the definition and calculation of the edge betweenness, `walktrap.community`, `fastgreedy.community`, `leading.eigenvector.community` for other community detection methods. `as.dendrogram` for creating an R dendrogram object from the result of the clustering. See `community.to.membership` to create the actual communities after a number of edges removed from the network.

### Examples

```g <- barabasi.game(100,m=2)
eb <- edge.betweenness.community(g)

g <- graph.full(10) %du% graph.full(10)
g <- add.edges(g, c(0,10))
eb <- edge.betweenness.community(g)
E(g) [ eb\$removed.edges[1] ]
```

[Package igraph version 0.5 Index]