# cluster_leading_eigen

##### Community structure detecting based on the leading eigenvector of the community matrix

This function tries to find densely connected subgraphs in a graph by calculating the leading non-negative eigenvector of the modularity matrix of the graph.

- Keywords
- graphs

##### Usage

```
cluster_leading_eigen(graph, steps = -1, weights = NULL, start = NULL,
options = arpack_defaults, callback = NULL, extra = NULL,
env = parent.frame())
```

##### Arguments

- graph
- The input graph. Should be undirected as the method needs a symmetric matrix.
- steps
- The number of steps to take, this is actually the number of tries to make a step. It is not a particularly useful parameter. #'
- weights
- An optional weight vector. The
weight edge attribute is used if present. Supply here if you want to ignore the`NA`

weight edge attribute. - start
`NULL`

, or a numeric membership vector, giving the start configuration of the algorithm.- options
- A named list to override some ARPACK options.
- callback
- If not
`NULL`

, then it must be callback function. This is called after each iteration, after calculating the leading eigenvector of the modularity matrix. See details below. - extra
- Additional argument to supply to the callback function.
- env
- The environment in which the callback function is evaluated.

##### Details

The function documented in these section implements the

The heart of the method is the definition of the modularity matrix,
`B`

, which is `B=A-P`

, `A`

being the adjacency matrix of the
(undirected) network, and `P`

contains the probability that certain
edges are present according to the `P[i,j]`

element of `P`

is the probability that there is
an edge between vertices `i`

and `j`

in a random network in which
the degrees of all vertices are the same as in the input graph.

The leading eigenvector method works by calculating the eigenvector of the modularity matrix for the largest positive eigenvalue and then separating vertices into two community based on the sign of the corresponding element in the eigenvector. If all elements in the eigenvector are of the same sign that means that the network has no underlying comuunity structure. Check Newman's paper to understand why this is a good method for detecting community structure.

##### Value

`cluster_leading_eigen`

returns a named list with the following members:membership The membership vector at the end of the algorithm, when no more splits are possible. merges The merges matrix starting from the state described by the `membership`

member. This is a two-column matrix and each line describes a merge of two communities, the first line is the first merge and it creates community ,`N`

`N`

is the number of initial communities in the graph, the second line creates community`N+1`

, etc.options Information about the underlying ARPACK computation, see `arpack`

for details.

##### References

MEJ Newman: Finding community structure using the eigenvectors of matrices, Physical Review E 74 036104, 2006.

##### See Also

`modularity`

, `cluster_walktrap`

,
`cluster_edge_betweenness`

,
`cluster_fast_greedy`

, `as.dendrogram`

##### Examples

```
g <- make_full_graph(5) %du% make_full_graph(5) %du% make_full_graph(5)
g <- add_edges(g, c(1,6, 1,11, 6, 11))
lec <- cluster_leading_eigen(g)
lec
cluster_leading_eigen(g, start=membership(lec))
```

*Documentation reproduced from package igraph, version 1.0.0, License: GPL (>= 2)*