leading.eigenvector.community {igraph} | R Documentation

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

### Description

These functions try to find densely connected subgraphs in a graph by calculating the leading non-negative eigenvector of the modularity matrix of the graph.

### Usage

    leading.eigenvector.community(graph, steps = -1,
           options = igraph.arpack.default)
    leading.eigenvector.community.naive(graph, steps = -1,
           options = igraph.arpack.default)
    leading.eigenvector.community.step (graph, fromhere = NULL,
        membership = rep(0, vcount(graph)), community = 0,
        options=igraph.arpack.default)
    community.le.to.membership(merges, steps, membership)

### 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. For `community.le.to.membership` the number of merges to produce from the supplied `membership` vector.

`naive`

Logical constant, it defines how the algorithm tries to find more divisions after the first division was made. If `TRUE` then it simply considers both communities as separate graphs and then creates modularity matrices for both communities, etc. This method however does not maximize modularity, see the paper in the references section below. If it is `FALSE` then the proper method is used which maximizes modularity.

`fromhere` | An object returned by a previous call to `leading.eigenvector.community.step`. This will serve as a starting point to take another step. This argument is ignored if it is `NULL`.

`membership`

The starting community structure. It is a numeric vector defining the membership of every vertex in the graph with a number between 0 and the total number of communities at this level minus one. By default we start with a single community containing all vertices. This argument is ignored if `fromhere` is not `NULL`.
For `community.le.to.memberhip` the starting community structure on which `steps` merges are performed.

`community` | The id of the community which the algorithm will try to split.
`options` | A named list to override some ARPACK options.
`merges` | The merge matrix, possible from the result of `leading.eigenvector.community`.

### Details

The functions documented in these section implement the ‘leading eigenvector’ method developed by Mark Newman and published in MEJ Newman: Finding community structure using the eigenvectors of matrices, arXiv:physics/0605087. TODO: proper citation.

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 ‘configuration model’. In other words, a `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.

`leading.eigenvector.community` is the proper implementation of the proposed algorithm. `leading.eigenvector.community.naive` is considered worse, in this implementation a community found after a division is considered as a separate graph for further divisions. `leading.eigenvector.community.step` is the proper implementation, but makes only one step, by trying to split the specified community.

From igraph 0.5 these functions use ARPACK to calculate the eigenvectors. See `arpack` for details.

`community.le.to.memberhip` creates a membership vector from the result of `leading.eigenvector.community` or `leading.eigenvector.community.naive`. It takes `membership` and permformes `steps` merges, according to the supplied `merges` matrix.

### Value

`leading.eigenvector.community` and `leading.eigenvector.community.naive` return 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.
`membership` | The new membership vector after the split. If no split was done then this is the same as the input membership vector.
`split` | Logical scalar, if `TRUE` that means that the community was successfully splitted.
`eigenvector` | The eigenvector of the community matrix, or `NULL` if the `eigenvector` argument was `FALSE`.
`eigenvalue` | The largest positive eigenvalue of the modularity matrix.
`options` | Information about the underlying ARPACK computation, see `arpack` for details.
`membership` | A membership vector, a numerical vector indication which vertex belongs to which community. The communities are always numbered from zero.
`csize` | A numeric vector giving the sizes of the communities.

### Author(s)

Gabor Csardi csardi@rmki.kfki.hu

### References

MEJ Newman: Finding community structure using the eigenvectors of matrices, arXiv:physics/0605087

### See Also

`modularity`, `walktrap.community`, `edge.betweenness.community`, `fastgreedy.community`, `as.dendrogram`

### Examples

    g <- graph.full(5) %du% graph.full(5) %du% graph.full(5)
    g <- add.edges(g, c(0,5, 0,10, 5, 10))
    leading.eigenvector.community(g)

    lec <- leading.eigenvector.community.step(g)
    lec$membership
    # Try one more split
    leading.eigenvector.community.step(g, fromhere=lec, community=0)
    leading.eigenvector.community.step(g, fromhere=lec, community=1)

[Package igraph version 0.5.2 Index]