igraph (version 0.4.3)

leading.eigenvector.community: 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 = vcount(graph), naive = FALSE)
leading.eigenvector.community.step (graph, fromhere = NULL,
    membership = rep(0, vcount(graph)), community = 0, eigenvector = TRUE)

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.

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 comm

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 cont

community

The id of the community which the algorithm will try to split.

eigenvector

Logical constant, whether to include the eigenvector of the modularity matrix in the result.

Value

• leading.eigenvector.community 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.

code

FALSE

itemize

• membership

item

• split
• eigenvector
• eigenvalue

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.

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)