statGraph (version 1.0.6)

GIC: Graph Information Criterion (GIC)

Description

GIC returns the Kullback-Leibler divergence, L1 or L2 distance between an undirected graph and a given graph model using the exact or degree-based spectral densities.

Usage

GIC(Graph, model, p = NULL, dist = "KL", ...)

Value

A list with class 'statGraph' containing the following components:

method:

a string indicating the used method.

info:

a string showing details about the method.

data.name:

a string with the data's name(s).

value:

a real number corresponding to the Kullback-Leibler divergence, L1, or L2 distance between the observed graph and the graph model.

Arguments

Graph

the undirected graph (igraph object). If Graph has the attribute eigenvalues containing the eigenvalues of Graph, such values will be used to compute its spectral density.

model

either a list, a string, or a function describing a graph model:

A list that represents the spectral density of a model. It contains the components 'x' and 'y', which are numeric vectors of the same size. The 'x' component contains the points at which the density was computed and the 'y' component contains the observed density.

A string that indicates one of the following models: 'ER' (Erdos-Renyi random graph), 'GRG' (geometric random graph), 'KR' (k regular random graph), 'WS' (Watts-Strogatz model), and 'BA' (Barabási-Albert model). When the argument model is a string, the user must also provide the model parameter by using the argument p.

A function that returns a graph (igraph object) generated by a graph model. It must contain two arguments: the first one corresponds to the graph size and the second to the parameter of the model. The model parameter will be provided by the argument p of the GIC function.

p

the model parameter. The user must provide a valid parameter if the argument model is a string or a function. For the predefined models ('ER', 'GRG', 'KR', 'WS', and 'BA'), the parameter the probability to connect a pair of vertices, for the 'ER' model (Erdos-Renyi random graph);

the radius used to construct the geometric graph in a unit square, for the 'GRG' model (geometric random graph);

the degree k of a regular graph, for the 'KR' model (k regular random graph);

the probability to reconnect a vertex, for the 'WS' model (Watts-Strogatz model);

and the scaling exponent, for the 'BA' model (Barabási-Albert model).

dist

string indicating if you want to use the 'KL' (default), 'L1' or 'L2' distances. 'KL' means Kullback-Leibler divergence.

...

Other relevant parameters for graph.spectral.density.

References

Takahashi, D. Y., Sato, J. R., Ferreira, C. E. and Fujita A. (2012) Discriminating Different Classes of Biological Networks by Analyzing the Graph Spectra Distribution. _PLoS ONE_, *7*, e49949. doi:10.1371/journal.pone.0049949.

Silverman, B. W. (1986) _Density Estimation_. London: Chapman and Hall.

Sturges, H. A. The Choice of a Class Interval. _J. Am. Statist. Assoc._, *21*, 65-66.

Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. _Journal of the Royal Statistical Society series B_, 53, 683-690. http://www.jstor.org/stable/2345597.

Examples

set.seed(1)
G <- igraph::sample_gnp(n=50, p=0.5)

# Using a string to indicate the graph model
result1 <- GIC(G, 'ER', 0.5)
result1

# Using a function to describe the graph model
# Erdos-Renyi graph
model <- function(n, p) {
   return (igraph::sample_gnp(n, p))
}
result2 <- GIC(G, model, 0.5)
result2