NetworkDistance (version 0.3.6)

nd.gdd: Graph Diffusion Distance

Description

Graph Diffusion Distance (nd.gdd) quantifies the difference between two weighted graphs of same size. It takes an idea from heat diffusion process on graphs via graph Laplacian exponential kernel matrices. For a given adjacency matrix \(A\), the graph Laplacian is defined as $$L := D-A$$ where \(D_{ii}=\sum_j A_{ij}\). For two adjacency matrices \(A_1\) and \(A_2\), GDD is defined as $$d_{gdd}(A_1,A_2) = max_t \sqrt{\| \exp(-tL_1) -\exp(-tL_2) \|_F^2}$$ where \(\exp(\cdot)\) is matrix exponential, \(\|\cdot\|_F\) a Frobenius norm, and \(L_1\) and \(L_2\) Laplacian matrices corresponding to \(A_1\) and \(A_2\), respectively.

Usage

nd.gdd(A, out.dist = TRUE, vect = seq(from = 0.1, to = 1, length.out = 10))

Value

a named list containing

D

an \((N\times N)\) matrix or dist object containing pairwise distance measures.

maxt

an \((N\times N)\) matrix whose entries are maximizer of the cost function.

Arguments

A

a list of length \(N\) containing adjacency matrices.

out.dist

a logical; TRUE for computed distance matrix as a dist object.

vect

a vector of parameters \(t\) whose values will be used.

References

hammond_graph_2013NetworkDistance

Examples

## load data and extract a subset
data(graph20)
gr.small = graph20[c(1:5,11:15)]

## compute distance matrix
output = nd.gdd(gr.small, out.dist=FALSE)

## visualize
opar = par(no.readonly=TRUE)
par(pty="s")
image(output$D[,10:1], main="two group case", col=gray((0:32)/32), axes=FALSE)
par(opar)