Network-Valued Data Analysis

Network-valued data are data in which the statistical unit is a network itself. This is the data with which we can make inference on populations of networks from samples of networks. The nevada package proposes a specific nvd class to handle network-valued data. Inference from such samples is made possible though a 4-step procedure:

The package focuses for now on the two-sample testing problem and assumes that all networks from both samples share the same node structure.

There are two types of questions that one can ask:

  1. Is there a difference between the distributions that generated the two observed samples?
  2. Can we localize the differences between the distributions on the node structure?

The nevada package offers a dedicated function for answering each of these two questions:

The nvd class for network-valued data

In nevada, network-valued data are stored in an object of class nvd, which is basically a list of igraph objects. We provide:

The constructor simulates networks with 25 nodes.

Network representation

There are currently 3 possible matrix representations for a network. Let \(G\) be a network with \(N\) nodes.

Adjacency matrix

A \(N\) x \(N\) matrix \(W\) is an adjacency matrix for \(G\) if element \(W_{ij}\) indicates if there is an edge between vertex \(i\) and vertex \(j\): \[ W_{ij}= \begin{cases} w_{i,j}, & \mbox{if } (i,j) \in E \mbox{ with weight } w_{i,j}\\ 0, & \mbox{otherwise.} \end{cases} \]

In nevada, this representation can be achieved with repr_adjacency().

Laplacian matrix

The Laplacian matrix \(L\) of the network \(G\) is defined in the following way:

\[ L = D(W) - W, \] where \(D(W)\) is the diagonal matrix whose \(i\)-th diagonal element is the degree of vertex \(i\).

In nevada, this representation can be achieved with repr_laplacian().

Modularity matrix

The elements of the modularity matrix \(B\) are given by

\[ B_{ij} = W_{ij} - \frac{d_i d_j}{2m}, \] where \(d_i\) and \(d_j\) are the degrees of vertices \(i\) and \(j\) respectively, and \(m\) is the total number of edges in the network.

In nevada, this representation can be achieved with repr_modularity().

Choosing a representation for an object of class nvd

Instead of going through every single network in a sample to make its representation, nevada provides the repr_nvd() function which does exactly that for an object of class nvd.

x <- nvd(model = "gnp", n = 3)
repr_nvd(x, representation = "laplacian")
#> [[1]]
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#>  [1,]    7   -1    0    0    0    0    0    0    0     0     0     0     0
#>  [2,]   -1    5    0    0    0    0    0   -1    0     0     0     0    -1
#>  [3,]    0    0    7   -1    0    0    0    0    0    -1    -1    -1    -1
#>  [4,]    0    0   -1    6    0    0    0    0    0     0     0    -1     0
#>  [5,]    0    0    0    0    9    0   -1   -1    0     0     0    -1     0
#>  [6,]    0    0    0    0    0    5    0    0    0     0     0     0    -1
#>  [7,]    0    0    0    0   -1    0   11   -1   -1     0    -1    -1     0
#>  [8,]    0   -1    0    0   -1    0   -1    7   -1     0     0     0    -1
#>  [9,]    0    0    0    0    0    0   -1   -1    5     0     0     0     0
#> [10,]    0    0   -1    0    0    0    0    0    0     8     0     0     0
#> [11,]    0    0   -1    0    0    0   -1    0    0     0     8     0     0
#> [12,]    0    0   -1   -1   -1    0   -1    0    0     0     0    10    -1
#> [13,]    0   -1   -1    0    0   -1    0   -1    0     0     0    -1     6
#> [14,]   -1    0    0   -1   -1    0   -1    0    0    -1     0     0     0
#> [15,]   -1   -1    0    0   -1    0   -1    0    0     0    -1    -1     0
#> [16,]    0    0    0    0   -1   -1    0    0    0    -1     0     0     0
#> [17,]   -1    0    0    0    0    0   -1    0    0     0    -1     0     0
#> [18,]   -1    0    0    0    0    0    0   -1   -1     0     0    -1     0
#> [19,]    0    0   -1    0    0    0    0    0    0    -1    -1    -1     0
#> [20,]    0    0    0    0    0    0   -1   -1   -1    -1    -1     0     0
#> [21,]    0    0    0   -1   -1   -1    0    0    0     0     0     0     0
#> [22,]   -1    0    0   -1    0   -1   -1    0    0    -1    -1    -1    -1
#> [23,]   -1   -1    0    0   -1    0    0    0    0    -1     0     0     0
#> [24,]    0    0    0    0    0   -1   -1    0   -1     0     0     0     0
#> [25,]    0    0   -1   -1   -1    0    0    0    0    -1    -1    -1     0
#>       [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
#>  [1,]    -1    -1     0    -1    -1     0     0     0    -1    -1     0     0
#>  [2,]     0    -1     0     0     0     0     0     0     0    -1     0     0
#>  [3,]     0     0     0     0     0    -1     0     0     0     0     0    -1
#>  [4,]    -1     0     0     0     0     0     0    -1    -1     0     0    -1
#>  [5,]    -1    -1    -1     0     0     0     0    -1     0    -1     0    -1
#>  [6,]     0     0    -1     0     0     0     0    -1    -1     0    -1     0
#>  [7,]    -1    -1     0    -1     0     0    -1     0    -1     0    -1     0
#>  [8,]     0     0     0     0    -1     0    -1     0     0     0     0     0
#>  [9,]     0     0     0     0    -1     0    -1     0     0     0    -1     0
#> [10,]    -1     0    -1     0     0    -1    -1     0    -1    -1     0    -1
#> [11,]     0    -1     0    -1     0    -1    -1     0    -1     0     0    -1
#> [12,]     0    -1     0     0    -1    -1     0     0    -1     0     0    -1
#> [13,]     0     0     0     0     0     0     0     0    -1     0     0     0
#> [14,]     7     0     0     0     0    -1     0     0     0    -1     0     0
#> [15,]     0    11    -1    -1     0     0     0    -1     0    -1    -1     0
#> [16,]     0    -1     9    -1    -1     0     0    -1    -1    -1     0     0
#> [17,]     0    -1    -1     6     0     0     0     0    -1     0     0     0
#> [18,]     0     0    -1     0    10    -1    -1    -1     0    -1     0    -1
#> [19,]    -1     0     0     0    -1     8     0     0     0    -1    -1     0
#> [20,]     0     0     0     0    -1     0     8     0    -1     0     0    -1
#> [21,]     0    -1    -1     0    -1     0     0     6     0     0     0     0
#> [22,]     0     0    -1    -1     0     0    -1     0    12    -1     0     0
#> [23,]    -1    -1    -1     0    -1    -1     0     0    -1    11     0    -1
#> [24,]     0    -1     0     0     0    -1     0     0     0     0     5     0
#> [25,]     0     0     0     0    -1     0    -1     0     0    -1     0     9
#> attr(,"representation")
#> [1] "laplacian"
#> 
#> [[2]]
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#>  [1,]   11   -1    0    0    0   -1    0    0   -1    -1     0    -1    -1
#>  [2,]   -1    9    0   -1    0   -1    0    0    0     0     0    -1     0
#>  [3,]    0    0    9   -1    0   -1    0   -1    0     0    -1     0     0
#>  [4,]    0   -1   -1    6    0    0    0   -1    0     0     0     0     0
#>  [5,]    0    0    0    0    8    0   -1    0   -1    -1     0     0     0
#>  [6,]   -1   -1   -1    0    0   11   -1    0   -1     0     0     0    -1
#>  [7,]    0    0    0    0   -1   -1   10    0   -1     0     0    -1     0
#>  [8,]    0    0   -1   -1    0    0    0   10    0    -1     0    -1     0
#>  [9,]   -1    0    0    0   -1   -1   -1    0   11    -1    -1     0     0
#> [10,]   -1    0    0    0   -1    0    0   -1   -1     8     0     0    -1
#> [11,]    0    0   -1    0    0    0    0    0   -1     0     7    -1     0
#> [12,]   -1   -1    0    0    0    0   -1   -1    0     0    -1     9     0
#> [13,]   -1    0    0    0    0   -1    0    0    0    -1     0     0     7
#> [14,]   -1   -1    0    0   -1    0   -1   -1    0     0    -1     0     0
#> [15,]   -1    0    0    0   -1   -1    0    0   -1     0     0    -1     0
#> [16,]   -1   -1   -1    0    0    0    0    0   -1    -1     0    -1     0
#> [17,]   -1   -1   -1    0    0   -1   -1    0    0    -1     0    -1     0
#> [18,]    0    0   -1    0    0    0   -1    0    0     0    -1     0    -1
#> [19,]    0    0    0    0    0    0   -1    0   -1     0     0     0     0
#> [20,]    0    0    0    0   -1    0    0   -1    0     0    -1    -1    -1
#> [21,]    0   -1   -1   -1    0   -1    0    0    0     0     0     0     0
#> [22,]    0    0    0    0   -1   -1   -1   -1    0     0    -1     0     0
#> [23,]    0    0    0   -1    0   -1   -1   -1   -1     0     0     0     0
#> [24,]    0    0   -1   -1    0    0    0   -1    0    -1     0     0    -1
#> [25,]   -1   -1    0    0   -1    0    0   -1   -1     0     0     0    -1
#>       [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
#>  [1,]    -1    -1    -1    -1     0     0     0     0     0     0     0    -1
#>  [2,]    -1     0    -1    -1     0     0     0    -1     0     0     0    -1
#>  [3,]     0     0    -1    -1    -1     0     0    -1     0     0    -1     0
#>  [4,]     0     0     0     0     0     0     0    -1     0    -1    -1     0
#>  [5,]    -1    -1     0     0     0     0    -1     0    -1     0     0    -1
#>  [6,]     0    -1     0    -1     0     0     0    -1    -1    -1     0     0
#>  [7,]    -1     0     0    -1    -1    -1     0     0    -1    -1     0     0
#>  [8,]    -1     0     0     0     0     0    -1     0    -1    -1    -1    -1
#>  [9,]     0    -1    -1     0     0    -1     0     0     0    -1     0    -1
#> [10,]     0     0    -1    -1     0     0     0     0     0     0    -1     0
#> [11,]    -1     0     0     0    -1     0    -1     0    -1     0     0     0
#> [12,]     0    -1    -1    -1     0     0    -1     0     0     0     0     0
#> [13,]     0     0     0     0    -1     0    -1     0     0     0    -1    -1
#> [14,]     8    -1     0     0     0     0     0     0     0     0    -1     0
#> [15,]    -1     8     0     0     0     0    -1     0     0    -1     0     0
#> [16,]     0     0     9     0     0     0     0     0    -1    -1    -1     0
#> [17,]     0     0     0     9    -1     0     0     0     0     0    -1     0
#> [18,]     0     0     0    -1     8     0    -1     0    -1     0    -1     0
#> [19,]     0     0     0     0     0     3     0     0     0    -1     0     0
#> [20,]     0    -1     0     0    -1     0     8     0     0     0    -1     0
#> [21,]     0     0     0     0     0     0     0     6     0    -1    -1     0
#> [22,]     0     0    -1     0    -1     0     0     0     8     0    -1     0
#> [23,]     0    -1    -1     0     0    -1     0    -1     0     9     0     0
#> [24,]    -1     0    -1    -1    -1     0    -1    -1    -1     0    13    -1
#> [25,]     0     0     0     0     0     0     0     0     0     0    -1     7
#> attr(,"representation")
#> [1] "laplacian"
#> 
#> [[3]]
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#>  [1,]    7    0   -1   -1   -1   -1   -1    0    0     0     0     0    -1
#>  [2,]    0    4   -1    0    0    0   -1    0    0    -1     0     0     0
#>  [3,]   -1   -1   13    0   -1   -1    0    0   -1     0    -1    -1    -1
#>  [4,]   -1    0    0    8    0   -1    0    0   -1     0    -1     0     0
#>  [5,]   -1    0   -1    0   10    0    0   -1    0    -1    -1     0     0
#>  [6,]   -1    0   -1   -1    0   12    0    0    0     0     0     0    -1
#>  [7,]   -1   -1    0    0    0    0    7    0   -1    -1     0    -1     0
#>  [8,]    0    0    0    0   -1    0    0    8   -1     0     0     0    -1
#>  [9,]    0    0   -1   -1    0    0   -1   -1   13    -1    -1    -1     0
#> [10,]    0   -1    0    0   -1    0   -1    0   -1     8     0     0     0
#> [11,]    0    0   -1   -1   -1    0    0    0   -1     0     7     0     0
#> [12,]    0    0   -1    0    0    0   -1    0   -1     0     0     5     0
#> [13,]   -1    0   -1    0    0   -1    0   -1    0     0     0     0     9
#> [14,]    0   -1    0   -1   -1   -1   -1    0   -1     0     0     0     0
#> [15,]   -1    0   -1   -1    0   -1    0    0    0     0    -1     0     0
#> [16,]    0    0   -1    0    0    0    0    0    0    -1     0     0    -1
#> [17,]    0    0    0    0   -1   -1   -1    0   -1     0     0     0     0
#> [18,]    0    0    0    0   -1    0    0   -1    0     0    -1    -1    -1
#> [19,]    0    0    0    0   -1    0    0    0   -1     0     0     0    -1
#> [20,]    0    0   -1    0    0   -1    0    0   -1     0    -1     0     0
#> [21,]    0    0    0   -1    0   -1    0   -1    0     0     0     0    -1
#> [22,]    0    0    0    0   -1   -1    0   -1   -1    -1     0     0    -1
#> [23,]    0    0   -1   -1    0    0    0    0    0     0     0     0     0
#> [24,]    0    0   -1    0    0   -1    0   -1    0    -1     0     0     0
#> [25,]    0    0    0    0    0   -1    0   -1   -1    -1     0    -1     0
#>       [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
#>  [1,]     0    -1     0     0     0     0     0     0     0     0     0     0
#>  [2,]    -1     0     0     0     0     0     0     0     0     0     0     0
#>  [3,]     0    -1    -1     0     0     0    -1     0     0    -1    -1     0
#>  [4,]    -1    -1     0     0     0     0     0    -1     0    -1     0     0
#>  [5,]    -1     0     0    -1    -1    -1     0     0    -1     0     0     0
#>  [6,]    -1    -1     0    -1     0     0    -1    -1    -1     0    -1    -1
#>  [7,]    -1     0     0    -1     0     0     0     0     0     0     0     0
#>  [8,]     0     0     0     0    -1     0     0    -1    -1     0    -1    -1
#>  [9,]    -1     0     0    -1     0    -1    -1     0    -1     0     0    -1
#> [10,]     0     0    -1     0     0     0     0     0    -1     0    -1    -1
#> [11,]     0    -1     0     0    -1     0    -1     0     0     0     0     0
#> [12,]     0     0     0     0    -1     0     0     0     0     0     0    -1
#> [13,]     0     0    -1     0    -1    -1     0    -1    -1     0     0     0
#> [14,]     7    -1     0     0     0     0     0     0     0     0     0     0
#> [15,]    -1     7     0     0     0     0     0     0     0     0     0    -1
#> [16,]     0     0     5    -1     0    -1     0     0     0     0     0     0
#> [17,]     0     0    -1     9    -1    -1     0     0     0     0    -1    -1
#> [18,]     0     0     0    -1    10    -1     0     0    -1    -1    -1     0
#> [19,]     0     0    -1    -1    -1     8     0    -1     0     0    -1     0
#> [20,]     0     0     0     0     0     0     7    -1     0    -1    -1     0
#> [21,]     0     0     0     0     0    -1    -1     7     0     0     0    -1
#> [22,]     0     0     0     0    -1     0     0     0     7     0     0     0
#> [23,]     0     0     0     0    -1     0    -1     0     0     4     0     0
#> [24,]     0     0     0    -1    -1    -1    -1     0     0     0     8     0
#> [25,]     0    -1     0    -1     0     0     0    -1     0     0     0     8
#> attr(,"representation")
#> [1] "laplacian"

Distances between networks

It is possible to choose which distance consider in the analysis. Let \(G\) and \(H\) be two networks with \(N\) nodes each and suppose that \(X\) and \(Y\) are the matrix representations of \(G\) and \(H\), respectively. The user can currently choose among 4 distances: Hamming, Frobenius, spectral and root-Euclidean.

Hamming distance

\[ \rho_H(G,H)=\frac{1}{N(N-1)}\sum_{i \neq j}^N \bigl\arrowvert X_{i,j}-Y_{i,j} \bigr\arrowvert. \]

In nevada, this distance can be computed with dist_hamming().

Frobenius distance

\[ \rho_F(G,H) = \left\| X - Y \right\|_F^2 = \sum_{i \neq j}^N \bigl ( X_{i,j}-Y_{i,j} \bigr )^2. \]

In nevada, this distance can be computed with dist_frobenius().

Spectral distance

\[ \rho_S(G,H)=\sum_{i \neq j}^N \bigl ( \Lambda^X_{i,j}-\Lambda^Y_{i,j} \bigr )^2, \] where \(\Lambda^X\) and \(\Lambda^Y\) are the diagonal matrices with eigenvalues on the diagonal given by the spectral decomposition of the matrix representations of \(G\) and \(H\).

In nevada, this distance can be computed with dist_spectral().

Root Euclidean distance

\[ \rho_{RE}(G,H) = \left\| X^{1/2} - Y^{1/2} \right\|_F^2. \]

Note that this distance is not compatible with all matrix representations as it requires that the representation be semi-positive definite.

In nevada, this distance can be computed with dist_root_euclidean().

Computing a matrix of pairwise distances for an object of class nvd

Pre-computation of the matrix of pairwise distances for samples of networks alleviates the computational burden of permutation testing. This is why nevada provides the convenient dist_nvd() function which does exactly that for an object of class nvd.

x <- nvd(model = "gnp", n = 3)
dist_nvd(x, representation = "laplacian", distance = "hamming")
#>           1         2
#> 2 0.5066667          
#> 3 0.5000000 0.5233333

Test statistics

The nevada package has been designed to work well with the flipr package, which handles the permutation scheme once suitable representation, distance and test statistics have been chosen. The most efficient way to two-sample testing with network-valued data pertains to use statistics based on inter-point distances, that is pairwise distances between observations.

Available statistics

From flipr

A number of test statistics along this line have been proposed in the literature, including ours (Lovato et al. 2020). As these test statistics rely on inter-point distances, they are not specific to network-valued data. As such, they can be found in flipr. We adopt the naming convention that a test statistic function shall start with the prefix stat_. All statistics based on inter-point distances are named with the suffix _ip. Here is the list of test statistics based on inter-point distances that are currently available in flipr:

  • stat_student_ip() and its alias stat_t_ip() implement a Student-like test statistic based on inter-point distances proposed by Lovato et al. (2020);
  • stat_fisher_ip() and its alias stat_f_ip() implement a Fisher-like test statistic based on inter-point distances proposed by Lovato et al. (2020);
  • stat_bg_ip() implements the statistic proposed by Biswas and Ghosh (2014);
  • stat_energy_ip() implements the class of energy-based statistics as proposed by Székely and Rizzo (2013);
  • stat_cq_ip() implements the statistic proposed by S. X. Chen and Qin (2010);
  • stat_mod_ip() implements a statistic that computes the mean of inter-point distances;
  • stat_dom_ip() implements a statistic that computes the distance between the medoids of the two samples, possibly standardized by the pooled corresponding variances.

From nevada

There are also 3 statistics proposed in H. Chen, Chen, and Su (2018) that are based on a similarity graph built on top of the distance matrix:

There are also Student-like statistics available only for Frobenius distance for which we can easily compute the Fréchet mean. These are:

Write your own test statistic function

In addition to the test statistic functions already implemented in flipr and nevada, you can also implement your own function. Test statistic functions compatible with flipr should have at least two mandatory input arguments:

  • data which is either a concatenated list of size \(n_x + n_y\) regrouping the data points of both samples or a distance matrix of size \((n_x + n_y) \times (n_x + n_y)\) stored as an object of class dist.
  • indices1 which is an integer vector of size \(n_x\) storing the indices of the data points belonging to the first sample in the current permuted version of the data.

The flipr package provides a helper function use_stat(nsamples = 2, stat_name = ) which makes it easy for users to create their own test statistic ready to be used by nevada. This function creates and saves a .R file in the R/ folder of the current working directory and populates it with the following template:

#' Test Statistic for the Two-Sample Problem
#'
#' This function computes the test statistic...
#'
#' @param data A list storing the concatenation of the two samples from which
#'   the user wants to make inference. Alternatively, a distance matrix stored
#'   in an object of class \code{\link[stats]{dist}} of pairwise distances
#'   between data points.
#' @param indices1 An integer vector that contains the indices of the data
#'   points belong to the first sample in the current permuted version of the
#'   data.
#'
#' @return A numeric value evaluating the desired test statistic.
#' @export
#'
#' @examples
#' # TO BE DONE BY THE DEVELOPER OF THE PACKAGE
stat_{{{name}}} <- function(data, indices1) {
  n <- if (inherits(data, "dist"))
    attr(data, "Size")
  else if (inherits(data, "list"))
    length(data)
  else
    stop("The `data` input should be of class either list or dist.")

  indices2 <- seq_len(n)[-indices1]

  x <- data[indices1]
  y <- data[indices2]

  # Here comes the code that computes the desired test
  # statistic from input samples stored in lists x and y

}

For instance, a flipr-compatible version of the \(t\)-statistic with pooled variance will look like:

stat_student <- function(data, indices1) {
  n <- if (inherits(data, "dist"))
    attr(data, "Size")
  else if (inherits(data, "list"))
    length(data)
  else
    stop("The `data` input should be of class either list or dist.")

  indices2 <- seq_len(n)[-indices1]

  x <- data[indices1]
  y <- data[indices2]
  
  # Here comes the code that computes the desired test
  # statistic from input samples stored in lists x and y
  x <- unlist(x)
  y <- unlist(y)
  
  stats::t.test(x, y, var.equal = TRUE)$statistic
}

Usage

Naming conventions

Test statistics are passed to the functions test2_global() and test2_local() via the argument stats which accepts a character vector in which:

  • statistics from nevada expected to be named without the stat_ prefix (e.g. "original_edge_count" or "student_euclidean").
  • statistics from flipr are expected to be named without the stat_ prefix but adding the flipr: prefix (e.g., "flipr:student_ip").
  • statistics from any other package pkg are expected to be named without the stat_ prefix but adding the pkg: prefix.
x <- nvd(model = "gnp", n = 10)
y <- nvd(model = "pa" , n = 10)
test2_global(
  x = x, 
  y = y, 
  representation = "laplacian", 
  distance = "frobenius", 
  stats = c("flipr:student_ip", "flipr:fisher_ip"), 
  seed = 1234
)$pvalue
#> [1] 0.0009962984

Note that you can also refer to test statistic function from nevada using the naming "nevada:original_edge_count" as you would do for test statistics from flipr. This is mandatory for instance if you have not yet loaded nevada in your environment via library(nevada).

Using multiple test statistics

In permutation testing, the choice of a test statistic determines the alternative hypothesis, while the null hypothesis is always that the distributions that generated the observed samples are the same. This means that if you were to use the Student statistic stat_student_ip() for instance, then what you would be actually testing is whether the means of the distributions are different. If you’d rather be sensitive to differences in variances of the distributions, then you should go with the Fisher statistic stat_fisher_ip().

You can also be sensitive to multiple aspects of a distribution when testing via the permutation framework. This is achieved under the hood by the flipr package which implements the so-called non-parametric combination (NPC) approach proposed by Pesarin and Salmaso (2010) when you provide more than one test statistics in the stats argument. You can read this article to know more about its implementation in flipr. The bottom line is that, for example, you can choose both the Student and Fisher statistics to test simultaneously for differences in mean and in variance.

References

Biswas, Munmun, and Anil K Ghosh. 2014. “A Nonparametric Two-Sample Test Applicable to High Dimensional Data.” Journal of Multivariate Analysis 123: 160–71.
Chen, Hao, Xu Chen, and Yi Su. 2018. “A Weighted Edge-Count Two-Sample Test for Multivariate and Object Data.” Journal of the American Statistical Association 113 (523): 1146–55.
Chen, Song Xi, and Ying-Li Qin. 2010. “A Two-Sample Test for High-Dimensional Data with Applications to Gene-Set Testing.” The Annals of Statistics 38 (2): 808–35.
Lovato, Ilenia, Alessia Pini, Aymeric Stamm, Maxime Taquet, and Simone Vantini. 2021. “Multiscale Null Hypothesis Testing for Network-Valued Data: Analysis of Brain Networks of Patients with Autism.” Journal of the Royal Statistical Society: Series C (Applied Statistics), January. https://doi.org/10.1111/rssc.12463.
Lovato, Ilenia, Alessia Pini, Aymeric Stamm, and Simone Vantini. 2020. “Model-Free Two-Sample Test for Network-Valued Data.” Computational Statistics & Data Analysis 144 (April): 106896. https://doi.org/10.1016/j.csda.2019.106896.
Pesarin, Fortunato, and Luigi Salmaso. 2010. Permutation Tests for Complex Data: Theory, Applications and Software. John Wiley & Sons.
Székely, Gábor J, and Maria L Rizzo. 2013. “Energy Statistics: A Class of Statistics Based on Distances.” Journal of Statistical Planning and Inference 143 (8): 1249–72.