# Backbone Unit Tests

#### Zachary Neal, Michigan State University, zpneal@msu.edu

# Table of Contents

1. Introduction
2. `backbone_from_projection()`
   1. From matrix
   2. From igraph
3. `backbone_from_weighted()`
   1. From weighted matrix
   2. From weighted igraph
   3. From weighted projection
   4. From weighted igraph projection
4. `backbone_from_unweighted()`
   1. `.escore()`
   2. `.normalize()`
   3. `.filter()`
   4. Models
5. Null model functions
6. Utility functions

# Introduction

The “vignette” is designed solely to display the automated unit tests used to check the `backbone` package during development. To identify any failed unit tests, search in the vignette for the word “failed”.

For a general introduction to the backbone package, please see the Introduction to Backbone vignette. Or, for a complete example using empirical data, please see the Empirical Example of Backbone Extraction vignette.

    library(backbone)
    library(igraph)
    library(tinytest)

back to Table of Contents

# `backbone_from_projection()`

## Setup

    trace <- function(x){sum(diag(x))}
    matcube <- function(x){x%*%x%*%x}
    triangle_index <- function(x){(trace(matcube(x)) + trace(matcube(abs(x))))/(2 * trace(matcube(abs(x))))}

## From matrix

### SDSM

    B <- rbind(cbind(matrix(rbinom(250,1,.85),10),   #An example block incidence matrix
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10)))

    bb <- backbone_from_projection(B, model = "sdsm", return = "everything")  #Extract SDSM matrix, return everything
    expect_equal(length(bb),6)  #Returned object contains six elements
    #> ----- PASSED      : <-->
    #>  call| expect_equal(length(bb), 6)
    expect_true(is(bb$bipartite,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$bipartite, "matrix"))
    expect_true(is(bb$projection,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$projection, "matrix"))
    expect_true(is(bb$backbone,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$backbone, "matrix"))
    expect_true(is(bb$pvalues$upper,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$pvalues$upper, "matrix"))
    expect_true(is(bb$narrative,"character"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$narrative, "character"))
    expect_true(is(bb$call,"call"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$call, "call"))

    bb <- backbone_from_projection(Matrix::Matrix(B), model = "sdsm", return = "everything")  #Extract SDSM Matrix, return everything
    expect_equal(length(bb),6)  #Returned object contains six elements
    #> ----- PASSED      : <-->
    #>  call| expect_equal(length(bb), 6)
    expect_true(is(bb$bipartite,"Matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$bipartite, "Matrix"))
    expect_true(is(bb$projection,"Matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$projection, "Matrix"))
    expect_true(is(bb$backbone,"Matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$backbone, "Matrix"))
    expect_true(is(bb$pvalues$upper,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$pvalues$upper, "matrix"))
    expect_true(is(bb$narrative,"character"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$narrative, "character"))
    expect_true(is(bb$call,"call"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$call, "call"))

    bb <- backbone_from_projection(B, model = "sdsm", signed = TRUE)  #Extract SDSM matrix as signed
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### FDSM

    bb <- backbone_from_projection(B, model = "fdsm", signed = TRUE, trials = 250)  #Extract FDSM matrix as signed
    #> Constructing edges' Monte Carlo p-values
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=                                                                     |   1%  |                                                                              |=                                                                     |   2%  |                                                                              |==                                                                    |   2%  |                                                                              |==                                                                    |   3%  |                                                                              |===                                                                   |   4%  |                                           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    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### fixedrow

    bb <- backbone_from_projection(B, model = "fixedrow", signed = TRUE)  #Extract fixedrow matrix as signed
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### fixedcol

    bb <- backbone_from_projection(B, model = "fixedcol", signed = TRUE)  #Extract fixedcol matrix as signed
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### fixedfill

    bb <- backbone_from_projection(B, model = "fixedfill", signed = TRUE)  #Extract fixedfill matrix as signed
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### SDSM-EC

    B <- as.vector(B)
    make_prohibited <- sample(which(B==0), 5, replace = FALSE)  #Pick some missing edges to prohibit
    B[make_prohibited] <- 10
    make_required <- sample(which(B==1), 5, replace = FALSE)  #Pick some present edges to require
    B[make_required] <- 11
    B <- matrix(B, 30, 75)  #Reassemble as matrix
    bb <- backbone_from_projection(B, model = "sdsm", signed = TRUE)  #Extract SDSM matrix as signed, considering structural values
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

## From igraph

### SDSM

    B <- rbind(cbind(matrix(rbinom(250,1,.85),10),   #An example block incidence matrix
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10)))
    B <- graph_from_biadjacency_matrix(B)          #Convert to igraph
    V(B)$agent_attrib <- c(c(1:30),rep(NA,75))     #Add agent attribute
    V(B)$artifact_attrib <- c(rep(NA,30),c(1:75))  #Add artifact attribute

    bb <- backbone_from_projection(B, model = "sdsm", return = "everything")  #Extract SDSM igraph, return everything
    expect_equal(length(bb),6)  #Returned object contains six elements
    #> ----- PASSED      : <-->
    #>  call| expect_equal(length(bb), 6)
    expect_equal(class(bb$bipartite)[1],"igraph")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$bipartite)[1], "igraph")
    expect_true(is_bipartite(bb$bipartite))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_bipartite(bb$bipartite))
    expect_equal(class(bb$projection)[1],"igraph")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$projection)[1], "igraph")
    expect_false(is_directed(bb$projection))
    #> ----- PASSED      : <-->
    #>  call| expect_false(is_directed(bb$projection))
    expect_equal(class(bb$backbone)[1],"igraph")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$backbone)[1], "igraph")
    expect_false(is_directed(bb$backbone))
    #> ----- PASSED      : <-->
    #>  call| expect_false(is_directed(bb$backbone))
    expect_equal(class(bb$pvalues$upper)[1],"matrix")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$pvalues$upper)[1], "matrix")
    expect_equal(class(bb$narrative)[1],"character")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$narrative)[1], "character")
    expect_equal(class(bb$call)[1],"call")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$call)[1], "call")

    bb <- backbone_from_projection(B, model = "sdsm")                              #Extract SDSM igraph with defaults
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### FDSM

    bb <- backbone_from_projection(B, model = "fdsm", trials = 250)                #Extract FDSM igraph with defaults
    #> Constructing edges' Monte Carlo p-values
    #>   |                                                                              |                                                                      |   0%  |                                                                              |=                                                                     |   1%  |                                                                              |=                                                                     |   2%  |                                                                              |==                                                                    |   2%  |                                                                              |==                                                                    |   3%  |                                                                              |===                                                                   |   4%  |                                           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    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### fixedrow

    bb <- backbone_from_projection(B, model = "fixedrow")                          #Extract fixedrow igraph with defaults
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### fixedcol

    bb <- backbone_from_projection(B, model = "fixedrow")                          #Extract fixedcol igraph with defaults
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### fixedfill

    bb <- backbone_from_projection(B, model = "fixedfill")                         #Extract fixedcol igraph with defaults
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### SDSM-EC

    E(B)$weight <- NA
    E(B)$weight <- sample(c(1,11), length(E(B)$weight), replace = TRUE, prob = c(.9,.1))
    bb <- backbone_from_projection(B, model = "sdsm")                              #Extract SDSM igraph with defaults, considering structural values
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

# `backbone_from_weighted()`

## From weighted matrix

### Disparity

    W <- matrix(c(0,10,10,10,10,75,0,0,0,0,
                  10,0,1,1,1,0,0,0,0,0,
                  10,1,0,1,1,0,0,0,0,0,
                  10,1,1,0,1,0,0,0,0,0,
                  10,1,1,1,0,0,0,0,0,0,
                  75,0,0,0,0,0,100,100,100,100,
                  0,0,0,0,0,100,0,10,10,10,
                  0,0,0,0,0,100,10,0,10,10,
                  0,0,0,0,0,100,10,10,0,10,
                  0,0,0,0,0,100,10,10,10,0),10)

    bb <- backbone_from_weighted(W, model = "disparity", return = "everything")  #Extract disparity backbone, return everything
    expect_equal(length(bb),5)  #Returned object contains five elements
    #> ----- PASSED      : <-->
    #>  call| expect_equal(length(bb), 5)
    expect_true(is(bb$weighted,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$weighted, "matrix"))
    expect_true(is(bb$backbone,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$backbone, "matrix"))
    expect_true(is(bb$pvalues$upper,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$pvalues$upper, "matrix"))
    expect_true(is(bb$narrative,"character"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$narrative, "character"))
    expect_true(is(bb$call,"call"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$call, "call"))

    bb <- backbone_from_weighted(Matrix::Matrix(W), model = "disparity", return = "everything")  #Extract disparity backbone, return everything
    expect_equal(length(bb),5)  #Returned object contains five elements
    #> ----- PASSED      : <-->
    #>  call| expect_equal(length(bb), 5)
    expect_true(is(bb$weighted,"Matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$weighted, "Matrix"))
    expect_true(is(bb$backbone,"Matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$backbone, "Matrix"))
    expect_true(is(bb$pvalues$upper,"matrix"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$pvalues$upper, "matrix"))
    expect_true(is(bb$narrative,"character"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$narrative, "character"))
    expect_true(is(bb$call,"call"))
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb$call, "call"))

    bb <- backbone_from_weighted(W, model = "disparity")  #Extract disparity backbone
    expect_true(is(bb,"matrix"))                          #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    bb <- graph_from_adjacency_matrix(bb, mode = "undirected")
    expect_true(is_tree(bb))                      #Backbone is a tree
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_tree(bb))

### LANS

    bb <- backbone_from_weighted(W, model = "lans")       #Extract lans backbone
    expect_true(is(bb,"matrix"))                          #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    bb <- graph_from_adjacency_matrix(bb, mode = "undirected")
    expect_true(is_tree(bb))                      #Backbone is a tree
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_tree(bb))

### MLF

    bb <- backbone_from_weighted(W, model = "mlf")        #Extract mlf backbone
    expect_true(is(bb,"matrix"))                          #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    bb <- graph_from_adjacency_matrix(bb, mode = "undirected")
    expect_true(is_tree(bb))                      #Backbone is a tree
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_tree(bb))

### Global

    bb <- backbone_from_weighted(W, model = "global")     #Extract global backbone (unsigned)
    expect_true(is(bb,"matrix"))                          #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(table(bb)[1]==58 & table(bb)[2]==42)      #Contains 58 0s and 42 1s
    #> ----- PASSED      : <-->
    #>  call| expect_true(table(bb)[1] == 58 & table(bb)[2] == 42)
    bb <- backbone_from_weighted(W, model = "global", parameter = c(10,74))     #Extract global backbone (signed)
    expect_true(table(bb)[1]==12 & table(bb)[2]==78 & table(bb)[3]==10)      #Contains 12 -1s, 78 0s, and 10 1s
    #> ----- PASSED      : <-->
    #>  call| expect_true(table(bb)[1] == 12 & table(bb)[2] == 78 & table(bb)[3] ==
    #>  call| 10)

## From weighted igraph

### Disparity

    W <- graph_from_adjacency_matrix(W, mode = "undirected", weighted = TRUE)

    bb <- backbone_from_weighted(W, model = "disparity")  #Extract disparity backbone
    expect_true(is(bb,"igraph"))                          #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_true(is_tree(bb))                      #Backbone is a tree
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_tree(bb))

### LANS

    bb <- backbone_from_weighted(W, model = "lans")       #Extract lans backbone
    expect_true(is(bb,"igraph"))                          #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_true(is_tree(bb))                      #Backbone is a tree
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_tree(bb))

### MLF

    bb <- backbone_from_weighted(W, model = "mlf")        #Extract mlf backbone
    expect_true(is(bb,"igraph"))                          #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_true(is_tree(bb))                      #Backbone is a tree
    #> ----- PASSED      : <-->
    #>  call| expect_true(is_tree(bb))

### Global

    bb <- backbone_from_weighted(W, model = "global")     #Extract global backbone (unsigned)
    expect_true(is(bb,"igraph"))                          #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    bb <- as_adjacency_matrix(bb, sparse = FALSE)   #Get matrix
    expect_true(table(bb)[1]==58 & table(bb)[2]==42)      #Contains 58 0s and 42 1s
    #> ----- PASSED      : <-->
    #>  call| expect_true(table(bb)[1] == 58 & table(bb)[2] == 42)
    bb <- backbone_from_weighted(W, model = "global", parameter = c(10,74))     #Extract global backbone (signed)
    bb <- as_adjacency_matrix(bb, sparse = FALSE, attr = "sign")   #Get matrix
    expect_true(table(bb)[1]==12 & table(bb)[2]==78 & table(bb)[3]==10)      #Contains 12 -1s, 78 0s, and 10 1s
    #> ----- PASSED      : <-->
    #>  call| expect_true(table(bb)[1] == 12 & table(bb)[2] == 78 & table(bb)[3] ==
    #>  call| 10)

## From weighted projection

### Disparity

    W <- rbind(cbind(matrix(rbinom(250,1,.85),10),
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10)))
    W <- W%*%t(W)
    diag(W) <- 0

    bb <- backbone_from_weighted(W, model = "disparity", signed = TRUE, alpha = 0.5)  #Extract signed disparity matrix
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### LANS

    bb <- backbone_from_weighted(W, model = "lans", signed = TRUE, alpha = 0.5)  #Extract signed lans matrix
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### MLF

    bb <- backbone_from_weighted(W, model = "mlf", signed = TRUE, alpha = 0.5)  #Extract signed mlf matrix
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

### Global

    upper <- mean(W) + sd(W)             #Use mean + sd as positive edge threshold
    lower <- mean(W) - sd(W)             #Use mean - sd as negative edge threshold
    bb <- backbone_from_weighted(W, model = "global", parameter = c(lower, upper))  #Extract signed global matrix
    expect_true(is(bb,"matrix"))         #Returns as matrix
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "matrix"))
    expect_true(all(bb %in% c(-1,0,1)))  #Contains only -1, 0, 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(bb %in% c(-1, 0, 1)))
    expect_true(any(bb %in% c(-1)))      #Contains some negative edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(-1)))
    expect_true(any(bb %in% c(0)))       #Contains some missing edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(0)))
    expect_true(any(bb %in% c(1)))       #Contains some positive edges
    #> ----- PASSED      : <-->
    #>  call| expect_true(any(bb %in% c(1)))
    triangle_index(bb)
    #> [1] 1
    expect_true(triangle_index(bb)>.8)   #Is nearly balanced
    #> ----- PASSED      : <-->
    #>  call| expect_true(triangle_index(bb) > 0.8)

## From weighted igraph projection

### Disparity

    W <- rbind(cbind(matrix(rbinom(250,1,.85),10),
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10),
                     matrix(rbinom(250,1,.15),10)),
               cbind(matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.15),10),
                     matrix(rbinom(250,1,.85),10)))
    W <- graph_from_biadjacency_matrix(W)
    W <- bipartite_projection(W, which = "false")
    V(W)$agent_attrib <- c(c(1:30))     #Add agent attribute

    bb <- backbone_from_weighted(W, model = "disparity", return = "everything")  #Extract disparity igraph, return everything
    expect_equal(length(bb),5)  #Returned object contains five elements
    #> ----- PASSED      : <-->
    #>  call| expect_equal(length(bb), 5)
    expect_equal(class(bb$weighted)[1],"igraph")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$weighted)[1], "igraph")
    expect_equal(class(bb$backbone)[1],"igraph")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$backbone)[1], "igraph")
    expect_equal(class(bb$pvalues$upper)[1],"matrix")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$pvalues$upper)[1], "matrix")
    expect_equal(class(bb$narrative)[1],"character")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$narrative)[1], "character")
    expect_equal(class(bb$call)[1],"call")
    #> ----- PASSED      : <-->
    #>  call| expect_equal(class(bb$call)[1], "call")

    bb <- backbone_from_weighted(W, model = "disparity", alpha = 0.25)            #Extract unweighted disparity igraph
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### LANS

    bb <- backbone_from_weighted(W, model = "lans", alpha = 0.25)                 #Extract unweighted lans igraph
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### MLF

    bb <- backbone_from_weighted(W, model = "mlf", alpha = 0.25)                  #Extract unweighted mlf igraph
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

### Global

    threshold <- mean(E(W)$weight) + sd(E(W)$weight)              #Use mean + sd as edge threshold
    bb <- backbone_from_weighted(W, model = "global", parameter = threshold)      #Extract unweighted global igraph
    expect_true(is(bb,"igraph"))                                                  #Returns as igraph
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(bb, "igraph"))
    expect_identical(vertex_attr_names(bb), c("agent_attrib"))            #Contains correct vertex attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(vertex_attr_names(bb), c("agent_attrib"))
    expect_identical(edge_attr_names(bb), c("oldweight"))                 #Contains correct edge attributes
    #> ----- PASSED      : <-->
    #>  call| expect_identical(edge_attr_names(bb), c("oldweight"))
    expect_true(modularity(bb, c(rep(1,10), rep(2,10), rep(3,10))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(modularity(bb, c(rep(1, 10), rep(2, 10), rep(3, 10))) >
    #>  call| 0.5)

# `backbone_from_unweighted()`

## `.escore()`

    A <- matrix(sample(c(0:1), 100, replace = TRUE),10,10)  #A binary, square, symmetric matrix
    diag(A) <- 0
    A <- pmax(A, t(A))

    test <- backbone:::.escore(A, "random")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "betweenness")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0))  #All values are 0 or larger
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "triangles")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test%%1==0))  #All values are integers
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test%%1 == 0))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "jaccard")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "dice")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "quadrangles")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test%%1==0))  #All values are integers
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test%%1 == 0))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "quadrilateral")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "degree")
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test%%1==0))  #All values are integers
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test%%1 == 0))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "meetmin")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "geometric")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

    test <- backbone:::.escore(A, "hypergeometric")
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A == 0] == 0))

## `.normalize()`

    A1 <- matrix(sample(c(0,0,0,1,2,3), 100, replace = TRUE),10,10)  #A weighted, square matrix
    diag(A1) <- 0
    A2 <- pmax(A1, t(A1))  #A weighted, square, symmetric matrix

    test <- backbone:::.normalize(A1, "rank")
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test%%1==0))  #All values are integers
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test%%1 == 0))
    expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A1 == 0] == 0))
    test <- backbone:::.normalize(A2, "rank")
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test%%1==0))  #All values are integers
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test%%1 == 0))
    expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A2 == 0] == 0))

    test <- backbone:::.normalize(A1, "embeddedness")
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A1 == 0] == 0))
    test <- backbone:::.normalize(A2, "embeddedness")
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test>=0 & test<=1))  #All values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test >= 0 & test <= 1))
    expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A2 == 0] == 0))

## `.filter()`

    A1 <- matrix(sample(c(0:10), 2500, replace = TRUE),50,50)  #A weighted, square matrix
    diag(A1) <- 0
    A2 <- A1; A2[upper.tri(A2)] <- t(A1)[upper.tri(A1)]  #Symmetrize using lower triangle

    test <- backbone:::.filter(A1, "threshold", 2)
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test %in% c(0, 1)))
    expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A1 == 0] == 0))
    expect_true(all(test[A1 <= 2] == 0))  #If edge is below threshold, it is missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A1 <= 2] == 0))
    test <- backbone:::.filter(A2, "threshold", 2)
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test %in% c(0, 1)))
    expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A2 == 0] == 0))
    expect_true(all(test[A2 <= 2] == 0))  #If edge is below threshold, it is missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A2 <= 2] == 0))

    test <- backbone:::.filter(A1, "proportion", .5)
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test %in% c(0, 1)))
    expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A1 == 0] == 0))
    sum(test!=0) / sum(A1!=0)
    #> [1] 0.5797753
    expect_true((sum(test!=0) / sum(A1!=0)) > 0.3 & (sum(test!=0) / sum(A1!=0)) < 0.7)  #Should keep 30-70% of original edges on average
    #> ----- PASSED      : <-->
    #>  call| expect_true((sum(test != 0)/sum(A1 != 0)) > 0.3 & (sum(test !=
    #>  call| 0)/sum(A1 != 0)) < 0.7)
    test <- backbone:::.filter(A2, "proportion", .5)
    expect_true(isSymmetric(test))  #Output is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test))
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test %in% c(0, 1)))
    expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A2 == 0] == 0))
    expect_true((sum(test!=0) / sum(A2!=0)) > 0.3 & (sum(test!=0) / sum(A2!=0)) < 0.7)  #Should keep 30-70% of original edges on average
    #> ----- PASSED      : <-->
    #>  call| expect_true((sum(test != 0)/sum(A2 != 0)) > 0.3 & (sum(test !=
    #>  call| 0)/sum(A2 != 0)) < 0.7)

    test <- backbone:::.filter(A1, "degree", .5)
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test %in% c(0, 1)))
    expect_true(all(test[A1 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A1 == 0] == 0))
    test <- backbone:::.filter(A2, "degree", .5)
    expect_true(all(diag(test)==0))  #Diagonal contains 0s
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(diag(test) == 0))
    expect_true(all(test %in% c(0,1)))  #All values are 0 or 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test %in% c(0, 1)))
    expect_true(all(test[A2 == 0] == 0))  #If edge is missing in original, also missing in result
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test[A2 == 0] == 0))

## Models

    #skeleton (no particular structure expected in backbone)
    U <- igraph::sample_sbm(60, matrix(c(.75,.25,.25,.25,.75,.25,.25,.25,.75),3,3), c(20,20,20))  #Unweighted graph with three hidden communities

    test <- backbone_from_unweighted(U, model = "skeleton", parameter = .5, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    test2 <- backbone_from_unweighted(U, model = "skeleton", parameter = .3, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #gspar
    test <- backbone_from_unweighted(U, model = "gspar", parameter = .5, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "gspar", parameter = .3, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #lspar
    test <- backbone_from_unweighted(U, model = "lspar", parameter = .5, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "lspar", parameter = .3, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #simmelian
    test <- backbone_from_unweighted(U, model = "simmelian", parameter = .5, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "simmelian", parameter = .7, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #jaccard
    test <- backbone_from_unweighted(U, model = "jaccard", parameter = .3, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "jaccard", parameter = .5, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #meetmin
    test <- backbone_from_unweighted(U, model = "meetmin", parameter = .5, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "meetmin", parameter = .7, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #geometric
    test <- backbone_from_unweighted(U, model = "geometric", parameter = .3, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "geometric", parameter = .5, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #hyper
    test <- backbone_from_unweighted(U, model = "hyper", parameter = .6, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "hyper", parameter = .8, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #quadrilateral
    test <- backbone_from_unweighted(U, model = "quadrilateral", parameter = .3, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(igraph::modularity(test$backbone, c(rep(1,20), rep(2,20), rep(3,20))) > .5)  #Backbone has high modularity
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::modularity(test$backbone, c(rep(1, 20), rep(2,
    #>  call| 20), rep(3, 20))) > 0.5)
    test2 <- backbone_from_unweighted(U, model = "quadrilateral", parameter = .5, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Larger parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

    #degree
    U <- igraph::sample_pa(n = 60, m = 3, directed = FALSE)  #A dense, scale-free network
    test <- backbone_from_unweighted(U, model = "degree", parameter = .5, return = "everything")
    expect_true(length(test)==4)  #Returned object has four elements
    #> ----- PASSED      : <-->
    #>  call| expect_true(length(test) == 4)
    expect_true(is(test$narrative,"character"))  #Narrative element is character class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$narrative, "character"))
    expect_true(is(test$call,"call"))  #Call element is call class
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test$call, "call"))
    expect_true(all.equal(U,test$original))  #Original element matches starting graph
    #> ----- PASSED      : <-->
    #>  call| expect_true(all.equal(U, test$original))
    expect_false(igraph::is_weighted(test$backbone))  #Backbone is unweighted
    #> ----- PASSED      : <-->
    #>  call| expect_false(igraph::is_weighted(test$backbone))
    expect_true(igraph::gorder(test$backbone)==igraph::gorder(U))  #Backbone size matches original graph size
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::gorder(test$backbone) == igraph::gorder(U))
    expect_true(which.max(igraph::degree(U)) == which.max(igraph::degree(test$backbone)))  #Backbone preserves highest-degree node
    #> ----- PASSED      : <-->
    #>  call| expect_true(which.max(igraph::degree(U)) == which.max(igraph::degree(test$backbone)))
    expect_true(cor(igraph::degree(U),igraph::degree(test$backbone)) > 0.75)  #Backbone preserves degree distribution
    #> ----- PASSED      : <-->
    #>  call| expect_true(cor(igraph::degree(U), igraph::degree(test$backbone)) >
    #>  call| 0.75)
    test2 <- backbone_from_unweighted(U, model = "degree", parameter = .2, return = "everything")
    expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))  #Smaller parameter yields more sparsification
    #> ----- PASSED      : <-->
    #>  call| expect_true(igraph::edge_density(test$backbone) > igraph::edge_density(test2$backbone))

# Null model functions

## `.sdsm()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.sdsm(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$upper))))
    expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$lower))))
    expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$upper))
    expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$lower))
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.sdsm_ec()`

    M <- rbind(c(10,0,1),c(0,11,0),c(1,0,1))
    test <- backbone:::.sdsm_ec(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$upper))))
    expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$lower))))
    expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$upper))
    expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$lower))
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.fixedrow()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.fixedrow(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$upper))))
    expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$lower))))
    expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$upper))
    expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$lower))
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.fixedcol()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.fixedcol(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$upper))))
    expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$lower))))
    expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$upper))
    expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$lower))
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.fixedfill()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.fixedfill(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$upper))))
    expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$lower))))
    expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$upper))
    expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$lower))
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.fdsm()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.fdsm(M, signed = TRUE, missing_as_zero = TRUE, alpha = 0.05, mtc = "none", trials = 1000)
    #> Constructing edges' Monte Carlo p-values
    #>   |                                                                              |                                                                      |   0%  |                                                                              |                                                                      |   1%  |                                                                              |=                                                                     |   1%  |                                                                              |=                                                                     |   2%  |                                                                              |==                                                                    |   2%  |                                                                              |==                                                                    |   3%  |                                           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    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(is.na(diag(test$upper))))  #Upper-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$upper))))
    expect_true(all(is.na(diag(test$lower))))  #Lower-tail diagonal is missing
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(is.na(diag(test$lower))))
    expect_true(isSymmetric(test$upper))  #Upper-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$upper))
    expect_true(isSymmetric(test$lower))  #Lower-tail is symmetric
    #> ----- PASSED      : <-->
    #>  call| expect_true(isSymmetric(test$lower))
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.disparity()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.disparity(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.lans()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.lans(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

## `.mlf()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- backbone:::.mlf(M, signed = TRUE, missing_as_zero = TRUE)
    test$upper <- round(test$upper,3)
    test$lower <- round(test$lower,3)
    expect_true(is(test, "list") & length(test)==2)  #Output is a two-item list
    #> ----- PASSED      : <-->
    #>  call| expect_true(is(test, "list") & length(test) == 2)
    expect_true(all(test$upper[upper.tri(test$upper)]>=0 & test$upper[upper.tri(test$upper)]<=1))  #Upper-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$upper[upper.tri(test$upper)] >= 0 & test$upper[upper.tri(test$upper)] <=
    #>  call| 1))
    expect_true(all(test$lower[upper.tri(test$lower)]>=0 & test$lower[upper.tri(test$lower)]<=1))  #Lower-tail p-values between 0 and 1
    #> ----- PASSED      : <-->
    #>  call| expect_true(all(test$lower[upper.tri(test$lower)] >= 0 & test$lower[upper.tri(test$lower)] <=
    #>  call| 1))

# Utility functions

## `bicm()`

    M <- rbind(c(0,0,1),c(0,1,0),c(1,0,1))
    test <- round(bicm(M),3)
    expect_equal(test, rbind(c(.216,.216,.568),c(.216,.216,.568),c(.568,.568,.863)))  #BiCM probabilities
    #> ----- PASSED      : <-->
    #>  call| expect_equal(test, rbind(c(0.216, 0.216, 0.568), c(0.216, 0.216,
    #>  call| 0.568), c(0.568, 0.568, 0.863)))

## `fastball()`

    M <- matrix(rbinom(100*1000,1,0.5),100,1000)
    test <- fastball(M)
    expect_equal(rowSums(test), rowSums(M))  #Row sums match
    #> ----- PASSED      : <-->
    #>  call| expect_equal(rowSums(test), rowSums(M))
    expect_equal(colSums(test), colSums(M))  #Column sums match
    #> ----- PASSED      : <-->
    #>  call| expect_equal(colSums(test), colSums(M))

## `.retain()`

    upper <- rbind(c(.01,.02,.03),  #Unsigned
                   c(.05,.06,.07),
                   c(0.5,0.6,0.7))
    p <- list(upper = upper)
    test <- backbone:::.retain(p, alpha = 0.05, mtc = "none")
    expect_equal(test, rbind(c(0,1,1),
                             c(0,0,0),
                             c(0,0,0)))
    #> ----- PASSED      : <-->
    #>  call| expect_equal(test, rbind(c(0, 1, 1), c(0, 0, 0), c(0, 0, 0)))

    upper <- rbind(c(.01,.02,.03),  #Signed
                   c(.05,.06,.07),
                   c(0.5,0.6,0.7))
    lower <- rbind(c(0.5,0.6,0.7),
                   c(.05,.06,.07),
                   c(.01,.02,.03))
    p <- list(lower = lower, upper = upper)
    test <- backbone:::.retain(p, alpha = 0.1, mtc = "none")  #Higher alpha because this is a two-tailed test
    expect_equal(test, rbind(c(0,1,1),
                             c(0,0,0),
                             c(-1,-1,0)))
    #> ----- PASSED      : <-->
    #>  call| expect_equal(test, rbind(c(0, 1, 1), c(0, 0, 0), c(-1, -1, 0)))