Overview of the backbone package

Rachel Domagalski, Zachary Neal, Bruce Sagan

library(backbone)

Weighted Graphs and Backbones

Introduction

In a graph \(G\), edges are either present (i.e. \(G_{ij}=1\)) or absent (i.e. \(G_{ij}=0\)). However in a weighted or valued graph, edges can take a range of values that may capture such properties as the strength or capacity of the edge. Although weighted graphs contain a large amount of information, there are some cases (e.g. visualization, application of statistical models not developed for weighted graphs) where it is useful to reduce this information by focusing on an unweighted subgraph that contains only the most important edges. We call this subgraph the backbone of \(G\), which we denote as \(G’\). Extracting \(G’\) from \(G\) requires deciding which edges to preserve. This usually involves selecting a threshold \(T_{ij}\) such that edges are preserved if they are above the threshold (i.e. \(G_{ij}’=1\) if \(G_{ij} > T_{ij}\)), and omitted if they are below the threshold (i.e. \(G_{ij}’=0\) if \(G_{ij} < T_{ij}\)). It is also possible to extract a signed backbone by selecting upper \(T^+_{ij}\) and lower \(T^-_{ij}\) thresholds such that \(G_{ij}’=1\) if \(G_{ij} > T^+_{ij}\), \(G_{ij}’=-1\) if \(G_{ij} < T^-_{ij}\), and \(G_{ij}’=0\) if \(G_{ij} > T^-_{ij}\) and \(G_{ij} < T^+_{ij}\). The key to all backbone extraction methods lies in the selection of \(T\). The backbone package provides several different methods for selecting \(T\) and thus extracting \(G’\) from \(G\).

Example data

We outline the use of the backbone package with Davis, Gardner, and Gardner’s Southern Women Dataset (Davis, Gardner, and Gardner 1941), which can be accessed via (Repository 2006). This data takes the form of a bipartite graph \(B\) containing 18 women (rows) and 14 social events (columns) taking place over a nine month period. In \(B\), \(B_{ij} = 1\) if women \(i\) attended event \(j\), and otherwise is 0. Let’s take a look at the Davis dataset included in this package to see that it is bipartite.

data(davis) #load the dataset
op <- options(width = 100)
davis #view the dataset
#>           6/27 3/2 4/12 9/26 2/25 5/19 3/15 9/16 4/8 6/10 2/23 4/7 11/21 8/3
#> EVELYN       1   1    1    1    1    1    0    1   1    0    0   0     0   0
#> LAURA        1   1    1    0    1    1    1    1   0    0    0   0     0   0
#> THERESA      0   1    1    1    1    1    1    1   1    0    0   0     0   0
#> BRENDA       1   0    1    1    1    1    1    1   0    0    0   0     0   0
#> CHARLOTTE    0   0    1    1    1    0    1    0   0    0    0   0     0   0
#> FRANCES      0   0    1    0    1    1    0    1   0    0    0   0     0   0
#> ELEANOR      0   0    0    0    1    1    1    1   0    0    0   0     0   0
#> PEARL        0   0    0    0    0    1    0    1   1    0    0   0     0   0
#> RUTH         0   0    0    0    1    0    1    1   1    0    0   0     0   0
#> VERNE        0   0    0    0    0    0    1    1   1    0    0   1     0   0
#> MYRNA        0   0    0    0    0    0    0    1   1    1    0   1     0   0
#> KATHERINE    0   0    0    0    0    0    0    1   1    1    0   1     1   1
#> SYLVIA       0   0    0    0    0    0    1    1   1    1    0   1     1   1
#> NORA         0   0    0    0    0    1    1    0   1    1    1   1     1   1
#> HELEN        0   0    0    0    0    0    1    1   0    1    1   1     0   0
#> DOROTHY      0   0    0    0    0    0    0    1   1    0    0   0     0   0
#> OLIVIA       0   0    0    0    0    0    0    0   1    0    1   0     0   0
#> FLORA        0   0    0    0    0    0    0    0   1    0    1   0     0   0
options(op)

We see that our two sets of vertices are women and events attended.

A weighted graph \(G\) can be constructed from \(B\) via bipartite projection, where \(G = BB^T\) and \(G_{ij}\) contains the number of events that both woman \(i\) and woman \(j\) attended. Looking at the matrix of southern women and events attended above, we see that Evelyn and Charlotte have attended three of the same events. This means that \(G_{15} = 3\) in the projection, shown below.

davis%*%t(davis) #The projected davis dataset
#>           EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
#> EVELYN         8     6       7      6         3       4       3     3    3
#> LAURA          6     7       6      6         3       4       4     2    3
#> THERESA        7     6       8      6         4       4       4     3    4
#> BRENDA         6     6       6      7         4       4       4     2    3
#> CHARLOTTE      3     3       4      4         4       2       2     0    2
#> FRANCES        4     4       4      4         2       4       3     2    2
#> ELEANOR        3     4       4      4         2       3       4     2    3
#> PEARL          3     2       3      2         0       2       2     3    2
#> RUTH           3     3       4      3         2       2       3     2    4
#> VERNE          2     2       3      2         1       1       2     2    3
#> MYRNA          2     1       2      1         0       1       1     2    2
#> KATHERINE      2     1       2      1         0       1       1     2    2
#> SYLVIA         2     2       3      2         1       1       2     2    3
#> NORA           2     2       3      2         1       1       2     2    2
#> HELEN          1     2       2      2         1       1       2     1    2
#> DOROTHY        2     1       2      1         0       1       1     2    2
#> OLIVIA         1     0       1      0         0       0       0     1    1
#> FLORA          1     0       1      0         0       0       0     1    1
#>           VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
#> EVELYN        2     2         2      2    2     1       2      1     1
#> LAURA         2     1         1      2    2     2       1      0     0
#> THERESA       3     2         2      3    3     2       2      1     1
#> BRENDA        2     1         1      2    2     2       1      0     0
#> CHARLOTTE     1     0         0      1    1     1       0      0     0
#> FRANCES       1     1         1      1    1     1       1      0     0
#> ELEANOR       2     1         1      2    2     2       1      0     0
#> PEARL         2     2         2      2    2     1       2      1     1
#> RUTH          3     2         2      3    2     2       2      1     1
#> VERNE         4     3         3      4    3     3       2      1     1
#> MYRNA         3     4         4      4    3     3       2      1     1
#> KATHERINE     3     4         6      6    5     3       2      1     1
#> SYLVIA        4     4         6      7    6     4       2      1     1
#> NORA          3     3         5      6    8     4       1      2     2
#> HELEN         3     3         3      4    4     5       1      1     1
#> DOROTHY       2     2         2      2    1     1       2      1     1
#> OLIVIA        1     1         1      1    2     1       1      2     2
#> FLORA         1     1         1      1    2     1       1      2     2

In this vignette, we demonstrate using the backbone package to extract the backbone of \(G\), which involves deciding whether to preserve an edge between Evelyn and Charlotte in \(G’\), and similarly for all other edges in \(G\).

General Backbone Methods

In this section, we will describe backbone methods that can be applied to any weighted graph, whether the weights are present in a natively unipartite graph, or are the product of a bipartite projection (as is the case in our example data). All of the methods described can accept inputs of matrices, sparse matrices, igraph objects, edgelists, and network objects. For the sake of these examples, we use matrices.

Universal Backbone: universal( )

The simplest approach to backbone extraction applies a single threshold \(T\) to all edges, and is achieved using the universal() function. The universal() function allows the user to extract a binary backbone by selecting a single threshold \(T\), or extract a signed backbone by selecting upper and lower thresholds \(T^+\) and \(T^-\).

The universal( ) function has four parameters,

The function universal() returns a backbone object containing the backbone graph, with either signed (or binary) edge weights, and a data frame called summary, containing the model name (universal threshold), number of rows in M, skew of row sums of M, number of columns of M, skew of column sums of M, and running time. The universal() function can be used in a variety of different ways, demonstrated in the following examples. Using the davis dataset, if we input the projected matrix G <- davis%*%t(davis), we can use the universal threshold on the weighted matrix G. If we set an upper threshold of 0, then if two women have attended any event together (co-attendance > 0), there will be an edge between the two. We can plot this graph with the igraph package.

G <- davis%*%t(davis) #projected davis dataset, a weighted graph
universal_bb <- universal(G, upper = 0)
#> Warning in universal(G, upper = 0): The input data is treated as unipartite
universal_bb$backbone
#>           EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
#> EVELYN         0     1       1      1         1       1       1     1    1
#> LAURA          1     0       1      1         1       1       1     1    1
#> THERESA        1     1       0      1         1       1       1     1    1
#> BRENDA         1     1       1      0         1       1       1     1    1
#> CHARLOTTE      1     1       1      1         0       1       1     0    1
#> FRANCES        1     1       1      1         1       0       1     1    1
#> ELEANOR        1     1       1      1         1       1       0     1    1
#> PEARL          1     1       1      1         0       1       1     0    1
#> RUTH           1     1       1      1         1       1       1     1    0
#> VERNE          1     1       1      1         1       1       1     1    1
#> MYRNA          1     1       1      1         0       1       1     1    1
#> KATHERINE      1     1       1      1         0       1       1     1    1
#> SYLVIA         1     1       1      1         1       1       1     1    1
#> NORA           1     1       1      1         1       1       1     1    1
#> HELEN          1     1       1      1         1       1       1     1    1
#> DOROTHY        1     1       1      1         0       1       1     1    1
#> OLIVIA         1     0       1      0         0       0       0     1    1
#> FLORA          1     0       1      0         0       0       0     1    1
#>           VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
#> EVELYN        1     1         1      1    1     1       1      1     1
#> LAURA         1     1         1      1    1     1       1      0     0
#> THERESA       1     1         1      1    1     1       1      1     1
#> BRENDA        1     1         1      1    1     1       1      0     0
#> CHARLOTTE     1     0         0      1    1     1       0      0     0
#> FRANCES       1     1         1      1    1     1       1      0     0
#> ELEANOR       1     1         1      1    1     1       1      0     0
#> PEARL         1     1         1      1    1     1       1      1     1
#> RUTH          1     1         1      1    1     1       1      1     1
#> VERNE         0     1         1      1    1     1       1      1     1
#> MYRNA         1     0         1      1    1     1       1      1     1
#> KATHERINE     1     1         0      1    1     1       1      1     1
#> SYLVIA        1     1         1      0    1     1       1      1     1
#> NORA          1     1         1      1    0     1       1      1     1
#> HELEN         1     1         1      1    1     0       1      1     1
#> DOROTHY       1     1         1      1    1     1       0      1     1
#> OLIVIA        1     1         1      1    1     1       1      0     1
#> FLORA         1     1         1      1    1     1       1      1     0
universal_bb$summary
#>                           Model Summary
#> Input Class                      matrix
#> Model               Universal Threshold
#> Number of Rows                       18
#> Mean of Row Sums               40.72222
#> SD of Row Sums                 13.55948
#> Skew of Row Sums               -0.22725
#> Number of Columns                    18
#> Mean of Column Sums            40.72222
#> SD of Column Sums              13.55948
#> Skew of Column Sums            -0.22725
#> Running Time (secs)                0.96
graph <- igraph::graph_from_adjacency_matrix(universal_bb$backbone, mode = "undirected")
op <- par(mar=c(0,0,0,0))
lo <- igraph::layout_(graph, igraph::with_fr())
plot(graph, vertex.label = 1:18, layout = lo)

par(op)

We can also use the universal() function on the original bipartite data. When inputting bipartite data, we set parameter bipartite = TRUE. The bipartite matrix will be multiplied by its transpose before the threshold is applied. Below, we input the bipartite matrix davis with the same threshold values as before, returning the same backbone matrix.

universal_bb <- universal(davis, upper = 0, bipartite = TRUE)
universal_bb$summary
#>                           Model Summary
#> Input Class                      matrix
#> Model               Universal Threshold
#> Number of Rows                       18
#> Mean of Row Sums                4.94444
#> SD of Row Sums                  2.12747
#> Skew of Row Sums                0.13747
#> Number of Columns                    14
#> Mean of Column Sums             6.35714
#> SD of Column Sums               3.58645
#> Skew of Column Sums             0.77915
#> Running Time (secs)                   0
graph <- igraph::graph_from_adjacency_matrix(universal_bb$backbone, mode = "undirected")
op <- par(mar=c(0,0,0,0))
plot(graph, vertex.label = 1:18, layout = lo)

par(op)

To create a signed backbone, we can apply both an upper and lower threshold value. For instance, we could choose to retain a positive edge if the women attended more than 4 events together, and a negative edge if they attended less than 2 events together (co-attendance of 0 or 1 events). We can do this with the following code. Note that the returned backbone matrix now has both \(+1\) and \(-1\) values.

universal_bb <- universal(davis, upper = 4, lower = 2, bipartite = TRUE)
universal_bb$backbone
#>           EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
#> EVELYN         0     1       1      1         0       0       0     0    0
#> LAURA          1     0       1      1         0       0       0     0    0
#> THERESA        1     1       0      1         0       0       0     0    0
#> BRENDA         1     1       1      0         0       0       0     0    0
#> CHARLOTTE      0     0       0      0         0       0       0    -1    0
#> FRANCES        0     0       0      0         0       0       0     0    0
#> ELEANOR        0     0       0      0         0       0       0     0    0
#> PEARL          0     0       0      0        -1       0       0     0    0
#> RUTH           0     0       0      0         0       0       0     0    0
#> VERNE          0     0       0      0        -1      -1       0     0    0
#> MYRNA          0    -1       0     -1        -1      -1      -1     0    0
#> KATHERINE      0    -1       0     -1        -1      -1      -1     0    0
#> SYLVIA         0     0       0      0        -1      -1       0     0    0
#> NORA           0     0       0      0        -1      -1       0     0    0
#> HELEN         -1     0       0      0        -1      -1       0    -1    0
#> DOROTHY        0    -1       0     -1        -1      -1      -1     0    0
#> OLIVIA        -1    -1      -1     -1        -1      -1      -1    -1   -1
#> FLORA         -1    -1      -1     -1        -1      -1      -1    -1   -1
#>           VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
#> EVELYN        0     0         0      0    0    -1       0     -1    -1
#> LAURA         0    -1        -1      0    0     0      -1     -1    -1
#> THERESA       0     0         0      0    0     0       0     -1    -1
#> BRENDA        0    -1        -1      0    0     0      -1     -1    -1
#> CHARLOTTE    -1    -1        -1     -1   -1    -1      -1     -1    -1
#> FRANCES      -1    -1        -1     -1   -1    -1      -1     -1    -1
#> ELEANOR       0    -1        -1      0    0     0      -1     -1    -1
#> PEARL         0     0         0      0    0    -1       0     -1    -1
#> RUTH          0     0         0      0    0     0       0     -1    -1
#> VERNE         0     0         0      0    0     0       0     -1    -1
#> MYRNA         0     0         0      0    0     0       0     -1    -1
#> KATHERINE     0     0         0      1    1     0       0     -1    -1
#> SYLVIA        0     0         1      0    1     0       0     -1    -1
#> NORA          0     0         1      1    0     0      -1      0     0
#> HELEN         0     0         0      0    0     0      -1     -1    -1
#> DOROTHY       0     0         0      0   -1    -1       0     -1    -1
#> OLIVIA       -1    -1        -1     -1    0    -1      -1      0     0
#> FLORA        -1    -1        -1     -1    0    -1      -1      0     0

We can also choose a threshold that is a multiple of some function, such as mean, max, or min. The function is applied to the edge weights, and then multiplied by the upper and lower thresholds. Any \(G_{ij}\) values above the upper threshold are counted as a positive \(+1\) value in the backbone, and any below the lower threshold are counted as a negative \(-1\) value in the backbone. The following code will return a backbone where the positive edges indicate two women attended more than 1 standard deviation above the mean number of events and negative edges indicate two women attended less than 1 standard deviation below the mean number of events.

universal_bb <- universal(davis, 
                          upper = function(x)mean(x)+sd(x), 
                          lower=function(x)mean(x)-sd(x), 
                          bipartite = TRUE)

Here, the davis matrix has first been projected. Then, the standard deviation of the \(G_{ij}\) entries is calculated and added to (or subtracted from) to the mean of the \(G_{ij}\) values. This value is then used to threshold the projected matrix for the positive (or negative) entries.

Bipartite Projection Backbone Methods

(Neal 2014)

The methods described above can be applied to any weighted graph \(G\). In this section we describe methods that are designed for weighted graphs that are the result of bipartite projections. They differ from other methods because they take into account the information contained in the original bipartite graph \(B\). Specifically, these methods are conditioned on the bipartite graph’s two degree sequences: the row vertex degrees (i.e. row marginals) and column vertex degrees (i.e. column marginals). We compare the values of \(G_{ij} = (BB^T)_{ij}\) to the probability distributions that describe \(G^*_{ij} = (B^*B^{*T})_{ij}\) for all bipartite graphs \(B^*\) that satisfy the row and column vertex degree restrictions we choose.

The backbone package implements three ways to construct the distribution of times \(G^*_{ij}\): the hypergeometric distribution using hyperg(), the stochastic degree sequence model using sdsm(), and the fixed degree sequence model using fdsm().

For two of the models, hyperg() and sdsm(), the probability distribution is known for the chosen restrictions. The hyperg() model fixes the row marginals but allows the column marginals to vary. The distribution of the \(G^*_{ij}\) values in this case is given by the hypergeometric distribution. The sdsm() model approximately fixes the row and column marginals. This scenario is described by the Poisson binomial distribution.

Since the distribution that describes having fixed row and column marginals, as in fdsm() is unknown, we use the following algorithm: 1. Construct a random bipartite graph \(B^*\) that preserves (to varying extents, depending on the method) one or both degree sequences (Strona, Ulrich, and Gotelli 2018). 2. Project \(B^*\) (i.e. \(B^{*} B^{*T}\)) to obtain a random weighted bipartite projection \(G^*\) 3. Repeat steps 1 and 2 \(N\) times to build a distribution of \(G^*_{ij}\) 4. Compute the proportion of times \(G^*{ij}\) is equal to or above the corresponding entry \(G_{ij}\) and the proportion of times \(G_{ij}^*\) is equal to or below the corresponding entry \(G_{ij}\).

In all cases, after finding or empirically generating the distribution, the backbone can then be extracted for a given \(\alpha\) level using the backbone.extract() function. In this section, we first describe backbone.extract(), then illustrate its use in the context of hyperg(), fdsm(), and sdsm().

Extracting the Backbone: backbone.extract( )

The hyperg(), fdsm(), and sdsm() functions return two matrices: a positive matrix containing the probability that (or in the case of fdsm(), the proportion of times that) \(G^*_{ij}\) was greater than or equal to \(G_{ij}\), and a negative matrix containing the number of times \(G^*_{ij}\) was less than or equal to \(G_{ij}\). The backbone.extract() function allows the user to take these positive and negative matrices and return a binary or signed backbone.

The backbone.extract() function has six parameters: matrix, signed, a significance test value alpha, fwer, class, and narrative. The matrix parameter takes in the entire backbone object which is the output of hyperg(), fdsm(), or sdsm(). If the signed parameter is set to TRUE (the default) a signed backbone is returned, if FALSE a binary backbone is returned.

One can adjust the precision of the significance test, alpha, to refine their backbone results. The value of alpha should be between 0 and 1. The default is alpha=0.05. The statistical test is two-tailed with an area of alpha/2 in each tail.

Extracting the backbone of a bipartite projection involves applying this significance test to each of the N(N-1)/2 edges in the projection. Because each of these tests is independent, this can inflate the familywise error rate beyond the desired alpha. The fwer parameter, which is set to NULL by default, offers two ways to correct for this. When fwer = bonferroni, the classical Bonferroni correction is applied. When fwer = holm, the more powerful Holm-Bonferroni correction is applied.

If an entry in the positive matrix is less than or equal to the alpha/2 value, it is considered a +1 edge in the backbone. If an entry in the negative matrix is less than or equal to the alpha/2 value, it is considered a -1 edge in the backbone. All other values are 0 in the backbone graph. The backbone.extract() function will return a backbone graph of the same class and input parameter class. This can be one of “original”, “matrix”, “sparseMatrix”, “igraph”, “network”, or “edgelist”. If “original”, the backbone graph returned is of the same class as the data inputted in one of sdsm(), fdsm(), or hyperg().

When narrative is set to TRUE, backbone.extract() will provide text describing the generated backbone graph that could be included in a manuscript. This text includes citations for the applied backbone methods.

We demonstrate this function’s use in the following sections.

Hypergeometric Backbone: hyperg( )

The hypergeometric distribution compares an edge’s observed weight, \(G_{ij}\) to the distribution of weights expected in a projection obtained from a random bipartite network where the row vertex degrees are fixed, but the column vertex degrees are allowed to vary. This method of backbone extraction was developed in (Tumminello et al. 2011) and later in (Neal 2013), which showed that the distribution of \(G^*_{ij}\) when only vertex degrees are fixed is given by the hypergeometric distribution. For documentation on the hypergeometric distribution, see stats::phyper.

The hyperg() function has one parameter,

The hyperg() function returns a backbone object of the following:

Following the hyperg() function, the user must use the backbone.extract() function to find the backbone at a given significance value alpha.

hyperg <- hyperg(davis)
#> Finding the distribution using hypergeometric distribution
hyperg_bb <- backbone.extract(hyperg, signed = TRUE)

The Stochastic Degree Sequence Model: sdsm( )

The stochastic degree sequence model compares an edge’s observed weight, \(G_{ij}\) to the distribution of weights expected in a projection obtained from a random bipartite network where both the row vertex degrees and column vertex degrees are approximately fixed. This method of backbone extraction was developed in (Neal 2014). The construction of \(B^*\) involves a series of steps:

  1. The \(\beta\) parameters in \(Pr(B_{ij}=1) = \beta_0 +\beta_1 B_i + \beta_2 B_j +\beta_3 (B_i \times B_j)\) are estimated using a binomial regression (e.g. logit, probit, complementary log-log, etc.), where \(B_i\) and \(B_j\) are the row vertex and column vertex degrees in \(B\), respectively.
  2. The fitted parameters are used to compute the predicted probability that \(B_{ij}=1\).
  3. A matrix \(G^*\) is constructed via the Poisson-Binomial distribution, where the \((i,j)\) entry of \(G\) is the probability of an edge weight begin above or below the observed value in the projection of \(B\).

The sdsm( ) function has four parameters,

If the ‘model’ parameter is one of c(‘logit’, ‘probit’, ‘cauchit’, ‘log’, ‘cloglog’), then this model is used as a ‘link’ function for a binary outcome model conditioned on the row degrees and column degrees, as described by and . If the ‘model’ parameter is ‘oldlogit’, then a logit link function is used but the model is conditioned on the row degrees, column degrees, and their product. If ‘model = lpm’, a linear probability model is used. If ‘model = chi2’, a chi-squared model is used. If ‘trials’>0, and ‘model = curveball’, probabilities are computed by using function trials times. The proportion of each cell being 1 is used as its probability. If ‘model = polytope’, the function is used to find a matrix of probabilities that maximizes the entropy function, with same row and column sums.

The probability of edge weights being above or below the observed values are computed using the Poisson Binomial distribution. These values are approximated using a Refined Normal Approximation (RNA method). This method is described by and used under GPL-2 license.

The sdsm() function returns a backbone object of the following:

sdsm <- sdsm(davis)
#> Finding the distribution using SDSM with polytope model.

The backbone package allows for two different types of family-wise error rate correction: Holm-Bonferroni and Bonferroni. To use Holm-Bonferroni correction, add parameter fwer = "holm" to backbone.extract(), and to use Bonferroni correction, add fwer = "bonferroni". Note in this case, the Holm-Bonferroni is too restrictive and leaves us with no edges in our backbone graph.

sdsm_bb <- backbone.extract(sdsm, signed = FALSE, alpha = 0.1, fwer = "bonferroni") 
sdsm_bb
#>           EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
#> EVELYN         0     0       0      0         0       0       0     0    0
#> LAURA          0     0       0      0         0       0       0     0    0
#> THERESA        0     0       0      0         0       0       0     0    0
#> BRENDA         0     0       0      0         0       0       0     0    0
#> CHARLOTTE      0     0       0      0         0       0       0     0    0
#> FRANCES        0     0       0      0         0       0       0     0    0
#> ELEANOR        0     0       0      0         0       0       0     0    0
#> PEARL          0     0       0      0         0       0       0     0    0
#> RUTH           0     0       0      0         0       0       0     0    0
#> VERNE          0     0       0      0         0       0       0     0    0
#> MYRNA          0     0       0      0         0       0       0     0    0
#> KATHERINE      0     0       0      0         0       0       0     0    0
#> SYLVIA         0     0       0      0         0       0       0     0    0
#> NORA           0     0       0      0         0       0       0     0    0
#> HELEN          0     0       0      0         0       0       0     0    0
#> DOROTHY        0     0       0      0         0       0       0     0    0
#> OLIVIA         0     0       0      0         0       0       0     0    0
#> FLORA          0     0       0      0         0       0       0     0    0
#>           VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
#> EVELYN        0     0         0      0    0     0       0      0     0
#> LAURA         0     0         0      0    0     0       0      0     0
#> THERESA       0     0         0      0    0     0       0      0     0
#> BRENDA        0     0         0      0    0     0       0      0     0
#> CHARLOTTE     0     0         0      0    0     0       0      0     0
#> FRANCES       0     0         0      0    0     0       0      0     0
#> ELEANOR       0     0         0      0    0     0       0      0     0
#> PEARL         0     0         0      0    0     0       0      0     0
#> RUTH          0     0         0      0    0     0       0      0     0
#> VERNE         0     0         0      0    0     0       0      0     0
#> MYRNA         0     0         0      0    0     0       0      0     0
#> KATHERINE     0     0         0      0    0     0       0      0     0
#> SYLVIA        0     0         0      0    0     0       0      0     0
#> NORA          0     0         0      0    0     0       0      0     0
#> HELEN         0     0         0      0    0     0       0      0     0
#> DOROTHY       0     0         0      0    0     0       0      0     0
#> OLIVIA        0     0         0      0    0     0       0      0     0
#> FLORA         0     0         0      0    0     0       0      0     0

The Fixed Degree Sequence Model: fdsm( )

The fixed degree sequence model compares an edge’s observed weight, \(G_{ij}\), to the distribution of weights expected in a projection obtained from a random bipartite network where both the row vertex degrees and column vertex degrees are fixed. This method of backbone extraction was developed in (Zweig and Kaufmann 2011), however the challenge lies in randomly sampling from the space of \(B^*\) with fixed degree sequences. The fdsm() function uses the curveball algorithm (Strona et al. 2014), which is proven to do so (Carstens 2015).

The fdsm( ) function has four parameters,

The fdsm() function returns a backbone object of the following:

We can find the backbone using the fixed degree sequence model as follows:

fdsm <- fdsm(davis, trials = 100, dyad=c(1,5))
#> Approximating the distribution using Curveball FDSM
#> Estimated time to complete is 4.5 secs
fdsm$dyad_values
#>   [1] 3 2 2 2 4 3 3 3 2 2 2 4 3 2 1 3 2 2 4 2 3 2 3 2 4 3 2 3 3 1 3 3 2 2 2 2 3
#>  [38] 4 1 3 4 4 2 1 3 2 3 2 1 4 2 2 3 2 3 3 2 1 3 3 2 3 3 2 3 2 3 2 3 2 2 4 2 2
#>  [75] 4 0 4 3 2 2 2 3 3 3 2 2 3 2 2 2 4 4 2 2 2 3 2 3 3 3
fdsm_bb <- backbone.extract(fdsm, signed = TRUE, alpha = 0.1)
fdsm_bb
#>           EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
#> EVELYN         0     0       0      0         0       0       0     0    0
#> LAURA          0     0       0      1         0       0       0     0    0
#> THERESA        0     0       0      0         0       0       0     0    0
#> BRENDA         0     1       0      0         0       0       0     0    0
#> CHARLOTTE      0     0       0      0         0       0       0     0    0
#> FRANCES        0     0       0      0         0       0       0     0    0
#> ELEANOR        0     0       0      0         0       0       0     0    0
#> PEARL          0     0       0      0         0       0       0     0    0
#> RUTH           0     0       0      0         0       0       0     0    0
#> VERNE          0     0       0      0         0       0       0     0    0
#> MYRNA          0     0       0      0         0       0       0     0    0
#> KATHERINE      0    -1       0     -1        -1       0       0     0    0
#> SYLVIA        -1    -1       0     -1         0       0       0     0    0
#> NORA          -1    -1       0     -1         0       0       0     0    0
#> HELEN         -1     0       0      0         0       0       0     0    0
#> DOROTHY        0     0       0      0         0       0       0     0    0
#> OLIVIA         0     0       0      0         0       0       0     0    0
#> FLORA          0     0       0      0         0       0       0     0    0
#>           VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
#> EVELYN        0     0         0     -1   -1    -1       0      0     0
#> LAURA         0     0        -1     -1   -1     0       0      0     0
#> THERESA       0     0         0      0    0     0       0      0     0
#> BRENDA        0     0        -1     -1   -1     0       0      0     0
#> CHARLOTTE     0     0        -1      0    0     0       0      0     0
#> FRANCES       0     0         0      0    0     0       0      0     0
#> ELEANOR       0     0         0      0    0     0       0      0     0
#> PEARL         0     0         0      0    0     0       0      0     0
#> RUTH          0     0         0      0    0     0       0      0     0
#> VERNE         0     0         0      0    0     0       0      0     0
#> MYRNA         0     0         1      0    0     0       0      0     0
#> KATHERINE     0     1         0      1    0     0       0      0     0
#> SYLVIA        0     0         1      0    0     0       0      0     0
#> NORA          0     0         0      0    0     0       0      0     0
#> HELEN         0     0         0      0    0     0       0      0     0
#> DOROTHY       0     0         0      0    0     0       0      0     0
#> OLIVIA        0     0         0      0    0     0       0      0     1
#> FLORA         0     0         0      0    0     0       0      1     0

The fdsm_props$dyad_values output is a list of the \(G_{1,5}^*\) values for each of the 100 trials, which in these data corresponds to the number of parties Evelyn and Charlotte would be expected to simultaneously attend if: (a) the number of parties attended by Evelyn was fixed, (b) the number of parties attended by Charlotte was fixed, and (c) the number of attendees at each party was fixed. Because we have provided both a positive and negative matrix, backbone.extract() returns a signed backbone matrix by conducting a two-tailed significance test in which alpha is \(0.05\) on each end of the distribution.

References

Carstens, C. J. 2015. “Proof of Uniform Sampling of Binary Matrices with Fixed Row Sums and Column Sums for the Fast Curveball Algorithm.” Physical Review E 91 (4). https://doi.org/10.1103/PhysRevE.91.042812.

Davis, Allison, Burleigh B Gardner, and Mary R Gardner. 1941. Deep South: A Social Anthropological Study of Caste and Class. University of Chicago Press. https://doi.org/10.1177/0002716242220001105.

Neal, Zachary. 2013. “Identifying Statistically Significant Edges in One-Mode Projections.” Social Network Analysis and Mining 3 (4): 915–24. https://doi.org/10.1007/s13278-013-0107-y.

———. 2014. “The Backbone of Bipartite Projections: Inferring Relationships from Co-Authorship, Co-Sponsorship, Co-Attendance and Other Co-Behaviors.” Social Networks 39 (October): 84–97. https://doi.org/10.1016/j.socnet.2014.06.001.

Repository, UCI Network Data. 2006. “Southern Women Data Set.” https://networkdata.ics.uci.edu/netdata/html/davis.html.

Strona, Giovanni, Domenico Nappo, Francesco Boccacci, Simone Fattorini, and Jesus San-Miguel-Ayanz. 2014. “A Fast and Unbiased Procedure to Randomize Ecological Binary Matrices with Fixed Row and Column Totals.” Nature Communications 5 (June): 4114. https://doi.org/10.1038/ncomms5114.

Strona, Giovanni, Werner Ulrich, and Nicholas J. Gotelli. 2018. “Bi-Dimensional Null Model Analysis of Presence-Absence Binary Matrices.” Ecology 99 (1): 103–15. https://doi.org/10.1002/ecy.2043.

Tumminello, Michele, Salvatore Miccichè, Fabrizio Lillo, Jyrki Piilo, and Rosario N. Mantegna. 2011. “Statistically Validated Networks in Bipartite Complex Systems.” PLOS ONE 6 (3): e17994. https://doi.org/10.1371/journal.pone.0017994.

Zweig, Katharina Anna, and Michael Kaufmann. 2011. “A Systematic Approach to the One-Mode Projection of Bipartite Graphs.” Social Network Analysis and Mining 1 (3): 187–218. https://doi.org/10.1007/s13278-011-0021-0.