Table of Contents
This vignette uses empirical data on bill sponsorship behaviors in
the 108th U.S. Senate to illustrate the use of the
backbone package to extract sparse, unweighted networks
from weighted projections, weighted networks, and unweighted networks. For a more general
introduction to the backbone package, please see the Introduction to Backbone vignette.
The backbone package can be cited as:
Neal, Z. P. (2025). backbone: An R package to Extract Network Backbones. CRAN. https://doi.org/10.32614/CRAN.package.backbone
The backbone package, and the igraph
package that is also used in this example, can be loaded using:
In the U.S. Congress, the first step in the passage of new legislation is the introduction of a bill in the chamber. When a bill is introduced, it has one or more legislators who act as “sponsors” and express their initial support. Examining bill sponsorship is useful because although formal votes are taken on only a few bills, all bills have sponsors. Patterns of co-sponsorship (i.e., legislators who sponsor bills together) can provide insight into the structure of collaboration and political alliances (Neal 2014).
This example uses data on bill sponsorship activity in the
108th U.S. Senate, which ran from 3 January 2003 until 3
January 2005. The data was generated using
incidentally::incidence.from.congress(session = 108, types = "s", format = "igraph")
(Neal
2022). It is bundled with the backbone package and can be
loaded and described using:
data(senate108)
senate108
#> IGRAPH 42f8dc2 UN-B 3135 19060 --
#> + attr: name (v/c), type (v/l), party (v/c), state (v/c), last (v/c),
#> | id (v/c), color (v/c), introduced (v/c), title (v/c), area (v/c),
#> | sponsor.party (v/c), partisan (v/n), status (v/c)
#> + edges from 42f8dc2 (vertex names):
#> [1] Sen. Akaka, Daniel K. [D-HI]--S144 Sen. Akaka, Daniel K. [D-HI]--S1612
#> [3] Sen. Akaka, Daniel K. [D-HI]--S1379 Sen. Akaka, Daniel K. [D-HI]--S504
#> [5] Sen. Akaka, Daniel K. [D-HI]--S1709 Sen. Akaka, Daniel K. [D-HI]--S1946
#> [7] Sen. Akaka, Daniel K. [D-HI]--S2949 Sen. Akaka, Daniel K. [D-HI]--S685
#> [9] Sen. Akaka, Daniel K. [D-HI]--S1786 Sen. Akaka, Daniel K. [D-HI]--S2011
#> [11] Sen. Akaka, Daniel K. [D-HI]--S2001 Sen. Akaka, Daniel K. [D-HI]--S1143
#> + ... omitted several edgessenate108 is a bipartite network stored as an
igraph object. The network contains 100 Senators (agents)
and 3035 bills (artifacts), where senators are connected to the bills
they sponsored. In addition to the network’s structure, the object also
contains attributes of both the Senators (e.g., party affilition) and
bills (e.g., title).
The network’s indicence matrix captures which Senators sponsored which bills:
as_biadjacency_matrix(senate108)[1:5,1:5]
#> S144 S1612 S1379 S504 S1709
#> Sen. Akaka, Daniel K. [D-HI] 1 1 1 1 1
#> Sen. Allard, Wayne [R-CO] 0 0 0 0 0
#> Sen. Allen, George [R-VA] 0 0 1 0 0
#> Sen. Alexander, Lamar [R-TN] 0 0 0 1 0
#> Sen. Baucus, Max [D-MT] 1 0 1 0 0For example, Senators Akaka and Baucus both sponsored S.144, while Senators Allard, Allen, and Alexander did not. Two features of the indicence matrix – the row sums and column sums – are particularly important:
rowSums(as_biadjacency_matrix(senate108))[1:5]
#> Sen. Akaka, Daniel K. [D-HI] Sen. Allard, Wayne [R-CO]
#> 188 120
#> Sen. Allen, George [R-VA] Sen. Alexander, Lamar [R-TN]
#> 203 99
#> Sen. Baucus, Max [D-MT]
#> 188
colSums(as_biadjacency_matrix(senate108))[1:5]
#> S144 S1612 S1379 S504 S1709
#> 13 11 76 38 20The row sums represent the Senator degrees, and thus indicate the number of bills that each Senator sponsored. For example, Sen. Allen sponsored more than twice as many bills as Sen. Alexander. The column sums represent the bill degrees, and thus indicate the number of of sponsors on each bill. For example, bill S1379 was quite popular with 76 sponsors, while bill S1612 was unpopular with only 11 sponsors.
From these data, it is possible to construct a weighted projection, which is a unipartite network in which Senators are connected to each other if they co-sponsored bills together, where the edge weights record the number of bills they co-sponsored:
projection <- bipartite_projection(senate108, which = "false")
V(projection)$name <- V(projection)$last #Use only last names as node labels
as_adjacency_matrix(projection, attr = "weight")[1:5,1:5]
#> 5 x 5 sparse Matrix of class "dgCMatrix"
#> AKAKA ALLARD ALLEN Alexander BAUCUS
#> AKAKA . 14 38 19 37
#> ALLARD 14 . 41 22 24
#> ALLEN 38 41 . 37 31
#> Alexander 19 22 37 . 17
#> BAUCUS 37 24 31 17 .
plot(projection, vertex.label = NA, vertex.frame.color = NA, vertex.size = 3, edge.width = E(projection)$weight^.1, edge.color = rgb(0,0,0,.1), main = "Weighted Projection")
The weighted projection indicates that, for example, Allard and Allen sponsored 41 bills together, while Akaka and Allard sponsored only 14 bills together. Although these edge weights may provide some information into the frequency of collaboration between Senators, or the strength of their alliance, the density and weights make this network difficult to analyze and visualize. Thus, it can be useful to instead focus on a backbone, which retains only the most “important” edges in a sparser unweighted network.
Because this network is a weighted projection, and the original
bipartite network is available, we can extract the backbone using
backbone_from_projection(). The backbone extraction models
implemented in this function are powerful because they take advantage of
information that is contained in the bipartite network (e.g., the row
sums and column sums), but that are missing in the weighted projection
itself.
This function implements multiple models, but the Stochastic Degree Sequence Model (SDSM) (Neal, Domagalski, and Sagan 2021) offers a balance of statistical power and computational efficiency. The SDSM performs a statistical test on each edge weight, asking whether it is statistically significantly larger than expected if the indicence matrix were random, but with the same expected row and column sums. In this context, for each edge it asks: “Did these two Senators sponsor more bills together than would be expected if they each sponsors bills randomly, but if they sponsored the same number of bills on average, and if each bill received the same number of sponsors on average?”
We can extract the SDSM backbone at the conventional
alpha = 0.05 level of statistical significance, then plot
it:
bb1 <- backbone_from_projection(senate108, model = "sdsm", alpha = 0.05, narrative = TRUE)
#> We used the backbone package for R (v3.0.0; Neal, 2025) to extract the unweighted backbone of the weighted projection of a bipartite network containing 100 agents and 3035 artifacts. An edge was retained in the backbone if its weight was statistically significant (alpha = 0.05) using the stochastic degree sequence model (SDSM; Neal, Domagalski, and Sagan, 2021), which reduced the number of edges by 82.93%.
#>
#> Neal, Z. P. 2025. backbone: An R Package to Extract Network Backbones. CRAN. https://doi.org/10.32614/CRAN.package.backbone
#>
#> Neal, Z. P., Domagalski, R., and Sagan, B. (2021). Comparing Alternatives to the Fixed Degree Sequence Model for Extracting the Backbone of Bipartite Projections. Scientific Reports, 11, 23929. https://doi.org/10.1038/s41598-021-03238-3
layout <- layout_nicely(bb1) #Get layout for backbone (we'll use it later)
plot(bb1, vertex.label = NA, vertex.frame.color = NA, vertex.size = 3, edge.color = rgb(0,0,0,.1), layout = layout, main = "SDSM Backbone")
Importantly, we must supply the original bipartite network,
senate108, to this function. Because we specified
narrative = TRUE, this function also displays a narrative
description of the backbone extraction process with the associated
references; this text can be used in a manuscript to ensure a complete
description. Unlike the weighted projection, the SDSM backbone clearly
captures the partisan polarization known to structure interactions in
the U.S. Senate. Republicans (red nodes) primarily collaborate with
other Republicans, and Democrats (blue nodes) primarily collaborate with
other Democrats. Additionally, it illustrates that there are a few
bipartisan Senators (nodes bridging the two communities), as well as a
few conservative-leaning Democrats (blue nodes in the red community) and
liberal-leaning Republicans (red nodes in the blue community).
We can also extract a signed SDSM backbone by specifying
signed = TRUE.
bb1_signed <- backbone_from_projection(senate108, model = "sdsm", alpha = 0.1, narrative = TRUE, signed = TRUE)
#> We used the backbone package for R (v3.0.0; Neal, 2025) to extract the signed backbone of the weighted projection of a bipartite network containing 100 agents and 3035 artifacts. An edge was retained in the backbone if its weight was statistically significant (alpha = 0.1) using the stochastic degree sequence model (SDSM; Neal, Domagalski, and Sagan, 2021), which reduced the number of edges by 52.67%.
#>
#> Neal, Z. P. 2025. backbone: An R Package to Extract Network Backbones. CRAN. https://doi.org/10.32614/CRAN.package.backbone
#>
#> Neal, Z. P., Domagalski, R., and Sagan, B. (2021). Comparing Alternatives to the Fixed Degree Sequence Model for Extracting the Backbone of Bipartite Projections. Scientific Reports, 11, 23929. https://doi.org/10.1038/s41598-021-03238-3
E(bb1_signed)$color <- "green" #Make all edges green
E(bb1_signed)$color[which(E(bb1_signed)$sign == -1)] <- rgb(1,0,0,.05) #Make negative edges transparent red
plot(bb1_signed, vertex.label = NA, vertex.frame.color = NA, vertex.size = 3, layout = layout, main = "SDSM Signed Backbone")
A signed backbone retains statistically significantly strong edges as
positive and retains statistically significantly weak edges as negative.
We use signed = 0.1 because extracting a signed backbone
implies a two-tailed test; this makes the positive edges in the signed
backbone comparable to the edges in a non-signed backbone extracted at
the alpha = 0.05 level earlier. The signed backbone
highlights an even stronger form of polarization, illustrating not only
that collaboration (green edges) exists within party, but that avoidance
(red edges) occurs between parties.
In some cases, we may have a weighted projection, but do not have
access to the original bipartite data. In other cases, we may have a
weighted network where the edge weights were measured directly, and were
not generated via projection. In these cases, we can extract the
backbone using backbone_from_weighted(). The backbone
extraction models implemented in this function consider local variation
in degrees and edge weights, and thereby can preserve key structural
features even in networks where edge weights heterogeneous or
multi-scalar.
This function implements multiple models, but the Disparity Filter (Serrano, Boguná, and Vespignani 2009) is among the most widely used, and is typically chosen as the default unless there are specific reasons to use a different model. We can extract the Disparity Filter backbone, then plot it:
bb2 <- backbone_from_weighted(projection, model = "disparity", alpha = 0.2, narrative = TRUE)
#> We used the backbone package for R (v3.0.0; Neal, 2025) to extract the unweighted backbone of a weighted network containing 100 nodes. An edge was retained in the backbone if its weight was statistically significant (alpha = 0.2) using the disparity filter (Serrano et al., 2009), which reduced the number of edges by 83.29%.
#>
#> Neal, Z. P. 2025. backbone: An R Package to Extract Network Backbones. CRAN. https://doi.org/10.32614/CRAN.package.backbone
#>
#> Serrano, M. A., Boguna, M., & Vespignani, A. (2009). Extracting the multiscale backbone of complex weighted networks. Proceedings of the National Academy of Sciences, 106, 6483-6488. https://doi.org/10.1073/pnas.0808904106
plot(bb2, vertex.label = NA, vertex.frame.color = NA, vertex.size = 3, edge.color = rgb(0,0,0,.1), main = "Disparity Filter Backbone")
Notably, we supply the weighted network projection (not
the bipartite network senate108) to this function. Because
the weighted network is missing information that was contained in the
original bipartite network (i.e., the Senator and bill degrees), it is
more difficult for the Disparity Filter to identify important edges. To
adjust for this, we use a more liberal alpha = 0.2 level of
statistical significance.
The Disparity Filter backbone also captures the partisan polarization in the U.S. Senate. Here, it is reflected in a dense community of collaborating Democrats, and a more diffuse community of collaborating Republicans. In the 108th U.S. Senate, Republicans held the majority. Thus, this pattern is consistent with prior findings that members of the majority can stake out more extreme positions and exhibit “strategic disloyalty”, while members of the minority party must “circle the wagons” to consolidate their power (Kirkland and Slapin 2017; Neal 2020).
In some cases, we might only have an unweighted network, but one that
is too dense to analyze or visualize. In these cases, we can extract the
backbone using backbone_from_unweighted(). The backbone
extraction models implemented in this function follow a “recipe” of
first assigning each edge a score based on their role in the structure,
normalizing those edge scores, then filtering based on the scores.
This function implements multiple models, but the Local Sparsification (L-Spar) model (Satuluri, Parthasarathy, and Ruan 2011) is one option that is designed to preserve clustering, which we suspect exists in the U.S. Senate in the form of partisan polarization. This model scores edges using the Jaccard coefficient of its endpoints’ neighborhoods, normalizes these scores by ranking them from the perspective of each node, then chooses edges to retain based on the degree of each node and a sparisfication `parameter’.
Suppose we did not have the weighted projection, but instead only had an unweighted network in which Senators were connected if they sponsored 25 or more of the same bills:
unweighted <- delete_edges(projection, which(E(projection)$weight < 25)) #Delete low-weight edges
unweighted <- delete_edge_attr(unweighted, "weight") #Delete edge weights to obtain an unweighted network
unweighted <- delete_vertices(unweighted, which(degree(unweighted) < 1)) #Delete isolated nodes
edge_density(unweighted) #Compute density
#> [1] 0.698927
plot(unweighted, vertex.label = NA, vertex.frame.color = NA, vertex.size = 3, edge.color = rgb(0,0,0,.25), main = "Unweighted Network")
The unweighted network is so dense that it would be difficult to analyze and uncover any meaningful patterns, or to visualize and see any meaningful structure. However, we can extract the L-Spar backbone from this unweighted network, then plot the backbone:
bb3 <- backbone_from_unweighted(unweighted, model = "lspar", parameter = .5, narrative = TRUE)
#> We used the backbone package for R (v3.0.0; Neal, 2025) to extract the unweighted backbone of an unweighted network containing 98 nodes. Edges were selected for retention in the backbone using Satuluri et al's (2011) Local Sparsification backbone model (filtering parameter = 0.5), which reduced the number of edges by 79.68%.
#>
#> Neal, Z. P. 2025. backbone: An R Package to Extract Network Backbones. CRAN. https://doi.org/10.32614/CRAN.package.backbone
#>
#> Satuluri, V., Parthasarathy, S., & Ruan, Y. (2011, June). Local graph sparsification for scalable clustering. In Proceedings of the 2011 ACM SIGMOD International Conference on Management of data (pp. 721-732). https://doi.org/10.1145/1989323.1989399
plot(bb3, vertex.label = NA, vertex.frame.color = NA, vertex.size = 3, edge.color = rgb(0,0,0,.25), main = "L-Spar Backbone")
Even based on the very limited information still left in this unweighted network (after the bipartite original has been lost, and the weights in the projection have been lost), the L-Spar backbone still captures the partisan structure of the U.S. Senate.
This brief empirical example illustrates the utility of the backbone package. There are a couple broad lessons to consider when extracting a network backbone:
Different backbone extraction models use different amounts of
information to decide which edges to retain. Use a backbone extraction
model that incorporates all of the available information. For example,
if the original network is a weighted projection, then extract the
backbone from the original bipartite network or hypergraph using
backbone_from_projection(). Only extract a backbone using
backbone_from_unweighted() if no information about edge
weights is available.
The backbone extraction model and its specifications impact the
backbone that is obtained. Therefore, clearly identify and cite the
backbone model that was used, and completely describe both the original
network and how the model was specified. This is simplified by
specifying narrative = TRUE, which automatically generates
and displays such a description.