Table 1.
Backbone extraction and sparsification models implemented by existing packages.
Fig 1.
Comparison of original and backbone networks using S3 plot generic.
Table 2.
Backbone extraction models implemented in backbone.
Fig 2.
Extracting a backbone from a weighted network.
(A) A toy weighted network with a multi-scale hub-and-spoke structure, where stronger edges are represented by thicker lines. (B) A backbone extracted from (A) using the mean edge weight as a global threshold, which only preserves high-weight edges. (C) A backbone extracted from (A) using the disparity filter, which preserves the hub-and-spoke structure.
Fig 3.
Extracting a backbone from a weighted projection.
(A) A toy bipartite network with three embedded communities. (B) The weighted projection of (A), where stronger edges are represented by thicker lines. (C) A backbone extracted from (B) using the stochastic degree sequence model (SDSM), which correctly preserves the known three communities. (D) A backbone extracted from (B) using the fixed degree sequence model (FDSM), which correctly preserves the known three communities.
Fig 4.
Extracting backbones from unweighted networks.
(A) An toy unweighted network with three embedded communities. (B) A backbone extracted from (A) using Local Sparsification (L-Spar), which preserves the three communities. (C) A toy unweighted network with embedded high-degree hubs. (D) A backbone extracted from (C) using Local Degree, which preserves the high-degree hubs.
Fig 5.
Backbones of bill co-sponsorship in the 108th U.S. Senate, where nodes representing Republican Senators are red and nodes representing Democratic Senators are blue.
If the source data represent a bipartite network (A1), then a backbone of the weighted projection can be extracted using a model such as the stochastic degree sequence model (SDSM, A2). If the source data represent a weighted network (B1), then a backbone can be extracted using a model such as the disparity filter (B2). If the source data represent an unweighted network, then a backbone can be extracted using a model such as local sparsification (L-Spar, C2). All backbones correctly capture the Senate’s partisan polarization, where Senators’ collaborations are clustered by party affiliation.
Table 3.
Comparison of models and empirical results.
Fig 6.
Advanced backbones of bill co-sponsorship in the 108th U.S. Senate, where nodes representing Republican Senators are red and nodes representing Democratic Senators are blue.
(A) A backbone extracted using the stochastic degree sequence model (SDSM), correcting for multiple tests by controlling the false discovery rate (FDR) at 0.05. (B) A signed backbone extracted using the stochastic degree sequence model (SDSM), where statistically significantly strong edges are preserved as positive (green) and statistically significantly weak edges are preserved as negative (red).