Build one network, or one network per time window, from bipartite links

The first step of the networkflow pipeline. Research data rarely come as a network: they come as a table of links between two kinds of entities, such as articles and the references they cite, authors and the papers they sign, or accounts and the hashtags they use. These functions project this two-mode (bipartite) relation onto the first kind of entity: two source_id entities (e.g. two articles) are linked when they share target_id entities (e.g. references).

build_network() builds one network from the whole table.
build_dynamic_networks() builds one network per time window (time_variable and time_window are required), which is the starting point of a dynamic analysis. It is build_network() repeated on slices of the corpus, with a few more arguments for time.

Write $a_{ij} = 1$ when source $i$ is linked to target $j$. The raw strength of the tie between sources $i$ and $k$ is the number of targets they share, $$c_{ik} = \sum_j a_{ij} a_{kj} = |R_i \cap R_k|,$$ with $R_i$ the set of targets of $i$. When sources are citing documents and targets their references, this is bibliographic coupling (Kessler 1963) . Swapping the roles (cited documents as source_id, citing documents as target_id) gives a co-citation network (Small 1973) .

Raw counts favour sources with many targets (articles with long bibliographies), and most pairs share a target or two by chance. The single method argument chooses how this problem is answered, among two families:

a similarity measure ("angle", "strength", "similarity", "jaccard", "vladutz") normalises the counts, as implemented in biblionetwork, and the pairs sharing at least edges_threshold targets are kept, with their weight;
a statistical backbone ("sdsm", "fdsm", "fixedfill", "fixedrow", "fixedcol") keeps only the ties that are stronger than expected under a null model of random co-occurrence, as implemented in the backbone package (Neal 2014, 2022) .

There is no default method: the choice changes the network, so it must be explicit. With verbose = TRUE the function says which method it uses and where it comes from. (The arguments projection_method, cooccurrence_method and model of earlier versions are replaced by method and give an error that explains the translation.)

Usage

build_dynamic_networks(
  nodes,
  directed_edges,
  source_id,
  target_id,
  document_id = NULL,
  projection_on = c("source", "target"),
  method,
  edges_threshold = 1,
  alpha = NULL,
  backbone_args = list(),
  time_variable = NULL,
  time_in = c("nodes", "edges"),
  time_unit = NULL,
  time_window = NULL,
  window_step = NULL,
  compute_size = FALSE,
  keep_singleton = FALSE,
  filter_components = FALSE,
  ...,
  verbose = TRUE
)

Arguments

nodes

A data frame with one row per source entity (e.g. one row per article) and its metadata (authors, year, title...). Not modified.

directed_edges

A data frame of bipartite links, one row per source_id -> target_id link (e.g. article -> cited reference). Duplicated links are counted once. Not modified.

source_id

Character scalar, the column identifying the entities that become the nodes of the network. It must exist in both nodes and directed_edges, with one row per value in nodes.

target_id

Character scalar, the column of directed_edges with the entities that create the links (e.g. references).

document_id

Optional column of directed_edges identifying the document through which each link exists. With it, source_id entities are not documents but entities that publish documents (authors, journals, institutions) and a reference cited by several documents of an entity counts several times, following biblionetwork::entity_network(). Only for the similarity measures.

projection_on

"source" (default) links the source_id entities that share target_id entities (bibliographic coupling: articles sharing references). "target" links the target_id entities that share source_id entities (co-citation: cited documents linked when they are cited by the same documents). With "target", nodes is keyed by target_id.

method

Required, no default. How the ties between sources are built; the origin of each method is given in the details. One of

the similarity measures of biblionetwork (biblionetwork::biblio_coupling(method = ), or biblionetwork::entity_network() with document_id): "angle", "strength", "similarity", "jaccard", "vladutz";
the null models of the statistical backbones of backbone::backbone_from_projection(): "sdsm", "fdsm", "fixedfill", "fixedrow", "fixedcol". They need alpha.

edges_threshold

For the similarity measures, the minimum number of shared targets for two sources to be linked (1 by default). Ignored by the backbone methods.

alpha

For the backbone methods, the significance level of the test of each tie (e.g. 0.05). Lower values keep fewer edges. Required for them, ignored by the similarity measures.

backbone_args

Optional list of further arguments for backbone::backbone_from_projection() (e.g. mtc = "holm" to correct for multiple tests). alpha given here overrides the argument of the function.

time_variable

Name of a column with the time of each node (or of each link, with time_in = "edges"): a number (e.g. a year), a Date, a POSIXct, or an ISO date written as text ("2001-03", "2001-03-15"). Years written as text ("2001") are numbers. Required by build_dynamic_networks() (without time, use build_network()).

time_in

"nodes" (default): time_variable is a column of nodes and a window selects the nodes of the period. "edges": time_variable is a column of directed_edges (e.g. the year of the citing document) and a window selects the links of the period, the nodes being those with a link in the window. Use "edges" for co-citation, whose period is the one of the citing documents, and for entities without a date of their own (authors).

time_unit

For dates only: the unit in which windows are counted, "year" (default), "quarter", "month", "week" (starting on Mondays) or "day". For numbers, the unit is the one of the variable and time_unit must stay NULL.

time_window

Whole number, the width of each window in time_units (or in units of the variable): 10 for ten years, 12 with time_unit = "month" for a year. Required by build_dynamic_networks().

window_step

Optional whole number, the number of units between the starts of two windows. NULL (default) gives successive windows (window_step = time_window); 1 slides the window one unit at a time (1975-1984, 1976-1985...); 3 with time_unit = "month" and a window of 12 gives a yearly window every quarter.

compute_size

Logical. If TRUE, adds a node_size column: the number of times each node is a target of the sources of the same window (for a coupling network, the number of citations received within the corpus). It requires nodes to contain the target_id column.

keep_singleton

Logical. If FALSE (default), nodes without any edge are removed.

filter_components

Logical. If TRUE, keeps only the main component(s) with filter_components().

...

Arguments passed to filter_components() (e.g. nb_components).

verbose

Logical. If TRUE (default), reports the steps.

Value

A tbl_graph (undirected) if time_variable is NULL; otherwise a list of tbl_graphs named after their window (e.g. "1975-1984"), in chronological order, whose nodes have a time_window column. Edges have a weight column for the structured method.

Details

The arguments to choose, case by case. projection_on says which side of the links becomes the nodes, and method says how the ties between them are built:

You want	Function and main arguments
A network of the whole corpus, ties weighted by a similarity measure (bibliographic coupling)	`build_network(method = "similarity", edges_threshold = )` (or another measure)
The same, with only the ties unlikely under a null model	`build_network(method = "sdsm", alpha = 0.05)` (or another backbone model)
One such network per period	`build_dynamic_networks(time_variable = , time_window = )`, plus `window_step` and `time_unit` if needed, with any `method`
Cited works linked when cited together (co-citation)	`projection_on = "target"`; for windows, `time_in = "edges"` with the time of the citing documents in `directed_edges`
Authors, journals or institutions linked through the references of their documents	`document_id =` the column of the document, with a similarity measure
Any other two-mode table (accounts and hashtags, members and boards)	the same arguments: nothing is specific to citations

The two families of method. They do not use the same arguments:

	Similarity measures	Statistical backbones
`method`	`"angle"`, `"strength"`, `"similarity"`, `"jaccard"`, `"vladutz"`	`"sdsm"`, `"fdsm"`, `"fixedfill"`, `"fixedrow"`, `"fixedcol"`
Idea	normalise the number of shared targets, then keep the pairs sharing at least `edges_threshold` targets	test, for each pair, whether the number of shared targets is larger than in random networks that keep some properties of the data, and keep the significant pairs
Edges	weighted (`weight`)	unweighted (each edge means "more than expected by chance")
Arguments used	`edges_threshold`, `document_id`	`alpha` (required), `backbone_args`
Ignored	`alpha`, `backbone_args`	`edges_threshold`; `document_id` is an error
Cost	fast, sparse matrices: seconds for tens of thousands of documents	a test for each pair: on a Scopus sample, about 20 seconds and 2 GB for 1,000 documents with `"sdsm"`; `"fdsm"` simulates random networks and needs `backbone_args = list(trials = )` to finish (about a minute for 100 documents with 1,000 trials)
Computed by	biblionetwork	the `backbone` package

Where the methods come from. The names are those of the packages that compute them, so that the documentation of each method can be found there:

`method`	What it is	Origin	Function behind
`"angle"`	cosine of the lists of targets	(Sen and Gan 1983)	`biblionetwork::coupling_angle()`
`"jaccard"`	shared targets over the union of the lists	(Sen and Gan 1983)	`biblionetwork::coupling_jaccard()`
`"strength"`	shared targets weighted by their rarity	(Shen et al. 2019)	`biblionetwork::coupling_strength()`
`"similarity"`	idf-weighted cosine	(Shen et al. 2019)	`biblionetwork::coupling_similarity()`
`"vladutz"`	Vladutz-Cook formula	(Vladutz and Cook 1984) , as reported by (Shen et al. 2019)	`biblionetwork::coupling_vladutz()`
`"sdsm"`, `"fdsm"`, `"fixedfill"`, `"fixedrow"`, `"fixedcol"`	null models of random bipartite networks	(Neal 2014, 2022; Domagalski et al. 2021)	`backbone::backbone_from_projection(model = )`

Similarity measures. With $L_i = |R_i|$ the number of targets of source $i$, $N$ the number of sources and $f_j$ the number of sources linked to target $j$, the five measures computed by biblionetwork (through biblionetwork::biblio_coupling(), or biblionetwork::entity_network() with document_id) are:

"angle", the cosine of Sen and Gan (1983) : $$w_{ik} = \frac{|R_i \cap R_k|}{\sqrt{L_i L_k}};$$
"jaccard", the relative coupling strength of Sen and Gan (1983) , a Jaccard index: the shared targets over the union of the two lists, $$w_{ik} = \frac{|R_i \cap R_k|}{|R_i \cup R_k|}.$$ Like the angle it only counts, but it is never above it and penalises more the pairs whose lists have very different lengths;
"strength" (Shen et al. 2019) , where a shared target counts less when it is linked to many sources, in the spirit of an inverse document frequency: $$w_{ik} = \frac{\sum_{j \in R_i \cap R_k} \log(N / f_j)}{L_i L_k};$$
"similarity", the idf-weighted cosine of Shen et al. (2019) : each source is described by a vector with one component per target, equal to $\sqrt{\log(N / f_j)}$, and the weight is the cosine of the vectors of two sources, with $S_i = \sum_{j \in R_i} \log(N / f_j)$ the weighted size of the bibliography: $$w_{ik} = \frac{\sum_{j \in R_i \cap R_k} \log(N / f_j)}{\sqrt{S_i S_k}}.$$ Like the coupling angle, it lies between 0 and 1; like the coupling strength, it discounts the targets linked to many sources;
"vladutz", the formula of Vladutz and Cook (1984) reported by Shen et al. (2019) , the ancestor of the coupling strength, with a much steeper discount ($1/f_j$ instead of $\log(N/f_j)$), so that it is almost driven by the rarest targets: $$w_{ik} = \frac{4}{L_i + L_k} \sum_{j \in R_i \cap R_k} \frac{1}{f_j}.$$

The measures fall into two families: those that only count the shared targets (angle, Jaccard) and those that give more weight to rare targets (strength, similarity, Vladutz-Cook). The angle and the Jaccard index give very similar rankings of the pairs in practice (Pearson correlation of 0.956 in the corpus of Shen et al. (2019) ), whereas the Vladutz-Cook formula departs markedly from the others there (correlations of about 0.6 to 0.7). It is provided mainly for comparison and replication. See the biblionetwork documentation of each measure (coupling_angle(), coupling_jaccard(), coupling_strength(), coupling_similarity(), coupling_vladutz()) for the intuition behind them. (Versions 0.1.1 and earlier of biblionetwork computed the similarity wrongly: biblionetwork 0.2.0 or later is required.)

Targets linked to a single source cannot create a tie and are dropped before the computation. The threshold applies to the raw count $|R_i \cap R_k|$, not to the weight: edges_threshold = 2 keeps the pairs of articles that share at least two references, a common choice to remove ties that one shared classic is enough to create. The measures differ in how they treat very cited targets and long target lists; in practice, clusters are often robust to the choice, but it is worth checking on one's own data.

Statistical backbones. Instead of fixing a threshold, the backbone approach asks, for each pair, whether the observed number of shared targets is larger than in random bipartite networks that preserve some properties of the observed one: the degrees of sources and targets on average ("sdsm", the stochastic degree sequence model) or exactly ("fdsm"), or only the total number of links ("fixedfill") (Neal 2014; Domagalski et al. 2021) . Edges are kept when the test is significant at level alpha; the resulting network is unweighted. This is slower, but each edge can be read as a tie unlikely to be due to chance.

Time windows. Time is counted in whole units: the values of a numeric variable, or years, quarters, months, weeks or days for dates. A window starting at unit $s$ and $w$ units wide (time_window) gathers the nodes (or links) with $s \le t < s + w$, and the network is computed as if the corpus were limited to them. A new window starts every window_step units: successive windows (the default) give independent snapshots, a step of 1 slides the window and smooths the evolution of the networks, at the price of many networks sharing most of their nodes. The last window may be shorter than the others when the period is not a multiple of the step (a warning is given); windows are labelled by their first and last unit ("1975-1984", "2001-01 to 2001-12"). The choice of the window is substantive: long enough for each network to have enough nodes (a few dozen at least) to show a structure, and short enough for changes to be visible. For documents that follow each other quickly (central bank statements, news, posts), months or weeks make sense, with a window wide enough to contain enough documents.

Beyond bibliographic coupling. Nothing in the function is specific to citations. The two sides of the links can be any two kinds of entities:

Co-citation (Small 1973) : with projection_on = "target", cited documents are linked when the same documents cite them. Windows should follow the citing documents: put their year in directed_edges and use time_in = "edges".
Author (or journal, institution) coupling (Zhao and Strotmann 2008) : with document_id, authors are linked when the documents they sign cite the same references.
Co-authorship and other co-occurrences: authors and the documents they sign, accounts and the hashtags they use, members and the boards they sit on: any two-mode table.

Identifiers of references. Every measure assumes that one work has one identifier. With raw bibliographic data, the same reference is often written in several ways (spelling, abbreviations, missing years), each spelling becomes a different target_id, and two documents citing the same work in different ways no longer share it. This matters most for co-citation, where cited works are the nodes: a work split into several spellings is split into several nodes, each with a fraction of its citations. It matters less when the database gives an identifier to each cited reference (e.g. the identifiers of Scopus, OpenAlex or Web of Science), but even then references that the database could not match are missing or duplicated. Clean and match references before building networks, and check the most cited nodes for duplicates.

References

Domagalski R, Neal ZP, Sagan B (2021). “Backbone: An R Package for Extracting the Backbone of Bipartite Projections.” PLOS ONE, 16(1), e0244363. doi:10.1371/journal.pone.0244363 .

Kessler MM (1963). “Bibliographic Coupling between Scientific Papers.” American Documentation, 14(1), 10–25. doi:10.1002/asi.5090140103 .

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

Neal ZP (2022). “backbone: An R Package to Extract Network Backbones.” PLOS ONE, 17(5), e0269137. doi:10.1371/journal.pone.0269137 .

Sen SK, Gan SK (1983). “A Mathematical Extension of the Idea of Bibliographic Coupling and Its Applications.” Annals of Library Science and Documentation, 30(2), 78–82.

Shen S, Zhu D, Rousseau R, Su X, Wang D (2019). “A Refined Method for Computing Bibliographic Coupling Strengths.” Journal of Informetrics, 13(2), 605–615. doi:10.1016/j.joi.2019.01.012 .

Small H (1973). “Co-Citation in the Scientific Literature: A New Measure of the Relationship between Two Documents.” Journal of the American Society for Information Science, 24(4), 265–269. doi:10.1002/asi.4630240406 .

Vladutz G, Cook J (1984). “Bibliographic Coupling and Subject Relatedness.” Proceedings of the American Society for Information Science, 21, 204–207.

Zhao D, Strotmann A (2008). “Author Bibliographic Coupling: Another Approach to Citation-Based Author Knowledge Network Analysis.” In Proceedings of the American Society for Information Science and Technology, volume 45(1), 1–10. https://asistdl.onlinelibrary.wiley.com/doi/full/10.1002/meet.2008.1450450292.

Examples

nodes <- subset(Nodes_stagflation, source_type == "Stagflation")

# Bibliographic coupling networks of the articles explaining the 1970s
# stagflation, one per decade
networks <- build_dynamic_networks(
  nodes = nodes,
  directed_edges = Ref_stagflation,
  source_id = "source_id",
  target_id = "target_id",
  time_variable = "source_year",
  time_window = 13,
  method = "similarity",
  edges_threshold = 1,
  verbose = FALSE
)
names(networks)
#> [1] "1975-1987" "1988-2000" "2001-2013"
networks[[1]]
#> # A tbl_graph: 67 nodes and 405 edges
#> #
#> # An undirected simple graph with 1 component
#> #
#> # Node Data: 67 × 8 (active)
#>    source_id source_author  source_year source_label source_title source_journal
#>    <chr>     <chr>          <chr>       <chr>        <chr>        <chr>         
#>  1 37095547  ATESOGLU-H     1980        ATESOGLU-H-… INFLATION A… "JOURNAL OF P…
#>  2 46282251  ATESOGLU-H     1982        ATESOGLU-H-… WAGES AND S… "JOURNAL OF P…
#>  3 42623802  BLAAS-W        1982        BLAAS-W-1982 INSTITUTION… "JOURNAL OF E…
#>  4 5240516   BLINDER-A      1979        BLINDER-A-1… ECONOMIC PO… ""            
#>  5 25091445  BLINDER-A      1981        BLINDER-A-1… THE 1971-19… ""            
#>  6 31895841  BLINDER-A      1982        BLINDER-A-1… THE ANATOMY… "INFLATION: C…
#>  7 1172921   BRANSON-W      1980        BRANSON-W-1… INTERNATION… ""            
#>  8 47749045  BRONFENBRENNE… 1976        BRONFENBREN… ELEMENTS OF… "ZEITSCHRIFT …
#>  9 14490177  BRUNNER-K      1980        BRUNNER-K-1… STAGFLATION… "JOURNAL OF M…
#> 10 6119013   BRUNO-M        1979        BRUNO-M-197… MACRO-ECONO… ""            
#> # ℹ 57 more rows
#> # ℹ 2 more variables: source_type <chr>, time_window <chr>
#> #
#> # Edge Data: 405 × 5
#>    from    to weight Source   Target    
#>   <int> <int>  <dbl> <chr>    <chr>     
#> 1    14    21 0.116  1021902  1111111122
#> 2    15    21 0.0259 1021902  1111111128
#> 3    15    22 0.0681 10585974 1111111128
#> # ℹ 402 more rows

# Co-citation: cited documents linked when the same documents cite them,
# with windows on the year of the citing documents
citations <- merge(
  transform(Ref_stagflation[, c("source_id", "target_id")],
            source_id = as.character(source_id), target_id = as.character(target_id)),
  nodes[, c("source_id", "source_year")],
  by = "source_id"
)
cited <- Ref_stagflation[!duplicated(Ref_stagflation$target_id),
                         c("target_id", "target_author", "target_year")]
cocitation <- build_dynamic_networks(
  nodes = transform(cited, target_id = as.character(target_id)),
  directed_edges = citations,
  source_id = "source_id",
  target_id = "target_id",
  projection_on = "target",
  time_in = "edges",
  time_variable = "source_year",
  time_window = 13,
  method = "angle",
  edges_threshold = 2,
  compute_size = TRUE,
  verbose = FALSE
)
cocitation[[1]]
#> # A tbl_graph: 122 nodes and 354 edges
#> #
#> # An undirected simple graph with 3 components
#> #
#> # Node Data: 122 × 5 (active)
#>    target_id  target_author                   target_year time_window node_size
#>    <chr>      <chr>                                 <int> <chr>           <int>
#>  1 1021902    FRIEDMAN, M.                           1977 1975-1987           4
#>  2 10585974   FRYE-J                                 1980 1975-1987           2
#>  3 1059321    TOBIN-J                                1972 1975-1987           5
#>  4 10990368   PERRY-G                                1970 1975-1987           3
#>  5 1111111171 SOLOW-R                                1980 1975-1987           2
#>  6 1111111317 HAVRILESKY-T                           1975 1975-1987           2
#>  7 1111111343 PERLOFF, J.M. AND WACHTER, W.H.        1979 1975-1987           5
#>  8 1111111507 DENISON E.                             1978 1975-1987           3
#>  9 1111111520 RASCHE-R                               1977 1975-1987           4
#> 10 1111111579 HAHN-F                                 1974 1975-1987           2
#> # ℹ 112 more rows
#> #
#> # Edge Data: 354 × 5
#>    from    to weight Source     Target    
#>   <int> <int>  <dbl> <chr>      <chr>     
#> 1     1     3  0.447 1021902    1059321   
#> 2     3     4  0.516 1059321    10990368  
#> 3     6     7  0.632 1111111317 1111111343
#> # ℹ 351 more rows

# Authors linked by the references of the documents they sign
by_author <- merge(
  transform(Authors_stagflation, source_id = as.character(source_id)),
  citations, by = "source_id"
)
author_coupling <- build_network(
  nodes = data.frame(author_name = unique(by_author$author_name)),
  directed_edges = by_author,
  source_id = "author_name",
  target_id = "target_id",
  document_id = "source_id",
  method = "strength",
  edges_threshold = 2
)
#> ℹ Coupling of entities through their documents
#>   (`biblionetwork::entity_network()`) with "strength": ties weighted by the
#>   coupling strength, which counts a shared target less when many sources are
#>   linked to it, after Shen et al. (2019), between author_name entities sharing
#>   target_id entities.
#> ℹ Two author_name are linked when they share at least 2 target_id
#>   (`edges_threshold`).
#> ℹ Nodes without any edge are removed (`keep_singleton = FALSE`).
author_coupling
#> # A tbl_graph: 120 nodes and 2365 edges
#> #
#> # An undirected simple graph with 1 component
#> #
#> # Node Data: 120 × 1 (active)
#>    author_name 
#>    <chr>       
#>  1 ROMER-C     
#>  2 ROMER-D     
#>  3 ORPHANIDES-A
#>  4 FRIEDMAN-M  
#>  5 KILIAN-L    
#>  6 BARSKY-R    
#>  7 BLANCHARD-O 
#>  8 FRYE-J      
#>  9 GORDON-R    
#> 10 SARGENT-T   
#> # ℹ 110 more rows
#> #
#> # Edge Data: 2,365 × 6
#>    from    to  weight Source     Target     Weighting_method
#>   <int> <int>   <dbl> <chr>      <chr>      <chr>           
#> 1    16   119 0.00355 ALBANESI-S BALL-L     strength        
#> 2     6   119 0.00267 ALBANESI-S BARSKY-R   strength        
#> 3    78   119 0.00548 ALBANESI-S BILBIIE-FO strength        
#> # ℹ 2,362 more rows

# \donttest{
# Statistical backbone: keep the ties that random networks with the same
# degrees would rarely produce
if (requireNamespace("backbone", quietly = TRUE)) {
backbones <- build_dynamic_networks(
  nodes = nodes,
  directed_edges = Ref_stagflation,
  source_id = "source_id",
  target_id = "target_id",
  time_variable = "source_year",
  time_window = 20,
  method = "sdsm",
  alpha = 0.05,
  backbone_args = list(mtc = "holm"),
  verbose = FALSE
)
}
#> Warning: The last window is shorter than the others.
#> ℹ The data end in "2013", but the last window ends in "2014".
# }