Characterise the role of nodes within and between clusters

Two nodes of the same cluster can play very different roles: one may be tied to almost every member of its cluster, another may sit at the border and connect several clusters. add_node_roles() computes the two measures of the "functional cartography" of Guimerà and Nunes Amaral (2005) and classifies each node into one of their seven roles.

With \(\kappa_i\) the weight of the edges of node \(i\) towards its own cluster \(s_i\), the within-module degree z-score measures how strongly \(i\) is connected to its cluster compared with the other members: $$z_i = \frac{\kappa_i - \bar{\kappa}_{s_i}}{\sigma_{\kappa_{s_i}}}.$$ With \(k_{is}\) the weight of the edges of \(i\) towards cluster \(s\) and \(k_i\) its total weight, the participation coefficient is close to 0 when all the ties of \(i\) stay within one cluster, and close to 1 when they are spread evenly across clusters: $$P_i = 1 - \sum_s \left( \frac{k_{is}}{k_i} \right)^2.$$

Usage

add_node_roles(
  graphs,
  cluster_id = "dynamic_cluster_leiden",
  weights = NULL,
  z_threshold = 2.5
)

Arguments

graphs

A tbl_graph or a list of tbl_graphs with a cluster column.

cluster_id

Character scalar, the node column with the clusters. The default, "dynamic_cluster_leiden", fits the output of merge_dynamic_clusters(); use "cluster_leiden" after add_clusters() alone.

weights

Edge weights, as in add_clusters() and the functions of tidygraph. NULL (default) uses the weight edge column if there is one, and treats the graph as unweighted otherwise. A character scalar names another edge column. NA ignores the weights.

z_threshold

Number, the within-module z-score above which a node is a hub (2.5, the value of Guimerà and Amaral).

Value

The same object as graphs with four new node columns: within_module_degree (\(\kappa_i\)), within_module_z (\(z_i\), 0 when all members of the cluster have the same \(\kappa\)), participation_coeff (\(P_i\), 0 for isolated nodes) and roles (the role).

Details

The roles combine the two measures, with the thresholds of Guimerà and Nunes Amaral (2005) :

Role	\(z_i\)	\(P_i\)
R1 ultra-peripheral	below z_threshold	below 0.05
R2 peripheral	below z_threshold	0.05 to 0.62
R3 non-hub connector	below z_threshold	0.62 to 0.80
R4 non-hub kinless	below z_threshold	above 0.80
R5 provincial hub	at least z_threshold	below 0.30
R6 connector hub	at least z_threshold	0.30 to 0.75
R7 kinless hub	at least z_threshold	above 0.75

Reading the roles. The two measures answer two questions: how central is this node in its own cluster? (\(z_i\)) and where do its ties go? (\(P_i\)). What a role means depends on what the nodes and the ties are. For the stagflation data of the package (documents linked by bibliographic coupling, Leiden clusters, resolution 1), the examples are:

• Provincial hub (R5): the central document of a literature, tied to most of its members and to almost nothing else. In a coupling network, it shares references with most documents of its cluster: Lansing's "Exploring the causes of the great inflation" (2000) in the cluster on the great inflation, Nordhaus's "Who's afraid of a big bad oil shock?" (2007) in the cluster on oil. In a co-citation network, the same role is played by the classics that are cited with most works of a literature.

• Connector hub (R6): central in its cluster and tied to other clusters: a document that is embedded in a literature while drawing on others. Pindyck's "Energy price increases and macroeconomic policy" (1980) is the only one: central in the cluster on productivity, oil and supply, it shares references with several other clusters.

• Kinless hub (R7): central in its cluster, with ties spread evenly over all clusters. It requires a cluster that is itself not very separate from the others; there is none in the stagflation example.

• Non-hub connector (R3): a document whose ties are spread over several clusters without being central in any: a bridge. Goldstein's "Downward price inflexibility, ratchet effects, and the inflationary impact of import price changes" (1977), or Blinder's "The supply-shock explanation of the great stagflation revisited" (2013), which belongs to the oil cluster but shares references with the other clusters, are such bridges. In a co-citation network, they are works used by several literatures.

• Non-hub kinless (R4): ties spread almost evenly over all clusters, which can mark an outlier as well as a bridge. In the example, the only one, Nitzan's "Regimes of differential accumulation: mergers, stagflation and the logic of globalization" (2001), shares references with five clusters in almost equal parts (17 to 20% of its tie weight each) and is below the average of its own cluster for within-module degree: it is tied to everybody a little, and to nobody much.

• Peripheral (R1, R2): all or most ties stay within the cluster. R1 (participation below 0.05) are documents whose ties all stay in their own cluster, such as Phelps's "Commodity-supply shock and full-employment monetary policy" (1978), in the cluster on imports.

The same classification means other things for other nodes. For authors coupled through their references (see build_dynamic_networks()), a connector is an author who cites several literatures, a hub an author who cites like most of the group. For networks that are not bibliographic, the roles describe how much each node holds its community together (hubs) and how much it links communities (connectors).

Cautions. (1) Roles depend on the clusters, hence on the algorithm and on the resolution: on the stagflation data, the mean participation coefficient is 0.31, 0.39 and 0.52 for Leiden resolutions of 0.75, 1 and 1.5, because smaller clusters leave more ties between clusters. Check that the nodes you interpret keep their role when the resolution changes a little. (2) The thresholds of Guimerà and Nunes Amaral (2005) come from metabolic networks. Projected networks are dense (here, a quarter of the pairs of documents are tied) and most nodes have a participation above 0.05: R2 is the usual role, and hubs are rare. (3) With the weights of similarity measures, long bibliographies and well-cited references influence the weights (see build_dynamic_networks()). (4) A role is a position in a network, not a judgement on the importance or the quality of a document; reading the document is necessary.

The original measures count edges; here they use the edge weights (strengths), which is the natural generalisation for weighted projected networks. The graph is treated as undirected. The thresholds were calibrated on biological networks: in small bibliometric networks, few nodes pass \(z \geq 2.5\), and the measures are best read as continuous indicators (who are the central members of a literature, which articles bridge two literatures) rather than as categories.

References

Guimerà R, Nunes Amaral LA (2005). “Functional Cartography of Complex Metabolic Networks.” Nature, 433(7028), 895–900. doi:10.1038/nature03288 .

Examples

network <- build_network(
  nodes = subset(Nodes_stagflation, source_type == "Stagflation"),
  directed_edges = Ref_stagflation,
  source_id = "source_id",
  target_id = "target_id",
  method = "similarity",
  edges_threshold = 1
)
#> ℹ Coupling (`biblionetwork::biblio_coupling()`) with "similarity": ties
#>   weighted by the idf-weighted cosine of the two lists of targets, after Shen
#>   et al. (2019), between source_id entities sharing target_id entities.
#> ℹ Two source_id are linked when they share at least 1 target_id
#>   (`edges_threshold`).
#> ℹ Nodes without any edge are removed (`keep_singleton = FALSE`).
network <- add_clusters(
  network,
  clustering_method = "leiden",
  objective_function = "modularity",
  seed = 123,
  verbose = FALSE
)

network <- add_node_roles(network, cluster_id = "cluster_leiden")
table(tidygraph::as_tibble(network, active = "nodes")$roles)
#> 
#>  R1 ultra-peripheral        R2 peripheral R3 non-hub connector 
#>                   12                  125                    9 
#>    R5 provincial hub 
#>                    1