Core-periphery clustering algorithms

These functions identify nodes belonging to (some level of) the core of a network:

• node_is_universal() identifies whether nodes are adjacent to all other nodes in the network.

• node_is_core() identifies whether nodes belong to the core of the network, as opposed to the periphery.

• node_in_core() categorizes nodes into two or more core/periphery categories based on their coreness.

• node_coreness() returns a continuous measure of how closely each node resembles a typical core node.

• node_kcoreness() assigns nodes to their level of k-coreness.

node_is_universal(.data)

node_is_core(.data, method = c("degree", "eigenvector"))

node_kcoreness(.data)

node_coreness(.data)

node_in_core(.data, groups = 3, cluster_by = c("bins", "quantiles", "kmeans"))

Arguments

.data

An object of a manynet-consistent class:

• matrix (adjacency or incidence) from {base} R

• edgelist, a data frame from {base} R or tibble from {tibble}

• igraph, from the {igraph} package

• network, from the {network} package

• tbl_graph, from the {tidygraph} package

method

Which method to use to identify cores and periphery. By default this is "degree", which relies on the heuristic that high degree nodes are more likely to be in the core. An alternative is "eigenvector", which instead begins with high eigenvector nodes. Other methods, such as a genetic algorithm, CONCOR, and Rombach-Porter, can be added if there is interest.

groups

Number of categories to create. Must be at least 2 and at most the number of nodes in the network. Default is 3.

cluster_by

Method to use to create the categories. One of "bins" (equal-width bins), "quantiles" (quantile-based bins), or "kmeans" (k-means clustering). Default is "bins".

Universal/dominating node

A universal node is adjacent to all other nodes in the network. It is also sometimes called the dominating vertex because it represents a one-element dominating set. A network with a universal node is called a cone, and its universal node is called the apex of the cone. A classic example of a cone is a star graph, but friendship, wheel, and threshold graphs are also cones.

Core-periphery

This function is used to identify which nodes should belong to the core, and which to the periphery. It seeks to minimize the following quantity: $$Z(S_1) = \sum_{(i<j)\in S_1} \textbf{I}_{\{A_{ij}=0\}} + \sum_{(i<j)\notin S_1} \textbf{I}_{\{A_{ij}=1\}}$$ where nodes \(\{i,j,...,n\}\) are ordered in descending degree, \(A\) is the adjacency matrix, and the indicator function is 1 if the predicate is true or 0 otherwise. Note that minimising this quantity maximises density in the core block and minimises density in the periphery block; it ignores ties between these blocks.

Core-periphery categories

This function categorizes nodes based on their coreness into a specified number of groups. The groups are labeled as "Core", "Semi-core", "Semi-periphery", and "Periphery" depending on the number of groups specified. The categorization can be done using different methods: equal-width bins, quantile-based bins, or k-means clustering.

References

On core-periphery partitioning

Borgatti, Stephen P., & Everett, Martin G. 1999. "Models of core /periphery structures". Social Networks, 21, 375–395. doi:10.1016/S0378-8733(99)00019-2

Lip, Sean Z. W. 2011. “A Fast Algorithm for the Discrete Core/Periphery Bipartitioning Problem.” doi:10.48550/arXiv.1102.5511

On k-coreness

Seidman, Stephen B. 1983. "Network structure and minimum degree". Social Networks, 5(3), 269-287. doi:10.1016/0378-8733(83)90028-X

Batagelj, Vladimir, and Matjaz Zaversnik. 2003. "An O(m) algorithm for cores decomposition of networks". arXiv preprint cs/0310049. doi:10.48550/arXiv.cs/0310049

On core-periphery categorization

Wallerstein, Immanuel. 1974. "Dependence in an Interdependent World: The Limited Possibilities of Transformation Within the Capitalist World Economy." African Studies Review, 17(1), 1-26. doi:10.2307/523574

See also

Other memberships: member_brokerage, member_cliques, member_community_hier, member_community_non, member_components, member_equivalence

Examples

node_is_universal(create_star(11))
#>   V1    V2    V3    V4    V5    V6    V7    V8    V9    V10   V11  
#> 1 TRUE  FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
node_is_core(ison_adolescents)
#>   Betty Sue   Alice Jane  Dale  Pam   Carol Tina 
#> 1 FALSE TRUE  TRUE  FALSE TRUE  FALSE FALSE FALSE
#ison_adolescents %>% 
#   mutate(corep = node_is_core()) %>% 
#   graphr(node_color = "corep")
node_kcoreness(ison_adolescents)
#> ▃▁▁▁▅ 
#>   Betty   Sue Alice  Jane  Dale   Pam Carol  Tina
#> 1     1     2     2     2     2     2     1     1
node_coreness(ison_adolescents)
#> ▃▁▁▁▃ 
#>   Betty   Sue Alice  Jane  Dale   Pam Carol  Tina
#> 1 0.181  0.89     1 0.562  0.82 0.619 0.071     0
node_in_core(ison_adolescents)
#> 3 groups
#>   Betty     Sue   Alice Jane           Dale  Pam            Carol     Tina     
#> 1 Periphery Core  Core  Semi-periphery Core  Semi-periphery Periphery Periphery