Degree Assortativity Coefficient

Computes the degree assortativity coefficient, measuring the tendency of nodes to connect to other nodes with similar degree. Positive values indicate assortative mixing (high-degree nodes connect to high-degree nodes), negative values indicate disassortative mixing.

Usage

assortativity(x, directed = NULL, type = NULL, digits = NULL, ...)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object.
directed: Logical or NULL. If NULL (default), auto-detect from matrix symmetry. Set TRUE to force directed, FALSE to force undirected.
type: Character string specifying which degree correlation to compute, or NULL (default) to choose automatically: "out-in" for directed networks and "degree" for undirected ones. For a directed network the accepted values are "out-in", "in-in", "out-out" and "in-out"; for an undirected network the only accepted value is "degree". Any other value raises an error.
digits: Integer or NULL. Round result to this many decimal places. Default NULL (no rounding).
...: Currently unused; directed is already an explicit argument above and to_igraph accepts no others.

Value

An object of class "cograph_assortativity" with components:

coefficient: Numeric scalar: the assortativity coefficient in $[-1, 1]$.
type: Character: the degree type used.
directed: Logical: whether the network was treated as directed.
n_nodes: Integer: number of nodes.
n_edges: Integer: number of edges.
network: The original input network.

Details

The degree assortativity coefficient is defined as the Pearson correlation coefficient between the degrees of nodes at either end of each edge (Newman 2002):

$$r = \frac{\sum_{jk} jk(e_{jk} - q_j q_k)}{\sigma_q^2}$$

where $e_{jk}$ is the fraction of edges connecting degree-$j$ to degree-$k$ vertices, $q_k$ is the excess degree distribution, and $\sigma_q^2$ its variance.

Because the Pearson correlation is invariant to subtracting a constant, the implementation computes the correlation of the raw (rather than excess) degrees at the two ends of each edge, counting every undirected edge in both orientations; this is numerically identical to the formula above.

For directed networks, the coefficient is the Pearson correlation between the source-end and target-end degrees over each edge in its stored orientation, with the degree mode at each end chosen by type (Foster et al. 2010).

The coefficient is NA when the network has no edges or when either degree vector has zero variance.

References

Newman, M.E.J. (2002). Assortative mixing in networks. Physical Review Letters, 89(20), 208701. doi:10.1103/PhysRevLett.89.208701

Foster, J.G., Foster, D.V., Grassberger, P., & Paczuski, M. (2010). Edge direction and the structure of networks. PNAS, 107(24), 10815-10820. doi:10.1073/pnas.0912671107

Examples

# Assortative network (high-degree connect to high-degree)
adj <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 1, 1, 0,
  1, 1, 0, 0, 1,
  1, 1, 0, 0, 1,
  0, 0, 1, 1, 0
), 5, 5)
rownames(adj) <- colnames(adj) <- LETTERS[1:5]
cograph::assortativity(adj)
#> Assortativity (Degree)
#> =================================== 
#>   Coefficient: -0.1667 
#>   Interpretation: disassortative 
#>   Nodes: 5   Edges: 7 
#>   Directed: FALSE