Modularity Vitality

Contribution of a node to the modularity of a fixed partition (Magelinski, Bartulovic & Carley 2021): $$V_Q(i) = Q(G, C) - Q(G - i,\; C \setminus \{i\}),$$ the drop in Newman modularity when node $i$ is deleted and the remaining nodes keep their communities. Positive values mark community hubs (removing them weakens the modular structure); negative values mark bridges (removing them sharpens it). Weighted graphs use edge weights; directed graphs use the Leicht-Newman directed modularity, as igraph does.

Usage

centrality_modularity_vitality(x, membership = NULL, ...)

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
membership: Community labels, one per node (integer, factor, or character). Required; without it the function warns and returns NA. Obtain one from detect_communities.
...: Additional arguments passed to centrality.

Value

Named numeric vector, one value per node. NaN where deleting the node leaves a graph with no edges.

Details

All $n$ vitalities are computed in closed form from one matrix product, without recomputing modularity $n$ times.

Conditions

Raises an error of class cograph_bad_membership when membership is not one non-missing label per node.

References

Magelinski, T., Bartulovic, M., & Carley, K. M. (2021). Measuring node contribution to community structure with modularity vitality. IEEE Transactions on Network Science and Engineering, 8(1), 707-723.

Examples

# Two triangles joined by one bridge edge (C -- D)
adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_modularity_vitality(adj, membership = c(1, 1, 1, 2, 2, 2))
#>           A           B           C           D           E           F 
#>  0.13714286  0.13714286 -0.01785714 -0.01785714  0.13714286  0.13714286