Summarize Topological Properties of a Graph — summarize_graph

Computes a comprehensive set of global topological metrics for an input graph, including basic structure, connectivity, spectral properties, and complexity. Supports both igraph objects and data frames representing edge lists.

Usage

summarize_graph_metrics(
  graph,
  weights = NULL,
  weight_type = c("strength", "distance")
)

Arguments

graph: An igraph object or a data frame with columns from and to representing an edge list.
weights: Optional edge weights: NULL (default) to ignore them, the name of an edge attribute, or a numeric vector of length igraph::ecount(graph). See netkit-weights.
weight_type: Either "strength" (default) or "distance". See netkit-weights.

Value

A one-row data.frame, each column a graph-level metric.

Metrics that are mathematically undefined for the input are NaN rather than substituted values, as returned by the underlying igraph and ineq functions. On degenerate graphs this is expected: an edgeless graph has no paths (Average_path_length), no connected triples (Clustering_coefficient), no degree variance (Degree_assortativity) and a zero mean degree (Gini_degree).

Details

Metrics computed:

Number of nodes and edges
Directed TRUE/FALSE
Graph density
Diameter and average path length of the largest connected component
Clustering coefficient (transitivity)
Degree assortativity
Average degree and betweenness centrality
Number of connected components and size of the largest connected component
Number of single nodes
Algebraic connectivity (second-smallest Laplacian eigenvalue)
Degree entropy (Shannon entropy of the degree distribution)
Gini coefficient of node degrees
Modularity of the community structure (via Louvain algorithm)

When weights are supplied, the metrics switch to their weighted definitions: Avg_degree becomes mean vertex strength; Diameter, Average_path_length and Avg_betweenness are computed on edge costs; Clustering_coefficient uses Barrat's weighted transitivity; Degree_assortativity correlates strengths rather than degrees; Degree_entropy and Gini_degree describe the strength distribution; and the Laplacian behind Algebraic_connectivity and the Louvain run behind Modularity both use edge strengths. Two columns are added, Is_weighted and Avg_strength, so a weighted and an unweighted summary can be row-bound.

Edge weights

Functions that can use edge weights take two arguments:

weights: NULL (the default) to ignore edge weights; the name of an edge attribute, such as "weight"; or a numeric vector with one value per edge, in igraph::E(graph) order.
weight_type: "strength" (the default) if a larger value means a more tightly connected pair – confidence scores, correlations, co-expression, read counts, interaction scores. "distance" if a larger value means further apart – costs, dissimilarities, reaction times.

Declaring which you have is not bookkeeping. igraph reads the weight attribute implicitly and gives it opposite meanings in different functions: a cost in igraph::betweenness(), igraph::distances(), igraph::diameter() and igraph::mean_distance(), but a strength in igraph::cluster_louvain() and the other community detection algorithms. Attaching a confidence score and letting that happen implicitly therefore inverts every path-based metric – a high-confidence interaction is treated as a long distance – while community detection reads the same numbers the way you intended.

netkit resolves weights and weight_type once per call and derives both a strength and a distance vector from them, so each metric receives the one it needs. A "strength" is converted to a distance by reciprocal (\(1/w\)); a "distance" is converted to a strength by reflection (\(\max(w) - w + \min(w)\)), which keeps a zero distance finite.

References

Newman, M. E. J. (2010). Networks: An Introduction. Oxford University Press.
Estrada, E. (2012). The Structure of Complex Networks: Theory and Applications. Oxford University Press.
Latora, V., Nicosia, V., & Russo, G. (2017). Complex Networks: Principles, Methods and Applications. Cambridge University Press.
Louvain modularity method: Blondel, V. D., Guillaume, J. L., Lambiotte, R., & Lefebvre, E. (2008). Fast unfolding of communities in large networks. J. Stat. Mech., 2008(10), P10008.
Barrat, A., Barthelemy, M., Pastor-Satorras, R., & Vespignani, A. (2004). The architecture of complex weighted networks. PNAS, 101(11), 3747-3752.

Examples

g <- igraph::sample_gnp(60, 0.08, directed = FALSE)
summarize_graph_metrics(g)
#>   Nodes Edges Is_directed Is_weighted   Density Diameter Average_path_length
#> 1    60   145       FALSE       FALSE 0.0819209        6            2.768927
#>   Clustering_coefficient Degree_assortativity Avg_degree Avg_strength
#> 1             0.06280788           0.09418808   4.833333     4.833333
#>   Avg_betweenness Components Single_nodes LCC_size LCC_percent
#> 1        52.18333          1            0       60           1
#>   Algebraic_connectivity Degree_entropy Gini_degree Modularity
#> 1              0.4074096       2.991815   0.2351724  0.3614507

# Weighted: declare what the attribute means, because igraph alone would read
# it as a cost for path metrics and as a strength for modularity.
igraph::E(g)$confidence <- runif(igraph::ecount(g), 0.1, 1)
summarize_graph_metrics(g, weights = "confidence", weight_type = "strength")
#>   Nodes Edges Is_directed Is_weighted   Density Diameter Average_path_length
#> 1    60   145       FALSE        TRUE 0.0819209 19.11682            5.570991
#>   Clustering_coefficient Degree_assortativity Avg_degree Avg_strength
#> 1             0.05964938            0.1441752   4.833333     2.562503
#>   Avg_betweenness Components Single_nodes LCC_size LCC_percent
#> 1        62.91667          1            0       60           1
#>   Algebraic_connectivity Degree_entropy Gini_degree Modularity
#> 1             0.08446972       5.906891   0.2743675  0.4163293