Estrada Index

A graph-level spectral invariant derived from subgraph centrality: $$EE(G) = \sum_{i=1}^{n} e^{\lambda_i}$$ where \(\lambda_i\) are the eigenvalues of the adjacency matrix. The Estrada index equals the total number of closed walks in the graph, weighted by walk length: \(EE(G) = \sum_k M_k / k!\) where \(M_k\) is the number of closed walks of length \(k\). It is the sum of subgraph centralities across all nodes.

Usage

estrada_index(x)

Arguments

x

Network input (matrix, igraph, network, cograph_network, tna object).

Value

A single numeric value — the Estrada index of the graph.

Details

Matches networkx.estrada_index at machine epsilon (max relative difference ~5e-15 across random test graphs).

References

Estrada, E. (2000). Characterization of 3D molecular structure. Chemical Physics Letters, 319(5-6), 713-718.

See also

centrality_subgraph for the per-node equivalent (sum of subgraph_centrality(x) equals estrada_index(x)).

Examples

# Karate club
g <- igraph::make_graph("Zachary")
estrada_index(g)
#> [1] 1041.247