Peeling a network by its local hubs

An interactive version of the local-edge decomposition (LED) from Jeong, Yu, and Lee, Phys. Rev. E 111, 054302 (2025). Slide the level control to strip the network shell by shell, and compare it with k-core decomposition.

Showing levels ≥ 0 of 0

Network

Decomposition

Local-edge (LED) k-core

Whole network Within communities

color by

Level Hub centrality Core / periphery Community

How LED works

A node's hub centrality is the share of its neighbours with strictly smaller degree:

h_i = N_{<k_i} / k_i, E_ij = h_i h_j

At each level, compute h in the current backbone and cut every edge whose product E is zero, meaning every edge touching a locally smallest node. The largest piece left is the next backbone; the nodes that fall off are that level's shell. Repeat until nothing is left.

Because h is relative, a modest node that dominates its own neighbourhood survives as long as a global hub does.

Edge pruning of the current backbone

Giant-component size G as edges are removed in order of increasing importance. With the hub-centrality product, the curve breaks exactly where the zero-product edges run out.

Where each level's neighbours sit

For nodes at each level, the share of their links to higher, lower or same levels (Fig. 5 in the paper). Grey columns are empty levels.

Core–periphery score

Treat levels ≥ L_B as the core. Densities are on a log scale; S_cp (Eq. 4) peaks at the boundary closest to an ideal core–periphery split.