Who depends on whom in a weighted network?

Each node keeps only its effective number of neighbors, set by the entropy of how it spreads its weight. Edges kept by one end only reveal hidden, one-sided dependency; edges kept by both ends are mutual.

After M. J. Lee, E. Lee, B. Lee, H. Jeong, D.-S. Lee & S. H. Lee, Uncovering hidden dependency in weighted networks via information entropy, Phys. Rev. Research 3, 043136 (2021).

Network

Color nodes by

Rényi order α

Network-level measures

Measure	This network	Null [100 shuffles]

The null model reshuffles the weights across the existing edges, keeping the binary network intact (paper, Sec. III B).

Node inspector

Karate club against random networks

Network	⟨k⟩	⟨κ→⟩	e	r	M	corr(ρ, τ)

How the subnetwork is built

Normalize from each end. Node i sees edge (i, j) as the share w̃_ij = w_ij / s(i) of its strength. Because strengths differ, w̃_ij ≠ w̃_ji in general.
Measure how concentrated that share is. The Rényi entropy S_α(i) = ln(Σ_j w̃_ij^α) / (1 − α) is near ln k(i) for even weights and near 0 when one neighbor dominates.
Turn it into a count. The effective out-degree k̃^→(i) = e^S_α(i) is rounded to the nearest integer, and any neighbors tied with the last one kept are also kept: κ^→ = ⌊k̃^→ + 0.5⌋ + ζ.
Keep the top neighbors as arrows. Node i draws i → j to its top κ^→(i) neighbors by weight, meaning “i depends on j.” Where both ends keep the edge, it is reciprocal.
Summarize. Edge density e = Σκ^→/Σk, reciprocity r = Σκ^↔/Σκ^→, local reciprocity ρ = κ^↔/κ^→, attraction ratio τ = κ^←/κ^→, and the mutuality M: the Pearson correlation between w̃_ij and w̃_ji.

Local reciprocity against attraction ratio

Each point is a node. The dashed line is ρ = τ, the upper bound for ρ when τ < 1 (Eq. 14). Points far to the right are nodes many others depend on. Select a point to inspect it.