Gene Co-Expression Network: Soft-Thresholding for Scale-Free Topology

Co-Expression Network

Soft-thresholding power β 6

Module coherence 65%

Edge display threshold 0.20

Live Network Stats

Scale-free fit R²0.00

Mean connectivity ⟨k⟩0.00

Modules detected0

How it works

48 simulated genes are measured across 36 synthetic samples, each gene assigned to one of 4 hidden co-expression modules driven by a shared latent factor plus individual noise (the module coherence slider sets the noise level). From this expression matrix a Pearson correlation cor_ij is computed for every gene pair.

unsigned:  a_ij = |cor_ij|^β
signed:    a_ij = ((1 + cor_ij) / 2)^β

This is the WGCNA soft-thresholding step: raising the correlation to a power β suppresses weak, noisy edges much faster than strong ones, pushing the weighted network toward a scale-free topology — a few highly-connected hub genes, many weakly-connected ones. The standard diagnostic is a log-log fit of the connectivity distribution:

p(k) ≈ k^-γ  →  log₁₀ p(k) ≈ -γ·log₁₀ k + c

The R² readout is the linear-regression fit quality of that log-log relationship — WGCNA practice picks the smallest β that pushes R² above ≈0.8. ⟨k⟩ is the mean weighted connectivity (Σ_j a_ij averaged over genes), and Modules detected counts connected components once edges below the display threshold are dropped, i.e. which genes still cluster together at this power.

β slider — re-weights the existing correlation matrix instantly; low β leaves a dense "hairball", high β prunes it into sparse hub-and-spoke modules.
Module coherence — regenerates the expression data with more or less within-module noise, changing how naturally clustered the raw correlations are.
Unsigned / Signed — unsigned treats strong negative and positive correlation as equally "connected"; signed keeps only positive co-expression, which is what most real WGCNA studies use.
Edge display threshold — a purely visual cutoff so the module structure stays readable; it also feeds the module-counting readout.

Real-world relevance: this soft-power correction is exactly how tools like WGCNA turn noisy RNA-seq co-expression data into the biological modules used to find hub genes and candidate biomarkers.

Module A Module B Module C Module D

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