Correction to: Scientific Reports https://doi.org/10.1038/s41598-025-17794-5, published online 26 September 2025

In the original PDF version of this Article, four equations did not display correctly.

In the Methods section,

“First, IEq not found inter-links are randomly chosen and cut to free-ends.”

now reads:

“First, \((1-1/{m}_{o})M\times w\) inter-links are randomly chosen and cut to free-ends.”

In addition, in the Results section, under the subheading ‘Various modular networks with continuously changing degree distributions’,

“power-law with exponential cut-off: IEq not found”

now reads:

“power-law with exponential cut-off: \(P(k)\approx C{k}^{-(1/2-{\mu }^{2})}\text{exp}(-2\mu \sqrt{k})\)”

And,

“For the actual rewiring rate w or the corresponded modularity Q, we obtain the linear relations IEq not found and \(Q=\alpha w+\beta\) as estimated in Table 1 by using the least squares method over 100 realizations of networks with each combination of parameters \(\nu\), \({w}^{\prime}\), and \({m}_{o}\).”

now reads:

“For the actual rewiring rate w or the corresponded modularity Q, we obtain the linear relations \(w=aw+b\) and \(Q=\alpha w+\beta\) as estimated in Table 1 by using the least squares method over 100 realizations of networks with each combination of parameters \(\nu\), \({w}^{\prime}\), and \({m}_{o}\).”

Finally, in the caption of Table 1,

“Estimated values for the linear relations IEq not found and \(Q=\alpha w+\beta\).”

now reads:

“Estimated values for the linear relations \(w=aw+b\) and \(Q=\alpha w+\beta\).”

The PDF version of the original Article has been corrected.