Small-world propensity

Recently developed by Muldoon2016 and utilized by Luppi2021 on LSD.

$\varphi$ quantifies the extent tha ta network displays small-world organization by taking into account deviation of the network’s empirically observed clustering coefficient $c_{obs}$ and characteristic path length ($L_{obs}$) from equivalent lattice and random networks

$$\begin{array}{c}\varphi =1-\sqrt{\frac{{\Delta }_C^2+{\Delta }_L^2}{2}}\end{array}$$ $$\begin{array}{c}{\Delta }_C=\frac{C_{latt}-C_{obs}}{C_{latt}-C_{rand}}\end{array}$$ $$\begin{array}{c}{\Delta }_L=\frac{L_{obs}-C_{obs}}{L_{latt}-L_{rand}}\end{array}$$

${\Delta }_C$ and ${\Delta }_L$ quantify the fractional deviation of the empirically observed clustering coefficient & characteristic path length form the corresponding null models according to the definition of a small-world network

• Lattice network for clustering coefficient
• Random network for characteristic path length

Both measures of fractional deviation are bound between 0 and 1 to account for the possibility of empirical networks exceeding the corresponding null models.

• if negative, set to 0
• if exceeding unity, set to 1

Limitations

Not intended to determine, in absolute terms, whether a network exhibits small-world structure

• better suited to compare (on a continuous scale) the degree of small-world organisation exhibited by diff. networks

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Related

small-world organization

References