 Solution

ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2025/paper-2/6/a/solution

Average the four local clustering coefficients and the four supplied mean graph distances, using the same weighting convention for the reference networks:

C = \frac{0.8 + 0.7 + 0.6 + 0.8}{4} = 0.725, L = \frac{3.6 + 3.7 + 3.6 + 3.8}{4} = 3.675.

The two reference values average to

C_{rand} = 0.35

and

L_{rand} = 3.6

. The normalized small-worldness is the relative clustering divided by the relative path length,

σ = \frac{C / C _{rand}}{L / L _{rand}} = \frac{0.725/0.35}{3.675/3.6} \approx 2.02915 .

This uses the standard ratio-of-ratios definition, with the supplied degree-matched random graphs as the chosen reference ensemble.^[ref]

 New to topics? Read the docs here!