  Quantitative descriptions of network structure in big data can provide
fundamental insights into the function of interconnected complex systems.
Small-world structure, commonly diagnosed by high local clustering yet short
average path length between any two nodes, directly enables information flow in
coupled systems, a key function that can differ across conditions or between
groups. However, current techniques to quantify small-world structure are
dependent on nuisance variables such as density and agnostic to critical
variables such as the strengths of connections between nodes, thereby hampering
accurate and comparable assessments of small-world structure in different
networks. Here, we address both limitations with a novel metric called the
Small-World Propensity (SWP). In its binary instantiation, the SWP provides an
unbiased assessment of small-world structure in networks of varying densities.
We extend this concept to the case of weighted networks by developing (i) a
standardized procedure for generating weighted small-world networks, (ii) a
weighted extension of the SWP, and (iii) a stringent and generalizable method
for mapping real-world data onto the theoretical model. In applying these
techniques to real world brain networks, we uncover the surprising fact that
the canonical example of a biological small-world network, the C. elegans
neuronal network, has strikingly low SWP in comparison to other examined brain
networks. These metrics, models, and maps form a coherent toolbox for the
assessment of architectural properties in real-world networks and their
statistical comparison across conditions.
