Erica Klarreich in Quanta:
A paper posted online last month has reignited a debate about one of the oldest, most startling claims in the modern era of network science: the proposition that most complex networks in the real world — from the World Wide Web to interacting proteins in a cell — are “scale-free.” Roughly speaking, that means that a few of their nodes should have many more connections than others, following a mathematical formula called a power law, so that there’s no one scale that characterizes the network.
Purely random networks do not obey power laws, so when the early proponents of the scale-free paradigm started seeing power laws in real-world networks in the late 1990s, they viewed them as evidence of a universal organizing principle underlying the formation of these diverse networks. The architecture of scale-freeness, researchers argued, could provide insight into fundamental questions such as how likely a virus is to cause an epidemic, or how easily hackers can disable a network.
Over the past two decades, an avalanche of papers has asserted the scale-freeness of hundreds of real-world networks. In 2002, Albert-László Barabási — a physicist-turned-network scientist who pioneered the scale-free networks paradigm — wrote a book for a general audience, Linked, in which he asserted that power laws are ubiquitous in complex networks.