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As Benjamin Franklin once joked, death and taxes are universal. Scale-free networks may not be, at least听听from CU Boulder.

The research challenges a popular two-decade-old theory that networks of all kinds, from Facebook and Twitter to the interactions of genes in yeast cells, follow a common architecture that mathematicians call 鈥渟cale-free.鈥澨�

Such networks fit into a larger category of networks that are dominated by a few hubs with many more connections than the vast majority of nodes鈥攖hink Twitter where for every Justin Bieber (105 million followers) and Kim Kardashian (60 million followers) out there, you can find thousands of users with just a handful of fans.

In research published this week in the journal听Nature Communications, CU Boulder鈥檚 Anna Broido and Aaron Clauset set out to test that trendy theory. They used computational tools to analyze a huge dataset of more than 900 networks, with examples from the realms of biology, transportation, technology and more.

Their results suggest that death and taxes may not have much competition, at least in networks. Based on Broido and Clauset鈥檚 analysis, close to 50 percent of real networks didn鈥檛 meet even the most liberal definition of what makes a network scale-free.

Those findings matter, Broido said, because the shape of a network听determines a lot about its properties, including how susceptible it is to targeted attacks or disease outbreaks.

鈥淚t鈥檚 important to be careful and precise in defining things like what it means to be a scale-free network,鈥� said Broido, a graduate student in the听Department of Applied Mathematics.听

Clauset, an associate professor in the听Department of Computer Science听and the听BioFrontiers Institute, agrees.

鈥淭he idea of scale-free networks has been a unifying but controversial theme in network theory for nearly 20 years,鈥� he said. 鈥淩esolving the controversy has been difficult because we lacked good tools and broad data. What we鈥檝e found now is that there is little evidence for classically scale-free networks except in a few specific places. Most networks don鈥檛 look scale-free at all.鈥�

Power law

Deciding whether or not a network is 鈥渟cale-free,鈥� however, can be tricky. Many types of networks look similar from a distance.听

But Scale-free networks are special because the patterns of connections coming into and out of nodes follows a precise mathematical form called a power law distribution.

鈥淚f human height followed a power law, you might expect one person to be as tall as the Empire State Building, 10,000 people to be as tall as a giraffe, and more than 150 million to be only about 7-inches-tall,鈥� Clauset said.听

Beginning in the late 1990s, a handful of researchers made a bold claim that all real-world networks follow a universal structure represented by such giraffe- and inch-sized disparities.

There was just one problem: 鈥淭he original claims were mostly based on analyzing a handful of networks with very rough tools,鈥� Clauset said. 鈥淭he idea was provocative, but also, in retrospect, quite speculative.鈥�

To take scale-free networks out of the realm of speculation, he and Broido turned to the听. This archive, which was assembled by Clauset鈥檚 research group at CU Boulder, lists data on thousands of networks from every scientific domain. They include the social links between Star Wars characters, interactions among yeast proteins, friendships on Facebook and Twitter, airplane travel and more.

Their findings were stark. By applying a series of statistical tests of increasing severity, the researchers calculated that only about 4 percent of the networks they studied met the strictest criteria for being scale free, meaning the number of connections that each node carried followed a power-law distribution. These special networks included some types of protein networks in cells and certain kinds of technological networks.听

A multitude of shapes

But not all researchers use those exact requirements to decide what makes a scale-free network, Broido said. To account for these alternative definitions, she and Clauset adapted their tests to account for each of the variations.

鈥淲herever you鈥檙e coming from, one of our definitions should be close to what you鈥檙e thinking,鈥� Broido said.

Despite the added flexibility, most networks still failed to show evidence even for weakly scale-free structure. Roughly half of all biological networks and all social networks, for example, didn鈥檛 look like anything close to a scale-free network, no matter how flexible the definitions were made.

Far from being a let-down, Clauset sees these null findings in a positive light: if scale-free isn鈥檛 the norm, then scientists are free to explore new and more accurate structures for the networks people encounter every day. 听

鈥淭he diversity of real networks presents a mystery,鈥� he said. 鈥淲hat are the common shapes of the networks? How do different kinds of networks assemble and maintain their structure over time? I鈥檓 excited that our findings open up room to explore new ideas.鈥�