Understanding complexity is one of the most fascinating challenges in current science, with direct implications to everyday life. In a paper (All Scale-Free Networks are Sparse) chosen as an “Editor’s Selection” in the prestigious physics journal, Physical Review Letters, and discussed in an article (Few and Far Between) in Physics Viewpoint, University of Houston physics professor Kevin E. Bassler and his colleagues at the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, identify fundamental constraints on how complex systems can be organized. They show that large scale-free networks, a type commonly observed and studied, can exist only if they have a relatively low number of connections. This finding has profound implications.

Many complex systems are organized as networks, consisting of a set of links between elements (nodes). A central property, known to have immense impact on the function and failure of networks, is the so-called degree distribution that specifies how many other nodes a given node connects to. 

Networks having a so-called scale-free distribution are of particular interest, as they describe many systems of social, technological and biological origin.  “For example, complex networks describe interactions among proteins in a cell, coordinate communication among the neurons in our brain, and govern how individuals in a society connect,” Bassler said.

Bassler’s results reveal the reason why some types of scale-free networks have never been observed. Their existence is precluded by fundamental mathematical constraints. “By ruling out types of networks that cannot be realized, this study provides an important insight into the organization of complex systems observed in nature, ranging from genetic regulation, swarms of insects, flocks of birds, and traffic, to the Internet,” Bassler said.