In their cover article “Networks in motion” (Physics Today, April 2012, page 43), Adilson Motter and Réka Albert present an interesting gallery of networks (their figure 1) comprising human brains, social networks, and internet service providers. Their gallery provides much food for thought for mathematicians and physicists interested in percolative structures. Many of the factors controlling the percolative expansion of networks and their possible phase transitions remain unknown and are an active area of research.

Compacted networks, a very important class of classical percolative networks, are not mentioned in the Motter and Albert article. Compacted networks are all around us, the most often overlooked example being window glass. They are self-organized by short-range connectivity rules (which can also be found in Motter and Albert’s examples), but in addition they have been compacted by long-range forces, seldom if ever discussed in models. In the window glass example, valence-bond rules govern short-range connectivity, but there are also long-range van der Waals forces that cause the glass density to be usually only about 10% lower than related crystalline densities.

The behavior of many physical systems is governed by delicate balances between short- and long-range forces, so the existence of compacted networks will not come as a surprise to most readers of Physics Today. What may come as a surprise is that quantitative theories of compacted networks are already being used by industry to design new specialty glasses, like extraordinarily damage-resistant Gorilla glass.1 

Self-organized percolation appears in many contexts. For instance, combined charge and rigidity percolation explains many features of high-temperature superconductors, including limits on the transition temperature.2 In the biosciences, the classical compacted globular structures of protein folds are determined by the competition between hydrophobic and hydrophilic forces. A new theory explains evolutionary trends of influenza virus in terms of those forces.3 Based on ideas of self-organized criticality4 and derived from a bioinformatic study5 of self-similarities in 5526 segments from the Protein Data Bank, it successfully predicts the frequency of disease mutations and may have important applications for the use of mutation-prolific viruses to treat disease.

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