As stresses and links increase, clusters expand until, suddenly and unpredictably, a quake breaks out. To model earthquakes, seismologists create percolation networks that match the scale and density of observed cracks, and then they account for stresses by adjusting the probability of cracks connecting up. Geologists use a version of percolation theory to study the sizes of clusters in fractured rock, which is relevant to the extraction of oil by fracking and to the occurrence of earthquakes. Using percolation theory, Broadbent and Hammersley predicted that in an idealized rock, the oil will switch from flowing through only small regions to suddenly permeating almost the entire rock when the density of cracks passes a certain threshold. Unsurprisingly, oil flows farther through rock that is more fractured. The percolation network of a rock layer consists of little holes in its structure, represented as nodes, along with the channels or cracks that allow fluid to flow between them, represented as edges. They abstracted the study of percolation in chemistry, which describes a fluid filtering through a material, such as oil seeping through porous rock or water filtering through ground coffee. In 1957 British mathematicians Simon Ralph Broadbent and John Michael Hammersley first framed percolation theory as a purely mathematical problem. As the world becomes increasingly connected through complex layers of links that transport people, provide them with energy by means of electrical grids or connect them via social media-and sometimes spread disease among them-percolation theory becomes ever more pertinent. Real-world networks are limited in extent, are often messy and require challenging calculations-but they, too, have transitions, albeit more rounded ones. Infinite networks are generally the only ones with truly sharp phase transitions. Mathematicians typically study idealized networks-symmetric in geometry and infinite in extent-because they are the ones amenable to theoretical calculations. The question that scientists struggle to answer is: When? What is the equivalent, for any given network, of the zero degrees Celsius at which ice melts or the 100 degrees C at which water boils? At what point does a meme go viral, a product dominate a market, an earthquake begin, a network of cell phones achieve full connectivity or a disease become a pandemic? Percolation theory provides insight into all these transitions. The fundamental insight of percolation theory, which dates back to the 1950s, is that as the number of links in a network gradually increases, a global cluster of connected nodes will suddenly emerge. Each node represents an object such as a phone or a person, and the edges represent a specific relation between two of them. Percolation theory examines the consequences of randomly creating or removing links in such networks, which mathematicians conceive of as a collection of nodes (represented by points) linked by “edges” (lines). Credit: Jen Christiansen ( graphic) Wee People font, ProPublica and Alberto Cairo ( figure drawings) Scientists describe such a rapid change in a network's connectivity as a phase transition-the same concept used to explain abrupt changes in the state of a material such as the melting of ice or the boiling of water. ![]() But full east-to-west or north-to-south communication appears all of a sudden as the density of users passes a critical and sharp threshold. As users join a new network, isolated pockets of connected phones slowly emerge. How many people scattered throughout Hong Kong need to be connected via the same mesh network before we can be confident that crosstown communication is possible? Credit: Jen Christiansen ( graphic) Wee People font, ProPublica and Alberto Cairo ( figure drawings)Ī branch of mathematics called percolation theory offers a surprising answer: just a few people can make all the difference. But for any two phones to communicate, they need to be linked via a chain of other phones. The collections of linked phones, known as mesh networks or mobile ad hoc networks, enable a flexible and decentralized mode of communication. These apps let a missive hop silently from one phone to the next, eventually connecting the sender to the receiver-the only users capable of viewing the message. For fear of state surveillance or interference, tech-savvy protesters in Hong Kong avoided the Internet by using software such as FireChat and Bridgefy to send messages directly between nearby phones. In fact, it typically goes on a long journey through a cellular network or the Internet, both of which rely on centralized infrastructure that can be damaged by natural disasters or shut down by repressive governments. When you hit “send” on a text message, it is easy to imagine that the note will travel directly from your phone to your friend's.
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