Unless you’re totally disconnected from the world around you, you’ve noticed by now that everything in the world around you is connected — in a network.
It’s not like the old days, when the “networks” were designated by acronyms for companies producing TV shows. Networks are more diverse now, consisting of any system of nodes connected by links: Web pages and hyperlinks, electric power plants and transmission lines, actors in movies with Kevin Bacon. Since the late 1990s, scientists have been analyzing every sort of network they can think of or think up: networks of terrorists, networks of disease transmission, networks of genes, proteins or nerve cells, banking networks, social networks, transportation networks. Every-thing that isn’t utterly isolated is part of some network or another. Studying networks, figuring out how they work and how they fail, can lead to more efficient systems for travel or communication and better defenses against terrorism and epidemics.
So far, though, most such research has focused on individual networks, apart from others. In real life, networks themselves are generally connected to other networks. Some scientists have begun to realize that the real goal in network science shouldn’t just be understanding individual networks, but deciphering the dynamical interactions in networks of networks. It’s almost always the case that any one network’s success hinges on the support of other networks it’s connected to, as Jianxi Gao of Boston University and colleagues point out in the January issue of Nature Physics.
“In interacting networks, the failure of nodes in one network generally leads to the failure of dependent nodes in other networks, which in turn may cause further damage to the first network, leading to cascading failures and catastrophic consequences,” Gao and collaborators write.
Of course, understanding networks of networks builds on the advances made so far in understanding properties of individual networks, such as their typical “small-world” nature. That realization was the big breakthrough of the 1990s, when mathematicians showed how real-world networks differed from the graphs of dots and lines used to describe networks in textbooks. In those days “graph theory” analyzed the mathematics of random networks — networks in which any two nodes were as likely to be linked as any two other nodes (so that the nodes had random numbers of links). On the other extreme, some experts liked to study completely regular networks, in which every node had precisely the same number of links as every other node.
That all made for interesting math, but most actual networks are neither random nor regular — they are somewhere in between. Take the World Wide Web. A very few nodes (such as Google) have an enormous number of connections; most nodes are barely linked to the rest of the Web at all. Links don’t add up at random; they depend on things like the content and quality of the page and how long it has been around.
In such networks, the huge Google-like hubs make it possible to navigate quickly from any one site to another in very few steps. That “small-world” feature, found in many networks, has implications for how to protect a network from failure, or how to most effectively attack a network (say of disease transmission or terrorists). Scientists have developed a mathematical toolkit for describing such networks, guiding efforts to predict how a given network would respond to such actions as the disabling of a large hub.
Only recently, though, have researchers worked on math for coping with networks of networks. For a simple example, think of the interaction between the power grid network and the communications network. If part of the power network goes out, communications nodes might crash too. And then loss of communications nodes can lead to further breakdowns in the power network.
“The failure of even a small number of elements within a single network may trigger a catastrophic cascade of events that destroys the global connectivity,” proclaim Gao and his coauthors: H. Eugene Stanley (also of Boston University), Sergey Buldyrev of Yeshiva University in New York City and Shlomo Havlin of Bar-Ilan University in Israel.
Early work on developing NON (network of networks) math has already shown signs that properties conferring stability to single networks may work in the opposite direction in networks of networks. So developing NON math further may have important implications for all kinds of social and economic issues. Just as people gradually became aware that networks are everywhere, it will soon be more obvious that networks of networks are everywhere.
Perhaps the most dramatic NON example of all is the human body itself: nervous system, cardiovascular system, respiratory system — all networks, all linked. So you, and everybody else in your social network, are, in fact, a network of networks.