In a map of a city it is easy to spot the links that, if cut, have the largest effects on connectivity, especially if there is a river flowing through it. Such a link is called a sparsest cut, and in a typical graph, it is much harder to spot. But there are many instances in which we want to make those cuts, such as when we wish to decompose a network into simpler subgraphs with minimal effect on connections. For instance, a Laplacian-based method has been used to extract features from images. The surface of an object can be approximated by a mesh. The spectrum of this mesh can be used by a computer to identify distinct regions of the image without any human input.
Cheeger’s work was important because it related something that was known to be very difficult to compute (sparsest cuts) to something that is much easier to compute (eigenvalues). It should be noted that Cheeger’s theorem is an estimate, not a precise equation. However, the connection has proved robust enough to make the spectrum of the Laplacian an important tool in graph partitioning.
Spectral methods—a class of techniques to numerically solve dynamical (or constantly evolving) systems—can be used to design new networks as well as to study existing ones. For example, networks with the largest possible eigenvalues, called “expander graphs,” have some remarkable properties. They convey information very rapidly, so it is easy to route messages from one point to another. They have no bottlenecks, so they cannot be disabled by the failure of a few nodes or links. Unfortunately, they are not so easy to design, so research in this area is currently a high priority.
Finally, although eigenvalues are easier to compute than sparsest cuts, even they get difficult when you are faced with monster networks consisting of millions or billions of nodes. Until recently, algorithms for finding the eigenvalues did not scale well and were not practical for such large networks. However, it now appears that this barrier has been breached.
In 2004, a method was devised for solving equations involving graph Laplacians that was significantly faster (in theory) than any previously known method. Like many great scientific advances, the strategy was wonderful, completely unexpected, and yet in retrospect completely natural. The idea is to replace the Laplacian of the graph with the Laplacian of a well-chosen subgraph. If this is done carefully, the spectrum is almost undisturbed yet the graph becomes much sparser. Imagine ripping out 95 percent of the streets in Manhattan and having everybody’s commute time remain almost the same! That is roughly what this method managed to do.
There is still plenty of room for practical improvement, and some substantial improvements have already been made. The long-standing dream of finding sparsest cuts in approximately linear time—in other words, being able to understand the structure of a network almost as quickly as one can read in the data—now appears much closer.
Until recently, algorithms for finding the eigenvalues did not scale well and were not practical for such large networks. However, it now appears that this barrier has been breached.