Probability Theory is one of the most important tools for studying networks. Most things Probability Theory tries to explain are about average or typical observations. When studying networks we use Probability Theory to answer questions like “what is an average degree in a network” or “what is the average time it takes to travel from A to B”. Sometimes, however, we want to know something about very rare events. For instance, we might ask “what is the probability that our electrical grid will be so overloaded that it breaks down?”
For such questions we can use Large Deviations Theory. The most important lesson that Large Deviations teaches us is that if something improbable must happen, then it will happen in the most probable of all improbable ways. I could go on to explain how Large Deviations Theory works, but fortunately I don’t have to because Nautilus has an article by David Steinsaltz, a Statistics professor at Oxford, that explains this amazing mathematical idea with improbable clarity.