Classical network science shares an origin with topology and graph theory: all there cite Swiss mathematician Leonard Euler's "Seven Bridges of Koningsberg" problem as a common ancestor. Euler, an undisputed titan of the mathematical enlightenment, had made strides in applied mathematics, in Fourier series, in the estimation of pi, in physics, logic, astronomy, and the calculus, and today is the eponym of the mathematical constant, e, the base of the natural logarithm. Our interest in Euler is because of a rather ordinary riddle that allegedly captivated the minds of mid-18th century Eastern Prussians: the "Seven Bridges of Koningsberg" Problem. The city of Konigsberg had been a cultural and economic hub for centuries in eastern Germany. It was the home and workplace of philosopher Immanuel Kant. Centuries later it served as a backdrop for some of Europe's most brutal struggles: the Napoleonic Wars, World War I and World War II. Geographically, the port city was first split and then forked by the Pregel River en route to the Baltic Sea. This produced two islands and divided the city itself into four major landmasses. The four landmasses were connected by a series of seven bridges: three of the landmasses each had three bridges, the remaining had five. It looked like this:

The mathematical riddle asks the question: Is there a path by which we can cross each bridge only once and return to the starting point? If you try, you'll soon find there isn't one. The Prussians knew there was no possible path, but it was uniquely Euler who codified the problem and answered the question definitively. Euler considered a simplification of the problem, one in which the components could be mapped by their connectednses. If we consider each landmass, and the bridge that connects them, as represented by a circle and a line, we see a simplification of the problem:

Since a full trip requires that each landmass is entered and exited, the mapping implicitly requires that each landmass be connected by an even number of bridges. Only when this assumption fulfilled is it possible to create a complete tour, today known as a Eulerian cycle. There is no Eulerian cycle in Koningsberg.

Today Euler's Koningsberg is gone. After repeated bombings by the Soviet and British Air Forces in World War II, the city was left in ruin. Nearly all of the bridges, along with the architecture, history and culture of a centuries-old city were reduced to rubble. The Germans were expelled after the war, and the Soviets seized the city. Today it is the Russian Oblast of Kaliningrad, where three re-built and two restored bridges connect the city.

Euler's consideration of the mathematical properties of the connectedness of the components is his great stylization that is credited as the first theorems in both graph theory and topology. This connected approach does not concern the entities themselves, landmasses and bridges, but instead represent a mapping of abstract entities. The answer to the Seven Bridges problem is that this mapping in aggregate holds the answer. The question can be analyzed simply in its representational form:

Discussing the Euler problem in some detail reflects a general theme of concentration on the connectedness between objects just as heavily as their composition. Topologically, it extends the idea that the fundamental "shape" of things may stay relatively or connectedly the same even while graphically undergoing a transformation. Using these techniques, we can begin to pose more questions regarding the underlying topologies of our analysis, to see if we can apply these principles and obtain insight into the overall system. Simply investigating the components is that it only tells you only one part of the story, that of properties. The connectivity, as we will find out, tells a story of its own.