http://mathforum.org/isaac/problems/bridges1.html

I wanted my first blog posting to get back to the roots of graph theory because the mathematics behind graphs and the number theory behind the mathematics is what fascinates me the most.  “The Seven Bridges of Konigsberg” and Euler numbers are one of the base principles when studying the math and number theory behind networks.  The website does an excellent job of explaining the basis of the problem, but I would like to show how it relates to the class. 

As it says on the web site, it is not possible to cross all of the bridges of the original problem only once.  This is similar to phenomena in class with triadic closure, directed connectedness of graphs and even the Braess paradox.  If we consider a node on each far end of a bridge, using either directed or undirected edges, there is no way to make the graph connected without having two edges between nodes (pending the condition that we can’t use an edge twice).  Having this ‘extra edge’ creates an increase in travel time that would be experienced to create this Braess paradox in the directed graph.   It also creates a lack of need for a bridge, as shown in the website.  At least one of the bridges is creating excessive travel time for the people of Konigsberg.  We can also view the number and direction of edges as a need to form strong/weak ties, or to look into power in social networks.  Certain edges must be traveled more than others, some must have power and some must be strong ties.

I am sure that most people in this class have had enough math classes to know about this, so I am keeping the blog brief.  It truly amazes me how much can be inferred about networks and about math from looking at such a basic map of a town.  This one example has been the basis for numerous proofs and a prime example of almost everything we have learned thus far in INFO 204.