May-28-2017

Chapter 2-1:  Seven Bridges of Königsberg

The Seven Bridges of Königsberg is a historically notable problem in mathematics.

The problem was to devise a walk through the city that would cross each of those bridges once and only once.

Its resolved by Leonhard Euler in 1736 and he laid the foundations of graph theory and prefigured the idea of topology.

Key points: 

1. Page 12: Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all bridges can have only one starting and one end point. 

2. Page 12: Graphs or networks have properties, hidden in their construction, that limit or enhance our ability to do things with them.

3. Page 12: The construction and structure of graphs or networks is the key to understanding the complex world around us. Small changes in the topology, affecting only a few of the nodes or links, can open up hidden doors, allowing new possibilities to emerge.

Chapter 2-2: The Random Network Theory (1959) Page 23.

1. In a random network, there is a population of n nodes. One by one we add links between nodes. We do this by randomly picking one of the nodes, and linking them randomly with another node. www.openabm.org/book/33102/113-random-network

2. Page 22: If the random network is large, almost all nodes will have approximately the same number of links. 

3. Page 22: The histogram a the degree for nodes in random network follows a Poisson distribution, which has a prominent peak, indicating that the majority of nodes have the same number of links as the average node does. On the two sides of the peak the distribution rapidly dinimishes, making signicant deviations form the average extrmely rare.