The book begins with a river in Prussia. In the 18th century, there was a series of bridges in Konigsberg that connected the banks of either side of the river to each other an island in the middle. A popular topic of discussion in cafes at the time asked if you could cross each bridge only once and come back to where you started. A man named Euler finally solved this problem. It's not significant that he solved it. The significance is in how he solved it.
Euler abstracted the bridges into a series of nodes and links. The nodes were the landmasses the bridges connected and the links were the bridges themselves. Using this graph, Euler discovered that the only way such a feat is possible is if the nodes with an odd number of links are either the starting or ending point in the traversal. That means there can only be two such nodes. Unfortunately, all the nodes on the Konigsberg bridge graph had an odd number of links so the answer to the problem is no.
This is important because Euler representing a real-life construct and as graph which had inherent properties through which its behavior could be deduced. Considering that graphs are just small networks, this discovery is significant.
Linked discusses "scale free networks," which are networks that consist of a huge number of small nodes connected to a small number of huge hubs via links. Scale-free networks can be found in naturally occurring systems such as the food web and social networks and in man-made systems such as the Web. Despite their dissimilarities, they all share some important properties.