Lecture 11: Large-scale structure of networks

In this lecture, various aspects and natural phenomenon of large scale networks were observed and discussed. In particular, we were interested in components within networks, their relationships among nodes, and shortest paths. One question arised in which we asked "Why would there be a giant component in a network?" We can answer this by considering the purpose of a network: to connect to others. Networks exist to provide communication between all inhabitants residing in these networks. Further, we examined the possibility of a network containing 2 largest components. We deduced this to be impossible as the probability of randomly added edges to the nwtwork is far too high in which a network can only contain either 1 or no largest component.

Another discussion in this lecture was the consideration of shortest paths and the small world effect. Analyzing shortest paths are a commonly computed measurement of networks. When considering shortest paths, we must also discuss the total number of nodes within a network and the small world effect. Essentially, as the number of nodes grow within random networks, the average path length, l, between nodes do not undergo a dramatic change. That is, most random networks have their average path lengths related to a logarithmic scale: l ~ log(n). It is because the fact that l does not change very much with respect to a dramatic increase in the nodes in a network that we define the small world effect. For example, in the current state of the internet, nodes can typically reach other nodes within 10 to 12 hops. Similarly, any human on Earth should be able to reach another human within 6 aquaintances. The small world effect essentially says we are all closely connected than we think. Further, if a network is a power law network, we will get l ~ log(log(n)).

Lastly, our lecture concluded with a discussion on hop distribution, i.e., how many nodes can be reached in n hops. When considering distribution, most centrality measures are similar to the power law except betweeness centrality (on the AS-level). If we examine the graph, we will notice a straight, veritcal line at the beginning of the plot. This indicates all the nodes (ASes) and their betweeness centrality. These nodes are stub networks such as universities that are connected to the network but do not necessarily have anyone passing through them and thus, not serving as a shortest path to anyone--they are on the edge of the network. Further, we observe the internet to lack redundancy as there are not many triangles in which the cluster coefficient is actually less than its expected value.