Fragile Correctness of Social Network Analysis

Draft version of the paperGraph techniques are widely used in social network analysis. However, there are some disputable applications where results are obtained from the graphs using paths longer than one and are not simply applicable to objects in the initial domain. The author provides several examples of such usages and tries to recover roots of an incorrect application of graphs

The random k-matching-free process

Let $\mathcal{P}$ be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty $n$-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that $\mathcal{P}$ is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the $k$-matching-free process, where $\mathcal{P}$ is the property of not containing a matching of size $k$. We are able to analyse the behaviour of this process for a wide range of values of $k$; in particular we prove that if $k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to terminate in a $k$-matching-free graph with the maximum possible number of edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds on $k$ are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime