6 - Measuring Graphs

Summary

On our adventurous journey, we formalised in the previous chapter the generation of various finite random graph models in terms of suitable algebraic objects and obtained representations of these models which reside beyond their classical or algorithmic descriptions. This approach naturally paves the way for a more rigorous investigation of the vast plethora of graph-theoretical measures that typically are, or only can be, considered in tedious and demanding numerical studies, or under stringent limitations in asymptotic assessments. In this chapter, we will exemplify how the properties of the algebraic objects governing the generation of graphs can be exploited, and how parametrised expressions for a variety of graph measures can be obtained. Here we must restrict our gaze into the sheer limitless realm of possibilities to a few selected directions. By highlighting some of the differences to already available results from a conceptual and mathematical vantage point, we will continue to argue for the necessity of a study of networks at finite scales, for which our operator graph-theoretical framework presents itself as one viable approach.