What do the bridges of Königsberg, synaptically connected neurons in our brains and the galaxies illuminating the dark voids of our universe have in common? All of these real-world phenomena can be described as collections of discrete discernible objects which are interlinked to form weblike structures called networks. This chapter will introduce the mathematical representation of such networks, and familiarise us with the basic concepts, ideas and terminology of a vast and ever-growing research field whose roots date back to the work of Leonard Euler. By taking a closer look at a number of concrete network models - specifically the random graph models which prominently feature as descriptive vessels for many natural phenomena - and briefly exploring some deep-rooted conceptual limitations of these models, we hope not only to motivate the need for a rigorous mathematical framework for the study of networks at finite scales, but also to accentuate the potential advantages of a more dynamical vantage point from which to view networks and their defining characteristics in later chapters of this book.

Over the course of three centuries, the field of graph theory has matured from its initial conception as an abstract model for solving a rather specific mathematical problem into a powerful vessel for describing countless real-world phenomena, with applications now reaching far beyond applied mathematics. However, its static formalisation puts a number of limitations in place which hamper an advantageous utilisation of graph-theoretical concepts in circumstances which require a more dynamical perspective. Is it possible to overcome these limitations by challenging the classical notion of a graph? In this chapter, we will propose exactly such a challenge by considering a graph’s nodes and their relations as the result of operations performed on a set of suitable objects. This subtle yet consequential change in the conception of a graph not only delivers a more dynamic vantage point, but eventually generalises the very notion of a graph by structurally equating it with an abstract algebra. This chapter will introduce the basic notions and formalisations of an operator graph-theoretical framework and candidly argue for its potential merits and usefulness.

When thinking of city maps, we instinctively envision a network of links along which an ever-changing flow of traffic is carried. Such an idealised description, however, is not limited to the maps we are all familiar with. From the interactions between atoms and subatomic particles to the gravitational forces which act between the billions of galaxies stretching across the known universe, from the transmission of electrical signals in our brains to the complexity of social interactions between people, most if not all phenomena we encounter, consciously or not, find a natural representation in the form of networks. Indeed, it can be argued that the abstract notion of interacting objects resides at the very heart of our conceptual understanding of nature as it touches upon the very fabric of physical reality with its finite and discrete makeup. How can we leverage the mathematical study of interconnected objects, the theory of networks and graphs, in our quest of understanding nature, and what are its limitations?

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.

Navigating across sometimes treacherous waters, we demonstrated in the previous chapter the construction and utilisation of a small selection of graph observables for measuring various properties of finite random graphs. Together with the generation of operator representations for arbitrary graph models, we should now be in possession of a sufficiently equipped toolset with which to further explore and characterise on rigorous algebraical grounds the plethora of graph models in the applied graph-theoretical literature. However, our adventurous journey would not be complete without touching upon another crucial aspect exhibited by many real-world networks - their dynamic nature. In this final chapter, we will explore with one hopefully light-hearted, playful example - the game of chess - how to formulate such dynamical aspects in our operator graph-theoretical language. As we will witness here, the construction of graphs that describe possible moves of chess pieces at any position during a game, and the transformations that lead to changes of such positional chess graphs, pose a formidable challenge for not only computational algorithms.

The first part of this book led us on a journey from one of the undoubtedly most cherished fields of applied mathematics, classical graph theory, across the ghastly depths of an inherently dynamic formalisation of physical reality in terms of mappings and operators, to an inspired attempt at a fusion of both of these perspectives. With the backing of a conceptional and notational framework at hand, it is now time to put this attempt to a test, as we continue our adventurous journey with an excursion into the endless realm of applications. This second part of our journey will start in this chapter with an exploration of graph generators and their operator graph-theoretical formulation. In this undertaking, we will focus primarily on the generation of random graphs as such models enjoy, in one way or another, widespread and prominent employment throughout almost all fields of science and technology. Only the last section will see the exemplary generation of an exact graph model, the finite square grid graph, as preparation for a closer inspection of an intriguing yet unsolved problem at the very heart of condensed matter physics in the next chapter.