Bridging the gap between graphs and networks

Abstract

Network science has become a powerful tool to describe the structure and dynamics of real-world complex physical, biological, social, and technological systems. Largely built on empirical observations to tackle heterogeneous, temporal, and adaptive patterns of interactions, its intuitive and flexible nature has contributed to the popularity of the field. With pioneering work on the evolution of random graphs, graph theory is often cited as the mathematical foundation of network science. Despite this narrative, the two research communities are still largely disconnected. In this commentary, we discuss the need for further crosspollination between fieldsbridging the gap between graphs and networksand how network science can benefit from such influence. A more mathematical network science may clarify the role of randomness in modeling, hint at underlying laws of behavior, and predict yet unobserved complex networked phenomena in nature.The mathematics of a networked reality Behind the history of mankind's scientific progress there is a story of mathematical theory building. One where scientists not only uncover the behavior of empirical phenomena through experiments and data analysis, but develop theories that abstract such behavior mathematically 1 . This process amounts to scientific understanding when mathematics is "sufficiently isomorphic" to reality 2 ; when theory includes the mechanisms deemed most important and, despite ignoring some features, still allows for logical inferences that accurately describe the phenomena of interest 3 . A sufficient similarity in structure between mathematics and reality allows scientists to forecast events, control natural systems, and even predict behavior that has not yet been confirmed empirically 4 .Up until recent times, scientific efforts towards a mathematical representation of reality have mostly focused on elements of the "simplest" natural systems at both the smallest and largest scales: the physical, chemical, and biological entities comprising the microscopic world, as well as thermodynamic systems and the large-scale structure of the Universe. The success of this

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“…a) Basic graph-based paradigm: Basic graph-based paradigm is based on the graph theory and it can be seen as a set of selection principles for microscopic laws of behaviour in network science [183] which typically involves a simplified graph representation and analysis of networks just concerned with nodes and their connections, e.g. [184], [185].…”

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