Centrality measures for graphons: Accounting for uncertainty in networks

Graphs provide a natural mathematical abstraction for systems with pairwise interactions, and thus have become a prevalent tool across various scientific domains. However, as uncertainty permeates data-acquisition methods, and the size of relational datasets continues to grow, traditional graph-based approaches are increasingly replaced by more flexible modeling paradigms. A promising framework in this regard is that of graphons, which provide an overarching class of non-parametric random graph models. While the theory of graphons is already well developed, some prominent tools in network analysis still have no counterpart within the realm of graphons. In particular, node centrality measures, which have been successfully employed in various applications to reveal important nodes in a network, have so far not been defined for graphons. A key motivation for closing this gap is that centrality measures defined at the modeling level of graphons will be inherently robust to stochastic variations of specific graph realizations. In this work we introduce formal definitions of centrality measures for graphons and establish their connections to centrality measures defined on finite graphs. In particular, we build on the theory of linear integral operators to define degree, eigenvector, and Katz centrality functions for graphons. We further establish concentration inequalities showing that these centrality functions arise naturally as limits of their analogous counterparts defined on sequences of converging graphs of increasing size. Moreover, we provide high-probability bounds on the distance between the graphon centrality function and the centrality measures realized in any sampled graph. We discuss and exemplify several strategies for computing graphon centrality functions and illustrate the aforementioned concentration inequalities through numerical experiments.