Scaling limits of critical directed random graphs

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Past work by Addario-Berry, Broutin and Goldschmidt has established the scaling limit of components of a critical undirected Erdos--Renyi graph to a sequence of metric spaces. These metric spaces are constructed by using tilted Brownian excursions to code real trees, then adding a finite number of point identifications. Further work has shown these limit objects exhibit universality. Work by authors Bhamidi and Sen and authors Conchon--Kerjan and Goldschmidt has established that the components of a critical configuration model (under finite moment conditions on the degree distribution) have the same scaling limit, and work by Bhamidi, Sen and Wang have shown the same is true for rank-one inhomogeneous random graphs. Real life networks, however, are usually directed. The relationships in networks like Twitter, financial transactions, the world wide web and disease transmission are all asymmetrical. Hence directed graphs provide a more realistic model of real-world networks, yet they remain relatively unstudied compared to their undirected counterparts. Luzack and Seierstad established a phase transition for the existence of a giant strongly connected component in the directed Erdos--Renyi model. In the critical regime of this phase transition, Goldschmidt and Stephenson have recently shown the strongly connected components (SCCs) can be scaled into a sequence of random weighted multi-digraphs. The goal of this research project is to show universality of these limit objects. Cooper and Frieze have shown the existence of the phase transition in the directed configuration model, and we have made progress in characterising the scaling limit of the SCCs in the critical regime. In particular with the appropriate choice of parameters, the limiting object is the same as that for the Erdos--Renyi model. We conjecture that the same will be true for certain classes of directed inhomogeneous random graphs. Extending results from the Erdos--Renyi model is important to apply these results practically. While the Erdos--Renyi model is analytically simple to work with, it is not an accurate model for real networks. For example, the degrees in real networks often exhibit a power law. This is not present in Erdos--Renyi random graphs, but the configuration model can be made to exhibit this property. Moreover, showing universality of the scaling limit means these results are less sensitive to model misspecification. The work by Conchon--Kerjan and Goldschmidt mentioned previously also looked at configuration models when the size-biased degree distribution is in the domain of attraction of a general alpha-stable Levy distribution rather than just the Gaussian distribution. This yielded a family of universality classes in a similar way to how alpha-stable Levy processes generalise Brownian motion. If successful in studying the directed configuration model with sized biased degree distributions in the domain of attraction of a Gaussian distribution, another goal of this research project would be to look at the alpha-stable case. Further there are edges between SCCs of a directed graph which we ignore when studying the scaling limit of the SCCs. This contrasts with the undirected case where all edges are included in a component. The condensation of a directed graph is a natural proxy for the edges not used in SCCs, thus another future research avenue could be studying the scaling of condensations of critical random digraphs.This project falls within the EPSRC Mathematical Analysis, Statistics and Applied Probability, and Logic and Combinatorics Research Areas. The project is supervised by Prof. Christina Goldschmidt the work is joint with Serte Donderwinkel.