Geometric Inhomogeneous Random Graphs

For the theoretical study of real-world networks, we propose a model of scale-free randomgraphs with underlying geometry that we call geometric inhomogeneous random graphs (GIRGs).GIRGs generalize hyperbolic random graphs, which are a popular model to test algorithms forsocial and technological networks. Our generalization overcomes some limitations of hyperbolicrandom graphs, which were previously restricted to one dimension. Nevertheless, our model istechnically much simpler, while preserving the qualitative behaviour of hyperbolic random graphs.We prove that our model has the main properties that are associated with social and technologicalnetworks, in particular power law degrees, a large clustering coefficient, a small diameter, an ultra-small average distance, and small separators. Notably, we determine the average distance of tworandomly chosen nodes up to a factor 1 + o(1). Some of the results were previously unknown evenfor the hyperbolic case.To make it possible to test algorithms on large instances of our model, we present an expected-linear-time sampling algorithm for such graphs. In particular, we thus improve substantially onthe best known sampling algorithm for hyperbolic random graphs, which had runtime Ω(n).Moreover, we show that with the compression schemes developed for technological graphs like theweb graph, it is possible to store a GIRG with constantly many bits per edge in expectation, andstill query the i-th neighbor of a vertex in constant time.For these reasons, we suggest to replace hyperbolic random graphs by our new model in futuretheoretical and experimental studies. Acknowledgement We thank Hafsteinn Einarsson, Tobias Friedrich and Anton Krohmer for helpful discussions.Supported by an ETH Zurich Postdoctoral Fellowship, [email protected]‡[email protected]§[email protected]