Robustness in sparse random-like graphs

Sparse, highly connected, random-like graphs are currently a focal point in discrete mathematics, theoretical computer science and network science, motivated by the insight that networks of this type are ubiquitous in computing, biology, economics, physics, social science, etc., and by the theoretical challenges of this setting.

Random graph models used in statistical modelling of real-world networks include the Molloy-Reed model for scale-free graphs and the Watts-Strogatz model (designed to simultaneously exhibit the small-world phenomenon and formation of hubs, cited 43000 times).

Barabási devotes a chapter of his classic network science book to robustness of random graphs, specifically addressing robustness against adversarial attack. Extremal combinatorics includes the fundamental study of sparse networks.

Our line of enquiry is sparse Ramsey theory, concerning sparse graphs that are robust in a strong sense, with respect to adversarial edge-partitioning. This notion is also of interest in theoretical computer science. An archetypal robust graph in Ramsey theory is the Erdős-Rényi random graph.

Our project addresses some limitations of this paradigm.This projects aims to (1) solve important open problems in Ramsey theory, shedding light on a surprising synergy between structural and Ramsey-type properties of graphs, (2) open up new frontiers in sparse Ramsey theory by using a random Cayley graph (RCG) as a much-needed alternative sparse robust graphs to the Erdős-Rényi model, (3) advance the essential tools in random graph theory (sparse regularity framework, concentration bounds, embedding and colouring techniques) by taking them into an entirely new algebraic setting, (4) illuminate potentially useful classes of expander graphs with both structure and randomness.