The shape of randomness
I’m happy to announce that the results of a fun project with Alice Patania, Giovanni Petri and Francesco Vaccarino are now available in a short paper in Physical Review E. Special thanks to the YRNCS who funded this collaboration through their Bridge Grant initiative.
Our paper, titled “Construction of and efficient sampling from the simplicial configuration model”, builds on recent work by Owen T. Courtney and Ginestra Bianconi. More specifically, we propose an efficient and provably correct MCMC algorithm for the maximally random ensemble of simplicial complexes with given degree and dimension sequences. This algorithm comes in handy when we use simplicial complexes as an abstraction for the structure of complex systems (as has been done more and more often recently). By randomizing a simplicial complex that encodes the structure of some system X, we get to tell what connection patterns in X are explained by simple local properties (degrees and dimensions), and what patterns are surprising. In our paper, we this method to show that the homology groups of a few real-world systems are decidedly not random, and we suggest some mechanisms that could explain these differences. But one could use the method to investigate different property; this will be dictated by application!
The next step will be to develop correlated variant of the ensemble (think of the degree-correlated CM) and to delve into combinatorial results (e.g., rigorously prove the traversability of the associated simplicial complex space).