Maximum entropy for economic and brain networks: network reconstruction, early-warning signals, and module detection

In many cases of practical relevance, one needs to construct ensembles of random networks, or random time series, that obey specified constraints. In these cases, the maximum entropy construction is a natural recipe to generate randomness, however the presence of several heterogeneous constraints leads to important differences with respect to the traditional construction. For instance, in order to reliably estimate the risk of collapse of a financial system, one needs to infer the network of linkages between banks and/or firms, but this network is empirically unaccessible due to confidentiality. One therefore has to reconstruct the network from partial, publicly available information about individual financial institutions. I will discuss various maximum-entropy network reconstruction methods, highlighting the importance of capturing the heterogeneity of the constraints correctly. I will also discuss how ensembles of reconstructed networks can be used as benchmarks to detect early-warning signals of upcoming crises in empirical interbank networks. Then, I will describe maximum-entropy ensembles of constrained time series, and use their properties to empirically identify communities of correlated stocks in financial markets and functional modules of correlated neurons in the brain. I will conclude showing that, in all the cases considered, the presence of an extensive number of constraints leads to a surprising breaking of the equivalence between canonical and microcanonical ensembles, with important consequences for the statistical physics of systems with many constraints.