Maximum likelihood thresholds via graph rigidity

DIB was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship from the US NSF Grant DMS-1802902. SD was partially supported by the Austrian Science Fund (FWF): P31888. AN was partially supported by the Heilbronn Institute for Mathematical Research. SJG was partially supported by US NSF Grant DMS-1564473. MS was partially supported by US NSF Grant DMS-1564480 and US NSF Grant DMS-1563234.The maximum likelihood threshold (MLT) of a graph G is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{ö}s-Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{ö}s-Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4. We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.Peer reviewe

Second Generation General System Theory: Perspectives in Philosophy and Approaches in Complex Systems

Following the classical work of Norbert Wiener, Ross Ashby, Ludwig von Bertalanffy and many others, the concept of System has been elaborated in different disciplinary fields, allowing interdisciplinary approaches in areas such as Physics, Biology, Chemistry, Cognitive Science, Economics, Engineering, Social Sciences, Mathematics, Medicine, Artificial Intelligence, and Philosophy. The new challenge of Complexity and Emergence has made the concept of System even more relevant to the study of problems with high contextuality. This Special Issue focuses on the nature of new problems arising from the study and modelling of complexity, their eventual common aspects, properties and approaches—already partially considered by different disciplines—as well as focusing on new, possibly unitary, theoretical frameworks. This Special Issue aims to introduce fresh impetus into systems research when the possible detection and correction of mistakes require the development of new knowledge. This book contains contributions presenting new approaches and results, problems and proposals. The context is an interdisciplinary framework dealing, in order, with electronic engineering problems; the problem of the observer; transdisciplinarity; problems of organised complexity; theoretical incompleteness; design of digital systems in a user-centred way; reaction networks as a framework for systems modelling; emergence of a stable system in reaction networks; emergence at the fundamental systems level; behavioural realization of memoryless functions