Constrained Markovian dynamics of random graphs

We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the `mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology.Comment: 28 pages, 6 figure

A rational method to determine cutoff scores

A new method is proposed to set multiple standards in performance tests. The method combines three sources of information coming from three different data collections. The first is an empirical definition of mastery of an item; the second consists of parameter estimates of the items in an Item Response Theory (IRT) model, and the third source is a collection of experts' judgments on the relation between item mastery and level of performance. These judgments are given as an answer to very simple questions. The method is not iterative, and the experts are not required to judge borderline persons. The standard setting procedure is simple and can be carried out without a computer