Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks

An accurate test for homogeneity of odds ratios based on Cochran's Q-statistic

Background: A frequently used statistic for testing homogeneity in a meta-analysis of K independent studies is Cochran's Q. For a standard test of homogeneity the Q statistic is referred to a chi-square distribution with K - 1 degrees of freedom. For the situation in which the effects of the studies are logarithms of odds ratios, the chi-square distribution is much too conservative for moderate size studies, although it may be asymptotically correct as the individual studies become large. Methods: Using a mixture of theoretical results and simulations, we provide formulas to estimate the shape and scale parameters of a gamma distribution to t the distribution of Q. Results: Simulation studies show that the gamma distribution is a good approximation to the distribution for Q. Conclusions: : Use of the gamma distribution instead of the chi-square distribution for Q should eliminate inaccurate inferences in assessing homogeneity in a meta-analysis. (A computer program for implementing this test is provided.) This hypothesis test is competitive with the Breslow-Day test both in accuracy of level and in power

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Graph isomorphism and genotypical houses

This paper will introduce a new method, known as small graph matching, anddemonstrate how it may be used to determine the genotype signature of a sample ofbuildings. First, the origins of the method and its relationship to other ?similarity? testingtechniques will be discussed. Then the range of possible actions and transformations willbe established through the creation of a set of rules. Next, in order to fully explain thismethod, a technique of normalizing the similarity measure is presented in order to permitthe comparison of graphs of differing magnitude. The last stage of this method ispresented, this being the comparison of all possible graph-pairs within a given sampleand the mean-distance calculated for all individual graphs. This results in theidentification of a genotype signature. Finally, this paper presents an empiricalapplication of this method and shows how effective it is, not only for the identification ofa building genotype, but also for assessing the homogeneity of a sample or sub-samples