We show that the number of points with pairwise different sets of neighbors in a graph is O(2) where r is the rank of the adjacency matrix. We also give an example that achieves this bound.

This chapter discusses the aggregate dynamics of population systems that maintain a consistent regularity through time in the sizes of their elements. City size distributions represent the classic example with the size of cities being inversely proportional to their rank. This is enshrined in Zipf’s (1949) rank-size rule which provides one of the strongest and most exact relationships throughou...

Effectively monitoring the size of a city in real time enables scientific planning urban development. Models that utilize distribution and variations generally use population data as inputs, which cannot be obtained timely rapid manner. However, night-time light (NTL) remote sensing may an alternative method. A case study was carried out on Yangtze River Delta (YRD) China, rank–size rule, law p...

Sample size analysis is a key part of the planning phase any research. So far, however, hardly literature focuses on sample methods for two-sample linear rank tests, although these have optimal properties different distributions. This article provides new method tests location shift alternatives based score-generating functions. Results show slightly anti-conservative behavior, no severe risk a...

Background A pilot study can be an important step in the assessment of an intervention by providing information to design the future definitive trial. Pilot studies can be used to estimate the recruitment and retention rates and population variance and to provide preliminary evidence of efficacy potential. However, estimation is poor because pilot studies are small, so sensitivity analyses for ...

Experimental design requires estimation of the sample size required to produce a meaningful conclusion. Often, experimental results are performed with sample sizes which are inappropriate to adequately support the conclusions made. In this paper, two factors which are involved in sample size estimation are detailed--namely type I (alpha) and type II (beta) error. Type I error can be considered ...