Connectivity comparison of uniform quantization-based graph transformation and its application in spectrum sensing

Abstract

In the uniform quantization-based graph transformation of random sequences, the connectivity of the graph is an important feature and can be considered as a basis for designing graph domain processing algorithms. However, the current research on graph transformation focuses mostly on analysing the majorization orders of different input distributions to achieve a comparison of graph connectivity. Generally, calculating majorization order between two candidate probability density functions requires two inversions and one integration for each probability density function, and obtaining an analytical solution is extremely difficult. Even if an analytical solution can be obtained, considering the impact of function transformations such as normalization, analysing the reservation of majorization order is challenging. However, the dispersive order calculation only involves one inversion and differentiation of the distribution functions, and it implies majorization under certain conditions. Consequently, the objectives of this study were to investigate the dispersive order and its order reservation between two random sequences with different probability distributions and to compare the differences in connectivity between the graphs generated under the same conditions, such as the number of uniform quantization levels and sample size. In addition, the properties of complete graphs generated from random sequences were evaluated, including the influence of the probability vector of vertices on the graphs, the sample size and number of vertices on the probability of generating complete graphs. As an application, a graph-based spectrum sensing algorithm based on the block-maxima of the power spectrum was formulated. Extensive simulations showed that the proposed algorithm outperforms the existing graph-based sensing algorithms both in detection performance and the computational complexity.