A few applications and critical phenomena on networks

Abstract

Statistical mechanics is all about interactions. Without interactions, the collective behavior of many objects is the simple sum of that of each object and there is no surprising emergent behavior stemming from collectivity. Despite the explosive development of classical and quantum mechanics dealing with the fundamental force of nature and the characteristic of each separate interaction, the topological structure of interactions had been somewhat considered as a minor topic of research, except for the Euclidean dimension. The situation abruptly changed at the beginning of twenty-first century, as physicists, along with researchers from all kinds of disciplines, started to realize that there are much more complex structure of interactions and the researches on the interaction structures or $\It{network science}$ have ``emerged`` and prospered to this day, as results from ``interactions`` or collaborations among many scientists. This thesis is the record of our participation to those interactions. First, we study the statistical properties of the sampled scale-free networks, deeply related to the proper identification of various real-world networks. We exploit three methods of sampling and investigate the topological properties such as degree and betweenness centrality distribution, average path length, assortativity, and clustering coefficient of sampled networks compared with those of original networks. It is found that the quantities related to those properties in sampled networks appear to be estimated quite differently for each sampling method. We explain why such a biased estimation of quantities would emerge from the sampling procedure and give appropriate criteria for each sampling method to prevent the quantities from being overestimated or underestimated. The next topic is about our efforts to construct the relatedness networks. Recently, massive digital records have made it possible to analyze an enormous amount of data in various research fields...