It is well known that many real world networks have a power-law degree distribution (the scale-free property). However, there are no rigorous results for continuous-time quantum walks on such realistic graphs. In this paper, we analyse the space–time behaviour of continuous-time quantum walks and random walks on the threshold network model, which is a reasonable candidate model having the scale-free property. We show that the quantum walker exhibits localisation at the starting point, although the random walker tends to spread uniformly.

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

An associative memory (AM) system using fuzzy sets is presented in this paper. Vectors to be stored in the AM are transformed into fuzzy sets. Fuzzy rules are constructed from the fuzzified vectors to associate vector when a stimulus vector is input to the AM system. Computational experiments are carried out to evaluate performance of the AM system. The experiments show that the AM system is responsive to feature of the stimulus vectors. In addition, the AM system performs more than 95% correct association rate until 20% of the binary valued elements of the stored vectors are flipped randomly from either zero or one.