The brain has various functions such as memory, learning, awareness, thinking and so on. These functions are produced by the activity of neurons that are connected to each other in the brain. There are many models to reproduce the memory of the brain, and the Hopfield model is one of the most studied (Hopfield, 1982). The Hopfield model was proposed to reproduce associative memory, and it has been studied extensively by physicists because this model is similar to the Ising model of spin glasses. This model was studied circumstantially, for example, the storage capacity was analysed by the replica method (Amit, 1989; Hertz et al., 1991). However, in these studies, the neural networks are completely connected, i.e. each neuron is connected to all other neurons. It was not clear how the properties of the model depend on the connections of neurons until recently (Tosh & Ruxton, 2006a, 2006b).
In recent years the study of complex networks has been paid much attention. A network consists of nodes and links. A node is a site or point on the network such as a neuron; the nodes are connected by links such as an axon or synapse of a neuron. Several characteristic network structures have been proposed, and the small-world and the scale-free networks have been studied heavily in recent years. Small-world networks have the properties that the characteristic path length is very short, and simultaneously the clustering coefficient is large (Watts & Strogatz, 1998).