Random walks on the neural network of C.elegans

In this article, we study random walks on the neural network of C.elegans. We define a random variable Y
sh
(z) as the steps the first arriving random walker takes to arrive at the target h when all z walkers start out from the source s simultaneously. Some characteristics concerned with Y
sh
(z) are discussed. Firstly, we find that as walk number z increases, 〈Y
sh
(z)〉 drops drastically in the interval z ∈ [1; 50]. Secondly, we observe that the probability for multiple random walkers taking optimal paths with short path length can be very high, for example, Prob{Y
sh
(N) ≤ 2d
sh
} = 0:84 on average. Thirdly, we look into the spreading processes on the network. We notice that when z = N, the walkers altogether will visit about 90% nodes within 8 steps. Lastly, we demonstrate different roles played by central nodes and periphery nodes. Central nodes can get information more quickly and with higher probability than periphery nodes. All these facts imply that the neural network of C.elegans has an optimized topological structure, which enables dynamic processes, such as transportation, information dissemination, random routing, etc, function efficiently on it.