Performance of networks of artificial neurons: The role of clustering

The performance of the Hopfield neural network model is numerically studied on various complex networks, such as the Watts-Strogatz network, the Barabási-Albert network, and the neuronal network of Caenorhabditis elegans. Through the use of a systematic way of controlling the clustering coefficient, with the degree of each neuron kept unchanged, we find that the networks with the lower clustering exhibit much better performance. The results are discussed in the practical viewpoint of application, and the biological implications are also suggested.

Figure 1

Exchange of two edges. Two edges [$A\text{−}B$ and $C\text{−}D$ in (a)] are randomly picked and then rewired to have different end vertices [ $A\text{−}D$ and $C\text{−}B$ as in (b)]. The edge exchange keeps the degree of each vertex unchanged.

Figure 2

Efficiency of the Hopfield neural network on the C. elegans neuronal network, BA network, and WS network at the rewiring probabilities $P=0,0.1$, and $1.0$. The overlap $m$ between the stored pattern and the neuron state measures the performance of each network and is plotted as a function of the clustering coefficient $\gamma \equiv \left(\sum {\gamma }_v\right)∕N$. The big filled circles are for the original networks before the edge exchange MC is applied (see text for details). It should be noted that at all points on each curve, not only the degree distribution but also the actual degree of each neuron is exactly identical. Performance of each network monotonically decreases as the clustering becomes stronger.

Figure 3

Networks are generated from the finite-temperature edge-exchange MC for the Hamiltonian $H=-{\sum }_v{\gamma }_v$. (a) Clustering coefficient $\gamma$ versus the temperature $T$. (b) The overlap $m$ versus $T$. From (a) and (b), $m$ as a function of $\gamma$ is obtained in (c).