How links are formed according to the distance in transaction network of 800,000 firms

We studied how links are formed according to the distance in transaction network of 800,000 firms. We investigated the inter-firm transaction network of $800,000$ firms in Japan. They are almost all firms in Japan. As far as we know, scale-freeness, hierarchy, and degree-degree correlation were discovered as fundamental characteristics in the network before the present paper. We discovered another fundamental characteristic as to distance. Distance plays an important role in the network. The number of transaction links between two regions is proportional to the product of the sales (degrees) of the two regions divided by the distance between the two regions. This is of a form of the gravity equation. It implies that the further the firms are, the less likely it is that they are linked in the network.