Size and form in efficient transportation networks

Many biological processes, from cellular metabolism to population dynamics, are characterized by allometric scaling (power-law) relationships between size and rate(1-10). An outstanding question is whether typical allometric scaling relationships-the power-law dependence of a biological rate on body mass-can be understood by considering the general features of branching ne tworks serving a particular volume. Distributed networks in nature stern fr om the need for effective connectivity(11), and occur both in biological sy stems such as cardiovascular and respiratory networks(1-8) and plant vascul ar and root systems(1,9,10), and in inanimate systems such as the drainage network of river basins(12), Here we derive a general relationship between size and flow rates in arbitrary networks with local connectivity. Our theo ry accounts in a general way for the quarter-power allometric scaling of li ving organisms(1-10), recently derived(8) under specific assumptions for pa rticular network geometries. It also predicts scaling relations applicable to all efficient transportation networks, which we verify from observationa l data on the river drainage basins. Allometric scaling is therefore shown to originate from the general features of networks irrespective of dynamica l or geometric assumptions.