Scaling theory for information networks

Networks distribute energy, materials and information to the components of a variety of natural and human-engineered systems, including organisms, brains, the Internet and microprocessors. Distribution networks enable the integrated and coordinated functioning of these systems, and they also constrain their design. The similar hierarchical branching networks observed in organisms and microprocessors are striking, given that the structure of organisms has evolved via natural selection, while microprocessors are designed by engineers. Metabolic scaling theory (MST) shows that the rate at which networks deliver energy to an organism is proportional to its mass raised to the 3/4 power. We show that computational systems are also characterized by nonlinear network scaling and use MST principles to characterize how information networks scale, focusing on how MST predicts properties of clock distribution networks in microprocessors. The MST equations are modified to account for variation in the size and density of transistors and terminal wires in microprocessors. Based on the scaling of the clock distribution network, we predict a set of trade-offs and performance properties that scale with chip size and the number of transistors. However, there are systematic deviations between power requirements on microprocessors and predictions derived directly from MST. These deviations are addressed by augmenting the model to account for decentralized flow in some microprocessor networks (e.g. in logic networks). More generally, we hypothesize a set of constraints between the size, power and performance of networked information systems including transistors on chips, hosts on the Internet and neurons in the brain.