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In the 1980s an important goal of the emergent field of fractals was todetermine the relationships between their physical and geometrical properties.The fractal-Einstein and Alexander-Orbach laws, which interrelate electrical,diffusive and fractal properties, are two key theories of this type. Here wesettle a long standing controversy about their exactness by showing that theproperties of a class of fractal trees violate both laws. A new formula isderived which unifies the two classical results by proving that if one holds,then so must the other, and resolves a puzzling discrepancy in the propertiesof Eden trees and diffusion limited aggregates. The failure of the classicallaws is attributed to anisotropic exploration of the network by a randomwalker. The occurrence of this newly revealed behaviour means that numeroustheories, such as recent first passage time results, are restricted to anarrower range of networks than previously thought.

Generalisation of the fractal Einstein law relating conduction and diffusion on networks