Interdependent resistor networks with process-based dependency

Abstract

Studies of resilience of interdependent networks have focused on structural dependencies between pairs of nodes across networks but have not included the effects of dynamic processes taking place on the networks. Here we study the effect of dynamic process-based dependencies on a system of interdependent resistor networks. We describe a new class of dependency in which a node’s functionality is determined by whether or not it is actually carrying current and not just by its structural connectivity to a spanning component. This criterion determines its functionality within its own network as well as its ability to provide support-but not electrical current-to nodes in another network. We present the effects of this new type of dependency on the critical properties of σ and ${{B}_{\infty }}$ , the overall conductivity of the system and the fraction of nodes which carry current, respectively. Because the conductance of current has direct physical effects (e.g. heat, magnetic induction), the development of a theory of process-based dependency can lead to innovative technology. As an example, we describe how the theory presented here could be used to develop a new kind of highly sensitive thermal or gas sensor.

Keywords

• interdependent networks
• complex networks
• resistor networks
• phase transitions
• statistical physics