The study of interdependent networks holds significant importance in understanding the behavior of various systems in our world, as they often exhibit vastly different behavior compared to isolated networks. Interdependent networks are systems that exhibit two types of interactions – connectivity within each network, and dependency between different networks. Connectivity nodes represent the ability of some quantity to propagate through the network, for example water in a water system, electricity in a power network, etc. Dependency nodes represent the mutual dependence between nodes of different networks. Each node in a network is affected by the state of nodes in the other network. Water supply, transportation, fuel, and power plants are examples of interdependent systems.
For a decade, the study of interdependent networks had flourished, with several theoretical papers by Havlin and colleagues contributing to the field. However, these studies were primarily theoretical and focused on non physical systems.
Recently, the theory of an interdependent networks was implemented in a physical system by Frydman (our) group. In our research we focus on the manifestation of the theory of interdependent networks in a real-world physical system. For this purpose, we couple two disordered superconducting networks via a medium that is an electrical insulator but a heat conductor as shown in Fig. 1. Because of the disorder each node in the network has its own critical temperature, which allowed us to control the functionality of nodes in the networks.
Figure 1. Interdependent Superconducting Networks system sketch – Two networks (in green), separated by a thermally conducting insulating medium (in blue).
The experimental results presented in Fig. 2 can be summarized as follows. When measured independently and under identical conditions, each layer undergoes a continuous and broad superconductor–normal phase transition at some finite bulk critical threshold, Tc, whose value depends on the disorder of the sample and on the driving current, Ib, flowing through it. Since the layers have different levels of disorder – each segment has a different critical current and temperature – they exhibit different values of Tc. The broad SN transitions become sharper for increasing values of Ib but they always remain continuous and non-hysteretic (Fig. 2a). On the other hand, when a similar sufficiently large Ib flows simultaneously in both layers, thermal couplings set in between the networks and their SN transitions become mutually abrupt and hysteretic (Fig. 2b).
Figure 2. Experimental results of interdependent superconducting networks. a. Resistances measured in the isolated case for the top (red) and bottom (blue) networks, for increasing (filled symbols) and decreasing (empty symbols) values of the temperature, T . Both layers undergo continuous superconducting normal (SN) transitions at different critical temperatures. b. Resistance measured in interdependent superconducting networks while an identical current is driving in both networks. For Ib > 15μA, the layers become thermally coupled and undergo mutual first-order SN transitions