Interconnections between networks act like an external field in first-order percolation transitions

Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we study if and how the effects of such interconnections can be described as an external field for interdependent networks experiencing first-order percolation transitions. We find that the critical exponents $\gamma$ and $\delta$, related to the external field can also be defined for first-order transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents can be found even within a single model of interdependent networks, depending on the dependency coupling strength. Specifically, the exponent $\gamma$ in the first-order regime (high coupling) does not obey the fluctuation dissipation theorem, whereas in the continuous regime (for low coupling) it does. Nevertheless, in both cases they satisfy Widom's identity, $\delta - 1 = \gamma / \beta$ which further supports the validity of their definitions. Our results provide physical intuition into the nature of the phase transition in interdependent networks and explain the underlying reasons for two distinct sets of exponents.