Critical field-exponents for secure message-passing in modular networks

TL;DR

This paper investigates the critical behavior of secure message-passing in modular networks with vulnerabilities, revealing new universality classes and showing that inter-module links enhance security more effectively than intra-module links.

Contribution

It introduces a novel analogy between secure message-passing and magnetic spin-systems, defining new critical exponents and universality classes for the problem.

Findings

Interlinks act like a magnetic field, preventing phase transitions.

Critical exponents are δ=2, γ=1 for single vulnerability, δ=1, γ=0 for multiple vulnerabilities.

Increasing inter-module links improves security more than intra-module links.

Abstract

We study secure message-passing in the presence of multiple adversaries in modular networks. We assume a dominant fraction of nodes in each module have the same vulnerability, i.e., the same entity spying on them. We find both analytically and via simulations that the links between the modules (interlinks) have effects analogous to a magnetic field in a spin system in that for any amount of interlinks the system no longer undergoes a phase transition. We then define the exponents $\delta$, which relates the order parameter (the size of the giant secure component) at the critical point to the field strength (average number of interlinks per node), and $\gamma$, which describes the susceptibility near criticality. These are found to be $\delta=2$ and $\gamma=1$ (with the scaling of the order parameter near the critical point given by $\beta=1$). When two or more vulnerabilities are equally present in a module we find $\delta=1$ and $\gamma=0$ (with $\beta\geq2$). Apart from defining a previously unidentified universality class, these exponents show that increasing connections between modules is more beneficial for security than increasing connections within modules. We also measure the correlation critical exponent $\nu$, and the upper critical dimension $d_c$, finding that $\nu d_c=3$ as for ordinary percolation, suggesting that for secure message-passing $d_c =6$. These results provide an interesting analogy between secure message-passing in modular networks and the physics of magnetic spin-systems.