  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.
