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

Shekhtman, Louis M.; Danziger, Michael M.; Bonamassa, Ivan; Buldyrev, Sergey V.; Caldarelli, Guido; Zlatić, Vinko; Havlin, Shlomo (2018) Critical field-exponents for secure message-passing in modular networks. New Journal of Physics, 20 . ISSN 1367-2630

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## 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 δ, 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 γ, which describes the susceptibility near criticality. These are found to be δ = 2 and γ = 1 (with the scaling of the order parameter near the critical point given by β = 1). When two or more vulnerabilities are equally present in a module we find δ = 1 and γ = 0 (with β ≥ 2). 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 ν, 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.

Item Type: Article
Uncontrolled Keywords: secure message-passing; modular networks
Subjects: NATURAL SCIENCES > Mathematics
NATURAL SCIENCES > Physics
Divisions: Theoretical Physics Division
Projects:
Ruđer Bošković Institute: Twinning for a step forward of the Theoretical Physics Division-RBI-T-WINNINGFabrizio NESTI692194EK
Depositing User: Vinko Zlatić
Date Deposited: 07 May 2018 15:08