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

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 n = d 3 c 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.

Structural resilience of spatial networks with inter-links behaving as an external field

“…Similar to our earlier studies [18,27], we find that the parameter r, governing the fraction of interconnected nodes, has effects analogous to a magnetic field in a spin system, near criticality. This analogy can be seen through the facts that: (i) the non-zero fraction of interconnected nodes destroys the original phase transition point of the single module; (ii) critical exponents (defined below) of values derived from percolation theory can be used to characterize the effect of external field on S(p, r).…”

Generalized K-Core Percolation in Networks with Community Structure

“…This is similar to the paradigm established in [1,20]; however, the introduction of community structure requires a nontrivial refined treatment of the probability partition to derive solvable self-consistency equations. It is tempting to consider a version of multivariate generating functions that have been used to deal with component based resilience in modular networks [30,8,10,33]. However, we find it cumbersome with a monolithic multivariate version and choose to track the interconnections separately from intraconnections in Gk-core percolation.…”

Bond and site color-avoiding percolation in scale-free networks

“…Another example of the sensitivity to the definition of connectivity is the so called k-connected percolation [4,5] in which vertices are connected only if there exist at least k independent paths among them. Yet another example is color-avoiding percolation [6][7][8]. In these papers every vertex in a network is colored with one color out of a certain set of colors.…”

“…From a theoretical point of view, percolation on networks [9,10] exhibits a number of features which are rare or absent in more conventional models of percolation on lattices. Discontinuous phase transitions [11], phase transitions with Berezinskii-Kosterlitz-Thouless singularity [12], magnetic field effects [8], explosive percolation [13] or inequality of site and bond percolation [14] represent just some of the critical phenomena that naturally emerge in percolation on complex networks.…”

“…Recently some of the authors developed color-avoiding percolation (CAP) [6][7][8] as a tool to study the robustness of networks to simultaneous failures of certain classes of vertices. Classes are defined by their common vulnerabilities (for example the type of critical software they share), and a common vulnerability of a whole class is represented by its own color.…”