Tricritical Points in Random Combinatorics: the (2+p)-SAT case

The (2+p)-Satisfiability (SAT) problem interpolates between different classes of complexity theory and is believed to be of basic interest in understanding the onset of typical case complexity in random combinatorics. In this paper, a tricritical point in the phase diagram of the random $2+p$-SAT problem is analytically computed using the replica approach and found to lie in the range $2/5 \le p_0 \le 0.416$. These bounds on $p_0$ are in agreement with previous numerical simulations and rigorous results.Comment: 7 pages, 1 figure, RevTeX, to appear in J.Phys.

Damage detection via shortest-path network sampling

Large networked systems are constantly exposed to local damages and failures that can alter their functionality. The knowledge of the structure of these systems is, however, often derived through sampling strategies whose effectiveness at damage detection has not been thoroughly investigated so far. Here, we study the performance of shortest-path sampling for damage detection in large-scale networks. We define appropriate metrics to characterize the sampling process before and after the damage, providing statistical estimates for the status of nodes (damaged, not damaged). The proposed methodology is flexible and allows tuning the trade-off between the accuracy of the damage detection and the number of probes used to sample the network. We test and measure the efficiency of our approach considering both synthetic and real networks data. Remarkably, in all of the systems studied, the number of correctly identified damaged nodes exceeds the number of false positives, allowing us to uncover the damage precisely