Optimization of random searches on defective lattice networks

We study the general problem of how to search efficiently for targets randomly located on defective lattice networks—i.e., regular lattices which have some fraction of its nodes randomly removed. We consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths ${\ell }_j$ from the power-law distribution $P\left({\ell }_j\right)\sim {\ell }_j^{-\mu }$, with the exponent $\mu$ regulating the strategy of the search process. At each step ${\ell }_j$, the searcher moves in straight lines and constantly looks within a detection radius of vision $r_v$ for the targets along the way. If there is contact with a defect, the movement stops and a new step length is chosen. Hence, the presence of defects decreases the efficiency of the overall process. We study numerically how three different aspects of the lattice influence the optimization of the search efficiency: (i) the type of boundary conditions, (ii) the concentration of targets and defects, and (iii) the category or class of search—destructive, nondestructive, or regenerative. Motivated by the results, we develop a type of mean-field model for the problem and obtain an analytical approximation for the search efficiency function. Finally we discuss, in the context of searches, how defective lattices compare with perfect lattices and with continuous environments.