The counter-intuitive network features of optimal diffusive navigation.

Time is a critical factor in numerous physical systems that rely on complex networks to ensure navigability among their components. From human-made transportation networks that seek to optimize routes between stations, to transport of particles in porous media and biological networks, intricate structures and active motion mechanisms ensure low mean first passage times (MFPT) between the target regions. Yet, random walkers can get lost within complex networks and stray from the desired target. This leads to a fundamental question: What is the best architecture that makes a graph optimally searchable? Traditionally, it has been assumed that small-world architectures with numerous shortcuts between distant nodes are ideal. Counter-intuitively, for diffusive exploration of spatially embedded networks, we find that the addition of a link can worsen the total search time between all pairs of nodes, akin to the well-known Braess paradox where increasing a system's capacity reduces its efficiency. This highlights that the topological advantage gained from the extra pathway can be negated by the diffusive delay required to traverse the length of the shortcut, as dictated by the mean squared displacement of anomalous diffusion. We generalize developing an optimization scheme according to which each edge adapts its conductivity in order to minimize the total mean first passage time. This unveals a crossover in the optimal architecture: for super-diffusive motion, the optimal graph features long-range links, while for sub-diffusive propagation geometrical nearest connections are preferred. Ultimately, we explore how multiple propagation speeds can mitigate the impact of "Braessian edges" inspired by the ingenious strategies of intracellular transport networks that utilize diffusion over short distances and motor-driven ballistic motion over longer ones.