Optimizing the First-Passage Process on a Class of Fractal Scale-Free Trees

First-passage processes on fractals are of particular importance since fractals are ubiquitous in nature, and first-passage processes are fundamental dynamic processes that have wide applications. The global mean first-passage time (GMFPT), which is the expected time for a walker (or a particle) to first reach the given target site while the probability distribution for the position of target site is uniform, is a useful indicator for the transport efficiency of the whole network. The smaller the GMFPT, the faster the mass is transported on the network. In this work, we consider the first-passage process on a class of fractal scale-free trees (FSTs), aiming at speeding up the first-passage process on the FSTs. Firstly, we analyze the global mean first-passage time (GMFPT) for unbiased random walks on the FSTs. Then we introduce proper weight, dominated by a parameter w(w>0), to each edge of the FSTs and construct a biased random walks strategy based on these weights. Next, we analytically evaluated the GMFPT for biased random walks on the FSTs. The exact results of the GMFPT for unbiased and biased random walks on the FSTs are both obtained. Finally, we view the GMFPT as a function of parameter w and find the point where the GMFPT achieves its minimum. The exact result is obtained and a way to optimize and speed up the first-passage process on the FSTs is presented.

Appreciate the process, by taking a random walk

“Appreciate the process, by taking a random walk” offers a basic introduction to random walks—a mathematical formalization for a path as an object wanders away from a starting point—with a particular focus on biased random walks, including a real-life example of a random walk of an E. coli bacterium. The discussion is illustrated with hand-drawn sketches. Mathematics students and enthusiasts are encouraged to build in a productive bias in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Walking Drosophila navigate complex plumes using stochastic decisions biased by the timing of odor encounters

ABSTRACTInsects find food, mates, and egg-laying sites by tracking odor plumes swept by complex wind patterns. Previous studies have shown that moths and flies localize plumes by surging upwind at odor onset and turning cross- or downwind at odor offset. Less clear is how, once within the expanding cone of the odor plume, insects use their brief encounters with individual odor packets, whose location and timing are random, to progress towards the source. Experiments and theory have suggested that the timing of odor encounters might assist navigation, but connecting behaviors to individual encounters has been challenging. Here, we imaged complex odor plumes simultaneous with freely-walking flies, allowing us to quantify how behavior is shaped by individual odor encounters. Combining measurements, dynamical models, and statistical inference, we found that within the plume cone, individual encounters did not trigger reflexive surging, casting, or counterturning. Instead, flies turned stochastically with stereotyped saccades, whose direction was biased upwind by the timing of prior odor encounters, while the magnitude and rate of saccades remained constant. Odor encounters did not strongly affect walking speed. Instead, flies used encounter timing to modulate the rate of transitions between walks and stops. When stopped, flies initiated walks using information from multiple odor encounters, suggesting that integrating evidence without losing position was part of the strategy. These results indicate that once within the complex odor plume, where odor location and timing are unpredictable, animals navigate with biased random walks shaped by the entire sequence of encounters.