# Multitarget search on complex networks: A logarithmic growth of global mean random cover time

## TL;DR

This paper derives an exact formula for the expected time a random walker needs to visit multiple targets on complex networks, revealing a universal logarithmic growth pattern in multitarget search times across various random walk types.

## Contribution

The study introduces a precise expression for mean random cover time and demonstrates the universal logarithmic growth pattern in multitarget search on networks, extending previous results.

## Key findings

- Mean random cover time grows logarithmically with target number.
- Universal growth pattern confirmed across different random walk types.
- Optimal bias parameters minimize cover time and mean first passage time.

## Abstract

We investigate multitarget search on complex networks and derive an exact expression for the mean random cover time that quantifies the expected time a walker needs to visit multiple targets. Based on this, we recover and extend some interesting results of multitarget search on networks. Specifically, we observe the logarithmic increase of the global mean random cover time with the target number for a broad range of random search processes, including generic random walks, biased random walks, and maximal entropy random walks. We show that the logarithmic growth pattern is a universal feature of multi-target search on networks by using the annealed network approach and the Sherman-Morrison formula. Moreover, we find that for biased random walks, the global mean random cover time can be minimized, and that the corresponding optimal parameter also minimizes the global mean first passage time, pointing towards its robustness. Our findings further confirm that the logarithmic growth pattern is a universal law governing multitarget search in confined media.