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Comparative study of random walks with one-step memory on complex networks

Miroslav Mirchev, Lasko Basnarkov, Igor Mishkovski

TL;DR

The experiments show that biasing based on inverse degree, persistence and local two-hop paths can lead to smaller searching times and these biasing approaches can be combined to achieve a more robust random search strategy.

Abstract

We investigate searching efficiency of different kinds of random walk on complex networks which rely on local information and one-step memory. For the studied navigation strategies we obtained theoretical and numerical values for the graph mean first passage times as an indicator for the searching efficiency. The experiments with generated and real networks show that biasing based on inverse degree, persistence and local two-hop paths can lead to smaller searching times. Moreover, these biasing approaches can be combined to achieve a more robust random search strategy. Our findings can be applied in the modeling and solution of various real-world problems.

Comparative study of random walks with one-step memory on complex networks

TL;DR

The experiments show that biasing based on inverse degree, persistence and local two-hop paths can lead to smaller searching times and these biasing approaches can be combined to achieve a more robust random search strategy.

Abstract

We investigate searching efficiency of different kinds of random walk on complex networks which rely on local information and one-step memory. For the studied navigation strategies we obtained theoretical and numerical values for the graph mean first passage times as an indicator for the searching efficiency. The experiments with generated and real networks show that biasing based on inverse degree, persistence and local two-hop paths can lead to smaller searching times. Moreover, these biasing approaches can be combined to achieve a more robust random search strategy. Our findings can be applied in the modeling and solution of various real-world problems.

Paper Structure

This paper contains 14 sections, 17 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mean first passage time of random walks with one-step memory on complex networks
  3. Graph search algorithms based on random walks
  4. Classical random walks without memory
  5. Uniform random walk (U-RW)
  6. Inverse degree random walk (ID-RW)
  7. Random walks with memory
  8. Forward random walk with memory (F-RWM)
  9. Inverse degree random walk with memory (ID-RWM)
  10. Two-hops random walk with memory (2H-RWM)
  11. Persistent random walk with memory (P-RWM)
  12. Persistent inverse degree random walk with memory (PID-RWM)
  13. Results
  14. Conclusion

Figures (4)

  • Figure 1: GrMFPT in (a) BA, (b) WS, (c) ER, and (d) ER directed networks, with $100$ nodes and varied average node degree $\langle k \rangle$ for $7$ different random walks. The lines are theoretical values (T) and the markers numerical estimates (N).
  • Figure 2: Kullback-Leibler divergence of the stationary occupation probability of $7$ different random walks from a uniform density in (a) BA, (b) WS, (c) ER, and (d) ER directed networks, with $100$ nodes and varied average node degree $\langle k \rangle$.
  • Figure 3: GrMFPT in WS networks with (a) $k=4$, and (b) $k=6$, composed of $100$ nodes with varied rewiring probability for seven different random walks. The lines are theoretical values (T) and the markers numerical estimates (N).
  • Figure 4: Visualization in Gephi of four real networks topologies, where a darker color indicates a larger node degree.