Random search processes, Levy fllights, and random walks on complex networks

• Applicant: Professor Dr. Ralf Metzler
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2016 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 316131235

• Final Report Year: 2020

Final Report Abstract

In this project we studied the concept of random search processes in various directions, achieving interesting results. While not all problems proposed in this project have been completely solved substantial progress has been made in all topics. In our view the project successfully produced a nice range of novel and useful results. Concretely, we solved the one-dimensional problem of composite Lévy flight search processes and random search of multiple targets. The same has been achieved on a comb structure for a target located in the main backbone channel. It has been shown that for searchers starting close to the target the preferable search strategy is Brownian motion. For faraway targets, quite intuitively, Lévy or combined Brownian-Lévy search is more efficient. Such search processes could be of interest to investigate the search for food by biological organisms or search along DNA chains, but also influence search strategies in computer algorithms. The studies of search processes were extended to complex networks, such as Bethe lattices and Cayley trees, and the findings could be helpful for the optimisation of the search strategy, to prolong the survival time, or to speed up the search process. Moreover, a multi-hopper model was introduced as a more efficient network search strategy. We are confident that these investigations will lead to further studies of anomalous kinetics in network structures. New generalised continuous time random walk models and generalised diffusion and diffusion-wave equations have been shown to create a useful framework for the modeling of multi-scaling time behaviour and crossover dynamics in complex systems. We showed that such generalised diffusion equations may give the same results for the mean squared displacement as those obtained from the generalised Langevin equations. However, the processes are entirely different and thus provide siimilar flexibility of the dynamic description for different physical situations. Specifically, we applied the generalised Langevin equation approach of stochastic particle motion driven by so-called tempered fractional Gaussian noise to successfully, quantitatively describe the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer mebrane systems. The introduced single theoretical model was shown to give excellent results not only for the short and long time limit behaviour of the particle but also for intermediate times. In the framework of stochastic search we further pushed the quantification in terms of the entire distribution of first-passage and reaction times as well as the concept of the search reliability, i.e., the probability that a given search algorithm is ultimately successful.

Publications

• Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities, Phys. Rev. X 7, 021002 (2017)
  A. V. Chechkin, F. Seno, R. Metzler and I. M. Sokolov
  
• Promoting cooperation by preventing exploitation: The role of network structure, Phys. Rev. E 96, 022315 (2017)
  Z. Utkovski, V. Stojkoski, L. Basnarkov and L. Kocarev
  
• Random multi-hopper model: super-fast random walks on graphs, J. Complex Networks 6, 382 (2017)
  E. Estrada, J. C. Delvenne, N. Hatano, J. L. Mateos, R. Metzler, A. P. Riascos and M. T. Schaub
  
• From continuous time random walks to the generalized diffusion equation, Fract. Calc. Appl. Anal. 21, 10 (2018)
  T. Sandev, R. Metzler and A. Chechkin
  
• Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion, New J. Phys. 20, 043044 (2018)
  V. Sposini, A. V. Chechkin, F. Seno, G. Pagnini and R. Metzler
  
• Random Search with Resetting: A Unified Renewal Approach, Phys. Rev. Lett. 121, 050601 (2018)
  A. Chechkin and I. M. Sokolov
  
• Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control, Commun. Chem. 1, 96 (2018)
  D. S. Grebenkov, R. Metzler and G. Oshanin
  
• First-passage and first-hitting times of Lévy flights and Lévy walks, New J. Phys. 21, 103028 (2019)
  V. V. Palyulin, G. Blackburn, M. A. Lomholt, N. W. Watkins, R. Metzler, R. Klages and A. V. Chechkin
  
• The Role of Multiplex Network Structure in Cooperation through Generalized Reciprocity, Physica A 531, 121805 (2019)
  V. Stojkoski, Z. Utkovski, E. Andre and L. Kocarev
  
• Anomalous diffusion and random search in xyz-comb: exact results, J. Stat. Mech. 2020, 053203 (2020)
  E. K. Lenzi, T. Sandev, H. V. Ribeiro, P. Jovanovski, A. Iomin and L. Kocarev