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Abstract: Large-scale networks represent a broad spectrum of systems in nature, science, and technology. Computer networks such as the World Wide Web and the Internet, social networks such as Twitter and Facebook, and knowledge-sharing online platforms such as Wikipedia exert considerable influence on our everyday lives. Many of these networks are very large and evolve with time, making investigation of their statistical properties a challenging task. I will describe a novel methodology, based on random walks, for the inference of statistical properties of complex networks with weighted or unweighted edges [1]. I will show how this formalism can yield reliable estimates of various network statistics, such as the network size, after only a small fraction of network nodes has been explored. I will introduce two novel algorithms for partitioning network nodes into non-overlapping communities - a key step in revealing network modularity and hierarchical organization [2]. These clustering tools will be applied to various benchmarks, including a large-scale map of roads and intersections in the state of Colorado. Finally, I will demonstrate how these ideas can be extended to computing various thermodynamic quantities in discrete systems such as spin glasses from small non-equilibrium samples of states. In summary, random walks can be used to reveal modular organization and global structure of complex networks and infer key statistical mechanics quantities that are otherwise difficult to estimate computationally.

References

1. Kion-Crosby, W.B. and Morozov, A.V. (2018) Phys Rev Lett 121, 038301

2. Ballal, A., Kion-Crosby, W.B. and Morozov, A.V. (2022) Phys Rev Res, in the press