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Smoothed analysis of dynamic networks

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Abstract

In this paper, we generalize the technique of smoothed analysis to apply to distributed algorithms in dynamic networks in which the network graph can change from round to round. Whereas standard smoothed analysis studies the impact of small random perturbations of input values on algorithm performance metrics, our proposed dynamic network version of smoothed analysis studies the impact of random perturbations of the underlying changing network topologies. Similar to the original application of smoothed analysis, our goal is to study whether known strong lower bounds in these models are robust or fragile: do they withstand small (random) perturbations, or do such deviations push the graphs far enough from a precise pathological instance to enable much better performance? Fragile lower bounds are likely not relevant for real-world deployment, while robust lower bounds represent a true difficulty caused by dynamic behavior. We apply this technique to three standard dynamic network problems with known strong worst-case lower bounds: random walks, flooding, and aggregation. We prove that these bounds provide a spectrum of robustness when subjected to smoothing—some are fragile (random walks), some are moderately fragile (flooding), and some are robust (aggregation).

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Notes

  1. Edit distance, in this paper, is the number of edge additions/deletions needed to transform one graph to another, assuming they share the same node set.

  2. The qualification that we are discussing “simple” random walks is important. As elaborated in Sect. 7, there are random walk variations that can perform better than simple walks in dynamic settings.

  3. The inequality can easily be proven by induction over the exponent: assuming the product so far satisfies \(p \ge 1/2\), we have \(p(1-x) \le p-x/2\).

  4. This is another technical difference between the aggregation problem and the other two problems studied in this paper. For flooding and random walks, the dynamic graphs were considered to be of indefinite size. The goal was to analyze the process in question until it met some termination criteria. For aggregation, however, the length of the dynamic graph matters as this is a problem that requires an algorithm to aggregate as much as it can in a fixed duration that can vary depending on the application scenario. An alternative version of this problem can ask how long an algorithm takes to aggregate to a single node in an infinite length dynamic graph. This version of the problem, however, is less interesting, as the hardest case is the graph with no edges, which when combined with smoothing reduces to a standard random graph style analysis.

  5. We can deal with odd n and/or \(\ell \) not a power of 2 by suffering only a constant factor cost to our final performance. For simplicity of presentation, we maintain these assumptions for now.

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Authors and Affiliations

  1. Johns Hopkins University, Baltimore, MD, USA

    Michael Dinitz

  2. Georgetown University, Washington, DC, USA

    Jeremy T. Fineman & Calvin Newport

  3. National University of Singapore, Singapore, Singapore

    Seth Gilbert

Authors
  1. Michael Dinitz
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  2. Jeremy T. Fineman
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  3. Seth Gilbert
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  4. Calvin Newport
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Corresponding author

Correspondence to Calvin Newport.

Additional information

Michael Dinitz was supported in part by NSF Grants CCF-1464239 and CCF-1535887. Jeremy T. Fineman was supported in part by NSF Grants CCF-1218188 and CCF-1314633. Seth Gilbert was supported in part by Singapore MOE Tier 2 ARC Project 2014-T2-1-157. Calvin Newport was supported in part by NSF Grant CCF-1320279.

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Dinitz, M., Fineman, J.T., Gilbert, S. et al. Smoothed analysis of dynamic networks. Distrib. Comput. 31, 273–287 (2018). https://doi.org/10.1007/s00446-017-0300-8

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  • DOI: https://doi.org/10.1007/s00446-017-0300-8

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