## Abstract

Spectral measures have long been used to quantify the robustness of real-world graphs. For example, spectral radius (or the principal eigenvalue) is related to the effective spreading rates of dynamic processes (e.g., rumor, disease, information propagation) on graphs. Algebraic connectivity (or the Fiedler value), which is a lower bound on the node and edge connectivity of a graph, captures the “partitionability” of a graph into disjoint components. In this work we address the problem of modifying a given graph’s structure under a given budget so as to maximally improve its robustness, as quantified by spectral measures. We focus on modifications based on *degree-preserving* edge rewiring, such that the expected load (e.g., airport flight capacity) or physical/hardware requirement (e.g., count of ISP router traffic switches) of nodes remain unchanged. Different from a vast literature of measure-independent heuristic approaches, we propose an algorithm, called EdgeRewire, which *optimizes* a specific measure of interest directly. Notably, EdgeRewire is *general* to accommodate six different spectral measures. Experiments on real-world datasets from three different domains (Internet AS-level, P2P, and airport flights graphs) show the effectiveness of our approach, where EdgeRewire produces graphs with both (i) higher robustness, and (ii) higher attack-tolerance over several state-of-the-art methods.

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## Notes

- 1.
A spanning tree is a subgraph over all nodes, containing \((n-1)\) edges and no cycles.

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## Acknowledgments

The authors thank the anonymous reviewers for their useful comments. This material is based upon work supported by the ARO Young Investigator Program under Contract No. W911NF-14-1-0029, NSF CAREER 1452425, IIS 1408287 and IIP1069147, a Facebook Faculty Gift, an R&D grant from Northrop Grumman Aerospace Systems, and Stony Brook University Office of Vice President for Research. Any conclusions expressed in this material are of the authors’ and do not necessarily reflect the views, either expressed or implied, of the funding parties.

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Responsible editors: Joao Gama, Indre Zliobaite, Alipio Jorge, Concha Bielza.

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Chan, H., Akoglu, L. Optimizing network robustness by edge rewiring: a general framework.
*Data Min Knowl Disc* **30, **1395–1425 (2016). https://doi.org/10.1007/s10618-015-0447-5

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### Keywords

- Graph robustness
- Edge rewiring
- Robustnesss measures
- Graph spectrum
- Optimization algorithms
- Attack tolerance