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Closed-Form Formulas for Cluster Sizing for Two-Level Hierarchical Networks with Source Routing

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  • Eric Rosenberg

    (Seton Hall University)

Abstract

In a two-level hierarchical network, nodes are grouped into clusters. A node in one cluster sees the entire topology of that cluster, but only a summarized view of other clusters. With source routing, the source node of a connection does an initial route computation based on its view of the network, and additional route computations are performed as the connection makes its way towards the destination. We study the problem of choosing the number of clusters to minimize the total complexity of all the path computations required to compute a path from a source node to a destination node. We show that, under mild conditions, for all sufficiently large networks two-level hierarchical routing is superior to flat network routing. Utilizing the duality theory of geometric programming, we provide a closed-form solution estimate for the number of clusters to minimize the total routing complexity. We also provide a closed-form expression for a lower bound on the minimal total complexity. We illustrate the modelling assumptions for a variety of networks. Computational results for a range of network sizes and parameters show that our solution estimate yields near-optimal results.

Suggested Citation

  • Eric Rosenberg, 2025. "Closed-Form Formulas for Cluster Sizing for Two-Level Hierarchical Networks with Source Routing," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-34, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02710-8
    DOI: 10.1007/s10957-025-02710-8
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    References listed on IDEAS

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    1. Chaoming Song & Shlomo Havlin & Hernán A. Makse, 2005. "Self-similarity of complex networks," Nature, Nature, vol. 433(7024), pages 392-395, January.
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