The Efficiency of Clusters on Networks and Their Robustness

Cluster structures are widespread in complex networks, yet conventional network-level measures do not distinguish the accessibility provided inside a cluster from that provided through its external links. This study asks how these two topological contributions can be measured consistently and how rapidly they deteriorate under different node-removal mechanisms. We define internal and external cluster efficiency by combining edge volume, relative cluster size, and a harmonic shortest-path factor based on the mean reciprocal distance rather than the reciprocal of an arithmetic mean distance. The measures are evaluated together with a Cluster Robustness Index (CRI) and Structural Resilience Entropy (SRE) on Barabási-Albert and Lancichinetti-Fortunato-Radicchi networks and five empirical network topologies under six attack strategies. The results show that smaller clusters are generally more vulnerable to attacks on central nodes, whereas larger and less centralized clusters retain more topological efficiency. A source-level audit also shows that replacing the reciprocal of the arithmetic mean distance by the arithmetic mean of reciprocal distances changes most normalized trends only slightly; the clearest quantitative changes occur in the weighted-average efficiency panels. Intra-cluster edge addition increases internal efficiency by 1-5% and CRI-based robustness by up to 11.9%, while cross-cluster edge rewiring improves external efficiency by up to 2.45% and CRI-based robustness by up to 2.33% under the tested targeted attacks. The reported quantities are structural proxies derived from unweighted network topology; they do not represent observed flow, transmission speed, recovery dynamics, or domain-specific functionality. The framework therefore supports cluster-level vulnerability diagnosis and topology-oriented reinforcement without making claims beyond the available network data.