A cycle-counting-ratio estimator for the β-model achieves minimax-optimal MSE and consistency under the weak conditions θ_max→0 and θ_t‖θ‖₁→∞, even at network densities near log n/n.
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Benders decomposition with combinatorial feasibility cuts, multiple-cut addition, clique-cover heuristic, and initial cut generation solves k-defensive domination instances on Erdos-Renyi, chordal, and Barabasi-Albert graphs that remain unsolved by classical formulations.
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Subgraph counting estimation for the $\beta$-model in sparse networks
A cycle-counting-ratio estimator for the β-model achieves minimax-optimal MSE and consistency under the weak conditions θ_max→0 and θ_t‖θ‖₁→∞, even at network densities near log n/n.