Optimizing edge weights in the inverse eigenvector centrality problem

In this paper we study the inverse eigenvector centrality problem on directed graphs: given a prescribed node centrality profile, we seek edge weights that realize it. Since this inverse problem generally admits infinitely many solutions, we explicitly characterize the feasible set of admissible weights and introduce six optimization problems defined over this set, each corresponding to a different weight-selection strategy. These formulations provide representative solutions of the inverse problem and enable a systematic comparison of how different strategies influence the structure of the resulting weighted networks. We illustrate our framework using several real-world social network datasets, showing that different strategies produce different weighted graph structures while preserving the prescribed centrality. The results highlight the flexibility of the proposed approach and its potential applications in network reconstruction, and network design or network manipulation.

💡 Research Summary

The paper tackles the inverse eigenvector centrality problem on directed, strongly‑connected graphs: given a desired centrality vector c (all entries positive) and a scalar spectral radius ρ > 0, find edge weights w that make c the eigenvector centrality of the weighted adjacency matrix. By exploiting the Perron‑Frobenius theorem, the authors show that the condition Aᵀ c = ρ c can be rewritten as a linear system B w = ρ c, where B is an m × |E| matrix whose columns correspond to arcs and contain the source‑node centrality value at the row of the head node. Because each column of B has exactly one non‑zero entry, the system is under‑determined and admits infinitely many positive solutions for any ρ > 0 and any positive c.

The existence proof proceeds by selecting, for each node j, one incoming arc (i*, j) and assigning a tiny ε > 0 to all other incoming arcs. The remaining weight on (i*, j) is then forced by the linear equation to be w_{i* j} = (ρ c_j − ε ∑{i≠i*} c_i)/c{i*}. Positivity of all weights is guaranteed if ε is chosen smaller than min_j ρ c_j / ∑_{i∈B_S(j)} c_i. Hence the feasible set ℱ = { w ≥ ε 1 | B w = ρ c } is non‑empty and convex.

Because the inverse problem is highly non‑unique, the authors introduce six optimization formulations to select a representative solution according to different design criteria:

• (P1) Minimum ℓ₁‑norm – minimize ‖w − 1‖₁, encouraging most weights to stay close to the unweighted value 1 and concentrating adjustments on a few edges (sparsity of change).
• (P2) Minimum ℓ₂‑norm – minimize ‖w − 1‖₂², spreading the required modifications evenly across all edges (fairness).
• (P3) Minimum ℓ_∞‑norm – minimize the worst‑case deviation, suitable when a hard bound on any single weight change is required.
• (P4) Minimum linear cost – minimize Σ_{(i,j)∈E} β_{ij} w_{ij}, where β_{ij} encodes a physical or social cost of maintaining the link; this yields the most cost‑efficient allocation of interaction intensities.
• (P5) Minimum‑energy – a quadratic cost that penalizes large weights directly, often interpreted as minimizing total “energy” in a physical system.
• (P6) Minimum‑uncertainty – a formulation that incorporates variance or robustness considerations (details are in the paper).

All six problems share the same linear equality and non‑negativity constraints, making (P1)–(P3) convex programs (ℓ₁ and ℓ_∞ can be cast as linear programs, ℓ₂ as a quadratic program) and (P4)–(P6) linear or quadratic programs as well. Consequently, standard solvers (e.g., CVX, Gurobi) can compute optimal weights efficiently.

The authors also derive a priori bounds on ρ and ε that guarantee feasibility, showing how the choice of these parameters influences the size of the feasible set. In particular, ε must be small enough to keep all weights positive, while ρ must be compatible with the magnitude of the target centrality vector.

Empirical validation is performed on a synthetic 4‑node example and three real‑world social networks (Zachary’s Karate Club, a Facebook friendship network, and a Twitter retweet network). For each dataset the same target centrality c is imposed, and the six optimization models are solved. The resulting weighted graphs are compared using several structural metrics: distribution of edge weights, total cost Σβ_{ij}w_{ij}, clustering coefficient, average shortest‑path length, and degree‑strength correlations.

Key observations include:

• ℓ₁ solutions concentrate large weight adjustments on a few edges, producing a sparse pattern of change that highlights specific influential ties.
• ℓ₂ solutions keep most weights near 1, yielding a network that is structurally close to the original unweighted graph while still achieving the desired centralities.
• ℓ_∞ solutions bound the maximum deviation, ensuring that no single edge is altered beyond a preset threshold—useful when physical capacities or policy limits exist.
• Cost‑minimization (P4) steers weight allocation away from expensive links (high β_{ij}), effectively “pruning” costly relationships while preserving the centrality profile.
• Energy‑minimization (P5) and uncertainty‑minimization (P6) generate weight patterns that are smoother or more robust, respectively, illustrating how additional domain‑specific objectives can be incorporated.

Overall, the experiments demonstrate that while all models satisfy the same centrality constraints, the internal structure of the resulting weighted networks can differ dramatically depending on the chosen objective. This underscores the importance of aligning the optimization criterion with the practical goals of network reconstruction, design, or manipulation.

In the concluding section the authors acknowledge limitations: the framework assumes a single strongly‑connected component, requires a strictly positive target centrality, and depends on the choice of ε and ρ. Future work is outlined along several directions: extending to multiple strongly‑connected components, handling time‑varying or dynamic networks, incorporating non‑linear centrality measures such as PageRank or Katz, and developing stochastic or Bayesian formulations that account for uncertainty in the target centrality vector.

In sum, the paper provides a rigorous characterization of the feasible weight space for the inverse eigenvector centrality problem, proposes a versatile suite of convex optimization models to resolve the inherent non‑uniqueness, and validates the approach on real data, offering a valuable toolkit for researchers and practitioners interested in network reconstruction, optimal design, or strategic manipulation of influence structures.