A Complete Characterization of the Inverse Eigenvector Centrality Problem for Undirected Graphs
Pith reviewed 2026-05-07 13:02 UTC · model grok-4.3
The pith
A positive vector is the eigenvector centrality of a connected undirected graph with suitable weights if and only if it satisfies conditions on every stable set and its neighborhood.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We study the inverse eigenvector centrality problem on connected undirected graphs, namely, whether a given positive vector can be realized by assigning suitable edge weights. We provide a complete characterization in terms of stable sets and their neighborhoods, showing that the undirected case requires nontrivial global constraints absent in the directed setting.
What carries the argument
The set of conditions on stable sets and their neighborhoods that a candidate vector must obey for realizability as eigenvector centrality.
If this is right
- A vector is realizable precisely when it meets the stable-set and neighborhood conditions.
- Realizability can be checked directly from the vector and graph structure without solving for weights.
- The symmetry of undirected edges forces additional global restrictions beyond those needed for directed graphs.
- The characterization completely settles the decision version of the inverse problem for this setting.
Where Pith is reading between the lines
- The conditions supply a combinatorial certificate that could be turned into an efficient verification algorithm.
- Similar neighborhood constraints may appear when inverting other spectral quantities on undirected graphs.
- Network-design applications can now test in advance whether a target centrality profile is even possible under undirected topology.
Load-bearing premise
The input vector is strictly positive and the underlying graph is connected.
What would settle it
Exhibit a strictly positive vector that obeys every stable-set and neighborhood condition yet cannot be obtained as the eigenvector centrality of any edge weighting on a connected undirected graph, or the converse.
Figures
read the original abstract
We study the inverse eigenvector centrality problem on connected undirected graphs, namely, whether a given positive vector can be realized by assigning suitable edge weights. We provide a complete characterization in terms of stable sets and their neighborhoods, showing that the undirected case requires nontrivial global constraints absent in the directed setting.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the inverse eigenvector centrality problem on connected undirected graphs: given a strictly positive vector, determine whether there exist non-negative edge weights such that the vector is the eigenvector centrality (i.e., the principal eigenvector of the weighted adjacency matrix, normalized appropriately). It supplies a complete characterization of realizable vectors in terms of conditions on stable sets and their neighborhoods; these conditions encode the global constraints forced by symmetry that are absent from the directed case. Necessity and sufficiency are established under the maintained hypotheses of connectedness and strict positivity of the input vector.
Significance. The result is significant because it furnishes a definitive, graph-theoretic resolution of the inverse problem for the undirected setting. By isolating the precise combinatorial obstructions arising from symmetry, the characterization clarifies why the undirected case is strictly harder than its directed counterpart and supplies an explicit, checkable criterion for realizability. The manuscript delivers both necessity and sufficiency together with a reduction to a system of linear equations, constituting a self-contained theoretical advance in spectral graph theory and network centrality.
minor comments (2)
- [§2] The opening paragraphs of §2 would benefit from an explicit reminder of the normalization convention used for the centrality vector (e.g., whether it is required to sum to 1 or to have Euclidean norm 1).
- [Theorem 3.2] In the statement of the main characterization (Theorem 3.2), the phrase “non-negative edge weights” could be replaced by “non-negative real edge weights” for precision, since the zero-weight case is implicitly allowed.
Simulated Author's Rebuttal
We thank the referee for their positive summary of the manuscript, for recognizing the significance of the complete characterization, and for recommending acceptance. We appreciate the clear articulation of how the result isolates the combinatorial obstructions due to symmetry in the undirected setting.
Circularity Check
No significant circularity in the characterization
full rationale
The manuscript provides a complete characterization of realizable positive vectors for eigenvector centrality in undirected graphs via conditions on stable sets and their neighborhoods. This is a direct graph-theoretic result derived from the standard definition of eigenvector centrality under the maintained hypotheses of a connected graph and strictly positive input vector. The argument establishes necessity and sufficiency for existence of suitable non-negative edge weights without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The derivation is self-contained against the input equations and external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The graph is connected and undirected; the target vector is strictly positive.
Reference graph
Works this paper leans on
-
[1]
[1]Arcagni, A., Candila, V., and Grassi, R.A new model for predicting the winner in tennis based on the eigenvector centrality.Annals of Operations Research 325(2023), 615–632. [2]Benzi, M., and Guglielmi, N.Changing the ranking in eigenvector centrality of a weighted graph by small perturbations.Mathematics of Computation(2025). [3]Bonacich, P.Factoring ...
work page 2023
-
[2]
[9]Ma, Y., Jiang, Z.-Q., Dai, Y.-H., and W ang, W.-X.Understanding the circulation network of agro-products in china based on the freight big data.Annals of Operations Research 348(2025), 511–541. [10]Michini, C.The stable set problem: some structural properties and relaxations.4OR 11 (2013), 199–200. [11]Minc, H.Nonnegative Matrices. Wiley, New York, USA,
work page 2025
-
[3]
[12]Nemhauser, G. L., and Trotter Jr, L. E.Properties of vertex packing and indepen- dence system polyhedra.Mathematical Programming 6, 1 (1974), 48–61. [13]Newman, M.Networks, 2 ed. Oxford University Press, Oxford, UK, 07
work page 1974
-
[4]
11 [14]Nicosia, V., Criado, R., Romance, M., Russo, G., and Latora, V.Controlling centrality in complex networks.Scientific reports 2, 1 (2012),
work page 2012
-
[5]
To appear in Contemporary Mathematics
[15]Passacantando, M., and Raciti, F.Optimizing edge weights in the inverse eigenvector centrality problem, arXiv Preprint arXiv:2602.11772v1 (2026). To appear in Contemporary Mathematics. 12
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.