Graph recovery from partial information

Abstract: We introduce a Fourier-analytic framework for graph complexity and recoverability. For a graph G on N vertices labeled by Z_N, we define the Fourier ratio FR(f) of its edge indicator f. The key invariant FR_min(G), the minimum Fourier ratio over all vertex labelings, measures the optimal additive spectral compressibility of the graph. We establish a lower bound FR_min(G) >= E(G)/sqrt(2s), where E(G) is the graph energy and s is the number of edges. This bound is sharp for abelian Cayley graphs under the natural group labeling. Using a compressed sensing theorem, we show that once a labeling with small Fourier ratio is available, the edge map can be efficiently recovered from sparse random samples of adjacency entries. The algorithmic problem of finding such a labeling remains open. We compute FR_min(G) for complete, Turán, cycle, and circulant graphs. Cycles and circulant graphs are highly compressible, while random labelings yield large Fourier complexity. We develop a spectral-projector framework for harmonic graph recovery. For each Laplacian eigenvalue {λ}, its spectral projector Πλ satisfies FR_min(Πλ) >= sqrt(m(λ)), where m(λ) is the multiplicity. Fourier-compressible projectors are recoverable via Fourier-side l^1 minimization. For abelian Cayley graphs, the natural group labeling simultaneously minimizes both edge and harmonic complexity, with the harmonic complexity attaining the lower bound exactly. Since low-frequency projectors govern heat flow and random walks, this yields a mechanism for recovering large-scale geometric structure from sparse observations without full graph reconstruction. Finally, we formulate an asymptotic spectral synthesis principle showing that spectrally regular functions cannot concentrate on small exceptional sets, yielding asymptotic uniqueness and recovery results for incomplete graph data.

• The paper develops a Fourier-ratio framework showing that compressed-sensing recovery depends on additive organization revealed by vertex labeling, with sample complexity scaling roughly as r²/ε².
• It proves sharp lower bounds, including FR(f) ≥ √2 for nonzero loop-free graph indicators and FR_min(G) ≥ E(G)/√(2s), with equality in important abelian Cayley graph cases.
• The results extend to spectral projectors and diffusion geometry, but practical recovery remains conditional because finding an optimal low-Fourier-ratio labeling is computationally unresolved and random-relabeling claims are heuristic.

Overview

The paper "Graph recovery from partial information" by Vishal Gupta and Alex Iosevich (2606.15475) develops a Fourier-analytic framework for quantifying when a graph can be reconstructed from sparse random samples of its adjacency entries. The central object is the Fourier ratio of an edge indicator function $f:\mathbb{Z}_N^2\to\{0,1\}$, defined as $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$, together with the isomorphism-invariant quantity $\operatorname{FR}_{\min}(G)$ obtained by minimizing over all vertex labelings. The guiding thesis is that graph recoverability is governed not by combinatorial sparsity or degree distribution but by whether some labeling organizes the edge relation into additive coordinates in which the Fourier representation becomes concentrated.

The recovery mechanism rests on a compressed sensing theorem of Burstein, Iosevich, and Nathan: if $\operatorname{FR}(f)\le r$ and each entry of $f$ is observed independently with probability $p$ satisfying $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$, then Fourier-side $\ell^1$ minimization recovers $f$ with error at most $11.47\,\epsilon\|f\|_2$ with high probability. The sample complexity scales as roughly $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$0, so the Fourier ratio acts as an effective dimension controlling sampling cost.

Labeling dependence and the cycle graph case study

The Fourier ratio is highly sensitive to vertex labeling. Under the natural cyclic labeling of the cycle $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$1, the edge map's Fourier transform is supported on the dual line $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$2, and a direct computation gives $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$3. By contrast, the authors argue heuristically that a random relabeling destroys this additive organization and yields $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$4. Since sample complexity scales like $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$5, the same abstract graph transitions from requiring $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$6 samples to essentially $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$7—that is, full observation—purely through relabeling. The authors are explicit that the random-relabeling analysis is heuristic: edge locations generated by a random permutation are highly dependent, and a rigorous treatment would require concentration inequalities for random permutations.

The complete graph admits a uniform bound independent of $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$8: $$\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$$9. Notably, this is not extremal among simple graphs. For even $\operatorname{FR}_{\min}(G)$0, the parity-labeled complete bipartite graph $\operatorname{FR}_{\min}(G)$1 has Fourier transform supported on exactly two frequencies, giving $\operatorname{FR}_{\min}(G)$2. The paper proves a sharp extremal theorem: every nonzero loop-free edge indicator satisfies $\operatorname{FR}_{\min}(G)$3, with equality attained by parity-labeled $\operatorname{FR}_{\min}(G)$4. This result makes clear that Fourier ratio measures additive spectral compressibility rather than density—the relation $\operatorname{FR}_{\min}(G)$5 is spectrally more complicated than the single-mode parity relation $\operatorname{FR}_{\min}(G)$6.

Additional families with low complexity include Turán graphs (with $\operatorname{FR}_{\min}(G)$7 computed exactly), circulant graphs ($\operatorname{FR}_{\min}(G)$8 via support on one dual line), and graphs whose edges are unions of $\operatorname{FR}_{\min}(G)$9 affine relations ($\operatorname{FR}(f)\le r$0). The framework extends to weighted, directed, and signed graphs; signed weights can induce cancellation that pushes ratios toward the extremal bound $\operatorname{FR}(f)\le r$1.

Relation to classical spectral invariants

The paper shows that no strong two-sided estimate relates the Fourier ratio to relabeling-invariant quantities such as spectral radius or Laplacian gap. The cycle graph's spectral radius is always $\operatorname{FR}(f)\le r$2, while its Fourier ratio ranges from about $\operatorname{FR}(f)\le r$3 under natural labeling to $\operatorname{FR}(f)\le r$4 under random relabeling for $\operatorname{FR}(f)\le r$5. The only general lower bound from spectral data is weak: $\operatorname{FR}(f)\le r$6 where $\operatorname{FR}(f)\le r$7 is average degree. The distinction is conceptual: spectral radius measures expansion, whereas the Fourier ratio measures whether a labeling reveals hidden algebraic structure.

The numerical tables in the paper are presented as illustrative demonstrations of predicted scaling laws rather than outcomes of a systematic computational study—a caveat the authors state plainly. Small-scale empirical checks (e.g., averages over 50 random permutations for $\operatorname{FR}(f)\le r$8) support the scaling behavior, and simulated annealing plus Fiedler-vector orderings appear to recover near-optimal ratios for structured graphs, suggesting spectral embeddings may serve as approximate detectors of hidden additive organization.

Spectral projectors and harmonic complexity

The second major contribution analyzes the Fourier ratio of Laplacian spectral projector kernels $\operatorname{FR}(f)\le r$9. Two structural theorems anchor this section:

• Edge lower bound: $f$0, where $f$1 is the graph energy and $f$2 the number of edges. The proof uses unitary invariance of the nuclear norm under conjugation by the Fourier matrix, so $f$3 for every labeling. Equality holds for Cayley graphs on $f$4 under the natural group labeling.
• Harmonic lower bound: $f$5, where $f$6 is the eigenspace multiplicity. Equality again holds for abelian Cayley graphs, where the natural group labeling simultaneously minimizes both edge and harmonic complexity.

A combined corollary gives $f$7.

These bounds yield a sharp dichotomy between edge and harmonic complexity. The complete graph is Fourier-simple at the edge level ($f$8) but Fourier-complex harmonically ($f$9); the cycle exhibits the opposite profile, with every spectral projector bounded by $p$0 since all multiplicities are at most two. For abelian Cayley graphs both complexities collapse to functions of multiplicity alone.

Since low-frequency projectors dominate the heat kernel $p$1 for large $p$2, the projector recovery theorem implies that large-scale diffusion geometry, coarse connectivity, and spectral embeddings can be recovered from sparse samples without reconstructing the entire Laplacian. The paper formalizes this via a decomposition $p$3 into Fourier-compressible and delocalized sectors, drawing an analogy to structured-pseudorandom decompositions in additive combinatorics—though the conjecture that hidden additive structure forces many compressible projectors remains unproven.

For strongly regular graphs, the paper derives an explicit inequality linking $p$4 to $p$5, connecting harmonic and edge complexity analytically. An exhaustive search over all $p$6 labelings of the Petersen graph produces a notable negative finding: the labeling minimizing $p$7 does not minimize either $p$8 or $p$9, although a single labeling simultaneously minimizes both projector ratios. Whether simultaneous minimization across all eigenspaces always exists is left open.

Asymptotic spectral synthesis

The final theoretical component formulates a graph analogue of spectral synthesis. The main theorem shows that sequences of functions supported on exceptional sets of size $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$0, satisfying a uniform $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$1 summability condition over spectral components and a decay condition on a graph spectral Fourier ratio $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$2, must satisfy $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$3 for all bounded test functions, provided $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$4. In particular, no nonnegative sequence with nonvanishing mean can satisfy these hypotheses. A corollary gives asymptotic uniqueness: two signals agreeing off a small set and obeying the summability condition are indistinguishable in the normalized weak sense. In the exact band-limited case, recovery reduces to solving a discrete Helmholtz-type Dirichlet problem $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$5 whenever the restricted operator is invertible.

Limitations and open problems

Several limitations bear directly on the strength of the results. First, the definition of $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$6 involves optimization over all $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$7 labelings, and the paper provides no efficient algorithm for deciding whether a low-Fourier-ratio labeling exists or for finding one; the recovery guarantee is conditional on having such a labeling in hand. Second, the analysis of random relabelings is explicitly heuristic, lacking rigorous concentration estimates. Third, the numerical results are illustrative rather than systematic. Fourth, the recovery of diffusion geometry assumes that the Fourier-compressible projectors coincide with low-frequency ones—an assumption verified for structured families but not characterized in general.

The open problems section poses concrete questions: characterizing graphs admitting labelings with $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$8 for $$pN^2 \ge C\frac{r^2}{\epsilon^2}\log(r/\epsilon)^2\log N$$9; determining the computational complexity of computing $\ell^1$0 (possibly NP-hard); establishing a converse theory showing that $\ell^1$1 forces an affine/circulant description of the edge relation; proving the conjecture that Erdős–Rényi graphs have $\ell^1$2 close to $\ell^1$3; testing Fiedler-vector orderings as labeling heuristics; relating the Fourier ratio to bandwidth, treewidth, and expansion; stability under edit-distance perturbations; continuous analogues on $\ell^1$4; quantum extensions; applications to networked inventory systems; and connections between the Fourier ratio and Rényi entropy of order $\ell^1$5, noting the identity $\ell^1$6.

Conclusion

This paper establishes a coherent program in which graph recoverability from partial observations is controlled by additive spectral organization, quantified through the Fourier ratio and its labeling-minimized invariant. The sharp lower bounds $\ell^1$7 and $\ell^1$8, attained exactly for abelian Cayley graphs, together with the extremal bound $\ell^1$9 for simple graphs, give the framework precise analytical content. The principal unresolved issue is algorithmic: without efficient methods for detecting or constructing low-Fourier-ratio labelings, the recovery guarantees remain conditional statements about graphs with known hidden additive structure.