# Fiedler Non-Identifiability Under k-Hop Projection

## Setup

Let $G = (V,E)$ be a graph, $L$ its Laplacian, and $v_2$ the Fiedler vector
(second eigenvector of $L$). Define the Fiedler sign $\phi(x) = \operatorname{sign}(v_2(x))$.

Define a $k$-hop sampling operator $\mathcal{P}_k : G \to \tilde{G}_k$ that:
- samples seeds uniformly from $V$,
- expands each seed by $k$ hops of neighborhood closure,
- induces the subgraph on the resulting node set (edges preserved iff both endpoints are sampled).

$\mathcal{P}_k$ is **not spectrum-preserving**: it applies a local density filter
to the eigenspace before eigenvalue decomposition.

## Empirical Finding (cit-HepPh, $k=2$)

On the 30,985-node 2-hop induced subgraph of cit-HepPh (sampled from 34,546
nodes, 421,578 edges):

- $\phi$ is constant on 94.62% of computed nodes
- $\operatorname{Var}(\phi) \approx 0.0509$ (essentially collapsed)
- PR-kNN recovers $\phi$ at 94.40% accuracy — equal to the trivial majority
  predictor (94.56%)
- The minority class (B, 5.38%) has 99.7% PR-kNN error: the feature geometry
  contains no separable decision boundary for it
- $\lambda_2(\tilde{L}_2) \approx 0.20$, well-separated from $\lambda_1 \approx 0$,
  but the eigenvector aligns with a local density mode, not a global partition

## Non-Identifiability Theorem

The Fiedler sign $\phi$ is **non-identifiable** under $\mathcal{P}_k$ when
three conditions hold simultaneously:

### Condition A — Boundary dilution

$$
P_{\mathcal{P}_k}(x \in \partial C) \ll P(x \in C)
$$

Boundary nodes (where $\phi$ changes sign) are under-sampled because they
occupy bridge positions with lower $k$-hop closure probability. A bridge
node has $O(1)$ neighbors in each community, so its $k$-hop expansion
captures only one side of the partition.

### Condition B — Degree-conditioned closure

$$
\mathbb{E}[d(x) \mid x \in \tilde{G}_k] \not\approx \mathbb{E}[d(x)]
$$

The sampled subgraph over-represents high-degree hubs whose neighborhoods
are dense. Low-degree bridges are systematically excluded. Since degree
correlates with community core membership, the sampling distribution is
biased toward the dominant community's interior.

### Condition C — Spectral gap collapse under restriction

$$
\lambda_2(\tilde{L}_k) \ll \lambda_2(L)
$$

or equivalently the second eigenvector of $\tilde{L}_k$ aligns with the
constant vector: the restricted Laplacian's second mode describes local
density variation, not global partition structure. Operationally:

$$
\langle \tilde{v}_2, \mathbf{1} \rangle \approx \|\tilde{v}_2\|
$$

### Theorem (informal)

If (A) boundary dilution and (B) degree-conditioned closure both hold, then
$\tilde{L}_k$ almost surely admits a Fiedler vector whose sign structure
converges to a constant on the sampled support:

$$
\tilde{v}_2(x) \approx c \quad \forall x \in \tilde{G}_k
$$

and therefore $\phi$ is unrecoverable from $\tilde{G}_k$:

$$
\phi \not\approx \operatorname{sign}(\tilde{v}_2)
$$

*Proof sketch.* Under (A), the $k$-hop subgraph contains predominantly
interior nodes of the dominant community. Under (B), the spectral mass is
concentrated in high-degree regions. The Cheeger inequality gives:

$$
\lambda_2(\tilde{L}_k) \leq 2h(\tilde{G}_k)
$$

where $h$ is the Cheeger constant. With boundary dilution, $h(\tilde{G}_k)$
approaches the trivial cut (one community empty), driving $\lambda_2$ toward
the conductance of a single cluster's internal expansion rather than the
inter-cluster bottleneck. The corresponding eigenvector therefore tracks
local density rather than global division. $\square$

## Interpretation

The $k$-hop projection pipeline does not "lose signal" — it enters a regime
where the second eigenmode of the induced measure space is no longer the
second eigenmode of the original.

**Why PR and HITS survive.** PageRank and HITS operate on first-order
stationary mass flow:
- PR ≈ Perron eigenvector of the transition matrix (stationary distribution)
- HITS ≈ leading singular vectors of the adjacency matrix

These are stable under $\mathcal{P}_k$ because they depend on *aggregate
mass flow* through the $k$-hop subgraph, which approximates the global
stationary distribution under ergodic assumptions. The Fiedler vector
($\lambda_2$) is a *second-order* property describing *difference* between
communities, and collapses when boundary structure is suppressed.

## Regime Classification

| Regime | Fiedler behavior | Detection |
|--------|-----------------|-----------|
| (1) True spectral | Bimodal sign, meaningful partition | $\operatorname{Var}(\phi) \approx 0.25$ |
| (2) Sampling collapse | Unimodal sign, density mode | $\operatorname{Var}(\phi) \ll 0.1$ |
| (3) Propagation fake agreement | Majority prior amplified | PR-kNN accuracy ≈ majority baseline |

The cit-HepPh cross-validation under $\mathcal{P}_2$ operates in regimes
(2) → (3).

## Operational Rule

If on the computed subgraph:

$$
\frac{\max(n_{\text{neg}}, n_{\text{pos}})}{n_{\text{total}}} > 0.90
\quad\text{and}\quad
\operatorname{Var}(\phi) < 0.1
$$

flag Fiedler as **non-informative** under the current sampling regime and
exclude from any scoring or purity metric.

## References

- von Luxburg, U. (2007). A tutorial on spectral clustering. *Statistics and
  Computing*, 17(4), 395–416. — Fiedler vector as global partition signal.
- Chung, F. (1997). *Spectral Graph Theory*. CBMS Regional Conference Series.
  — Cheeger inequality and $\lambda_2$ as conductance.
- This repo: `snap_pist_spectral_crossval.py` — empirical collapse detection.
- This repo: `docs/specs/DP_RRC_RECEIPT_ENCODING_SPEC.md` — broader receipt
  encoding context for spectral observables.