Structural Integrity Score (SIS)

• Structural Integrity Score (SIS) is a set of quantitative measures that assess how well a system retains its ideal structural organization across diverse domains.
• The methodology varies by application, such as using count ratios in nanorod arrays, thermal connectivity in 3D nanophotonics, and multi-metric profiles in LLM knowledge graphs and quantum circuits.
• SIS frameworks provide actionable insights by benchmarking deviations from ideal models, aiding in structural diagnostics for fields including vascular biomechanics, photonics design, and scene composition.

Structural Integrity Score (SIS) denotes a family of quantitative measures of structural fidelity rather than a single universal formula. In the literature summarized here, the term is used for aligned nanorod arrays as the integrity fraction $IF$, for quantum circuits as a similarity score derived from weighted structural deviations, for 3D nanophotonic inverse design as heat-based connectivity objectives, and for scene composition as a broader framework that subsumes SCSSIM. Closely related formulations include the Relative Structural Integrity Index (RSII) for abdominal aortic aneurysm wall assessment and a network-based LLM evaluation protocol that explicitly does not define one fixed aggregate SIS scalar (Thöle et al., 2017, Ahmed et al., 29 Apr 2026, Kuster et al., 10 Jan 2025, Haque et al., 7 Aug 2025, Jamshidian et al., 14 Feb 2025, Boudourides, 2 Mar 2026).

1. Cross-domain scope and meanings

The same acronym labels distinct constructs whose shared concern is preservation of structural organization under observation, generation, optimization, or degradation. In nanorod arrays, SIS is a count ratio against an ideal hexagonal lattice. In abdominal aortic aneurysm analysis, the related RSII is a local, patient-specific normalization of strain-to-tension ratios. In LLM evaluation, “structural integrity” is operationalized as a multi-metric profile over nodes, edges, centrality, communities, and citations. In quantum circuits, SIS is a bounded similarity score over global circuit descriptors. In 3D nanophotonics, the supplied formulation uses integrated heat as a differentiable proxy for connectivity. In scene-composition analysis, SCSSIM is presented as a concrete instantiation of a broader SIS framework (Thöle et al., 2017, Jamshidian et al., 14 Feb 2025, Boudourides, 2 Mar 2026, Ahmed et al., 29 Apr 2026, Kuster et al., 10 Jan 2025, Haque et al., 7 Aug 2025).

Domain	Formal quantity	Interpretation
Nanorod arrays	$SIS=IF=N_{\rm img}/N_0$	apparent array elements relative to a defect-free lattice
AAA wall assessment	$RSII(x)=|SII(x)|/\langle |SII| \rangle$	local compliance hot-spot relative to the patient’s own mean
LLM knowledge graphs	no fixed scalar SIS	diagnostic profile over fabrication, overlap, centrality, modularity, and citations
Quantum circuits	$SIS=1-\Delta_{\rm struct}$	global structural agreement with a reference circuit
3D nanophotonics	$\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$ and derived normalized thermal objectives	proxy for material and void connectivity
Scene composition	SCSSIM and generalized SIS	preservation of Scene Composition Structure

This suggests that “structural integrity” is domain-relative: the reference may be an ideal lattice, a patient’s own wall-average response, a canonical graph, a reference circuit, prescribed connectivity sinks, or a reference image.

2. Nanorod arrays: SIS as integrity fraction

For aligned nanorod arrays, SIS is defined as the integrity fraction $IF$, namely the number of array elements recognized in a micrograph divided by the number expected in a defect-free array over the same area. Let $N_{\rm img}$ denote the number of array elements actually recognized in the micrograph and $N_0$ the number that an ideal hexagonal array would contain over image area $A$. Then

$$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$

where $SIS=IF=N_{\rm img}/N_0$0 is the lattice constant. The metric is therefore a real-space count-based normalization to an ideal lattice density rather than a direct measure of mechanical stress or deformation (Thöle et al., 2017).

The image-analysis workflow is explicit. A top-view SEM image is acquired in 8-bit grayscale; Gaussian blur may optionally be applied to suppress pixel-level noise, using

$SIS=IF=N_{\rm img}/N_0$1

with $SIS=IF=N_{\rm img}/N_0$2 px in the reported implementation. Threshold intensity $SIS=IF=N_{\rm img}/N_0$3 is swept from 0 to 255, binarization is performed, a minimum object size $SIS=IF=N_{\rm img}/N_0$4 is imposed to reject spurious white-pixel clusters, and the number of recognized objects $SIS=IF=N_{\rm img}/N_0$5 is recorded over the full $SIS=IF=N_{\rm img}/N_0$6 grid. Finite-difference slope estimates,

$SIS=IF=N_{\rm img}/N_0$7

lead to

$SIS=IF=N_{\rm img}/N_0$8

In well-preserved arrays, a triangular low-slope plateau appears at intermediate $SIS=IF=N_{\rm img}/N_0$9 and $RSII(x)=|SII(x)|/\langle |SII| \rangle$0; the local slope minimum on this plateau with the largest $RSII(x)=|SII(x)|/\langle |SII| \rangle$1 is taken as $RSII(x)=|SII(x)|/\langle |SII| \rangle$2. In highly collapsed arrays, where the plateau disappears, the recommended choice is the largest $RSII(x)=|SII(x)|/\langle |SII| \rangle$3 at a local slope minimum near the kink separating noise-dominated and object-dominated regions.

The interpretation is tightly constrained. Gaussian pre-filtering suppresses high-frequency pixel noise, and exploring nearby $RSII(x)=|SII(x)|/\langle |SII| \rangle$4 values yields an uncertainty estimate $RSII(x)=|SII(x)|/\langle |SII| \rangle$5, hence an uncertainty in SIS. Because $RSII(x)=|SII(x)|/\langle |SII| \rangle$6 depends only on known image area and lattice constant, the score is normalized across fields of view and resolutions. The examples given are length-dependent: for $RSII(x)=|SII(x)|/\langle |SII| \rangle$7, $RSII(x)=|SII(x)|/\langle |SII| \rangle$8, $RSII(x)=|SII(x)|/\langle |SII| \rangle$9, and $SIS=1-\Delta_{\rm struct}$0; for $SIS=1-\Delta_{\rm struct}$1, $SIS=1-\Delta_{\rm struct}$2; for $SIS=1-\Delta_{\rm struct}$3, $SIS=1-\Delta_{\rm struct}$4; and for $SIS=1-\Delta_{\rm struct}$5, $SIS=1-\Delta_{\rm struct}$6. The reported plot therefore shows rapid collapse of structural integrity for rods longer than approximately $SIS=1-\Delta_{\rm struct}$7. The score can also be repeated on time-series micrographs to obtain $SIS=1-\Delta_{\rm struct}$8 as a measure of structural degradation or self-healing.

The principal limitations are also explicit. The method assumes that the array retains hallmarks of the original lattice, including uniform tip brightness and roughly monodisperse sizes. For completely randomized collapse, $SIS=1-\Delta_{\rm struct}$9, the plateau may disappear and uncertainty grows. Large agglomerates are counted as single objects if they exceed the plateau’s $\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$0 range, so SIS primarily reports the fraction of intact or small-cluster elements. It does not by itself separate single rods from small clusters; such distinctions require further analysis of the size-frequency distribution $\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$1.

3. Vascular biomechanics: SII and RSII in abdominal aortic aneurysm assessment

In abdominal aortic aneurysm analysis, the paper defines a local Structural Integrity Index $\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$2 and a Relative Structural Integrity Index $\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$3, rather than a global SIS scalar. The construction combines image-derived wall strain with finite-element wall tension and is designed to be independent of wall material properties, thickness, and blood-pressure measurement conditions after normalization. Wall-tangent tension is defined by integrating tangential Cauchy stress components through the assumed wall thickness $\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$4,

$\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$5

circumferential wall strain is

$\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$6

and the local structural index is

$\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$7

The relative index is then

$\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$8

By construction, $\mathcal{L}_{\rm heat}=\int_{\mathcal D}u\,dV$9, and values much greater than 1 indicate hot-spots of unusually high compliance relative to the patient’s own aorta (Jamshidian et al., 14 Feb 2025).

The end-to-end pipeline begins with ECG-gated 4D-CTA, using 10 frames per heartbeat and patient systolic/diastolic blood pressure. AI-based segmentation with PRAEVAorta is followed by MATLAB post-processing, surface extraction, smoothing, and generation of a quadratic tetrahedral mesh with approximately $IF$0 million C3D10H elements and mean edge length $IF$1 mm. Wall tension is computed in Abaqus/Standard with linear isotropic materials, $IF$2 GPa, $IF$3, and ILT set $IF$4 more compliant; inlet and outlet faces are fixed, uniform internal pressure $IF$5 kPa is applied, and residual stress is incorporated using Fung’s Uniform Stress Hypothesis. Wall strain comes from total-variation-regularized deformable registration of systolic to diastolic frames, interpolation of the 3D displacement field at wall nodes, local normal estimation by least-squares plane fitting, radius estimation from a 1 mm neighborhood, and computation of $IF$6. The final outputs are maps of $IF$7, $IF$8, $IF$9, and $N_{\rm img}$0, together with summaries such as the $N_{\rm img}$1th-percentile RSII and the fraction of wall area above thresholds.

The reported validation uses $N_{\rm img}$2 AAA patients. The $N_{\rm img}$3th-percentile wall tension ranges from $N_{\rm img}$4 to $N_{\rm img}$5 N/mm with mean $N_{\rm img}$6 N/mm, the $N_{\rm img}$7th-percentile strain from $N_{\rm img}$8 to $N_{\rm img}$9 with mean $N_0$0, and the RSII $N_0$1th percentile from $N_0$2 to $N_0$3 with mean approximately $N_0$4. RSII maps show localized islands with $N_0$5 in maximal-diameter regions, and in some patients high RSII also occurs outside the bulge. For a healthy proximal aorta, the map is more uniform, approximately $N_0$6–$N_0$7, with no sharp islands. The interpretation given is that $N_0$8 indicates above-average local compliance for that patient, while $N_0$9 or above the $A$0th percentile marks a potential weakening hot-spot worth closer surveillance. Longitudinal use is proposed through follow-up scans that track changes in RSII distribution.

4. Knowledge graphs, citation networks, and structural hallucination

In the LLM-evaluation setting, “structural integrity” is not reduced to a single scalar SIS. The paper explicitly states that it does not specify a single aggregate SIS with fixed weights and instead treats the stress test as a multi-metric diagnostic profile. The protocol starts from an authoritative reference graph $A$1 and an LLM-generated graph $A$2. Fabrication is measured by

$A$3

Set-overlap similarity is measured by node- and edge-set Jaccard indices,

$A$4

Centrality preservation is evaluated over the shared node set using Spearman rank correlation,

$A$5

with $A$6, alongside upward-mobility analysis for nodes whose LLM rank greatly exceeds their reference rank. Community coherence is examined via Louvain-type community detection, modularity comparison, and overlap in community membership. Citation integrity is checked through external registries, with citation recall

$A$7

and citation omission defined as $A$8 (Boudourides, 2 Mar 2026).

The metrics are interpreted as structural, not merely factual, diagnostics. Fabrication rate and Jaccard overlap quantify node- and edge-level faithfulness, macro-$A$9 on cross-reference edges and citation recall quantify completeness, Spearman $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$0 tests whether the relative importance of entities is preserved, upward mobility identifies conceptual re-centering, modularity and community overlap test higher-order thematic substructures, and DOI validity plus external lookups test bibliographic grounding. The paper’s central claim is that structural fidelity cannot be inferred from local fluency alone.

Validation spans three domains. For Roget’s Thesaurus, node-set Jaccard is $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$1, fabrication rate is $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$2 of terms, and cross-reference macro-$$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$3. For Wikidata philosophers, hallucination rates exceed $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$4 across the six main structured fields, with country of citizenship giving the best $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$5 and influenced_by yielding $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$6; age/date fields are reported as more than $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$7 hallucinated. For Dimensions.ai publications, citation omission is $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$8, DOI reconstruction has near-zero $$SIS = IF = \frac{N_{\rm img}}{N_0}, \qquad N_0=\frac{2A}{\sqrt{3}\,d^2},$$9 of $SIS=IF=N_{\rm img}/N_0$00, times_cited has $SIS=IF=N_{\rm img}/N_0$01, date $SIS=IF=N_{\rm img}/N_0$02, type $SIS=IF=N_{\rm img}/N_0$03, and only $SIS=IF=N_{\rm img}/N_0$04 papers have any citations generated at all. A possible implication, explicitly framed in the supplied summary as illustrative rather than fixed, is that any future scalar SIS for LLM-generated knowledge graphs would need to combine fabrication, overlap, centrality consistency, community integrity, and citation recall rather than rely on any one of them.

5. Quantum circuits: SIS as global structural similarity

For quantum circuits, SIS is defined directly as a bounded similarity score between a test circuit $SIS=IF=N_{\rm img}/N_0$05 and a reference circuit $SIS=IF=N_{\rm img}/N_0$06. The formulation first computes an aggregate normalized structural deviation

$SIS=IF=N_{\rm img}/N_0$07

with

$SIS=IF=N_{\rm img}/N_0$08

in the reported experiments. The SIS is then

$SIS=IF=N_{\rm img}/N_0$09

The four components are the normalized relative differences in total gate count, circuit depth, two-qubit gate usage, and interaction-topology structure. Concretely,

$SIS=IF=N_{\rm img}/N_0$10

$SIS=IF=N_{\rm img}/N_0$11

where $SIS=IF=N_{\rm img}/N_0$12 is a DAG of gates and data dependencies and $SIS=IF=N_{\rm img}/N_0$13 is a normalized topology-deviation measure. SIS therefore ranges in $SIS=IF=N_{\rm img}/N_0$14, with 1 indicating perfect structural agreement (Ahmed et al., 29 Apr 2026).

The implementation is intended as a fast pre-execution check. Gate count, depth, and two-qubit count are extracted in $SIS=IF=N_{\rm img}/N_0$15, while DAG-topology deviation is reported as $SIS=IF=N_{\rm img}/N_0$16 or $SIS=IF=N_{\rm img}/N_0$17 depending on the algorithm; the paper uses a lightweight adjacency-difference measure yielding near-linear time, so overall SIS is stated as $SIS=IF=N_{\rm img}/N_0$18. Empirically, gate deletion and insertion anomalies cause SIS to drop monotonically with severity, whereas gate substitution and gate reordering leave SIS close to $SIS=IF=N_{\rm img}/N_0$19 across all severity levels. A threshold of $SIS=IF=N_{\rm img}/N_0$20 defines structural blind spots: circuits considered structurally indistinguishable from the reference despite injected anomalies.

The blind-spot statistics are central to the paper’s argument that structural integrity alone is insufficient. Across 569 test circuits with $SIS=IF=N_{\rm img}/N_0$21, SIS misses 100% of those anomalies by definition. The interaction-level metric IGS detects 413 of 569, or $SIS=IF=N_{\rm img}/N_0$22, while the behavioral metric OIS detects 534 of 569, or $SIS=IF=N_{\rm img}/N_0$23. Broken down by anomaly severity, the corresponding IGS/OIS detection rates are $SIS=IF=N_{\rm img}/N_0$24 at severity $SIS=IF=N_{\rm img}/N_0$25, $SIS=IF=N_{\rm img}/N_0$26 at severity $SIS=IF=N_{\rm img}/N_0$27, and $SIS=IF=N_{\rm img}/N_0$28 at severity $SIS=IF=N_{\rm img}/N_0$29. The reported conclusion is that structural similarity alone does not ensure behavioral equivalence; SIS is therefore useful as a global structural screen but not as a complete integrity guarantee.

6. 3D nanophotonics: heat-based SIS as a connectivity constraint

In 3D nanophotonic inverse design, the supplied summary uses SIS for a structural-integrity metric based on auxiliary steady-state heat-diffusion solves. The design region $SIS=IF=N_{\rm img}/N_0$30 is assigned a scalar temperature field $SIS=IF=N_{\rm img}/N_0$31 that satisfies

$SIS=IF=N_{\rm img}/N_0$32

or, as implemented,

$SIS=IF=N_{\rm img}/N_0$33

The conductivity mapping is

$SIS=IF=N_{\rm img}/N_0$34

with $SIS=IF=N_{\rm img}/N_0$35 and $SIS=IF=N_{\rm img}/N_0$36. Two source terms are defined: $SIS=IF=N_{\rm img}/N_0$37 Dirichlet heat-sink conditions $SIS=IF=N_{\rm img}/N_0$38 are imposed on selected boundary subsets $SIS=IF=N_{\rm img}/N_0$39, chosen to enforce required material connectivity for the material problem and to cover the remainder of $SIS=IF=N_{\rm img}/N_0$40 for the void problem, with homogeneous Neumann conditions elsewhere. The raw heat-based quantity is

$SIS=IF=N_{\rm img}/N_0$41

from which

$SIS=IF=N_{\rm img}/N_0$42

are obtained (Kuster et al., 10 Jan 2025).

The interpretation is that disconnected material islands or trapped voids cannot dissipate heat to prescribed sinks, so the integrated temperature grows large there. Structural integrity is thus encoded through thresholded normalized thermal objectives,

$SIS=IF=N_{\rm img}/N_0$43

When either normalized value becomes negative, the corresponding connectivity target is considered attained. These thermal terms are combined with a normalized electromagnetic objective

$SIS=IF=N_{\rm img}/N_0$44

passed through the softplus

$SIS=IF=N_{\rm img}/N_0$45

and aggregated with a binarization penalty in the total cost

$SIS=IF=N_{\rm img}/N_0$46

The optimization schedule uses 120 iterations in 6 stages of 20 steps each. During the first five stages, the Heaviside-projection steepness $SIS=IF=N_{\rm img}/N_0$47 is increased from 1 to 30, and in the final stage the binarization penalty is turned on. The thermal solver is an in-house FEM implementation on $SIS=IF=N_{\rm img}/N_0$48 only, using H8 elements at 40 voxels/$SIS=IF=N_{\rm img}/N_0$49, with two heat solves and two adjoint solves per iteration. Each thermal simulation is reported as $SIS=IF=N_{\rm img}/N_0$50–$SIS=IF=N_{\rm img}/N_0$51 cheaper than the FDFD EM solve, giving approximately $SIS=IF=N_{\rm img}/N_0$52–$SIS=IF=N_{\rm img}/N_0$53 overhead; a full 120-step run takes roughly $SIS=IF=N_{\rm img}/N_0$54–$SIS=IF=N_{\rm img}/N_0$55 days on an NVIDIA A100 GPU. Two validation cases are reported. For the focusing element, the EM-only design has $SIS=IF=N_{\rm img}/N_0$56, while the SIS-constrained design reaches $SIS=IF=N_{\rm img}/N_0$57 with both normalized thermal metrics negative. For the waveguide junction, the EM-only design has $SIS=IF=N_{\rm img}/N_0$58, and the SIS-constrained design $SIS=IF=N_{\rm img}/N_0$59, again with both normalized thermal metrics negative. The reported observation is that the heat-based integrity measure correlates perfectly with visual connectivity: positive normalized thermal objectives coincide with disconnected islands or cavities, whereas both negative values coincide with fully connected material and void.

7. Scene composition: SCSSIM and generalized SIS

For scene-composition evaluation, the primary defined metric is the SCS Similarity Index Measure (SCSSIM), which the supplied summary presents as a concrete instantiation of a broader SIS framework. The underlying object of interest is Scene Composition Structure (SCS), defined as the geometric relationships among objects and background, including relative positions, sizes, and orientations. SCSSIM is analytical and training-free and is based on Cuboidal Hierarchical Partitioning (CuPID). For an image $SIS=IF=N_{\rm img}/N_0$60, the sum of squared errors is

$SIS=IF=N_{\rm img}/N_0$61

where $SIS=IF=N_{\rm img}/N_0$62 is the RGB vector of pixel $SIS=IF=N_{\rm img}/N_0$63. A candidate cut partitions $SIS=IF=N_{\rm img}/N_0$64 into $SIS=IF=N_{\rm img}/N_0$65 and $SIS=IF=N_{\rm img}/N_0$66, with gain

$SIS=IF=N_{\rm img}/N_0$67

The strongest cut is the one with maximal gain $SIS=IF=N_{\rm img}/N_0$68, and the process recurses until a predetermined number $SIS=IF=N_{\rm img}/N_0$69 of cuts is reached (Haque et al., 7 Aug 2025).

The final similarity is built from normalized cumulative-gain curves. For the reference image $SIS=IF=N_{\rm img}/N_0$70, with gains $SIS=IF=N_{\rm img}/N_0$71 and total SSE $SIS=IF=N_{\rm img}/N_0$72,

$SIS=IF=N_{\rm img}/N_0$73

For the second image $SIS=IF=N_{\rm img}/N_0$74,

$SIS=IF=N_{\rm img}/N_0$75

The element-wise log-ratio is

$SIS=IF=N_{\rm img}/N_0$76

with mean

$SIS=IF=N_{\rm img}/N_0$77

A directional similarity is then

$SIS=IF=N_{\rm img}/N_0$78

and the symmetrized SCSSIM is

$SIS=IF=N_{\rm img}/N_0$79

With $SIS=IF=N_{\rm img}/N_0$80 and $SIS=IF=N_{\rm img}/N_0$81, the summary states that the measure is bounded in $SIS=IF=N_{\rm img}/N_0$82, symmetric, equals 1 only when $SIS=IF=N_{\rm img}/N_0$83, is invariant to noise and blur as non-compositional distortions, and decreases monotonically under compositional changes such as rotation, zoom, and pan.

The broader SIS formulation is presented as a generalization of this idea. SCSSIM is said to measure SCS preservation by comparing normalized cumulative gains from CuPID, whereas SIS extends the concept to alternative partitioning schemes, additional structural features, and weighting strategies. The example generalized form is

$SIS=IF=N_{\rm img}/N_0$84

where $SIS=IF=N_{\rm img}/N_0$85 is a block-wise feature vector, $SIS=IF=N_{\rm img}/N_0$86 is a block-wise structural similarity, and $SIS=IF=N_{\rm img}/N_0$87 is a block weight. The proposed extensions include oriented cuts, semantic-aware partitions, and block features such as centroid position, dominant orientation, and aspect ratio. The comparison with traditional metrics is explicit: pixel-level metrics such as PSNR, SSIM, and MS-SSIM are described as overly sensitive to additive noise or blur, while perception-based metrics such as LPIPS, CLIP Score, and FID may remain invariant to drastic compositional changes. In that framing, SCSSIM fills the gap by remaining nearly constant under non-compositional distortions while decreasing monotonically under genuine structural transformations.