Perception Gap Index (PGI) in Opinion Networks

Updated 13 July 2026

The Perception Gap Index (PGI) is defined on graphs to measure the divergence between a node’s local neighborhood average and the global opinion average.
It extends the binary majority illusion to continuous settings by using squared deviations to capture local-global misperceptions.
Matrix formulations and spectral graph theory reveal how graph connectivity, community structure, and low-degree nodes influence perception distortion.

The Perception Gap Index (PGI) is a graph-based measure that quantifies local-global opinion divergence, and can be viewed as a generalization of the majority illusion to continuous settings (Gehlot et al., 15 Nov 2025). In its named form, PGI is defined on an undirected graph with continuous node opinions and measures the extent to which the average opinion in each node’s closed neighborhood departs from the global average opinion. In adjacent literatures, the phrase “perception gap” is also used to describe mismatches between correct outputs and grounded evidence, between visual perception and reasoning, or between visual proxies and lived experience; however, those works generally do not introduce an explicit scalar called PGI, instead relying on conditional failure rates, benchmark accuracies, hierarchical evaluations, or agreement thresholds (Kang et al., 21 May 2026, Chen et al., 22 Sep 2025, Ye et al., 6 Jan 2026, Malekzadeh et al., 4 Nov 2025). This suggests that PGI is currently both a specific index in network science and a broader diagnostic motif across perception-sensitive evaluation settings.

1. Conceptual basis

PGI is defined for settings in which each node has a local view of opinions through its closed neighborhood $N[v_i] = N(v_i)\cup\{v_i\}$ , while the analyst can observe the network-wide opinion distribution. The central question is whether local exposure systematically misrepresents the global state. PGI is large when local neighborhood averages differ substantially from the global mean, so it formalizes a structural form of misperception associated with echo chambers, filter bubbles, and the majority illusion (Gehlot et al., 15 Nov 2025).

The importance of the index lies in its extension of a binary phenomenon to continuous opinions. The majority illusion is usually formulated for a binary attribute, where a rare trait may appear common locally because of network structure. PGI replaces that binary logic with squared deviations between local and global opinion averages, allowing the same intuition to be studied when opinions lie in $[-1,1]$ rather than in $\{0,1\}$ (Gehlot et al., 15 Nov 2025).

This framing makes PGI a structural measure rather than a purely behavioral one. It does not require strategic behavior, survey misreporting, or opinion change; the distortion arises from the topology of exposure itself. In that sense, PGI isolates how network organization can cause systematic local-global misalignment even when all node opinions are fully specified.

2. Formal definition and matrix representation

Let $G=(V,E)$ be an undirected graph with $n$ nodes, and let the opinion vector be

$S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$

For node $v_i$ , define

$\hat{s}_i=\sum_{v_k\in N[v_i]} s_k,$

and let $d_i=|N[v_i]|$ be the closed-neighborhood size. The Perception Gap is

$P(G,S)=\sum_{i=1}^{n}\left(\frac{\hat{s}_i}{d_i}-\frac{\sum_{j=1}^{n}s_j}{n}\right)^2.$

This expression compares each node’s local average opinion with the global average opinion and aggregates the squared discrepancies over all nodes (Gehlot et al., 15 Nov 2025).

The paper also reformulates PGI in matrix form to enable spectral analysis. Let $[-1,1]$ 0 be the adjacency matrix with conceptual self-loops, $[-1,1]$ 1 the diagonal degree matrix, $[-1,1]$ 2 the mean-centered opinion vector,

$[-1,1]$ 3

and $[-1,1]$ 4 the centering matrix. Then

$[-1,1]$ 5

and equivalently

$[-1,1]$ 6

An important invariant is

$[-1,1]$ 7

so mean-centering the opinions does not change the PGI (Gehlot et al., 15 Nov 2025).

The matrix formulation is significant because it places PGI within spectral graph theory. Once written as a quadratic form in the centered opinion vector, the index can be bounded, optimized, and compared across graph families using singular values and eigenvalues of normalized adjacency operators.

3. Spectral properties, random-opinion behavior, and community structure

The main theoretical message of the PGI framework is that better-connected graphs reduce worst-case perception distortion, whereas strong community structure increases vulnerability (Gehlot et al., 15 Nov 2025). For any graph $[-1,1]$ 8 and opinion vector $[-1,1]$ 9 with $\{0,1\}$ 0,

$\{0,1\}$ 1

where $\{0,1\}$ 2 are the largest and second-largest singular values of $\{0,1\}$ 3. For $\{0,1\}$ 4-regular graphs, this specializes to

$\{0,1\}$ 5

where $\{0,1\}$ 6 is the normalized adjacency matrix and $\{0,1\}$ 7 is its second-largest eigenvalue. The interpretation given in the paper is that smaller $\{0,1\}$ 8 means stronger connectivity or expansion and hence smaller worst-case PGI. The complete graph $\{0,1\}$ 9 is the limiting case: $G=(V,E)$ 0 for every opinion vector $G=(V,E)$ 1 (Gehlot et al., 15 Nov 2025).

Under random opinions, the index admits an exact expectation. If opinions are i.i.d. standard Gaussian with mean $G=(V,E)$ 2 and variance $G=(V,E)$ 3, then

$G=(V,E)$ 4

The same statement is reported to hold for any i.i.d. zero-mean, unit-variance distribution, including bounded distributions such as uniform on $G=(V,E)$ 5. A direct implication is that nodes with small degree contribute more to expected PGI, so networks with many low-degree nodes tend to have larger expected perception gap (Gehlot et al., 15 Nov 2025).

The stochastic block model analysis makes the role of community structure explicit. In a two-block SBM with within-block edge probability $G=(V,E)$ 6, between-block edge probability $G=(V,E)$ 7, and opposite block opinions $G=(V,E)$ 8, the paper derives a closed-form PGI and reports the large- $G=(V,E)$ 9 simplification

$n$ 0

This shows that PGI grows as the separation between communities increases. When $n$ 1 is large, local neighborhoods become more internally homogeneous, local opinion averages diverge more from the global mean, and the network becomes more vulnerable to perception distortion (Gehlot et al., 15 Nov 2025).

4. Minimization by link recommendation

The intervention problem studied for PGI is to add a fixed number of missing edges in order to reduce local-global misperception. Given a graph $n$ 2, an opinion vector $n$ 3, the set of missing edges $n$ 4, and a budget $n$ 5, the optimization problem is

$n$ 6

where $n$ 7 (Gehlot et al., 15 Nov 2025).

The paper proves a strong hardness result: there is no polynomial-time algorithm with a bounded approximation ratio for this problem unless $n$ 8. The proof proceeds by reduction from the Generalized Partition Problem. The authors also prove that the objective $n$ 9 is neither monotone nor supermodular. This rules out the standard approximation arguments that often justify greedy methods for graph intervention problems (Gehlot et al., 15 Nov 2025).

Because of this hardness, the paper proposes practical heuristics rather than approximation guarantees. The methods are Random, Greedy, Batch Greedy, and Random Batch Greedy. Greedy adds the missing edge with the largest immediate reduction in PGI at each step. Batch Greedy adds the top $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 0 edges at each round, with $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 1 in the reported experiments, to reduce computation. Random Batch Greedy ranks edges as in Batch Greedy and then randomly picks one edge from the top $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 2, introducing exploration while still focusing on strong candidates. Random serves as a baseline and performs poorly in experiments (Gehlot et al., 15 Nov 2025).

For exact-search acceleration, the paper introduces Maximum Potential Contribution (MPC) filtering. For each missing edge, MPC computes an upper bound on how much that edge could reduce PGI in the best-case context of already chosen adjacent edges. Edges whose best-case contribution cannot beat the greedy baseline are discarded, which shrinks the search space and makes brute-force exact search feasible on larger graphs than naive exhaustive search would allow. Empirically, Greedy, Batch Greedy, and Random Batch Greedy perform very similarly, and Greedy is reported as almost optimal in most tested cases (Gehlot et al., 15 Nov 2025).

5. PGI-like constructs in multimodal AI

Outside social networks, several papers analyze “perception gaps” without defining an explicit scalar called PGI. They instead separate apparent success from grounded perception.

Domain	Construct	Measurement basis
Personality perception in MLLMs	Prejudice Gap	PR, CR, IR, HR, and RGM
Geometric reasoning in MLLMs	visual perception gap	GeoPQA accuracy and model-human gap
Functional PET imaging in MLLMs	functional perception gap	hierarchical answer accuracy and CoT plausibility/accuracy divergence

In multimodal personality perception, the relevant construct is the Prejudice Gap, defined as the mismatch between a model’s ability to produce a correct personality rating and its ability to ground that rating in observable evidence. The paper formalizes Grounded Personality Reasoning (GPR) as a three-part chain over a video: rating, evidence-grounded reasoning, and cue grounding. It then defines task-success indicators $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 3 with default thresholds $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 4 and $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 5, and introduces four sample-level rates: $S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 6

$S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 7

It also defines Rating–Grounding Misalignment (RGM) as

$S = [s_1,\dots,s_n]^T,\qquad s_i\in[-1,1].$ 8

Across 27 MLLMs on MM-OCEAN, the paper reports that 51.3% of correct ratings are ungrounded, a mean HR of 10.4%, a best HR of 33.5%, and 0.0% for some models. It also reports that even at the proprietary frontier, about 15% of correct ratings remain ungrounded (Kang et al., 21 May 2026). These metrics are the closest direct analogue to a PGI in the sense of quantifying “right answer, wrong reason.”

In geometric reasoning, no explicit PGI is introduced. Instead, GeoPQA accuracy functions as a proxy for perceptual bottleneck severity. The paper argues that reinforcement learning for reasoning in MLLMs is upper-bounded by visual perception, and measures the “perception gap” as the discrepancy between model and human accuracy on carefully designed geometric perception questions. On the GeoPQA test set, reported accuracies are 68.2% for Qwen2.5-VL-3B-Instruct, 75.8% for Qwen2.5-VL-7B-Instruct, 80.0% for GPT-4o, and 91.0% for Human. The proposed two-stage RL framework improves geometric reasoning by 9.7% and geometric problem solving by 9.1% relative to direct reasoning training, supporting the paper’s claim that perceptual grounding constrains downstream reasoning (Chen et al., 22 Sep 2025).

In functional medical imaging, the relevant notion is the functional perception gap: the inability of current vision encoders to decode functional tracer biodistribution independent of morphological priors. PET-Bench is organized as a five-level hierarchy—Tracer Identification, Image Quality Assessment, Organ Recognition, Abnormality Detection, and Disease Diagnosis—to separate atomic perception from diagnostic reasoning. The paper reports mean zero-shot accuracy of 15.44% for image quality assessment, describes a Chain-of-Thought hallucination trap in which plausible rationales become decoupled from visual evidence, and proposes Atomic Visual Alignment (AVA), which improves diagnostic accuracy by up to 14.83% (Ye et al., 6 Jan 2026). Here again, the “gap” is operationalized through level-wise accuracy and the divergence between plausibility and grounding, not through a named PGI scalar.

A separate line of work uses PGI-like comparison to study alignment between visual proxies and reported experience. In urban-attractiveness analysis, Street View Imagery (SVI)-based predicted attractiveness is compared with PPGIS-marked attractive and unattractive places using strict and moderate agreement criteria. The spatial matching rule assigns to each PPGIS point the mean predicted attractiveness score from SVI images within a 50-meter radius buffer, with at least 5 images within the buffer. Under the moderate threshold, agreement reaches 67% for attractive and 77% for unattractive places; under the strict threshold, agreement drops to 27% and 29%, respectively. The paper attributes much of the mismatch to non-visual contextual variables, especially presence of people, noise, traffic volume, and speed limits (Malekzadeh et al., 4 Nov 2025).

This urban study is informative for PGI because it shows how a “perception gap” can be operationalized as alignment or misalignment between two data sources without defining a single universal index. The method uses agreement rates rather than a closed-form scalar and distinguishes between moderate and high-confidence alignment. A plausible implication is that, outside graph-based opinion analysis, perception-gap measurement often depends on domain-specific matching rules rather than a shared canonical formula.

A common misconception is that any paper mentioning “perception gap” necessarily defines or computes PGI. The provided document titled “Meltdown: Bridging the Perception Gap in Sustainable Food Behaviors Through Immersive VR” does not support such a reading. According to the available details, it is an ACM acmart template/sample document, does not define a perception gap, and does not mention a Perception Gap Index or any analogous metric (Xuan et al., 19 Apr 2025). This clarifies the current scope of the term: PGI is explicitly named and formally defined in the social-network setting, while other literatures more often speak of perception gaps through proxy benchmarks, hierarchical evaluations, or agreement-based diagnostics rather than through a single standardized index.