Whose Polarization Is It? Auditing Fairness in Network Polarization Measures

Stanisław Stępień; Michalina Janik; Akrati Saxena; Radosław Michalski

cs.SIOct 15, 2026

Whose Polarization Is It? Auditing Fairness in Network Polarization Measures

Authors: Stanisław Stępień, Michalina Janik, Akrati Saxena, Radosław Michalski

Organizations: Wroclaw University of Science and Technology, Wrocław, Poland · Leiden University, Leiden, The Netherlands

Abstract

A polarization score is supposed to describe a community, not its surroundings. Yet when a minority community is embedded in a larger network, its measured polarization can rise or fall with the size and structure of the majority around it. We ask whether polarization measures are therefore fair to minorities and develop an audit built around two criteria: measurement invariance (MI), which asks whether otherwise equivalent majorities and minorities receive equivalent scores, and structural exogeneity (SE), which asks whether a minority's score changes when its surrounding majority changes. Resolution sensitivity (RS) is treated as a prerequisite for meaningful fairness assessment. We audit seven polarization measure families spanning the three network-operationalizable dimensions of polarization, introducing Network-Weighted Distance (NWD) to fill an unserved extremeness-separation combination. Across controlled synthetic networks, mechanism-isolating diagnostic topologies, and real political and communication networks, no measure passes all applicable checks. Minorities are systematically scored as more polarized than equivalent majorities. We trace this dependence to two separate channels: majority structure can alter the opinions that opinion-reading measures consume, and, on a different path, opinion-orthogonal structural change can affect the arithmetic of structural measures. Reading opinions is thus double-edged, letting contamination in when structure moves opinions but shielding the score when structure changes and opinions do not. We close with practical guidance for reporting polarization scores without inheriting these biases.

Figures & tables

Figure 1: Coverage of the seven audited measures across the three Bramson dimensions: extremeness (D1), clustering (D2), and separation (D3). Each measure sits in the region of the dimensions it captures. Modularity appears here as a single measure (Q) but is audited in two operationalisations,

Q_{\text{opinion}}

and

Q_{\text{struct}}

Figure 2: The two fairness criteria and resolution-sensitivity prerequisite on Track A, per measure, ordered so opinion-reading measures sit above the two opinion-free ones (

Q_{\text{struct}}

, AEI). (A) Measurement invariance: Cohen’s

d

(majority

-

minority); red bars are negative (the minority scored higher), and bars past

\pm 0.3

(dashed) are Benjamini-Yekutieli-significant violations. (B) Structural exogeneity: semi-partial

R^{2}

of the minority score on majority parameters under Friedkin-Johnsen opinions, violations past

0.02

(dashed); under fixed bimodal opinions, the control condition, all measures are exogenous (

R^{2}\leq 0.01

). (C) Resolution sensitivity: detection rate by size ratio

r

, one cell per measure, darker is higher and bold marks detection

\geq 0.8

; the dashed box is the flagged region (

r<0.5

). AEI is degenerate on the single-community minority subgraphs, so its MI and SE entries are not substantively evaluable.

Measure	$R^{2}_{\text{maj}}$ (FJ)	Dominant predictor	SE (FJ)
$Q_{\text{opinion}}$	$0.00$	-	✓
$Q_{\text{struct}}$	$0.00$	-	✓
E-R	$0.070$	$p_{\text{out}}$	$\times$
$A$	$0.047$	$p_{\text{out}}$	$\times$
LE	$0.126$	$p_{\text{out}}$	$\times$
GED	$0.087$	$\log n_{\text{maj}}$	$\times$

Table 1: Structural Exogeneity, opinion-dynamics (Friedkin-Johnsen) channel: semi-partial

R^{2}_{\text{maj}}

of each measure’s minority-subgraph score on the majority parameters, the dominant majority predictor, and the violation verdict (Benjamini-Yekutieli-corrected). Under fixed (bimodal) opinions, the control condition, every measure is exogenous (

R^{2}\leq 0.01

, not shown). The Adaptive E-I Index is degenerate on single-community minority subgraphs.

Figure 3: Track B diagnostic audit: which measures fail which diagnostic constructions. Each cell records whether a measure follows the theoretically expected direction on the corresponding topology; red marks a mismatch, amber a calibration or baseline artifact, gray a case undefined for that measure, and hatching a case not evaluated. The columns are grouped as T1/T2 Structure vs. Opinion , T3/T4 Diversity vs. Conflict , T5/T6 Resolution vs. Null , and T7 Broker Dilution . NWD is evaluated only on T7, the purpose-built broker-dilution diagnostic for its walk-based mechanism; the remaining diagnostics were not assigned prespecified directional predictions for NWD. Full constructions, scores, and expected directions are in the appendix (Table 4 ).

Figure 4: T8 (Asymmetric Echo Chambers): mean majority

-

minority score difference across 100 replicate pairs. Asterisks mark Benjamini-Yekutieli-significant violations.

Measure	MI	SE ${}_{\text{FJ}}$	T8 size	RS	TB
$Q_{\text{opinion}}$	✓	✓	✓	$\times$	1
$Q_{\text{struct}}$	$\times$	✓	$\times$	$\times$	0
E-R	$\times$	$\times$	$\times$	✓	1
$A$	✓	$\times$	$\times$	$\times$	0
LE	$\times$	$\times$	$\times$	$\times$	3
GED	$\times$	$\times$	$\times$	$\times$	1

Table 2: Cross-measure audit overview, one column per check. MI (Measurement Invariance) and SE

{}_{\text{FJ}}

(Structural Exogeneity via the Friedkin-Johnsen channel) are the fairness checks; RS (Resolution Sensitivity) is a validity prerequisite, and T8 is a size-scaling diagnostic; TB is the Track B mismatch count. ✓ = pass,

\times

= violation, - = not meaningfully evaluable. † The Adaptive E-I Index is degenerate on the single-community minority subgraphs, so its MI and SE

{}_{\text{FJ}}

entries are definitional rather than substantive.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Diagnostic topologies (Track B). Rows are the three inverse pairs and two standalone diagnostics; within each inverse pair the two topologies probe a measure from opposite directions. Node color encodes opinion on a fixed scale (blue $-1$ , cream $0$ , dark red $+1$ ); layout and node size are illustrative. Construction and expected behaviour are given in Table 3 .

Topology	Construction	Isolates	Predicted behaviour	Criterion
Pair I - Structure without opinion vs. opinion without structure (probes Q, GED)
T1 Separated Consensus	Two dense communities, all $x=+0.5$	structure, no opinion	Q reports high (structural false positive); E-R and LE correctly $\approx 0$	SE
T2 Opinion-Separated Mixed	Dense E-R ( $p=0.7$ ), bimodal $\pm 1$ , edges opinion-blind	opinion, no structure	Q $\approx 0$ (miss); E-R max; LE high; GED should be nonzero	coverage
Pair II - Diversity vs. conflict (probes LE, GED)
T3 Diverse Community	One clique, $x\sim\mathcal{U}(-1,1)$	diversity, no separation	LE high (false positive from density $\times$ range); GED suppresses via its mesolevel term	coverage
T4 Disconnected Clusters	20 disconnected cliques, alternating $\pm 1$	distributional separation, no cross-edges	LE exactly $0$ (miss); E-R max	coverage

Appendix

Table 3: The eight diagnostic topologies (Track B), grouped into three inverse pairs and two standalone diagnostics. Each pair probes one measure from opposite directions; the final column records the role of each construction in the audit. Full generation parameters are given in the appendix.

Topology	Measure	Score	Expected	Direction check	Role
T1	$Q_{\text{opinion}}$	$0.0000$	ZERO	ok
T1	$Q_{\text{struct}}$	$0.4954$	HIGH	ok	MI
T1	E-R	$0.0000$	ZERO	ok
T1	$A$	-	N/A	N/A
T1	LE	$0.0000$	ZERO	ok
T1	GED	$0.0000$	ZERO	ok

Appendix

Table 4: Track B diagnostic results, reproduced from the pipeline output. Cell shading: green = score matched its theoretically expected direction; red = mismatch; amber = calibration finding (MOD/NONZERO) or undefined (N/A). Scores are on the pipeline’s normalized scale.

Topology	Measure	Score	Expected	Direction check	Role
T5	$Q_{\text{opinion}}$	$0.0303$	ZERO	MISMATCH	RS
T5	$Q_{\text{struct}}$	$0.0837$	ZERO	ok	RS
T5	E-R	$0.0241$	HIGH	ok
T5	$A$	$0.6238$	HIGH	ok
T5	LE	$0.0093$	HIGH	MISMATCH
T5	GED	$1444.8210$	MOD	MOD (baseline artifact)	RS

Appendix

Table 5: Track B diagnostic results (continued).

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