Calibrated Fusion for Heterogeneous Graph-Vector Retrieval in Multi-Hop QA

Andre Bacellar

cs.IRMar 30, 2026

Calibrated Fusion for Heterogeneous Graph-Vector Retrieval in Multi-Hop QA

Authors: Andre Bacellar

Abstract

Graph-augmented retrieval combines dense similarity with graph-based relevance signals such as Personalized PageRank (PPR), but these scores have different distributions and are not directly comparable. We study this as a score calibration problem for heterogeneous retrieval fusion in multi-hop question answering. Our method, PhaseGraph, maps vector and graph scores to a common unit-free scale using percentile-rank normalization (PIT) before fusion, enabling stable combination without discarding magnitude information. Across MuSiQue and 2WikiMultiHopQA, calibrated fusion improves held-out last-hop retrieval on HippoRAG2-style benchmarks: LastHop@10 increases from 69.1% to 71.0% on MuSiQue (15W/5L, p=0.041, n=514) and LastHop@5 from 51.7% to 53.6% on 2WikiMultiHopQA (11W/2L, p=0.023, n=491), both on independent held-out test splits. Against the official HippoRAG 2 pipeline on 2WikiMultiHopQA, calibrated fusion is ahead at LastHop@10 (+6.3pp, p<10^-3) and behind at LastHop@5 (-8.4pp), a cutoff-dependent cross-over we report in full. A theory-driven ablation shows that percentile-based calibration is directionally more robust than min-max normalization on both tune and test splits (1W/6L, p=0.125), while Boltzmann weighting performs comparably to linear fusion after calibration (0W/3L, p=0.25). These results suggest that score commensuration is a robust design choice, and the exact post-calibration operator appears to matter less on these benchmarks.

Figures & tables

Bench.	Strategy	LastHop	W	L	$p$
MuSiQue (@10)	Vector only	69.1	–	–	–
	True RRF	71.8	25	11	.029
	PhaseGraph (.7, dk30)	71.0	15	5	.041
2Wiki (@5)	Vector only	51.7	–	–	–
	True RRF	53.0	8	2	.109
	PhaseGraph (.4, dk20)	53.6	11	2	.023

Table 1: HippoRAG2 benchmark held-out test results. W/L vs. vector only on LastHop @10 for MuSiQue (

n{=}514

, terminal-hop passage) and LastHop @5 for 2Wiki (

n{=}491

). Configurations fixed on the tune split; MuSiQue test figures recomputed under the terminal-hop metric from a fresh-embedding re-run (footnote in Section 4 ).

Figure 1: Per-query last-hop outcomes: PhaseGraph vs. vector-only on 2Wiki HippoRAG2 test split (n

=

491). Left : all 491 queries; right : zoomed win/loss. The 11:2 asymmetry gives

p{=}0.022

( McNemar, 1947 ) .

Figure 2: Raw score distributions from 100 sample queries (2Wiki HippoRAG2). Vector cosine similarity and PPR scores occupy incomparable scales; direct fusion would be dominated by vector scores.

Figure 3: Effect of normalization. Top row : PIT maps both distributions to approximately uniform

[0,1]

, making them commensurable. Bottom row : min-max normalization preserves the power-law spike in PPR scores, producing unequal marginals. Dashed line: expected uniform count per bin.

Config	Axis	Tune		Test (held-out)
Config	Axis	LastHop	W/L	LastHop	W/L	$p$
Baseline (PIT + Boltzmann)	–	52.5	–	53.8	–	–
Min-max norm	Norm.	51.7	1/5	52.7	1/6	.125
Raw/max norm	Norm.	52.8	5/3	–	–	–
Linear fusion	Formula	52.5	2/2	53.2	0/3	.250
Fixed $T{=}0.3$	Temp.	52.1	2/4	–	–	–

Table 2: Theory ablation on 2WikiMultiHopQA HippoRAG2 (tune n

=

509, test n

=

491). W/L vs. baseline on LastHop @5. Test split evaluated once (held-out).

Strategy	LastHop	R@5	W	L	net
Vector only	58.1	88.0	–	–	–
Thermo .3, uncap.	51.0	78.8	36	81	$-$ 45
True RRF, uncap.	58.6	83.4	32	39	$-$ 7
Thermo .7, dk30	60.4	88.6	16	5	+11
+ Entity disambig. (synonym linking):
Vector only	60.1	89.4	–	–	–

Table 3: Full-corpus scaling (500 queries, 6,841 passages). W/L vs. vector only on LastHop @10.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Theory ablation: LastHop @5 on 2Wiki HippoRAG2 for baseline vs. normalization and fusion ablations. Bars show tune (n

=

509, lighter) and held-out test (n

=

491, solid). Annotations: W/L vs. baseline and

\Delta

LastHop on test bars.

Strategy	Recall	95% CI	$p$	OR
Vector only	39.4	[28.5, 51.5]	—	—
True RRF	74.2	[62.6, 83.3]	$2.4\text{e-}7$	4.43
Thermo	78.8	[67.5, 87.2]	$3.0\text{e-}8$	5.71
Thermo+Ising	80.3	[69.2, 88.2]	$1.5\text{e-}8$	6.27
Thermo+CE	9.1	[4.2, 18.4]	$3.6\text{e-}5$	0.15

Appendix

Table 4: LastHop @10 with Wilson 95% CIs, McNemar ( McNemar, 1947 )

p

-values (vs. vector only), and odds ratios. MuSiQue hard-slice,

n{=}66

Figure 5: Ising parameter sweep. Sharp threshold at blend

=

0.25 across all

(J,T)

pairs. Green: 27 wins, gold: 26, red gradient:

\leq

25.

Category	#Cfg	Best W	Max LastHop
Tier 1 (fusion)	102	26	78.8
Tier 2 (reranking)	277	27	80.3
Tier 3 (research)	288	27	80.3
Total	667	27	80.3

Appendix

Table 5: Configuration sweep summary across three tiers. Bold : best per column.

Config	F1 [95% CI]	EM
Vector only	.135 [.075, .203]	6.1%
Context stuff.	.183 [.099, .275]	15.2%
Thermo	.271 [.181, .369]	18.2%
Thermo+Ising	.307 [.213, .406]	21.2%

Appendix

Table 6: End-to-end QA on hard-slice (Claude Haiku 4.5, 66 queries). 95% bootstrap CIs (

n{=}10{,}000

Strategy	LastHop	W	L	$p$
Vector only	37.8	–	–	–
True RRF †	41.2	15	6	.078
Thermo (tune winner)	43.5	26	11	.020
Thermo (best recall)	44.3	24	7	.003
Thermo (safest)	43.1	15	1	$<$ .001

Appendix

Table 7: 2WikiMultiHopQA held-out test results (262 queries, isolated database, top-10). W/L vs. vector only on LastHop @10. † one query failed (n

=

261). Thermo configs: tune winner (

\alpha{=}0.4

, dk20), best recall (

\alpha{=}0.4

, dk50), safest (

\alpha{=}0.5

, dk50).

Strategy	LastHop	FullSup	W	L
Vector only	39.4	6.1	–	–
Legacy RRF	39.4	6.1	0	0
True RRF ∗∗∗	74.2	48.5	23	0
Thermo ∗∗∗	78.8	57.6	26	0
Log-linear ∗∗∗	78.8	57.6	26	0
Power mean ∗∗∗	78.8	57.6	26	0

Appendix

Table 8: Retrieval on MuSiQue hard-slice (66 queries, top-10). Bold : best; underline : tied at PIT-fusion ceiling (78.8%). ∗∗∗ :

p<0.001

vs. vector only ( McNemar, 1947 ) .

Figure 6: Fusion strategy ablation on MuSiQue hard-slice (

n{=}66

, LastHop @10 vs. vector-only). Four PIT-normalized strategies tie at the 26-win ceiling; only Ising coupled reranking breaks through to 27.

Method	all	tune	test	2-hop	3-hop	4-hop
HippoRAG 2 (pub.)	74.7	–	–	–	–	–
PropRAG (pub.)	77.3	–	–	–	–	–
Vector only	69.8	69.5	70.1	77.9	68.9	46.2
PhaseGraph (thermo)	69.8	69.6	70.0	77.9	69.0	46.2

Appendix

Table 9: MuSiQue R@5 on all 1000 HippoRAG2 benchmark queries (486 tune / 514 test). Thermo:

\alpha{=}0.7

, dk

=

30. Published baselines not directly comparable (different pipelines).

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Query-Aware Spreading Activation for Multi-Hop Retrieval over Knowledge Graphs

Illia Makarov, Mykola Glybovets

National University of Kyiv-Mohyla Academy Kyiv, Ukraine

Multi-Hop QA Graph Retrieval-Augmented Generation Graph-Based Retrieval Knowledge Graphs

Open paper AskPDF arXiv

Retrieval-augmented generation built on knowledge graphs (Graph RAG) outperforms flat passage retrieval on multi-hop question answering by leveraging graph structure. In most existing systems, however, the question only sets the seed nodes; the subsequent traversal becomes "query-blind", depending solely on the graph structure. The exception is QAFD-RAG, which implements query-aware traversal via a flow-diffusion solver with combined edge re-weighting. This architecture requires loading the full graph into Python memory and an iterative solver with a variable number of iterations complicating integration with the graph database. We propose a spreading-activation method that achieves the same query-aware traversal with a single per-step semantic gate: the step weight is the cosine similarity between the candidate entity's description and the question, and the number of iterations is fixed. The whole retrieval procedure - seed mapping, propagation, top-K selection and context assembly - is expressed as a single Cypher query executed in one round-trip to Neo4j; the graph never leaves the database. On MuSiQue our method matches QAFD-RAG by exact match (32.80 vs 33.50) and outperforms the strongest purely-structural baseline in our comparison, HippoRAG, by 5.3 EM and 3.4 F1; on 2WikiMultiHopQA HippoRAG and QAFD-RAG retain an advantage due to their phrase-node architectures. An ablation with the gate disabled confirms that the gate is the source of a simultaneous F1 gain of 3.6 to 7.4 points and a retrieval-latency reduction by a factor of 1.5 to 4.9.

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