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Graph-free Retrieval with Graph-Modified Embeddings
Federica Santi, Vasilije MarkoviÄ, Daniel Scott, Lazar ObradoviÄ
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Abstract
Abstract:Dense vector retrieval can miss evidence connected through intermediate entities, while direct graph retrieval requires access to the graph at query time. We compare five offline graph-based embedding transformations and strategies for using the resulting graph-modified embeddings alongside the originals, without query-time graph access. On a synthetic professional-network benchmark with complete supporting-evidence sets, direct replacement of the original embeddings reduces evidence recovery. Multi-stage strategies improve recovery by using graph-modified embeddings to guide the search while retaining the originals for semantic relevance. On questions requiring linked facts or comparisons among candidates, the strongest configuration improves recovery by up to 25.9% relative to plain vector retrieval. It also matches baseline recovery with 40% fewer returned nodes. Controls confirm that graph information contributes beyond multi-stage retrieval alone. Across the tested methods, retrieval performance depends more on how graph-modified embeddings are used than on how they are produced.
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- Source: https://arxiv.org/abs/2610.18410v1
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Graph-free Retrieval with Graph-Modified Embeddings Federica Santi ā thanks: Correspondence: federica@topoteretes.com. Vasilije MarkoviÄ Daniel Scott Lazar ObradoviÄ Affiliation: Cognee Inc. Abstract Dense vector retrieval can miss evidence connected through intermediate entities, while direct graph retrieval requires access to the graph at query time. We compare five offline graph-based embedding transformations and strategies for using the resulting graph-modified embeddings alongside the originals, without query-time graph access. On a synthetic professional-network benchmark with complete supporting-evidence sets, direct replacement of the original embeddings reduces evidence recovery. Multi-stage strategies improve recovery by using graph-modified embeddings to guide the search while retaining the originals for semantic relevance. On questions requiring linked facts or comparisons among candidates, the strongest configuration improves recovery by up to 25.9 % relative to plain vector retrieval. It also matches baseline recovery with 40 % fewer returned nodes. Controls confirm that graph information contributes beyond multi-stage retrieval alone. Across the tested methods, retrieval performance depends more on how graph-modified embeddings are used than on how they are produced. 1 Introduction Retrieval is a central component of retrieval-augmented generation (RAG), where external evidence is retrieved to support large language model (LLM) outputs [1]. It also supports LLM agents and long-term memory systems by identifying relevant documents, passages, or stored facts. Dense retrieval embeds queries and passages in a shared vector space and ranks candidates by vector similarity [2]. This approach is effective when supporting evidence is semantically close to the query, but may miss relevant passages connected through intermediate entities or multi-step relations. Knowledge graphs represent these connections explicitly, but using them for retrieval introduces two challenges. Traversing or reranking the graph for every query increases query-time complexity, while modifying embeddings according to graph structure may distort the semantic relationships captured by the original embeddings. This leads to the central question of this work: can graph structure be incorporated offline into graph-modified embeddings that improve retrieval without requiring graph access at query time or replacing the original embeddings? Prior work has either incorporated relational knowledge into embeddings [3, 4, 5], used graphs directly at query time [6, 7], or refined retrieval through textual feedback alone [8, 9, 10]. Our work connects these directions by examining both how graph-modified embeddings should be constructed and how they should be used in multi-stage retrieval. We construct graph-modified embeddings offline using several transformations, including local neighbourhood aggregation, random-walk endpoints, community centroids [11], and convex neighbour pull. We then compare retrieval strategies ranging from replacing the original embeddings with graph-modified embeddings to multi-stage strategies that combine both. At query time, every strategy uses only cosine-similarity searches over stored embeddings; the graph is neither traversed nor queried. Our central hypothesis is that graph-modified embeddings are more useful for guiding the expansion of retrieval than for measuring relevance to the original query. The best-performing strategies therefore assign distinct roles to the two sets of embeddings. An initial search over the original embeddings retrieves semantically relevant results (seeds). Their graph-modified embeddings are then aggregated to guide the selection of candidates that are both relevant to the query and related to the seeds through the graph. Finally, the original embeddings of the selected candidates are used to reformulate the query for another search over the original embeddings [8, 9, 10]. We refer to this use of graph-modified embeddings to guide retrieval as graph conditioning. To evaluate these strategies, we introduce a synthetic professional-network benchmark in which the corpus, graph, questions, answers, and supporting evidence are generated from the same underlying facts. The benchmark includes single-fact, relational and collection, and extremisation questions, together with exact gold answers and complete gold-evidence sets containing the intermediate and competing records required to establish each answer [12]. We select configurations on a development corpus and report results on separately generated validation data (Section 3.1). All methods use the same original embeddings, cosine similarity, and the same 40-node retrieval limit. Multi-stage controls without graph conditioning allow us to distinguish improvements produced by graph-modified embeddings from those produced by multi-stage retrieval alone. None of the graph-modified representations improves direct retrieval over the original embeddings in our experiments. The original embeddings recover 73.1%73.1\,\% of the gold evidence, whereas the five transformations recover between 52%52\,\% and 72.9%72.9\,\%. In contrast, multi-stage retrieval improves performance across all five transformations, and the choice of retrieval strategy has a greater effect than the transformation used to construct the graph-modified embeddings. The strongest configuration improves gold-evidence recovery relative to plain vector retrieval by 25.9%25.9\,\% on extremisation questions and by 10.0%10.0\,\% on relational and collection questions (19.2%19.2\,\% across the two groups). It also matches the gold-evidence recovery achieved by plain vector retrieval with 40 nodes after returning only 24, corresponding to 40%40\,\% fewer returned nodes. Contributions. ā¢ We provide a controlled comparison of multiple graph-based embedding transformations and retrieval strategies under a common cosine-similarity framework, without requiring graph access at query time. ā¢ We introduce a synthetic benchmark with questions of different relational complexity, exact gold answers, and complete question-specific supporting evidence. ā¢ We show that retrieval performance depends more on how graph-modified embeddings are used than on the particular transformation that produces them, and that they are most effective as intermediate signals within a multi-stage retriever. ā¢ We introduce a retrieval strategy in which graph-modified embeddings guide exploration while the original embeddings preserve semantic alignment with the query, and we quantify its performance. 2 Related Work Our work connects research on graph-modified embeddings, feedback-based and multi-hop retrieval, and graph-based retrieval. We also draw on evidence-level evaluation to assess whether these approaches recover the passages needed to answer a question. Encoding graph structure into embeddings. Retrofitting adapts pretrained embeddings using external relational knowledge. Faruqui et al. [3] introduced an objective that balances fidelity to the original embeddings with attraction between connected concepts. Counter-fitting adds attraction and repulsion constraints [4], while Functional Retrofitting introduces relation-specific transformations for heterogeneous knowledge graphs [5]. Retrofitting has also been applied beyond word embeddings, including multilingual sentence representations aligned through Abstract Meaning Representation [13] and sense representations specialised using WordNet relations [14]. Related approaches use entity descriptions and language models to construct representations for knowledge-graph completion, incorporating structural information through different training objectives or architectures [15, 16, 17, 18, 19]. Another line of work derives node representations from graph connectivity. DeepWalk learns embeddings from contexts generated by truncated random walks [20], while node2vec uses biased walks to capture different network neighbourhoods [21]. These methods are conceptually related to our random-walk transformation, which instead averages existing textual embeddings at walk endpoints without learning new representations from walk sequences. Simple Graph Convolution separates graph-based feature propagation from a learned classifier [22], providing a related perspective on our neighbourhood-based aggregation of existing embeddings. We draw on these ideas to study graph-modified embeddings for evidence retrieval, rather than intrinsic similarity, node classification, or knowledge-graph completion. In particular, we compare their use as direct search representations with their use as intermediate signals in a multi-stage retriever. Feedback-based and multi-hop dense retrieval. Pseudo-relevance feedback refines a query using the results of an initial retrieval stage and then performs a second search. NPRF (neural pseudo-relevance feedback) brought the idea into learned ranking, feeding the top-ranked documents of a first pass into a neural ranking model [8]. More recent approaches apply the same principle to dense retrieval, where the query is a vector rather than a set of terms, and differ in what that vector is allowed to be. ColBERT-PRF keeps a multi-vector query and extends it, extracting representative embeddings from the pseudo-relevant documents and adding them to the query representation before re-retrieval [9]. ANCE-PRF keeps a single query vector but learns an encoder that consumes the initially retrieved passages, leaving the document index unchanged [10]. Multi-hop retrieval extends this sequential approach to questions whose supporting evidence cannot be identified from the original query alone. Multi-Hop Dense Retrieval conditions successive searches on previously retrieved passages to recover chains of supporting evidence [23]. Other approaches use generated text to guide subsequent retrieval. IRCoT interleaves retrieval with intermediate chain-of-thought reasoning steps [24], while Iter-RetGen alternates retrieval and generation, using each generated output to inform the next retrieval iteration [25]. Our method uses offline graph-modified embeddings to guide intermediate retrieval, followed by a Rocchio-style update that blends the query with the mean embedding of retrieved nodes in the original space [26], without query-time graph access or LLM generation. Controls without graph conditioning (Section 4.5.2) isolate the contribution of graph conditioning. Graph-based retrieval. Graph-based systems use explicit relations to find evidence that semantic similarity may miss, such as education records linked to a company through its employees. GRAFT-Net reasons over question-specific subgraphs of facts and text [6]; PullNet expands them through iterative retrieval [7]; and HippoRAG propagates relevance through entity connections using Personalized PageRank [29]. These systems use the graph during query processing, whereas we encode its relations into embeddings offline and retrieve evidence through stored vectors alone. GraphRAG uses community summaries to answer global questions about a corpus [30]. Although related to our use of graph communities, it produces text for corpus-level synthesis, whereas we produce node-level vectors for retrieving question-specific evidence under a fixed budget. Evidence-level evaluation. Multi-hop question-answering benchmarks support evaluation beyond final-answer accuracy by identifying the evidence needed for reasoning. HotpotQA provides sentence-level supporting facts alongside questions and answers, including both bridge and comparison questions [27]. MuSiQue constructs multi-hop questions by composing single-hop questions whose reasoning steps depend on one another, explicitly addressing shortcuts that allow apparently multi-hop questions to be answered without combining the required evidence [28]. Ground-truth-subgraph benchmarks such as SynthKGQA and GTSQA further support the evaluation of graph retrieval independently of answer generation [12]. We adopt this evidence-centred perspective: our synthetic corpus and questions are generated from the same structured facts, enabling us to specify question-specific gold-evidence sets and measure their recovery directly. The benchmark includes relational and collection questions and extremisation questions, allowing us to examine retrieval of both indirectly connected evidence and the competing records required to establish an extremum. 3 Experimental Setup We evaluate how graph-based embedding transformations and retrieval strategies affect evidence recovery on a synthetic professional-network benchmark. 3.1 Dataset and Knowledge Graph We use separate development and validation corpora, generated by the same pipeline and containing synthetic resumes and organisational personnel rosters. Each corpus is represented as a tripartite graph with three node types: documents, chunks (textual evidence from those documents), and entities (people and organisations). Three relation types connect them: part_of links chunks to documents, about links documents to their primary entities, and mentions links chunks to entities referenced in their text. The schema admits only these three edge patterns. Each corpus contains approximately 1,200 chunks, 370 entities, 230 documents, and 3,700 edges. The development and main validation corpora are node-disjoint: none of the 1,759 node identifiers in the validation corpus occurs in the development corpus. We use Cognee11 1 An open-source AI memory framework combining knowledge graphs and vector search: https://github.com/topoteretes/cognee. to ingest and store the generated graph and manage its associated node embeddings. We select configurations on the development corpus and report results on the validation set of 977977 questions, including 9090 generated on a supplementary corpus from the same pipeline. Appendix A provides generation details, exact validation statistics, and the schema with an example (Figure 10). 3.2 Questions and Gold Evidence We generate questions and ground-truth (gold) answers from the structured facts underlying the dataset. Gold evidence includes every chunk required to establish an answer. The benchmark contains 11 question families, detailed in Appendix A. We group them into three categories: Single-fact questions: These ask for attributes directly associated with entities mentioned in the query and generally require a single evidence chunk or an immediate relation. For example, āWhat skills did a person use at a given employer?ā can be answered using a job chunk that mentions the person, employer, and skills. Relational and collection questions: These ask for attributes or entities connected to a named entity through intermediate entities. For example, āWhich universities were attended by the employees of an organisation?ā requires evidence linking the organisation to its employees and those employees to their universities. The target chunks may lie two or three relations from the named entity and share little surface content with the query. Extremisation questions: These compare candidates to identify a maximum and may then request another attribute of the selected candidate. Establishing the maximum requires the relevant records of all competing candidates. For example, āWhat certifications does the most experienced worker at a given company hold?ā requires experience records for the comparison and the winnerās certification evidence, which may be semantically unrelated to the comparison criterion. 3.3 Graph-Based Embedding Transformations We modify the original embeddings offline to incorporate information from related nodes in the graph. The five transformations differ in which nodes influence each modified vector and how their contributions are combined (Figure 1). One-hop neighbour blendTwo-hop neighbour blendConvex neighbour pullRandom-walk endpoint blendLouvain cluster blendthe node being modifiedsimilar to itnot similaraveraged inweighs morecommunity centroid Ī¼ the node lands Figure 1: Graph information used by the five transformations. Darker nodes are more similar to the source node; orange outlines mark embeddings included in an average, with thicker outlines indicating greater weight. Dashed regions and crosses show Louvain communities and their centroids. Original embeddings encode chunk text or, for document and entity nodes, their names. We embed nodes and queries with OpenAIās text-embedding-3-small, using 1,5361,536 dimensions. Each transformation produces one modified vector per node. We denote the original and modified embeddings of node i by ix_i and iz_i, respectively. Appendix B lists all evaluated transformations and hyperparameter configurations. One-hop neighbour blend (hop-1). For each node i, we consider its one-hop neighbours. We define SiS_i as the 20%20\,\% of these nodes with the lowest cosine similarity to the original embedding ix_i, selecting at least two whenever available. The modified embedding iz_i combines the original vector with the mean of the original embeddings of the selected neighbours: i=Ī±āi+(1āĪ±)ā1|Si|āājāSij,z_i= _i+(1-Ī±) 1|S_i| _jā S_ix_j, (1) where Ī±=0.2Ī±=0.2. Selecting the least semantically similar one-hop neighbours is intended to introduce graph-related information that is not already encoded in the source representation. Two-hop neighbour blend (hop-2). This method uses the same selection and blending procedure, applied to nodes at graph distance exactly 22. Convex neighbour pull. Let Z collect the modified node embeddings and E denote the graphās edge set. We minimise the classical retrofitting objective [3] with uniform anchor and edge weights: Jā”()=āiā„iāiā„22+ā(i,j)āEā„iājā„22.J(Z)= _i _i-x_i _2^2+ _(i,j)ā E _i-z_j _2^2. (2) The first term anchors each node to its original embedding, while the second pulls adjacent nodes closer together in the modified space. The anchoring term makes the objective strictly convex, ensuring a unique global minimiser to which the iterative optimisation procedure converges. Random-walk endpoint blend (walk). This transformation launches 100 random walks of exactly two steps from each node. The modified embedding is computed as the frequency-weighted mean of the endpointsā original embeddings. Louvain cluster blend. This transformation partitions the unweighted, undirected graph using Louvain community detection at resolution 0.50.5; we refer to Blondel et al. [11] for details of the community-construction algorithm. Each node i is then moved towards the centroid cā”(i) Ī¼_c(i) of its community cā”(i)c(i): i=Ī±āi+(1āĪ±)ācā”(i).z_i= _i+(1-Ī±) Ī¼_c(i). (3) Each centroid is the mean of its communityās original embeddings and remains fixed. The selected configuration uses Ī±=0.5Ī±=0.5, assigning equal weight to the nodeās original embedding and its community centroid. 3.4 Retrieval Strategies We compare retrieval strategies that use the original and graph-modified embeddings separately or together. These range from direct searches in either space to combinations of independent rankings and multi-stage searches guided by earlier results. We hold the output at 4040 distinct nodes to compare evidence recovery under the same retrieval budget (Figure 2). Plain baselineOne modified collection aloneAlternating budget splitTwo-stage fused retrievalSingle requeryChained requery Figure 2: Allocation of the 40-node retrieval budget. Each cell represents one returned node. Light blue and pink denote original- and modified-space searches; orange denotes fused scoring. The two darker blues denote successive searches with shifted queries in the original space. Vertical lines separate stages. All strategies use cosine similarity over stored embeddings of all node types, without graph access at query time. Later stages exclude previously returned nodes. Appendix C lists all evaluated methods and configurations. Direct retrieval. The first two rows of Figure 2 each use a single embedding collection. The plain baseline returns the nodes whose original embeddings are most similar to the query embedding q. One modified collection alone applies the same search to graph-modified embeddings. This comparison tests whether incorporating graph structure into the vectors makes them more useful for direct query matching. Alternating budget split. This strategy gives both spaces a share of the output. We search them independently using q and alternate between their rankings, skipping duplicate node identifiers. Two-stage fused retrieval. Here, the initial results guide the search for further evidence. The first stage retrieves 20 nodes (seeds) from the original space using q and computes m as the mean of their modified embeddings. The second stage pools candidates from an original-space search with q and a modified-space search with m, then ranks each candidate node n by the fused score s(n,Ī±)=(1āĪ±)cos(,n)+Ī±cos(,n).s(n,Ī±)=(1-Ī±) (q,x_n)+Ī± (m,z_n). (4) The fusion weight Ī±=0.5Ī±=0.5 gives equal weight to similarity to the original query and similarity to the seeds in the modified space. The 20 highest-scoring remaining nodes complete the output. Single requery. This strategy uses the fused-stage results to refine the query for another search in the original space. It retrieves 16 seeds, selects 12 nodes using the fused score in Equation (4), and averages the latterās original embeddings to obtain ĀÆ x. The shifted query ā²=0.7ā+0.3āĀÆq =0.7q+0.3 x (5) retrieves the final 12 nodes, giving a 16/12/1216/12/12 allocation. Chained requery. This strategy adds a second query update, so the first shifted search supplies evidence for the next. It uses 6/6/14/146/6/14/14 nodes for seed retrieval, fused scoring, and two shifted-query searches in the original space. Each update uses the mean original embedding of the preceding stageās results. Both use Equation (5) with the original query q; only the mean ĀÆ x changes. Controls without graph conditioning. To separate the contribution of graph conditioning from that of query shifting, we compare each requery strategy with two controls that change only its second stage. One sets the fusion weight in Equation (4) to Ī±=0Ī±=0; the other retains Ī±=0.5Ī±=0.5 but uses original embeddings for both the seed mean m and the candidates. Section 4.5.2 presents this comparison. 4 Experimental Results We first test graph-modified embeddings in direct retrieval, then compare strategies that combine original and modified representations. Finally, control experiments assess the contribution of graph conditioning. For each question, gold-evidence recovery is the fraction of its gold chunks returned; reported percentages average this fraction equally across questions. Document and entity nodes consume retrieval budget but do not count as recovered evidence. 4.1 Direct Retrieval with Graph-Modified Embeddings We compare the original embeddings with the five graph-modified collections as direct search representations across all 977977 validation questions. 0010102020303040405050606070708080Two-hop neighbour blendOne-hop neighbour blendConvex neighbour pullRandom-walk endpoint blendLouvain cluster blendOriginal embeddings52.452.458.558.571.671.671.971.972.972.973.173.1Gold chunks found (%) Figure 3: Gold-evidence recovery under direct retrieval across all 977977 validation questions. The dashed line denotes plain vector retrieval. All five transformations recover fewer gold chunks than the original embeddings, with the largest losses observed for the one-hop and two-hop neighbour blends. 4.2 Comparison of Retrieval Strategies Plain retrieval already recovers 98.9%98.9\,\% of the gold evidence on the 450450 single-fact questions, leaving little room to distinguish strategies. Unless otherwise stated, the remaining comparisons therefore focus on the 527527 difficult questions in the other two groups. Figure 4 compares the five embedding transformations across retrieval strategies. OnealoneSplitFusedSinglerequeryChainedrequery2020 %3030 %4040 %5050 %6060 %Gold chunks found (%)convexLouvainwalkhop-2hop-1Plain baseline Figure 4: Gold-evidence recovery across retrieval strategies on the 527527 difficult questions. The dashed line denotes plain retrieval. Performance varies most across embedding transformations under direct retrieval and budget splitting. Fused retrieval and requery outperform the plain baseline with all five transformations, with smaller differences between them. 4.3 Alternating Budget Split The alternating split assigns half of the retrieval budget to each embedding space. In the group-level plots, error bars show 95%95\,\% confidence intervals for changes relative to the baseline, estimated from 10,00010,000 paired bootstrap resamples of questions. 00252550507575100100hop-1convexLouvainplain baselinewalkhop-241.441.448.648.650.750.752.352.355.155.151.051.0Gold chunks found (%) (a) Gold-evidence recovery under the alternating budget split for each graph-modified embedding collection. The dashed line denotes the performance of plain vector retrieval. ā10-100010102020Extremisation59.6 %Relational andcollection42.9 %ā10.4-10.4-17.4 %+17.9+17.9+41.6 %Ī vs. the baseline (points) (b) Effect of the hop-2 alternating budget split across question groups. Baseline recovery is shown below each group label; bar annotations report the corresponding absolute and relative changes. Figure 5: Alternating budget split on the 527527 difficult questions. The alternating budget split reveals a clear trade-off. Panel (a) shows that only the walk and hop-2 representations outperform plain retrieval, with hop-2 performing best. However, Panel (b) shows that its gain is uneven: relational and collection questions improve by 41.6%41.6\,\%, whereas extremisation questions decline by 17.4%17.4\,\% relative to their baselines. The modified space helps reach semantically distant but graph-connected evidence, while assigning half of the budget to it reduces the broad candidate coverage required for extremisation. 4.4 Two-Stage Fused Retrieval We next test whether letting the original-space results guide the second stage yields more consistent gains than the independent budget split. 00252550507575100100plain baselineconvexhop-1Louvainhop-2walk54.054.054.554.555.455.455.655.656.056.051.051.0Gold chunks found (%) (a) Gold-evidence recovery of the two-stage fused retriever across graph-modified embedding collections. The dashed line denotes plain vector retrieval. ā5-500551010Extremisation59.6 %Relational andcollection42.9 %+1.3+1.3+2.2 %+7.6+7.6+17.6 %Ī vs. the baseline (points) (b) Effect of two-stage fused retrieval with the hop-2 representation across question groups. Baseline recovery is shown below each group label; bar annotations report absolute and relative improvements. Figure 6: Two-stage fused retrieval on the 527527 difficult questions. Two-stage fused retrieval improves over the baseline with every graph-modified collection, recovering 54.0%54.0\,\% to 56.0%56.0\,\% of the gold evidence against 51.0%51.0\,\% for plain retrieval. The narrow range suggests less sensitivity to the choice of transformation than under budget splitting. The walk representation performs best, gaining 5.05.0 percentage points (9.8%9.8\,\% relative). Panel (b) again uses hop-2 to compare directly with the alternating split. Its gain is concentrated on relational and collection questions: 7.67.6 points (17.6%17.6\,\% relative). The improvement on extremisation questions is small and statistically uncertain. 4.5 Requery Strategies and Graph Conditioning 4.5.1 Performance of Requery Strategies We examine whether using the fused-stage results to refine the query improves recovery beyond the two-stage schedule. 00252550507575100100Louvainconvexhop-1hop-2walkplain baseline56.356.3 (++10.410.4 %)56.756.7 (++11.211.2 %)57.557.5 (++12.712.7 %)58.158.1 (++13.913.9 %)58.658.6 (++14.914.9 %)55.955.9 (++9.69.6 %)58.258.2 (++14.114.1 %)59.159.1 (++15.915.9 %)59.259.2 (++16.116.1 %)60.860.8 (++19.219.2 %)51.051.0Gold chunks found (%)single requerychained requeryplain baseline Figure 7: Gold-evidence recovery of the single requery and chained requery schedules across graph-modified embedding collections. The dashed line denotes plain vector retrieval. Chained requery with the walk representation performs best, recovering 60.8%60.8\,\% of the gold evidence against 51.0%51.0\,\% for plain retrieval, a gain of 9.89.8 percentage points (19.2%19.2\,\% relative). The walk representation also performs best with single requery. Unlike the alternating split, this configuration improves both difficult question groups (Figure 8). Recovery increases by 15.415.4 points on extremisation questions (25.9%25.9\,\% relative) and by 4.34.3 points on relational and collection questions (10.0%10.0\,\% relative). 0055101015152020Extremisationquestionsbaseline 59.6 %Relational andcollectionbaseline 42.9 %+15.4+15.4 (+25.9 %)+4.3+4.3 (+10.0 %)Ī gold-evidence recovery vs. the plain baseline (points) Figure 8: Improvement of chained requery with the walk representation over plain retrieval, reported by question group. Annotations show absolute and relative gains. 4.5.2 Contribution of Graph Conditioning The fused stage selects the nodes used for subsequent query shifts, so graph conditioning can affect later searches as well as the second-stage results. We compare chained requery with two controls using the same 6/6/14/146/6/14/14 schedule. The first sets the fusion weight in Equation (4) to Ī±=0Ī±=0, reducing the second-stage score to cosā”(,n) (q,x_n). The second uses original embeddings for both seeds and candidates: 0.5ā cosā”(,n)+0.5ā cosā”(,n),0.5Ā· (q,x_n)+0.5Ā· (m,x_n), (6) where m is the mean original embedding of the first-stage results. These controls measure the benefit of the multi-stage schedule and query reformulation without graph conditioning. 661212202026263333404020203030404050506060234Nodes returnedGold chunks found (%)plain baselinecontrol (Ī±=0Ī±=0)control (original space)chained requery (walk) Figure 9: Cumulative gold-evidence recovery over the first k nodes returned in stage order, without global reranking. The full chained requery method uses the walk representation; both controls retain its 6/6/14/146/6/14/14 schedule but remove graph conditioning. Vertical dotted lines mark stage boundaries. The horizontal dashed line shows plain retrievalās recovery at 40 nodes. At the full budget, the controls recover 54.7%54.7\,\% and 55.5%55.5\,\% of the gold evidence, compared with 51.0%51.0\,\% for plain retrieval (Figure 9). Original-space fusion provides only a small improvement over query-only scoring in this configuration. The full method reaches 60.8%60.8\,\%, exceeding the stronger control by 5.35.3 points. Graph conditioning thus contributes slightly more than half of its 9.89.8-point gain over plain retrieval. The advantage emerges during the fused stage and persists through later searches. After twelve returned nodes, recovery is 39.6%39.6\,\%, against 31.4%31.4\,\% and 32.8%32.8\,\% for the controls. The full method also matches plain retrievalās 40-node recovery after returning only 24 nodes, or 40%40\,\% fewer returned nodes. 5 Conclusion Our results show that graph structure can improve evidence retrieval without graph access at query time. Graph-modified embeddings guide the search for additional evidence, while the original embeddings maintain semantic relevance to the query. This supports developing graph-modified embeddings specifically for multi-stage retrieval and evaluating their contribution to the complete retrieval process. Retrieval should adapt as evidence accumulates. The question and earlier results should guide when to use graph information, how to allocate the remaining search budget, and when enough evidence has been found to support an answer. Future evaluation should test the approach on natural corpora, additional embedding models, and graphs with missing or incorrect relations. It should also establish whether smaller retrieved contexts offset the cost of additional searches. The broader goal is to design retrieval with the final answer in mind, using semantic similarity and graph structure together to provide the evidence needed within practical computational limits. CRediT Author Statement Federica Santi: Conceptualization, Data Curation, Formal Analysis, Investigation, Methodology, Software, Validation, Visualization, WritingāOriginal Draft, WritingāReview & Editing. Vasilije MarkoviÄ: Conceptualization, Funding Acquisition, Project Administration, Resources, Supervision. 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The formulation of each embedding transformation, its parameter grid, and the implementation choices the main text states only in outline. Appendix C ā Retrieval strategies. The same for the retrievers: the stage schedules, their quotas, and the parameters that distinguish them. Appendix D ā Per-family results. Results for the six harder question families separately, whereas Section 4 reports them pooled into groups. Appendix A Graph Construction and Evaluation Dataset A.1 Graph Construction The corpus is a fully synthetic professional network of fictional candidates, organisations, resumes, employer rosters, and textual evidence. Generation is deterministic under a fixed seed and does not involve a language model: every fact is sampled from predefined source pools and then rendered into text by fixed templates. Because the graph is built before any question exists, no answer information can enter its construction. Candidate identity. A candidateās name is a firstālast pair drawn from two flat pools of 512 and 458 entries. A used_names set rejects any repeat and the draw is retried, so every person in the dataset has a distinct full name before the identity-collision checks performed at ingest. With roughly 2.3Ć1052.3Ć 10^5 available combinations and a few hundred candidates per graph, rejection is rare. Uniqueness is what allows the human-readable entity:person:<Name> identifier to serve as the canonical key: two candidates sharing a name would otherwise collapse into a single node and silently merge their evidence. What is generated. Each candidate is assigned a primary sector sampled uniformly from eleven, which governs certifications and supplies the default source for employer and role attributes. A candidate receives one or two education records, between one and four jobs, zero to two projects per job, zero to three certifications, and zero to two endorsements. Education degrees are sampled with weights 50%50\,\% Bachelorās, 35%35\,\% Masterās, and 15%15\,\% PhD, and universities come from a global pool that is not sector-scoped. Project descriptions reuse a subset of the skills attached to their job, so project evidence is correlated with the job rather than sampled independently. Controlled cross-sector bridges. With probability 0.20.2 a jobās role attributesātitle, skills, responsibilities, tenure, salary band, team size, and project nameāare drawn from a different sector, selected through a fixed affinity map, while the employer remains tied to the candidateās primary sector. This prevents the graph from collapsing into isolated sector-specific clusters and produces known bridges between otherwise sector-oriented regions, such as technology roles inside healthcare organisations. Dual-perspective evidence. Employment information is represented from two document perspectives. Resume-side chunks describe one personās education, jobs, projects, endorsements, and certifications. After all candidate histories are generated, a second pass groups jobs by employer: any employer with at least two employees receives a roster document and a roster chunk listing every employee and title, which mentions the organisation and each employee individually. The same employment fact therefore appears both in a personās job chunk and in an employerās roster chunk, so information about one entity is distributed across documents connected only through shared entities. This is what makes the corpus suitable for graph-retrieval experiments rather than reducible to independent flat documents. Schema. The graph is a plain JSON dictionary with three node types and three edge types. Every node exposes an embed_field naming the field embedded at ingest: name for entities and documents, text for chunks. The graph itself is therefore not embedded during generation. Category Type Semantics Node entity Person or organisation Node document Resume or roster Node chunk Textual evidence Edge part_of Chunk ā document Edge about Document ā entity Edge mentions Chunk ā entity For each person the generator creates an entity:person:<Name> node and a document:resume:<Name> node joined by about. Every education, job, project, endorsement, and certification record becomes a chunk, linked to the resume by part_of and to the person by mentions. Education chunks also mention a university entity and job and project chunks an employer entity; organisation nodes are created on first use and reused thereafter. Identifiers are deliberately human-readableāentity:person:<Name>, chunk:job:<Name>:<i>āand these are the canonical evaluation identifiers, preserved downstream in the vector store and in the result artefacts. The schema admits exactly three edge patterns and no others, so the graph is tripartite. One hop from a chunk reaches the entities it mentions and the document it belongs to; two hops return to other chunks through a shared entity. Figure 10 draws the schema and one instance from the validation corpus; the recurring chunkāentityāchunk route it shows is what the two-hop neighbour blend exploits. chunkdocumententitypart_of1,171about227mentions 2,267 āManish Henderson worked as a Social Services Manager at Eastgate Public Works Departmentā¦ā āAna Ramos worked as a Litigation Support Specialist at Eastgate Public Works Departmentā¦ā document:resume:Manish Hendersonentity:person:Manish Henderson entity:org:Eastgate Public Works Department part_ofaboutmentionsmentionsmentionsthe two coloured edges are one two-hop path: a chunk, a shared entity, another chunk Figure 10: The graph schema and one instance from the validation corpus. Shape marks the node type: a circle is a chunk, a rounded square a document, a square an entity. Counts on the schema edges are those of the validation graph. In the instance, a resume document is about a person, its chunks are part_of it, and each chunk mentions the entities named in its text. Because mentions always runs from a chunk to an entity, two hops lead from one chunk to another through an entity they share ā here two employees of the same organisation. Structured facts and gold supervision. Every generated fact is retained in two synchronised forms. Graph edges encode its structural relationships, and a human-readable rendering is stored in the corresponding chunk; the originating structured dictionary is preserved unchanged on that chunk as source. Text is therefore the representation exposed to embedding and retrieval, while source is the authoritative representation used to compute gold answers and supporting evidence. Question generation reads the structured copy, so exact gold evidence is obtained without an LLM reconstructing facts from prose. Determinism. The generator seeds Pythonās global random module once with SEED=42SEED=42 and proceeds strictly sequentially. The one hash-order-sensitive operationāthe union of skills across a candidateās jobs, used to sample endorsementsāis sorted before sampling, so process-dependent string hashing cannot affect the output. Under the same configuration and source pools the resulting graph.json is byte-identical across runs and machines, a property checked by tests/test_generation_determinism.py under different interpreter hash seeds. Development, validation, and test graphs. A separate generation mode partitions the source pools themselves into disjoint development, validation, and test vocabularies using SPLIT_SEED=7SPLIT\_SEED=7. Names, attributes, employers, and universities are split before the three graphs are generated, in the order development, validation, test, from a single seeded stream and a single shared used_names set. No person name is therefore repeated across the three graphs, and the later graphs are not a mirrored copy of the earlier draws. Hyperparameter selection, generalisation checks, and final reporting do not share the same underlying vocabulary, and the corpora are node-disjoint by construction rather than by filtering. Validation corpus. The validation graph used throughout this paper contains 1,759 nodes and 3,665 edges. Component Count Composition Entities 361 133 persons, 228 organisations Documents 227 133 resumes, 94 rosters Chunks 1,171 326 project, 319 job, 195 education, 133 endorsement, 104 certification, 94 roster mentions edges 2,267 chunk ā entity part_of edges 1,171 chunk ā document about edges 227 document ā entity Each node carries an original embedding in ā1536R^1536, and every modified collection preserves the same node identities, so a retrieved identifier means the same thing in every collection. A.2 Questions and Gold Answers Each question is paired with an exact answer and with the evidence required to derive it. Generation is deterministic under a fixed seed and uses no language model: Python functions traverse the graph, read the structured fact preserved in a chunkās source field, instantiate a fixed template, and record the nodes touched along the way. The generated chunk text is never parsed to recover an answerātext is what retrieval sees, while the structured record defines the ground truth. The stored record. Every generator returns the triple (q,Cq,aq)(q,C_q,a_q), where q is the natural-language question, CqC_q is the set of graph nodes touched while deriving the answer, and aqa_q is the answer computed from structured facts. Before persistence CqC_q is filtered to the evidence-bearing chunks, giving the gold-evidence set GqG_q, so each stored example is (q,aqgold,Gqgold)(q,a_q^gold,G_q^gold). Why only chunks count as targets. Filtering keeps only nodes whose type is in GOLDEN_NODE_TYPES = ("chunk",). Entities named in the question are query inputs, not retrieval targets: the retriever already has them. What it must recover is the text that carries the evidence. Rewarding a system for returning an entity the question itself names would measure nothing. Filtering happens at record-writing time only, and changes neither the question text, nor the computed answer, nor the sampling or reasoning depth. Question families. Eleven families span basic, compositional, and multi-hop reasoning. Basic families retrieve or aggregate local attributes, compare candidates, or select extrema; compositional families chain two basic operations and union their evidence; multi-hop families traverse explicit personāorganisationāuniversity relations over one, two, or three hops, alternating between two predefined chains at each level. Because every traversal step is recorded, the benchmark keeps the whole supporting chain and not only the node holding the final answer. Family Main operation Val. collect_local Read one attribute for one entity or entityāorganisation pair 90 collect_global Aggregate one attribute across all people linked to an organisation 90 comparison Compute a set intersection for two people 90 extremize Select an argmax over job facts, optionally filtered and grouped 77 extremize_collect Find an extremal person, then retrieve a second attribute 90 collect_extremize Restrict to an organisationās people, then apply an extremum 90 collect_global_collect Traverse through an organisation and retrieve a second attribute for each linked person 90 multi_collect Retrieve two attributes for one person in one question 90 hop1 One-hop organisation/person or university/person reasoning 90 hop2 Two-hop chains crossing entity types 90 hop3 Three-hop chains with aggregation or peer discovery 90 Validation set 977 One question of each family, taken from the validation set. Family Example question collect_local āWhat accreditations does Ethan Ito have?ā collect_global āWhat degrees has Union College awarded?ā comparison āWere Naomi Das and Emily Hong ever at the same company?ā extremize āWhich candidate held a single position for the most years?ā extremize_collect āWhich credentials has the most experienced grant management practitioner earned?ā collect_extremize āWho worked at Inkerman & Price for the most years?ā collect_global_collect āWhat certifications do employees of Juniper Health hold?ā multi_collect āWhat was Rajan Cookās tenure and what skills did they use at Xcel Freight Co?ā hop1 āWho attended University of Kentucky?ā hop2 āWhere did people who worked at Jettison Labs study?ā hop3 āWhich employer of Yale University graduates has the most total employees?ā Configuration-driven generation. Most basic families are parameterised rather than implemented separately. A configuration names the relevant chunk and entity prefixes, whether an organisation is required, one or more surface templates, and an answer_fn that extracts the target value from source. A generic routine resolves the linking chunk, applies the extraction function, and fills a template. Question semantics are therefore separated from traversal logic, and a new subtype costs a configuration entry rather than new generation code. Multi-hop generators are the exception: each chain is implemented directly, because successive hops cross different entity types. Sampling and budget. Candidate tuples come from type-specific samplersāpersonāorganisation, person pair, organisation, extremisationāeach relying on seeded shuffling. Families with several subtypes cycle through their configurations so that consecutive questions alternate subtype until the budget is met or the candidate pool is exhausted. The nominal budget is divided evenly across the eleven families, so a request for 1,000 questions asks for 90 of each. Two families cannot fill that quota, and the reason is structural rather than incidental. extremize realises 77 on the validation graph, because an argmax needs a competition to resolve. collect_global_collect realises a single question, because some of its subtypes require one organisation to satisfy more than one semantic role at once, and few organisations in a single graph do. The validation set reported in this paper therefore takes the 888 questions realised on the validation graph and completes the under-filled family with 90 collect_global_collect questions generated on a second corpus from the same pipeline, for 977 in total. Gold-evidence set size. Gold-evidence set size reflects the number of independent facts needed to establish an answer, not where the answer value happens to sit. A local extraction is usually one chunk: q0000 asks how long a person worked at an organisation and its gold-evidence set is the single job chunk holding that record. Aggregation scales with the organisation: a question about tenure lengths across a companyās staff carries one job chunk per employee. An argmax carries its competitorsāq0180 asks who is most experienced in a given skill and its gold-evidence set is the 15 job chunks that mention it, because the maximum cannot be verified without them. Compositional questions union the two stepsā evidence, and three-hop questions produce the largest sets, up to 15 chunks spanning employer-to-employee, employee-to-university, and university-to-peer links. Determinism and materialisation. Sampling and template choice draw from Pythonās global random generator, seeded once before generation. Each question receives a stable identifier of the form q0000, and questions.jsonl stores one deterministic JSON record per question: answer sets are serialised as sorted lists and gold-evidence node identifiers are sorted, so downstream evaluation joins on persistent identifiers rather than on text order or container iteration. Question embeddings are held in a separate vector collection, produced by the same embedding engine used for the graph-node collections, so query and document vectors share an embedding model by construction. Questions are stored as a plain model rather than as a graph data point, which keeps them out of any retrieverās default collection scan. Appendix B Retrofitting Strategies: Formulations and Implementation Details The following sections consolidate the recorded implementation and parameter choices. Counts refer to generated collections, not all combinations with all retrievers. Unless otherwise stated, the blending coefficient retains the original representation with weight Ī±. B.1 One-Hop Neighbour Blend hop1_neighbor_blend averages a subset of the nodeās direct neighbours and mixes that mean with the original vector. Parameter Meaning Values neighbor_pool_share Fraction of available direct neighbours used 0.2, 0.5, 1.0 alpha Weight retained from the original vector 0.0, 0.2, 0.5 neighbor_pick Select least- or most-similar eligible neighbours least_similar, most_similar min_neighbors Minimum selected count when the fraction is small 2 (fixed) The share is degree-relative: at 20 %, a node with 60 neighbours uses 12, while a node with five uses the minimum of two. A full Cartesian grid would contain 18 combinations. At share 1.0, however, both selection orders include the entire pool and produce byte-identical outputs; only one copy is retained, leaving 15 collections. B.2 Two-Hop Neighbour Blend hop2_only_neighbor_blend uses nodes at graph distance exactly two, excluding the source and its direct neighbours. It uses the same share, Ī±, selection, and minimum-neighbour grid as the one-hop neighbour blend. The additional parameter neighbor_vectors is fixed to original: the method always averages the original embeddings of two-hop nodes. Blending already-modified neighbour vectors was deliberately not implemented because its interaction with share-based selection was not specified. Along chunkāentityāchunk paths, two hops often map a chunk to another chunk through a shared entity. The method therefore captures other texts that discuss the same entities rather than the entities themselves. B.3 Convex Neighbour Pull This anchored neighbour-attraction formulation is related to retrofitting [3], applied here to corpus-node embeddings. convex_neighbor_pull jointly adjusts node embeddings by balancing fidelity to their original textual representations against agreement between graph neighbours. A useful physical interpretation is a system of springs: each node is attached to its original position and to every adjacent node. The modified embeddings are the equilibrium of this coupled system. Objective and uniqueness. Let ix_i be the original embedding and iz_i the unknown modified embedding. Let E contain unordered edge pairs, each counted once, and let Nā”(i)N(i) be the corresponding neighbour set. The objective is Jā”()=aāāiāiāiā22+bāā(i,j)āEāiājā22.J(Z)=a _i\|z_i-x_i\|_2^2+b _(i,j)ā E\|z_i-z_j\|_2^2. (7) Here a=anchor_weight>0a= anchor\_weight>0 and b=edge_weightā„0b= edge\_weightā„ 0. We use a rather than Ī± to distinguish this penalty weight from the blending coefficient used by other transformations. Squared Euclidean distances yield a differentiable quadratic objective with linear stationarity equations. The positive anchor term makes the objective strictly convex, so its minimiser is unique. The implementation rejects nonpositive anchor weights. With a=0a=0 and b>0b>0, any embedding configuration constant within each connected component has zero energy; the minimiser is no longer unique. Closed-form coordinate update. Setting the derivative with respect to one node to zero gives aā”(iāi)+bāājāNā”(i)(iāj)=0,a(z_i-x_i)+b _jā N(i)(z_i-z_j)=0, (8) and hence the weighted-average update iāaāi+bāājāNā”(i)ja+bā|Nā”(i)|.z_iā ax_i+b _jā N(i)z_ja+b|N(i)|. (9) The original vector has weight a, while each neighbour has weight b. Thus, total graph influence grows with degree: the denominator is a+3āba+3b for three neighbours and a+60āba+60b for sixty. An isolated node remains at its original vector. Although each coordinate update has a closed form, the global solution requires solving the coupled system. GaussāSeidel solver. The solver initialises Z with the original embeddings and visits nodes in sorted identifier order. Each update is written immediately, so later nodes in the same sweep use the newest available neighbour vectors. This is GaussāSeidel iteration, rather than a synchronous Jacobi update based entirely on the previous sweep. It requires no learning rate or gradient clipping, although the objective weights and stopping controls remain configuration choices. For this anchored quadratic, convergence is to the same unique minimiser; finite-sweep results and convergence speed can depend on update order. Sorted identifiers ensure a reproducible order. Stopping rule. During sweep t, the solver records the largest Euclidean displacement over all node updates, Ī“t=maxiā”āiafterāupdateāibeforeāupdateā2. _t= _i\|z_i^after\ update-z_i^before\ update\|_2. (10) The displacement is measured before overwriting each vector. Convergence is declared when Ī“t<10ā6 _t<10^-6; the criterion uses the maximum rather than the mean, so every node must satisfy the tolerance. On the reported development graph of 1,811 nodes, convergence took approximately forty sweeps, with a final maximum displacement of 9Ć10ā79Ć 10^-7. A safety cap of 200 sweeps returns the current configuration even if convergence has not been reached; the manifest records the sweep count and final displacement so that incomplete convergence remains visible. A separate early-stop setting terminates after a prescribed number of sweeps. Parameter Meaning Values anchor_weight Strength of the original-vector fidelity term 1.0 (fixed) edge_weight Strength of attraction along graph edges 0.5, 1.0 convergence_tol Stop when vector movement falls below this threshold 10ā610^-6 (fixed) max_sweeps Safety cap on full update sweeps 200 (fixed) stop_after_sweeps Optional forced early stop None, 1, 2, 3 Grid construction. Multiplying both weights by the same positive constant rescales the objective without changing its minimiser or update rule. Only their ratio matters, so a=1a=1 fixes the reference scale. The grid contains two runs to convergence, with bā0.5,1.0bā\0.5,1.0\, and three early-stop probes at b=1.0b=1.0 after one, two, or three sweeps. These are five configurations, not the full Cartesian product of the listed values. stop_after_sweeps=None uses the convergence criterion subject to the safety cap. Early-stopping probe. The probe tests whether stopping short of the optimum preserves useful structure that full smoothing might remove. In the reported comparison, F1 increases across the tested sweep counts and is highest at convergence: 1 sweep 2 sweeps 3 sweeps Converged F1 0.0933 0.1051 0.1120 0.1178 These measurements provide no evidence of a benefit from the tested early stops. They do not establish monotonic retrieval improvement at every unmeasured iteration. Retrieval behaviour. The separately reported usage comparison gives F1 of 0.1202 for standalone retrieval and 0.1510 when this representation is used in the single requery schedule. A random-walk endpoint configuration scores 0.1534, a numerical difference of 0.0024, without an iterative convergence phase. This is a comparison of the supplied scores, not a claim of statistical equivalence. It illustrates that minimising the graph-smoothing objective and maximising retrieval quality are different goals. The reported ranks (491 and 25 in an approximately 600-entry comparison) are omitted because the exact ranking population is not specified relative to the configurations documented here. B.4 Random-Walk Endpoint Blend For each node i whose embedding is being modified, walk_endpoint launches 100 random walks starting from that same node i. It forms the graph-context vector for i as a frequency-weighted mean of the endpoint embeddings; intermediate paths are discarded. Parameter Meaning Values walk_steps Number of graph transitions per walk 2, 3 alpha Original-vector retention 0.0, 0.5, 0.8 walks_per_node Walks launched from each node 100 (fixed) walk_seed Reproducibility seed 42 (fixed) Along alternating chunkāentity paths, two transitions can return to a chunk and three can end on an entity. Other node and edge types may change this pattern. Returns to the starting node are possible unless excluded by the transition rule; that rule remains to be verified. B.5 Louvain Cluster Blend Community detection uses the Louvain family introduced by Blondel et al. [11]; the subsequent embedding blend is the transformation evaluated here. louvain_cluster detects graph communities and moves each node towards the centroid of its community. Parameter Meaning Values resolution Controls community granularity; larger values produce smaller communities 0.5, 1.0, 2.0 alpha Original-vector retention 0.0, 0.5, 0.8 louvain_seed Reproducible partitioning seed 42 (fixed) At resolutions 0.5, 1.0, and 2.0, the graph contains respectively 20, 34, and 50 communities, with median sizes 85, 44, and 31. When Ī±=0Ī±=0, every member is mapped to its frozen original-space community centroid, eliminating within-community distinctions. B.6 Shared-Neighbour Pull shared_neighbor_pull adds an attraction term between nodes that are not directly connected but share graph neighbours. Parameter Meaning Values shared_pair_weight Strength of attraction between common-neighbour pairs 1.0, 3.0, 9.0 stop_after_sweeps Optional forced early stop None, 1, 2, 3 anchor_weight Original-vector fidelity term 1.0 (fixed) edge_weight Direct-edge attraction term 1.0 (fixed) convergence_tol Convergence threshold 10ā610^-6 (fixed) max_sweeps Safety cap 200 (fixed) The coefficient values are 1, 3, and 9 with direct-edge weight fixed to 1. Their aggregate influence also depends on the number and weighting of eligible shared-neighbour pairs. Three configurations converge fully; three early-stop probes use the weight-3 setting. Fixing the edge weight to one makes convex_neighbor_pull__e1p0 an exact zero-shared-pair control under the same solver. B.7 Gradient-Based Neighbour Pull gradient_neighbor_pull optimises embeddings so that vector similarity follows graph-hop distance. Parameter Meaning Values hop_target_curve Desired similarity decay over hop distance None, linear_steep, hyperbolic_sech step_size Magnitude of each optimisation update 0.25, 0.5, 1.0 max_shift Maximum displacement as a fraction of vector norm 0.5, 1.0, 2.0 steps Number of optimisation passes 30 (fixed) patch_hops Radius of overlapping graph patches 2 (fixed) hop_target_max_hop Hop distance beyond which target similarity is zero 4 (fixed) patch_seed Reproducibility seed 42 (fixed) The grid is intentionally sparse: one default configuration uses no explicit curve, step 0.5, and shift 1.0; the two named curves are crossed with two step sizes and two shift limits, producing eight additional configurations. This is the only transformation with a learning-rate-like step and an explicit displacement cap. B.8 Residual Representations A residual has no independent hyperparameters. For each of the 22 principal swept collections, it subtracts the normalised original vector from the normalised modified vector. The sole parameter, modified_collection, selects the source collection. Appendix C Retrieval Strategies: Formulations and Implementation Details Every retriever below is evaluated under the invariant final budget B=40B=40, and each is crossed with the derived collections. Two accessory grids re-run part of the shortlist at half the budget and over a chunk-only search domain. C.1 Graph-Agnostic Embedding Retrieval and One Modified Collection Alone MergedSearchRetriever performs standard nearest-neighbour retrieval over a chosen set of stored collections. Parameter Meaning Values collections The index set searched the three originals jointly, or one derived collection One run searches the three original collections jointly and defines plain_baseline; the others search one derived collection directly. C.2 Alternating Budget Split BudgetSplitRetriever searches the original and the modified space independently, then allocates the final positions between the two rankings. Parameter Meaning Values beta Share of the final budget given to the modified space 0.5, 0.2 At Ī²=0.5Ī²=0.5, 20 positions go to each space; at Ī²=0.2Ī²=0.2, 8 go to text and 32 to the modified space. Thirty hop-derived collections are crossed with both values, and the fixed-count collection is evaluated once at Ī²=0.5Ī²=0.5. C.3 Reranking RerankRetriever retrieves a deep pool from the original space and rescores it with a graph-derived signal, scoring each candidate as si=(1āmix)āsimā(,i)+mixāsimā(,i).s_i=(1- mix)\,sim(q,x_i)+ mix\,sim(m,z_i). (11) Parameter Meaning Values mix Weight of the graph-derived score against the original-space score 0.3, 0.5, 0.7 RERANK_OVERSAMPLE Candidate pool as a multiple of the budget 4 (fixed) The 22 modified collections are crossed with the three mix values. Mix zero is excluded because it exactly reproduces the text-only ranking. The oversampling factor fixes the pool at 4āB=1604B=160, and the seed count is derived as B/2=20B/2=20; neither is treated as a tunable parameter. C.4 Two-Stage Fused Retrieval and Requery Schedules MultiStageRetriever fills the budget sequentially, one stage at a time. Seven stage schedules are evaluated: Quotas Stage ranks Interpretation 16/12/12 original, fused, shifted Single requery: expansion followed by query-shifted text retrieval 6/6/14/14 original, fused, shifted, shifted Chained requery: a short opening, then two successive shifts 20/20 original, fused Two-stage fused retrieval: text seeds followed by modified-space expansion 16/8/8/8 original, fused, shifted, shifted The same chaining from a wide opening 6/12/22 original, fused, shifted Single requery from a short opening 6/6/14/14 original, fused, fused, fused The chained quotas with every later stage fused instead of shifted 20/20 original, shifted Graph-free control: the shift alone, with no fused stage Parameter Meaning Values mix Weight of the modified-space score in a fused stage 0.3, 0.5, 0.7 query_shift Weight of the previous stageās mean in the shifted query 0.3, 0.5 (bridge only) The shifted query is ā²=(1āquery_shift)ā+query_shiftāĀÆprevious.q =(1- query\_shift)q+ query\_shift\, x_previous. (12) The coefficient is a nominal weight: the norm of the mean determines its effective contribution, so a coefficient of 0.5 need not produce equal directional influence. The chained schedule carries two parameters the others do not: beyond the mix and query-shift grids it also varies which retrieved nodes a shifted stage anchors on and how far back it looks. The modified-space input ranges over 44 representations: 22 graph-modified collections and their 22 residuals. C.5 Output Ordering, Overlap, and Computational Scope Single requery (bridge) is formulated in Section 3.4. Its final output concatenates the three stage lists; it is not globally reranked. Stage-level exclusion uses node identity. Candidate-depth and refill rules must ensure that an implementation actually returns 40 distinct nodes, including when independent searches overlap. For the alternating budget split, the quota and overlap-resolution convention must be recovered from the implementation; taking two top lists alone does not guarantee 40 distinct nodes. The recorded search-call counts are one for plain retrieval, two for budget splitting, one for reranking, three for two-stage fused retrieval, four for single requery, and five for the four-stage chained schedules. They count vector-search operations rather than equal units of work. Reranking evaluates 160 candidates; modified collections require preprocessing and storage. No latency or hardware measurements are available. Appendix D Per-Family Results Section 4 reports the question families pooled into groups. The table below gives each of the six harder families separately, for the retrieval strategies described in Section 3.4. The five single-fact families are omitted: the baseline already recovers between 96.4%96.4\,\% and 100%100\,\% of their evidence, so no strategy can separate itself on them except by degrading them. Each column uses the representation the main text pairs with that schedule: the hop-2 neighbour blend for the alternating budget split and for two-stage fused retrieval, the walk-endpoint blend for both requery schedules. The baseline column is plain vector retrieval on the original collections. Gold chunks found (%) Group Family Base Split Fused Single Chained hop-2 walk Extremisation collect_extremize 72.4 71.6 86.6 87.5 88.1 extremize 61.9 42.5 54.5 68.4 75.0 extremize_collect 44.7 32.4 40.6 59.7 61.7 Relational and collection collect_global_collect 66.4 85.7 80.3 70.0 71.6 hop2 42.3 64.7 48.5 46.7 50.0 hop3 20.1 31.9 22.7 20.6 20.2 Two patterns are visible here that the group-level reporting averages away. The first is that the two leading strategies are complementary rather than ranked. On the three extremisation families the chained requery is best everywhere, by 13.2 to 17.0 points over the baseline; on the three relational and collection families the order reverses and the split is best on all three, by 11.8 to 22.4 points. Which strategy helps therefore depends on the operation a question requires ā covering a comparison set, or reaching along a relational chain ā and not on a single ordering of methods. The two failures are not symmetric, and the difference matters for anyone who must choose one schedule for a mixed workload. Outside its own group the split is actively harmful: it falls below the baseline on all three extremisation families, by 0.80.8, 12.312.3 and 19.319.3 points. The chained requery outside its own group is merely modest: it stays above the baseline on all three relational and collection families, by 5.25.2, 7.67.6 and 0.10.1 points, the last of which is indistinguishable from zero. Choosing the split therefore costs real evidence on the questions it does not suit, whereas choosing the chained schedule costs almost nothing. The second is that hop3 resists every strategy. Its baseline is the lowest in the benchmark at 20.1%20.1\,\% and the best column improves it by 11.811.8 points, while both requery schedules leave it where it was. The limit is not the budget: 40 retrieved nodes are ample for a median gold-evidence set of twelve chunks, and doubling the budget from 20 to 40 nodes gains the baseline only 1.61.6 points, because the extra nodes repeat content the ranking has already exhausted.