Chapter map

1. 1. Why collections are heterogeneous
2. 2. Cells as graph nodes
3. 3. Pairwise dataset comparisons
4. 4. Multiple plausible mappings
5. 5. Within- and between-sample edges
6. 6. A global joint graph
7. 7. Edge weights and uncertainty
8. 8. Community detection
9. 9. Recurrent versus unique communities
10. 10. Label propagation
11. 11. Information propagation beyond labels
12. 12. Large sample collections
13. 13. Trajectories in a joint graph
14. 14. Held-out dataset validation
15. 15. A student global-graph project
16. 16. Governance and careers

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1. Why collections are heterogeneous

The central mathematical idea is concrete. Multi-sample studies differ in donors, conditions, tissues and composition. One global transformation may be too rigid.

A concrete situation

Some populations recur while others are sample-specific.

What the mathematics does not license

Partial overlap creates weak-bridge risk. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Draw expected sample overlaps. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Cells as graph nodes

Here the geometry becomes operational. Each measured cell becomes a node; local similarity becomes a weighted edge.

A concrete situation

Within-dataset neighbours preserve sample geometry.

What the mathematics does not license

A graph is only as valid as its distance and features. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect degree and components. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Pairwise dataset comparisons

This step deserves a careful pause. Dataset pairs are compared to propose cross-sample correspondences.

A concrete situation

Study A can align with B through shared T cells.

What the mathematics does not license

Many pairs increase accidental-match opportunities. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record support for every pair. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Multiple plausible mappings

The useful question is not only how to calculate, but what the calculation assumes. Conos uses several possible inter-sample mappings rather than demanding one exact alignment.

A concrete situation

Alternative neighbours can represent uncertainty.

What the mathematics does not license

Too many weak mappings add noise. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare mapping strategies and overlap. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Within- and between-sample edges

A small example makes the mechanism visible. The global graph combines native sample structure with cross-sample links. Their relative weights shape communities.

A concrete situation

Within edges protect local trajectories; between edges expose recurrence.

What the mathematics does not license

Weight imbalance can isolate or overmix samples. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Plot edge-weight distributions by type. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. A global joint graph

This is where modelling choices begin to affect evidence. All pairwise and within-sample edges are assembled into one network.

A concrete situation

Indirect paths connect samples through shared populations.

What the mathematics does not license

One bridge can change global topology. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Run edge-removal component analysis. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Edge weights and uncertainty

The central mathematical idea is concrete. Weights summarise similarity or mapping support. Low-weight edges should not be interpreted like strong reciprocal links.

A concrete situation

Dense shared populations form many strong edges.

What the mathematics does not license

Density and sample size affect weight patterns. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish an edge ledger. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Community detection

Here the geometry becomes operational. Graph community algorithms identify recurrent groups with many internal connections.

A concrete situation

A community can contain the same cell type from many donors.

What the mathematics does not license

Resolution changes splits and merges. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare algorithms and resolution stability. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Recurrent versus unique communities

This step deserves a careful pause. Communities can be assessed for sample coverage, revealing recurring or sample-specific populations.

A concrete situation

A disease-only community may remain local.

What the mathematics does not license

Low coverage can reflect rarity or technical failure. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report composition and markers. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Label propagation

The useful question is not only how to calculate, but what the calculation assumes. Known labels on some nodes can spread through weighted graph paths to unlabelled cells.

A concrete situation

A reference sample can annotate other studies.

What the mathematics does not license

Weak bridges propagate wrong labels far. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Calibrate, abstain and audit paths. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Information propagation beyond labels

A small example makes the mechanism visible. Graph diffusion can transfer annotations, scores or other sample information.

A concrete situation

A pathway score can be compared across neighbours.

What the mathematics does not license

Propagation creates estimates, not measurements. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep source, path length and uncertainty. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Large sample collections

This is where modelling choices begin to affect evidence. Pairwise comparisons and graph size grow with datasets and cells, requiring computational design and pruning.

A concrete situation

Atlas-scale collections need sparse graphs.

What the mathematics does not license

Pruning can remove rare bridges. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Benchmark runtime and biological recall. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Trajectories in a joint graph

The central mathematical idea is concrete. Paths can reveal recurring developmental structure across samples.

A concrete situation

Shared progenitor branches connect donors.

What the mathematics does not license

Cross-edges can shortcut true time order. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check markers and within-sample order. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Held-out dataset validation

Here the geometry becomes operational. Construct settings without one dataset, then add it under locked rules.

A concrete situation

A new donor tests community and label transfer.

What the mathematics does not license

Final labels must not tune edges. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report community, marker and rejection outcomes. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. A student global-graph project

This step deserves a careful pause. Create three point clouds with partial overlap, build pairwise kNN edges and combine them with within-cloud edges.

A concrete situation

Known groups reveal communities and bridges.

What the mathematics does not license

Real data add sparse counts and many samples. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Remove one edge set and observe topology. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and careers

The useful question is not only how to calculate, but what the calculation assumes. Version features, pair mappings, edge weights, communities, labels and propagation. Joint graphs connect geometry, algorithms and biological review.

A concrete situation

Students can begin with adjacency matrices.

What the mathematics does not license

A community proves no mechanism or universal identity. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Create a graph model card with failure zones. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

A Conos audit starts with multiple datasets whose cell populations may overlap only partly. Pairwise alignments propose several plausible cross-dataset relations, which are combined with within-dataset structure into a global graph of all cells. Community detection and label propagation operate on this graph.

Record preprocessing, embeddings, pairwise k-nearest-neighbour parameters, matching strategy, edge weights, within- versus between-dataset balance, graph components, community method, resolution and label-propagation rules. A large graph can contain weak bridges that change global communities.

Validate pairwise edges, component stability, recurrent populations, unique states, held-out datasets, label propagation, trajectories and markers in original data. The graph supports comparison across heterogeneous collections; it does not require every dataset to share every cell type or make propagated labels ground truth.

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Frequently asked questions

What is Conos?

Conos is a graph-based method for joint analysis of collections of heterogeneous single-cell RNA-seq datasets.

What is a pairwise mapping?

It is a proposed relationship between cells in two datasets based on their local expression geometry.

Why use multiple plausible mappings?

Heterogeneous data can support several local correspondences; combining them reduces reliance on one global correction.

What does the global graph enable?

It supports recurrent community detection, visualisation, label propagation and comparison across samples.

Can Conos keep unique populations?

Yes, if graph construction and validation avoid forcing unsupported cross-dataset edges.

How should Conos be validated?

Audit pairwise edges, weak bridges, components, communities, label propagation, held-out datasets, markers and unique-state preservation.

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Primary source and fact-check note

Nikolas Barkas, Viktor Petukhov, Daria Nikolaeva et al., “Joint analysis of heterogeneous single-cell RNA-seq dataset collections,” Nature Methods 16, 695–698 (2019), DOI 10.1038/s41592-019-0466-z. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Scanorama dataset panoramas
• BBKNN balanced neighbour graphs
• Seurat anchor integration

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Pairwise-map registry

For every dataset pair, store the feature space, neighbour rule, candidate matches, reciprocal support, distances and weights. Summarise how many cells in each population receive cross-dataset links. A global graph can conceal a weak pairwise join; the registry lets reviewers trace every influential bridge back to its local evidence.

Edge-type balance

Separate within-dataset and between-dataset edges in diagnostics. Compare their weight distributions and contribution to node degree. If cross-dataset edges dominate, local trajectories may collapse; if within-dataset edges dominate, studies may remain disconnected. The balance is a modelling choice that should be tuned on training evidence and reported explicitly.

Bridge-removal test

Rank edges by weight and by their effect on connectivity. Remove weak or high-betweenness cross-dataset bridges in controlled steps, then recompute components and communities. If one edge joins two large biological regions, inspect its cells and markers. Global topology should not rest on an unsupported accidental match.

Community-stability table

Repeat plausible graph settings, community algorithms, resolutions and seeds. Align communities across runs and report stable cores, splits, merges and uncertain boundary cells. A community number is arbitrary; recurrence and external biological evidence matter. Keep sample composition and original markers beside every graph-defined group.

Label-propagation calibration

Hide labels from complete datasets or donors, propagate from the remaining graph and compare predictions with truth. Report accuracy by class, path length, source study and edge support. Add an abstention rule where neighbourhood evidence is weak. A propagated label is an inference whose provenance includes the path through the graph.

Unique-population guard

Create or identify a population found in only one dataset. Test whether it remains a local component or receives long weak links to unrelated cells. The correct graph may contain sample-specific structure. Perfect colour mixing is not the goal when the underlying biology differs.

Scaling audit

Pairwise comparisons grow with the number of datasets, while the cell graph grows with cells and retained edges. Record runtime, memory, candidate-edge counts and pruning loss. Sparse computation makes large collections possible, but pruning can disproportionately remove rare bridges, so benchmark biological recall as well as speed.

Graph claim boundary

Communities, paths and propagated scores organise relationships; they do not prove mechanism, lineage or diagnosis. Confirm claims with original expression, donors, markers and external studies. A precise report states which pairwise links were supported, which groups recurred and where the graph remained disconnected or uncertain.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Why collections are heterogeneous

Start with the mechanism: Multi-sample studies differ in donors, conditions, tissues and composition. One global transformation may be too rigid. The worked case is deliberately concrete: Some populations recur while others are sample-specific. Now expose the boundary: Partial overlap creates weak-bridge risk. A student should complete this evidence step: Draw expected sample overlaps. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 2: Cells as graph nodes

Start with the mechanism: Each measured cell becomes a node; local similarity becomes a weighted edge. The worked case is deliberately concrete: Within-dataset neighbours preserve sample geometry. Now expose the boundary: A graph is only as valid as its distance and features. A student should complete this evidence step: Inspect degree and components. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 3: Pairwise dataset comparisons

Start with the mechanism: Dataset pairs are compared to propose cross-sample correspondences. The worked case is deliberately concrete: Study A can align with B through shared T cells. Now expose the boundary: Many pairs increase accidental-match opportunities. A student should complete this evidence step: Record support for every pair. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 4: Multiple plausible mappings

Start with the mechanism: Conos uses several possible inter-sample mappings rather than demanding one exact alignment. The worked case is deliberately concrete: Alternative neighbours can represent uncertainty. Now expose the boundary: Too many weak mappings add noise. A student should complete this evidence step: Compare mapping strategies and overlap. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 5: Within- and between-sample edges

Start with the mechanism: The global graph combines native sample structure with cross-sample links. Their relative weights shape communities. The worked case is deliberately concrete: Within edges protect local trajectories; between edges expose recurrence. Now expose the boundary: Weight imbalance can isolate or overmix samples. A student should complete this evidence step: Plot edge-weight distributions by type. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 6: A global joint graph

Start with the mechanism: All pairwise and within-sample edges are assembled into one network. The worked case is deliberately concrete: Indirect paths connect samples through shared populations. Now expose the boundary: One bridge can change global topology. A student should complete this evidence step: Run edge-removal component analysis. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 7: Edge weights and uncertainty

Start with the mechanism: Weights summarise similarity or mapping support. Low-weight edges should not be interpreted like strong reciprocal links. The worked case is deliberately concrete: Dense shared populations form many strong edges. Now expose the boundary: Density and sample size affect weight patterns. A student should complete this evidence step: Publish an edge ledger. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 8: Community detection

Start with the mechanism: Graph community algorithms identify recurrent groups with many internal connections. The worked case is deliberately concrete: A community can contain the same cell type from many donors. Now expose the boundary: Resolution changes splits and merges. A student should complete this evidence step: Compare algorithms and resolution stability. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 9: Recurrent versus unique communities

Start with the mechanism: Communities can be assessed for sample coverage, revealing recurring or sample-specific populations. The worked case is deliberately concrete: A disease-only community may remain local. Now expose the boundary: Low coverage can reflect rarity or technical failure. A student should complete this evidence step: Report composition and markers. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 10: Label propagation

Start with the mechanism: Known labels on some nodes can spread through weighted graph paths to unlabelled cells. The worked case is deliberately concrete: A reference sample can annotate other studies. Now expose the boundary: Weak bridges propagate wrong labels far. A student should complete this evidence step: Calibrate, abstain and audit paths. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 11: Information propagation beyond labels

Start with the mechanism: Graph diffusion can transfer annotations, scores or other sample information. The worked case is deliberately concrete: A pathway score can be compared across neighbours. Now expose the boundary: Propagation creates estimates, not measurements. A student should complete this evidence step: Keep source, path length and uncertainty. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 12: Large sample collections

Start with the mechanism: Pairwise comparisons and graph size grow with datasets and cells, requiring computational design and pruning. The worked case is deliberately concrete: Atlas-scale collections need sparse graphs. Now expose the boundary: Pruning can remove rare bridges. A student should complete this evidence step: Benchmark runtime and biological recall. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 13: Trajectories in a joint graph

Start with the mechanism: Paths can reveal recurring developmental structure across samples. The worked case is deliberately concrete: Shared progenitor branches connect donors. Now expose the boundary: Cross-edges can shortcut true time order. A student should complete this evidence step: Check markers and within-sample order. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 14: Held-out dataset validation

Start with the mechanism: Construct settings without one dataset, then add it under locked rules. The worked case is deliberately concrete: A new donor tests community and label transfer. Now expose the boundary: Final labels must not tune edges. A student should complete this evidence step: Report community, marker and rejection outcomes. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 15: A student global-graph project

Start with the mechanism: Create three point clouds with partial overlap, build pairwise kNN edges and combine them with within-cloud edges. The worked case is deliberately concrete: Known groups reveal communities and bridges. Now expose the boundary: Real data add sparse counts and many samples. A student should complete this evidence step: Remove one edge set and observe topology. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 16: Governance and careers

Start with the mechanism: Version features, pair mappings, edge weights, communities, labels and propagation. Joint graphs connect geometry, algorithms and biological review. The worked case is deliberately concrete: Students can begin with adjacency matrices. Now expose the boundary: A community proves no mechanism or universal identity. A student should complete this evidence step: Create a graph model card with failure zones. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.