Algorithms and Applications: instructor model responses

Exact reciprocity calculations, null comparisons and a defensible analysis card.

These are synthetic worked answers. The census is small enough to enumerate exactly, so no normal approximation or random rewire sampler is involved.

1. Graph construction

At threshold one, all six listed ordered pairs become edges. The reciprocal pairs are {A,B} and {A,C}, so R = 2. Four of the six directed edges have a reverse: the reciprocated-edge fraction is 2R/m = 4/6 = 2/3.

At threshold two, B-to-A and B-to-C disappear. Four edges remain: A-to-B, A-to-C, C-to-A and C-to-D. Only {A,C} is reciprocal: R = 1, and 2R/m = 2/4 = 1/2. All four nodes remain by the stated inclusion rule. Contact multiplicity has been used for thresholding, not retained as a weight in the final binary graph.

The edge-fraction convention matches NetworkX’s reciprocity definition. A raw reciprocal-pair count is a different statistic; name which one is being used.

2. Exact null results

Threshold one: the mean is (480 + 2×360 + 3×20)/924 = 15/11 ≈ 1.364. Observed/expected is 2/(15/11) = 22/15 ≈ 1.467. The inclusive upper tail is (360+20)/924 = 95/231 ≈ 0.411. The result is above the mean but does not meet the prespecified 0.05 rule.

Threshold two: the mean is (240 + 2×15)/495 = 6/11 ≈ 0.545. Observed/expected is 11/6 ≈ 1.833, but the tail is (240+15)/495 = 17/33 ≈ 0.515. A larger enrichment ratio does not imply stronger evidence. The threshold changes m, so a six-edge null would no longer match this four-edge graph.

These tails count all null graphs at least as extreme in the prespecified direction. They are not probabilities that the null is true. Failure to meet the rule is not proof of random wiring or absence of reciprocity-related biology.

This null is the directed fixed-edge-count model described by the NetworkX G(n,m) documentation. It controls edge count, not each node’s degree, spatial opportunity or type composition. It is deliberately simple for teaching. The supplied data have no positions or types from which to construct a biologically informed spatial/type null.

3. Error scenario

Removing B-to-A from the threshold-one graph leaves five edges, one reciprocal pair {A,C}, and reciprocated-edge fraction 2/5 = 0.4. This is a scenario sensitivity check, not an uncertainty interval or measured error rate. If computing a new p-value, compare against the corresponding five-edge null rather than reusing the six-edge tail. The scenario shows endpoint dependence on one contact; it does not establish the probability that contact is wrong.

4. Example analysis card

Hypothesis and endpoint: the synthetic graph has an excess of unordered reciprocal pairs, R, relative to the specified fixed-edge-count null. The primary construction uses threshold one. The graph version is synthetic-four-neuron-graph-v1, with four labeled nodes, no self-loops and one edge per qualifying ordered pair.

Null and decision rule: all 924 labeled four-node/six-edge directed simple graphs are equally likely. It would be uninteresting under this null if the observed R were common among graphs with this edge count. Use the prespecified inclusive upper tail and 0.05 threshold, not a direction chosen after seeing the result.

Result: R = 2 versus a null mean of 15/11; upper tail 95/231. The teaching rule is not met. Threshold two also fails its rule. Removing the uncertain B-to-A edge lowers R to 1; this is a single sensitivity scenario with no assigned error probability.

Provenance: archive the worksheet or query source, commands and outputs, actual execution date, Python version and emitted source hash for the code route. Exact enumeration has no random seed. Do not claim a code run for a paper calculation.

Non-claim: this toy result establishes neither functional computation nor a general rule about biological circuits. A real extension needs validated object and partner identities, a defined population/boundary, measured error mechanisms and the covariates required by the intended null. Null constraints follow the scientific question; adding constraints automatically is not always the right choice.

Feedback guide

Score 0–2 each for graph/statistic definitions, null/tail calculation, interpretation, and provenance/error sensitivity. Two means explicit and correct, one a recoverable omission, zero absent or contradictory. Proficient means at least 6/8 and no zero for null/tail reasoning or interpretation. This is a local formative rubric, not a validated assessment instrument. A well-supported negative result earns full credit.

Teaching material: CC BY-SA 4.0, NeuroTrailblazers.