10 Relevance and significance answer different questions

10.1 Direction is not attention

A topic coordinate tells us how strongly an article lies along a direction. Eigen significance supplies another scalar: a story’s attention score, based on the original story-ranking method. Anthropology keeps that signal alongside the vector rather than silently using it as a weight in the equal-person PCA fit.

The source’s story score is allocated to its member articles and normalized by the scored source’s total in the relevant period. Let $\alpha_a$ be article $a$’s resulting nonnegative attention share. Recall that $z_{aj}$ is coordinate $j$ of the standardized article row $z_a$. Let $\operatorname{rank}_j(a)$ denote an attention-weighted topic score, not an ordinal position in a list. One selectable latest-article ranking uses

$$\operatorname{rank}_j(a)=|z_{aj}|\alpha_a.$$

The absolute coordinate measures topic strength on either pole; $\alpha_a$ measures allocated attention. The product deliberately answers a combined question. A reader can also choose topic-only ordering or inspect one pole separately.

For a tiny illustration, suppose article X has topic strength 2 and attention share 0.01, while article Y has topic strength 0.5 and attention share 0.10. Topic-only ordering puts X first. The combined scores are 0.02 and 0.05, so attention-weighted ordering puts Y first. Nothing about the topic coordinates changed.

10.1.1 Allocate an attention score before multiplying

Use a separate synthetic story with attention score 12 and three member articles. An equal allocation rule would give each article $12/3=4$ units. If that source’s scored total in the period is 40, each article’s attention share is $4/40=0.1$. Their shares sum to 0.3, the story’s share $12/40$. This illustrates one declared allocation rule; it does not assert that every source allocates every story equally.

The topic product then has two separate inputs. An article with standardized topic coordinate -2 and attention share 0.1 receives an either-pole score $|-2|(0.1)=0.2$. A positive-pole query would need an explicit sign restriction instead. Allocation, source normalization, pole selection, and multiplication each change the question, so each belongs in the ranking’s definition.

10.2 Person attention shares can overlap

If a 0.10-share article mentions A and B, that coverage can contribute to both people’s attention histories. Their combined credited share can exceed the article’s 0.10. These histories describe overlapping coverage, not a partition of total human importance.

Source normalization matters too. Averaging shares over active sources, including sources with no mention of a person, differs from pooling raw scores from a large and a small outlet. State the denominator before interpreting the percentage. The Eigen Times discussion of story energy and ranking provides the deeper article-level background; the person layer adds attribution and aggregation rather than redefining that original significance.

10.2.1 Compare two denominators numerically

Suppose two active sources have total attention scores 100 and 1,000. The selected person’s associated coverage receives 10 units in the first and zero in the second. Source-specific shares are $10/100=0.10$ and $0/1000=0$. Giving each source one vote yields $(0.10+0)/2=0.05$. Pooling the raw scores instead gives $(10+0)/(100+1000)=1/110$, approximately 0.00909. Omitting the zero-mention source would produce yet another quantity, 0.10.

None of these denominators can be recovered merely by reading a displayed percentage. State the active source set, period, attribution rule, and aggregation rule. The two-person credit on one 0.10-share article likewise totals $0.10+0.10=0.20$ because the histories overlap; it does not create another article or more source attention.

10.3 Today is a query with evidence requirements

The natural question “show today’s database news against today’s database CEOs” joins several tasks. News coordinates find topic-relevant articles. Dated role assertions identify people recorded as CEOs at the requested time. Supported person profiles locate those people in a historical coverage geometry. Qualified article-person associations establish which current articles actually concern them.

The current overlay can show relevant articles alongside historical profiles, but the audited overlay has zero newly qualified article-person joins. Proximity between an article and a profile is therefore a lead to inspect, not proof that the article concerns that person. Similarly, an old article calling someone a CEO does not establish that the role is current.

The Hacker News people model remains in its hn1 basis. New article embeddings can be measured in that fixed basis while carrying a scalar attention share from hn2. This is coherent only when the separation is explicit: the scalar is an overlay; it does not substitute a different set of coordinate axes. Model, corpus, dates, support, and role evidence remain visible parts of the query.

Check 9. Article X is more topic-aligned than article Y. Must X rank higher after attention weighting? Which separate evidence would be needed before calling either one news about a selected person?