Monograph II · Topical geometry
Site Radius, Topical Centroids and the Cost of Off-Topic Leaves
A site is a point cloud in embedding space. Its authority is a function of where the centroid sits and how tightly the cloud is packed. Every page you publish moves both numbers.
| Record | WO2020033805A1 |
|---|---|
| Subject | Site-level topic embeddings and focus measurement |
| Applicant | Google LLC |
| Content Warehouse module | QualityAuthorityTopicEmbeddingsVersionedItem |
| Attributes | siteFocusScore, siteRadius, siteEmbedding, pageEmbedding, versionId |
The Content Warehouse exposes a small module whose name is longer than its contents: QualityAuthorityTopicEmbeddingsVersionedItem. It holds five fields. Two are embeddings, one for the page and one for the site. One is a version identifier. The remaining two are scalars, siteFocusScore and siteRadius, and they are the most consequential eleven syllables in the leak for anyone who publishes at scale.1The documentation describes siteFocusScore as a number denoting how much a site is focused on one topic and siteRadius as the measure of how far pageEmbeddings deviate from siteEmbedding. Neither definition specifies the distance metric; cosine distance is assumed throughout this monograph because it is the convention for normalised text embeddings.
The site is a centroid
Every indexed page is mapped to a dense vector that encodes what it is about. The site embedding is, in the simplest reading, the mean of those page vectors: the centroid of the point cloud. The focus score measures how coherent the cloud is around that centroid, and the radius measures how far the outliers sit. A domain with 300 pages about commercial roofing in the Carolinas has a tight cloud, a confident centroid and a small radius. The same domain after publishing 2,000 programmatic pages about “roofing in [every town in America]” plus a blog on home décor has a centroid that has drifted toward generic home-improvement space and a radius that has roughly tripled.
siteEmbedding = (1/N) Σ_i pageEmbedding_i
siteRadius ≈ √( (1/N) Σ_i ‖pageEmbedding_i − siteEmbedding‖² )
siteFocusScore ≈ f( 1 / siteRadius , N_core / N )A working model, not the production formula. The radius is a root-mean-square distance from the centroid; focus falls as radius grows and as the share of on-topic pages N_core / N shrinks.
Why the leaf pages hurt
The damage done by off-topic leaves is quadratic, not linear, because the radius is a root-mean-square quantity. A page that sits twice as far from the centroid contributes four times as much to the variance. Thin location pages are the canonical offender: they share a template with the core pages, so their embeddings are close enough to be counted, but each one is padded with a different town name, a different set of nearby landmarks and nothing that the core topic predicts. Individually they drift a little. Collectively they drag the centroid and inflate the radius, and the pages that earn revenue find themselves farther from a centroid they used to define.2In a 2025 replication across 38 service-business domains, the sites that pruned or consolidated location pages below a measurable traffic threshold saw a median 22 percent lift on the retained pages within two crawls, with no new links acquired.
The WIPO filing adds a second, less obvious mechanism. The query also has an embedding, and a document’s relevance can be modulated by how close its site is to the query, not only how close the page is. A perfectly relevant page on a diffuse site competes against an equally relevant page on a focused site and loses, because the focused site’s centroid is nearer to the query. This is the mathematical form of what practitioners have long called topical authority.
From radius to dampener
The Content Warehouse groups a cluster of site-wide multipliers under the quality umbrella: siteAuthority, the nsr family, lowQuality, and the various demotion attributes that feed what the documentation calls the “final score.” A wide radius does not trigger these directly, but it degrades the inputs they depend on. Focus feeds authority; authority feeds the baseline every page on the domain starts from. The failure mode is slow and site-wide: no manual action, no sudden cliff, just a gradual loss of position on queries where a tighter competitor now sits closer to the query vector.
You cannot publish your way to authority on a topic by also publishing on every other topic.
Operating within the geometry
- Measure before you prune. Embed the site with any competent sentence encoder, compute the centroid and per-page distance, and plot the distribution. The tail is where the radius lives.
- Consolidate, do not merely delete. Pages that carry inbound links should 301 to the nearest on-topic page, so the equity returns to the cloud rather than leaving it.
- Grow along the principal axis. New content should extend the cloud in the direction it already points. A roofing site publishing on gutters moves the centroid a little; publishing on mortgages moves it a lot.
- Treat location pages as a radius budget. Each one must carry enough entity-specific, topic-predictive content to sit near the centroid, or it should not exist as a separate URL.
- Split when the business is genuinely two businesses. Two tight clouds on two domains outperform one diffuse cloud on a single domain, even with the equity divided.
The geometry is indifferent to intent. It does not know that the décor blog was a content-marketing experiment or that the location pages were meant to capture long-tail demand. It knows only where the points are. Publish accordingly.