1 · Full-text representation
For term \(q\) in text window \(d\), the page uses log-scaled TF–IDF:
\[
x_{q,d}=\left(1+\ln f_{q,d}\right)\left[\ln\!\left(\frac{N+1}{n_q+1}\right)+1\right].
\]
Here \(f_{q,d}\) is the term count, \(n_q\) is the number of documents containing the term, and \(N\) is the current set of book windows plus the IOAA task documents. New or revised TeX therefore changes the vector space automatically.
2 · How important a topic is in the book
Every substantive section, subsection, or subsubsection becomes a live topic. Let \(\mathbf v_A\) be the vector built from the complete body of topic \(A\). Its activation in book window \(d\) is
\[
a_{A d}=\max\!\left(0,\frac{\cos(\mathbf v_A,\mathbf x_d)-\tau}{1-\tau}\right).
\]
Topic prominence is the normalized amount of discussion across the whole current book:
\[
P(A)=\frac{\sum_d a_{A d}}{\max_B\sum_d a_{B d}}.
\]
Node size follows \(P(A)\). No IOAA marks enter this quantity.
3 · How closely book topics are related
Two topics are related when they become active in the same parts of the book. Using their activation profiles across all book windows,
\[
R(A,B)=\frac{\sum_d a_{A d}a_{B d}}{\sqrt{\sum_d a_{A d}^2}\sqrt{\sum_d a_{B d}^2}}.
\]
Blue/violet topic-to-topic lines use \(R(A,B)\). This makes the cluster a map of the current book first, before IOAA 2025 is added as an external comparison layer.
4 · Comparing the book with IOAA 2025
Each OM, OT, and OP scored task is represented by its question meaning and the skills needed to carry it out. Its similarity to book topic \(A\) is
\[
S(A,u)=\frac{\mathbf v_A\!\cdot\!\mathbf y_u}{\|\mathbf v_A\|\,\|\mathbf y_u\|}.
\]
Every task is treated equally. The green task links use \(S(A,u)\); marks are retained only as factual task metadata and never affect node size, link strength, or topic importance.
5 · Topic relevance and book alignment
For each book topic, external IOAA relevance is the mean of its strongest \(k\) task similarities:
\[
Q(A)=\frac{1}{k}\sum_{u\in\operatorname{Top}_k(A)}S(A,u).
\]
The green halo around a book node follows \(Q(A)\). A prominence-weighted book-to-IOAA alignment index is also reported:
\[
F_{\rm book}=\frac{\sum_A P(A)Q(A)}{\sum_A P(A)}.
\]
This asks whether the topics emphasized by the current book are also relevant to the 2025 observation tasks.
6 · Force-directed placement
The layout is ForceAtlas2/LinLog-inspired. Prominent topics repel more strongly, while book-relatedness and IOAA-similarity edges attract according to their own relevance weights:
\[
F^{\rm rep}_{ij}=k_r\frac{(1+P_i)(1+P_j)}{d_{ij}^2+\epsilon},\qquad
F^{\rm att}_{ij}=k_a\,w_{ij}\ln\!\left(1+\frac{d_{ij}}{d_0}\right).
\]
There are no meaningful axes. Read node size as book prominence, neighbourhood as conceptual relatedness, and green connections/halos as IOAA 2025 topical similarity.