Graphs and Networks Lecture 19 Graph Clustering: Spectral Methods and Normalized Cuts
Bibliographic record
Abstract
In this and the next lecture, we are going to consider approaches to clustering the vertices of a graph. I think that we understand reasonably well how to partition the vertices of a graph into two sets. However, in clustering, we want to divide the vertices of a graph into many sets. This problem is not nearly as well understood. There is quite a bit of disagreement over what one should be optimizing. Even once one has a measure of the quality of a clustering, it is usually computationally difficult to find a clustering that optimizes this measure. So, one typically uses a heuristic. The best heuristics typically combine two operations: a global optimization followed by local improvements. This lecture will probably just focus on the global optimizations, unless I have time time to implement some local improvement algorithms. Thealgorithms that workbestdependquiteabitontheareaofapplication. TheScientificComputing community has developed a number of algorithms for partitioning well-shaped meshes (Chaco, Metis and Scotch). Different, but related, algorithms have proved popular in Image Segmentation (Shi and Malik, Yu and Shi). A very different type of algorithm is popular with Phyisicists who now study social networks. We will see this type of algorithm next lecture. For now, let me recommend the survey of von Luxburg [Lux07]. 19.2 K-Means Before we get too into how one shouldcluster the vertices of a graph, lets take a moment to consider the seemingly easier problem of clustering vectors in IR d. Lets call the vectors x1,...,xn. One of the most popular measures of the quality of a partition of these vectors into clusters C1,...,Ck is the k-means objective function. It is k∑ a=1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.016 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".