Communities, clustering phase transitions, and hysteresis: Pitfalls in constructing network ensembles
Bibliographic record
Abstract
Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a "biased rewiring model" where clustering is enhanced by means of a fugacity as in the Strauss (or "triangle") model, but where the number of links attached to each node is strictly preserved. Similar models have been proposed previously in Milo [Science 298, 824 (2002)]. Our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k>2 we find a single first order transition. For all nonregular networks that we studied (including Erdös-Rényi, scale-free, and several real-world networks) multiple jumps resembling first order transitions appear. The jumps coincide with the sudden emergence of "cluster cores:" groups of highly interconnected nodes with higher than average degrees, where each edge participates in many triangles. Hence, clustering is not smoothly distributed throughout the network. Once formed, the cluster cores are difficult to remove, leading to strong hysteresis. To study the cluster cores visually, we introduce q-clique adjacency plots. Cluster cores constitute robust communities that emerge spontaneously from the triangle generating process, rather than being put explicitly into the definition of the model. All the quantities we measured including the modularity, assortativity, clustering and number of four and five-cliques exhibit simultaneous jumps and are equivalent order parameters. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to strong hysteresis which leads to broken ergodicity on realistic sampling time scales.
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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 | 0.005 | 0.026 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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".