Randomized Communication and the Implicit Graph Conjecture.
Bibliographic record
Abstract
The most basic lower-bound question in randomized communication complexity is: Does a given problem have constant cost, or non-constant cost? We observe that this question has a deep connection to the Implicit Graph Conjecture (IGC) in structural graph theory. Specifically, constant-cost communication problems correspond to a certain subset of hereditary graph families that satisfy the IGC: those that admit constant-size probabilistic universal graphs (PUGs), or, equivalently, those that admit constant-size adjacency sketches. We initiate the study of the hereditary graph families that admit constant-size PUGs, with the two (equivalent) goals of (1) giving a structural characterization of randomized constant-cost communication problems, and (2) resolving a probabilistic version of the IGC. For each family $\mathcal F$ studied in this paper (including the monogenic bipartite families, product graphs, interval and permutation graphs, families of bounded twin-width, and others), it holds that the subfamilies $\mathcal H \subseteq \mathcal F$ are either stable (in a sense relating to model theory), in which case they admit constant-size PUGs (i.e. adjacency sketches), or they are not stable, in which case they do not. We conjecture that this always holds, i.e. that constant-cost randomized communication problems correspond to the set of stable families that satisfy the IGC. As a consequence of our results, we also obtain constant-size adjacency sketches, and an $O(\log n)$ adjacency labeling scheme, for the induced subgraphs of arbitrary Cartesian products, as well as constant-size small-distance sketches for Cartesian products and stable families of bounded twin-width (including planar graphs, answering a question from earlier work).
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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 | 0.008 | 0.074 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.004 | 0.006 |
| Scholarly communication | 0.005 | 0.024 |
| Open science | 0.005 | 0.005 |
| Research integrity | 0.005 | 0.009 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
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".