Do coordinated rings look like tiny communities, or hide inside real ones?

Links: CS224W L02 (node embeddings, including spectral / Laplacian eigenmaps), L03–L04 (GNNs, the project's community task), notebooks/project/retweet_thread.py §L02–04 (community prediction on the same world). Atlas: community detection; social networks and information ecosystems.

Question

Community detection is the oldest structural tool of social network analysis, and moderation teams still use it first: "find the dense cluster, look inside". A coordinated ring is dense by construction, but it also points almost all of its retweets at a few popular accounts inside one political camp. Does modularity maximisation (and its spectral cousin) see a ring as its own community, or absorb it into the camp it imitates? And does the answer change when the ring has no internal ties at all (pure co-amplification)?

Hypotheses (written before running)

1. Absorption. At the default resolution (γ = 1), Louvain puts ring members in the same community as their ring's main target: at least 80% of members share it.

2. Resolution. For reciprocal rings there is a resolution in the sweep at which each ring becomes (almost) its own community: one community holds at least 70% of the ring and is at least 70% ring members. For co-amplification rings no resolution in the sweep does this.

3. Conductance. A reciprocal ring, taken as a node set, has much lower conductance than random same-size sets from its camp. A co-amplification ring has conductance close to 1 (no internal edges), no lower than the random sets.

4. Spectral localisation. Some of the lowest eigenvectors of the normalised Laplacian localise on reciprocal rings (at least 7 of an eigenvector's 10 largest entries in one ring). None do for co-amplification rings.

Setup

• Worlds: gl.retweet.make_world(seed, stealthy()) (reciprocal rings) and stealthy(co_amplify=True); seeds in config.yaml. Graph: the undirected retweet graph, edge weight = number of retweets either way.

• Louvain (networkx.community.louvain_communities) over a resolution sweep, several Louvain seeds.

• Conductance of each ring vs. 200 random same-size sets from the ring's camp.
• The lowest k eigenvectors of the normalised Laplacian of the largest connected component.

Run: uv run marimo edit experiments/rings-in-communities/rings_in_communities.py and press the run button (about two minutes). The notebook writes results.md and results.json.

Results (summary; full tables in results.md)

• Hypothesis 1 refuted. At γ = 1 reciprocal rings are not absorbed: each one is its own Louvain community in every run (only 11% of members share their target's community). Modularity's merge test explains it: merging a ring into its camp raises Q only for γ below γ* = e(S,T)·2m/(d(S)d(T)), and γ* is 0.30–0.71 for every ring. A stealthy ring's ties to its camp are fewer than the null model expects. Co-amplification rings sit mostly with their camp (64%), short of the 80% threshold.

• Hypothesis 2 supported. Reciprocal rings are isolated at almost every γ ≥ 1; co-amplification rings at most 26% of the time (γ = 20), though at high γ their modal community is 72% ring members: "ring plus targets" appears among ~400 small communities.

• Hypothesis 3 supported. Reciprocal ring conductance 0.55 against 0.98 for random same-size sets; co-amplification rings 0.99 (no internal ties), and adding their targets (0.95) does not make a cluster.

• Hypothesis 4 supported. In every reciprocal world, eigenvectors 4, 5 and 6 of the normalised Laplacian (right after the three community eigenvectors) each sit on one ring; an unsupervised localisation score gives AUC 0.997. Co-amplification worlds: no localised eigenvectors, AUC 0.62.

• A trap. The flag "my community has low conductance" has AUC 0.000 at γ = 1 in reciprocal worlds, perfectly inverted: the camps are better clusters than the rings. It works (0.998) only once γ breaks the camps up.

Reading: reciprocal rings are the easy case for community detection, with no labels needed. Co-amplification rings defeat it, which is consistent with rege-vs-wl experiment 3: what is left is a count over two hops, not a dense block.

Next steps

1. Co-amplification rings with more targets (diluting the shared neighbourhood) and more camouflage.
2. Bipartite (account–target) community detection or biclustering, the natural tool for "many accounts boost the same few".
3. Local spectral methods (personalised PageRank sweep cuts, L04) from one known ring member.

In the repository: experiments/rings-in-communities/ · README.md config.yaml results.json results.md rings_in_communities.py