Pith Number

pith:ZE7U6LTC

pith:2026:ZE7U6LTCFVT5P33IBBXSSGWOSC

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The interplay between network transitivity and community structure

Chengay S Wangchuk, Irin Rahman, Mingao Yuan, Minglian Lin

Clustering coefficient limits in geometric block models undergo phase transitions between weak and strong community regimes and prove non-monotonic in strength.

arxiv:2604.19067 v2 · 2026-04-21 · math.ST · stat.TH

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\pithnumber{ZE7U6LTCFVT5P33IBBXSSGWOSC}

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Claims

C1strongest claim

Both limits exhibit a phase transition; specifically, the functional forms of the limit functions differ between the weak and strong community structure strength regimes. For a GBM with balanced communities, the limits of the global and average clustering coefficients are identical, whereas these limits differ for unbalanced communities. In general, the clustering coefficients do not exhibit a monotonic relationship with community structure strength.

C2weakest assumption

The geometric block model with its edge probability rules based on community membership and geometry is the correct generative process, and the asymptotic analysis applies under the stated parameter regimes for weak versus strong community structure.

C3one line summary

Limits of clustering coefficients in the geometric block model show phase transitions between weak and strong community regimes, with non-monotonic behavior and differences for balanced versus unbalanced communities.

Receipt and verification

First computed	2026-06-01T01:02:39.965978Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c93f4f2e622d67d7ef68086f291ace90a241411d0653eaaff2086a6a1c3d56ed

Aliases

arxiv: 2604.19067 · arxiv_version: 2604.19067v2 · doi: 10.48550/arxiv.2604.19067 · pith_short_12: ZE7U6LTCFVT5 · pith_short_16: ZE7U6LTCFVT5P33I · pith_short_8: ZE7U6LTC

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZE7U6LTCFVT5P33IBBXSSGWOSC \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c93f4f2e622d67d7ef68086f291ace90a241411d0653eaaff2086a6a1c3d56ed

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8432d1f1b9085e06354edd6c7b8369cbb0e4e88a34c93323c507dff8be528a9a",
    "cross_cats_sorted": [
      "stat.TH"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.ST",
    "submitted_at": "2026-04-21T04:19:17Z",
    "title_canon_sha256": "08b698c53e36750ac6376123921d7266a425c0ce36a8ffb3650178dc26cb47f7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.19067",
    "kind": "arxiv",
    "version": 2
  }
}