Triadic Closure — How Social Networks Cluster

Network Growth

Triadic closure probability p 0.75

New links per joining node m 2

Growth rate 2.0 nodes/s

Network size cap 60 nodes

Live Metrics

Nodes N

Edges E

Avg. degree ⟨k⟩

0.00

Avg. clustering C̄

0.000

Random-graph baseline C ≈ ⟨k⟩/N

0.000

Node colour scales with degree (dark → warm = more connections = a local hub). Drag to orbit, scroll to zoom.

How it works

This is the Holme–Kim model (Holme & Kim, 2002) — a growth process that reproduces the one property plain random-attachment graphs get wrong about real social networks: your friends tend to know each other.

Each new node adds m edges:
 1st edge  → a uniformly random existing node u
 edge i>1  → with probability p: a random neighbour of the
             previously-attached node (closes a triangle)
             else: a uniformly random existing node

The triadic-closure step is exactly "friend of a friend": if you just connected to person A, there's a chance your next connection is one of A's existing contacts, not a stranger. That single rule is what pushes local clustering far above what pure random attachment produces.

Local clustering:  Cᵢ = 2·Tᵢ / (kᵢ(kᵢ−1))
  Tᵢ = triangles through node i,  kᵢ = degree of node i
Network average:   C̄ = (1/N) Σᵢ Cᵢ
Random-graph (Erdős–Rényi) expectation for the same size: C ≈ ⟨k⟩ / N

p = 0 — pure random attachment, no preference for closing triangles; C̄ stays close to the random-graph baseline.
p → 1 — nearly every extra edge closes a triangle; C̄ climbs sharply above the baseline while the network keeps growing at the same rate, matching what's measured in real friendship and collaboration graphs.
m — more edges per arriving node means a denser, more connected graph overall (higher ⟨k⟩), independent of the clustering effect.

This same triadic-closure mechanic underlies why real social graphs — unlike a purely random Erdős–Rényi graph of the same size and density — split into tightly-knit friend groups instead of staying uniformly sparse.

Edge (social tie) Hub (high degree)

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