Friendship Paradox Explorer

Hover nodes to inspect · drag to reposition

Network metrics

Mean degree
⟨k⟩ = Σkᵢ / N
—

Classical FP — mean-based

Alter-based FP  —
⟨k_friend⟩ = ⟨k²⟩/⟨k⟩
—
Expected degree following a random edge

Ego-based FP  —
⟨k_nn⟩ = Σ kₙₙ(i) / N
—
Average of each node's mean-neighbor degree

Difference (governed by degree–degree covariance)

⟨k_friend⟩ − ⟨k_nn⟩ = Cov(k,kₙₙ)/⟨k⟩ = —

Majority-type — going beyond averages

φ_global — mean-based majority  —
fraction {kᵢ < kₙₙ(i)}
—
Share of nodes whose degree is below their neighbors' mean

φ_local — median-based majority  —
fraction {hᵢ < ½}  where hᵢ = |{j∈𝒩: kⱼ < kᵢ}|/kᵢ
—
Share of nodes where most neighbors have higher degree

Node color · domination type

Not dominated (neither)

Mean-dominated only (kᵢ < kₙₙ)

Median-dominated only (hᵢ < ½)

Both mean- and median-dominated

Key insight: The classical FP (⟨k_friend⟩ ≥ ⟨k⟩ and ⟨k_nn⟩ ≥ ⟨k⟩) always holds, but says nothing about how many nodes are individually dominated. φ_global and φ_local can both fall below ½ even while the FP inequalities hold — and they can diverge from each other when neighbor-degree distributions are skewed.