The Triangle Friendship Paradox

Bishakh Bhattacharya; Frank den Hollander; Nitya Gadhiwala; Pradeeptha R Jain; Tejashree Subramanya

REVIEW 3 major objections 4 minor 1 cited by

The Triangle Friendship Paradox

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Triangle counts break the friendship paradox: the average bias can be negative.

desk verdict The negative-bias message holds up, but the configuration-model section is the softest part—nonstandard triangle convention plus an unverifiable finite-n formula. read the letter →

arxiv 2507.02627 v1 pith:RU3ZT7QJ submitted 2025-07-03 math.PR

Bishakh Bhattacharya , Nitya Gadhiwala , Frank den Hollander , Pradeeptha R Jain , Tejashree Subramanya This is my paper · ORCID

classification math.PR MSC 05C8060C0560F15

keywords generalizedfriendshipparadoxtrianglebiascountssparserandomgraphsconfigurationmodelgraphonslocalweakconvergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that the friendship paradox does not automatically hold when the attribute of interest is the number of triangles a vertex belongs to. For degree and wedge counts the average friendship-bias is always non-negative, but for triangle counts the paper constructs finite graphs where the average bias is negative. It proves non-negativity for partially completed star-graphs and for gluing two such graphs at a non-centre vertex, and it computes limits for sparse and dense random graphs: positive limits for sparse binomial random graphs, sparse configuration models and rank-1 graphons, but a negative limiting bias for some two-block graphons. The message is that triangle-based friendship bias is genuinely more delicate, because triangle counts depend on edges among a vertex's neighbours and not just on its degree.

What carries the argument

The central object is the triangle friendship-bias $\Delta_T[n]$ of (1.1)--(1.2), the average over vertices of the difference between the average triangle count of neighbours and the vertex's own triangle count. The argument uses a covariance representation, $\Delta_x[n]=\mathrm{Cov}(x_U,\kappa_U)$ with $\kappa_i=\sum_j A_{ij}/d_j$, to convert the sign of the bias into a correlation question; local weak convergence to pass from finite sparse graphs to rooted limiting graphs; and the graphon functional $\chi_T=\int_0^1 dx\,[D(x)^{-1}\int_0^1 dy\,\kappa(x,y)T(y)-T(x)]$ for dense random graphs. The rank-1 graphon case and a specific two-block graphon are the examples that settle when the paradox holds or fails in the dense regime.

What would settle it

Compute $\mathbb{E}[\Delta_T[n]]$ by exhaustive enumeration for a small sparse binomial random graph, say $n=10$ or $n=20$, and compare with the closed formula (1.5); any disagreement would refute Theorem 1.10(a). Alternatively, simulate the two-block graphon with $\alpha=0$, $\beta=1/4$, $\gamma=1/2$ and $p=10/33$ for large $n$ and check whether $n^{-2}\Delta_T[n]$ is negative as the paper predicts.

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Extended reading notes

Core claim

The central claim is that the average triangle friendship-bias, defined by $\Delta_T[n]=\frac1n\sum_i\bigl[\frac1{d_i}\sum_j A_{ij}t_j-t_i\bigr]$, can be negative, unlike the degree and wedge versions of the friendship paradox. The paper proves that for partially completed star-graphs the bias is non-negative with an explicit closed form; for the sparse binomial random graph $G(n,\lambda/n)$ it proves $\lim_{n\to\infty} n\,\mathbb{E}[\Delta_T[n]]=\lambda^2-\lambda+(\lambda-\tfrac12\lambda^3)e^{-\lambda}$, which is strictly positive; for sparse configuration models it proves the limit $\zeta(c_1,c_2,c_3)=\frac{(c_2-c_1)^2}{2c_1^4}[(c_3-c_1c_2)-3(c_2-c_1^2)]\ge 0$, with equality only in the regular case; and for dense rank-1 graphons it proves $\chi_T\ge 0$, while giving a two-block graphon with $\chi_T<0$. In both sparse models the weak law of large numbers fails, because triangle-free realisations occur with positive probability in the limit.

Load-bearing premise

For the configuration-model theorems, the paper counts triangles only from triples of distinct vertices and drops the self-loop term in the bias formula, a convention that would change the finite-$n$ expectation, the limiting value, and the nonnegativity conclusion if the standard multi-graph counting including self-loops were used.

Editorial extensions

If this is right

For locally tree-like sparse graphs the average triangle friendship-bias tends to zero, since triangle counts vanish in the local limit; nonzero bias requires visible triangle structure.
In sparse binomial random graphs the expected bias is positive for large $n$ and grows like a strictly increasing function of the edge density, but with positive limiting probability the bias is exactly zero because the graph is triangle-free.
In sparse configuration models the bias is asymptotically non-negative for all degree distributions satisfying the moment condition, with equality only for regular graphs.
For dense random graphs, the bias scales like $n^2$ and its sign is controlled by the graphon functional $\chi_T$; rank-1 weighted graphs always satisfy the paradox, while some two-block stochastic block models violate it.
Gluing two partially completed star-graphs together at a non-centre vertex preserves the non-negativity of the bias, but gluing four of them at a common vertex can produce a negative bias.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The covariance interpretation suggests a general design principle: for any attribute, the average friendship-bias has the sign of the correlation between that attribute and the neighbour-averaging weight $\kappa_i$, so triangle counts fail precisely when high-triangle vertices are not the vertices with large $\kappa_i$.
The negative two-block example implies that real networks with strong community structure could exhibit an anti-friendship paradox for triangles, where vertices inside dense communities have more triangles than their neighbours on average; this is a testable prediction for empirical network data.
The limiting formulas could be turned into a cheap statistical test: from one large graph, estimate $\chi_T$ by averaging local triangle-ratio quantities, and use its sign as a diagnostic for whether the triangle friendship paradox holds in that network.
The paper leaves heavy-tailed degree distributions, such as preferential attachment graphs, open; since triangles there concentrate in a small core, one might expect the bias to be positive, but that is an extrapolation beyond what the paper proves.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the central point is true and well supported: unlike degree or wedge bias, average triangle friendship-bias can be negative, and the paper goes beyond the known counterexample to give a real analysis. Second, the sparse configuration-model section is the weak link—not because the math is necessarily wrong, but because the main finite-n formula appears only as 'available on request' and the whole result rests on a nonstandard triangle-counting convention that changes the interpretation.

What's actually new: the exact finite-n expectation for the sparse ER graph and the ζ(λ) limit; the classification of partially completed star graphs and the gluing theorem; the graphon convergence result; and the rank-1 versus two-block dichotomy for dense graphs. The ER computation is checkable in the text and the asymptotics work out (modulo a typo: ζ(λ) ~ λ⁴/3, not λ⁴/2, as λ→0). The deterministic counterexample in Figure 2 is credited to [10], and the two-block graphon example in Theorem 1.13(b) survives scrutiny even though the messy χT expression has normalization errors: the rank-1 formula is off by a factor of 2, and the claimed −8.24 example doesn't reproduce from (4.3). These don't change the signs of the key examples, but they need correction.

The configuration-model issue is more serious. The paper explicitly adopts a convention where triangles are counted only from triples of distinct vertices, with self-loops excluded from t_i, yet the A_ii t_i term remains in (1.1). The authors are transparent about this, and the calculation is what it is under that convention, but the headline 'TFP holds for sparse configuration models' is then a statement about a modified statistic, not the standard CM triangle count. On top of that, the finite-n expectation in Theorem 1.11(a) is asserted after 'a tedious but straightforward computation (available from the authors on request)'. That's not verifiable in review. A referee can't check the main formula of a section from the paper alone.

Minor issue: Theorem 1.7 leaves the k=1 gluing case as an exercise. Who is this for? Probabilists and network scientists who work on friendship-paradox variants. It deserves a serious referee. My recommendation: send it out, but insist the authors supply the CM computation (in a supplement if not in the paper), fix the normalization errors in the graphon section, and reposition the CM result as valid under their stated convention rather than as the standard triangle count.

Referee Report

3 major / 4 minor

Summary. The paper studies the generalized friendship paradox for the attribute "number of triangles at a vertex". It defines the average triangle friendship-bias ΔT[n] in (1.1)-(1.2) and shows that, in contrast to degree- and wedge-based versions, this quantity is not always nonnegative. The negative claim is supported by a deterministic counterexample (Figure 2) and by two-block graphons (Theorem 1.13(b)). The paper also proves nonnegativity for partially completed star-graphs and their gluings (Theorems 1.6-1.7), derives an exact expectation and a limit ζ(λ) for sparse Erdős-Rényi graphs (Theorem 1.10), and gives a finite-n formula plus a nonnegative limit ζ(c1,c2,c3) for sparse configuration models under a stated triangle-counting convention (Theorem 1.11). Dense graphon limits are treated in Theorems 1.12 and 1.13.

Significance. The qualitative finding — that triangle-based generalized friendship paradox can fail — is a clean and useful addition to the friendship-paradox literature, and the deterministic counterexample is simple and convincing. The partially completed star-graph analysis and the sparse Erdős-Rényi computation are explicit and largely checkable from displayed sums, and the paper is honest about which computations are omitted or heuristic. The configuration-model theorem, however, is the least secure: its finite-n formula is supplied on request rather than proved, and its convention for counting triangles is nonstandard. If the configuration-model computation is provided and the convention is made precise in the theorem statement, the paper would be a solid contribution; in its current form the positive sparse-configuration-model claim is not independently verifiable.

major comments (3)

[Section 3.3, Theorem 1.11(a)] The finite-n formula is not derived in the manuscript. After the displayed sums below (3.13), the proof states that the formula follows by a 'tedious but straightforward computation (available from the authors on request)'. Since the limiting formula in Theorem 1.11(b) is obtained by taking the limit of this unstated expression, the nonnegativity claim for sparse configuration models cannot be checked from the paper. Please include the full algebra in an appendix or state Theorem 1.11 as conditional on a supplied computation.
[Paragraph before Theorem 1.11 and Eq. (3.9)] The configuration-model result is proved for a nonstandard triangle-counting convention. The text defines t_i with i≠j≠k, excluding repeated-vertex configurations, while retaining the A_ii t_i term in (1.1) because A_ii may be nonzero. Under the usual counting of triangles in a multigraph, which includes loops and repeated-vertex configurations, the finite-n expectation, the limit ζ(c1,c2,c3), and the nonnegativity conclusion are not established and will in general change. The theorem statement should either be explicitly conditional on the convention or should be reworked for the standard counting convention.
[Section 3.1, proof of Theorem 1.9] The proof claims that ΔT[n](G_n) = E[t_{U_n}(G_n) | G_n], which is false: the left-hand side is the average of the triangle friendship biases, not the average triangle count. Moreover, local convergence in probability of rooted graphs would make the average of a bounded local functional converge in probability to the deterministic expectation under the limiting rooted law, not to the random variable ΔT as stated. The theorem and its proof should be corrected; Theorems 1.10 and 1.11 do not rely on this proof.

minor comments (4)

[Section 1.3.2, after Theorem 1.10] The stated asymptotic ζ(λ) ∼ 1/2 λ^4 as λ ↓ 0 is inconsistent with (1.6); expanding (1.6) gives ζ(λ) = (1/3)λ^4 + O(λ^5).
[Remark 4.1] The numerical example with p=0.4, α=0.1, β=0.3, γ=0.8 does not reproduce χT = −8.24 from (4.3); evaluating (4.3) for these parameters gives a value of order −10^−4. Please correct the value or clarify the computation.
[Section 1.3.3, Dense Configuration Model] The sentence 'The formula in Theorem 1.10(a) is valid for any choice of degrees as long as n is finite' appears to refer to Theorem 1.11(a), since Theorem 1.10(a) is the sparse Erdős-Rényi formula, not a configuration-model formula.
[Definition 1.5 and Figure 4] The notation in Figure 4, G(2,2,{2,5}), is inconsistent with the caption text 'two bands of adjacent triangles of width 2 and 4'; the second band should be labelled 4 rather than 5 if the notation matches the definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all central results are derived from the stated definitions and model assumptions, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's Triangle Friendship Bias is defined in (1.1)-(1.2) as an explicit functional of the adjacency matrix and triangle counts, and each theorem is a direct computation or limit of that functional under a clearly stated graph model. The deterministic results (Theorems 1.6 and 1.7) are proved by summing the vertex-level biases according to vertex types. The sparse Erdős-Rényi result (Theorem 1.10) is an independent combinatorial expectation calculation leading to the closed-form ζ(λ). The configuration-model result (Theorem 1.11) is also a combinatorial computation under an explicitly stated convention about how triangles are counted; this convention is a modeling choice, not a parameter fitted to the conclusion. The dense graphon limit (Theorem 1.12) follows from the a.s. cut-metric convergence of the empirical graphon, and Theorem 1.13 is a direct substitution into (1.9)-(1.10). No fitted quantity is renamed as a prediction, and no load-bearing argument reduces to a self-citation. The only self-citations are background references (e.g., [7]) and are not used to force a result. The noted gaps, such as the finite-n configuration-model algebra being 'available from the authors upon request' and the stated normalization issues in some dense examples, are matters of completeness and correctness of presentation, not circularity. The central negative result is supported by explicit deterministic and two-block graphon counterexamples that do not depend on any fitted input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data fitting is involved: all formulas are closed-form consequences of the definitions. The only ad hoc element is the explicit configuration-model convention for triangles, which is stated transparently. The paper introduces a new deterministic graph family, partially completed star-graphs, but that is a mathematical object rather than a postulated physical entity.

assumptions (5)

standard math Local weak convergence framework and notation from van der Hofstad's book are used to define convergence and to justify Theorem 1.9.
Invoked in Section 1.3.2 and in the proof of Theorem 1.9; the theorem is restricted to graph sequences that converge locally in probability.
standard math Empirical graphons of G(n, κ) converge to κ in the cut-metric almost surely, and dominated convergence transfers this to the TFB limit.
Used in the proof of Theorem 1.12, Eq. (4.1), citing Athreya and Röllin [16].
standard math The number of triangles in sparse ERRG and sparse CM converges to a Poisson distribution in the limit.
Used in the proofs of Theorem 1.10(c) and 1.11(c) to lower-bound the probability that nΔT[n] equals zero.
ad hoc to paper For configuration-model results, triangle counts exclude self-loops and A_ii contributions; the TFB uses (1.1)-(1.2) with this convention.
Stated in the Convention paragraph before Theorem 1.11. This changes the model and is adopted to simplify computations and scaling.
domain assumption In the dense graphon theorems, κ is continuous and its degree density D(x) is strictly positive for every x.
Assumptions in Theorems 1.12 and 1.13; they ensure the denominators are nonzero and dominated convergence can be applied.

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Cite this review

Pith. "Pith review of The Triangle Friendship Paradox." pith.science (2026). https://pith.science/paper/RU3ZT7QJ

@misc{pith2026250702627,
  author       = {Pith},
  title        = {Pith review of: The Triangle Friendship Paradox},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RU3ZT7QJ}},
  note         = {Machine review of arXiv:2507.02627}
}

read the original abstract

We consider the generalised friendship paradox, focussing on the number of triangles at a vertex as the relevant attribute. We show that, contrary to the setting where the attribute is the number of edges at a vertex or the number of wedges at a vertex, the average friendship-bias of the number of triangles at a vertex is not always non-negative. We identify classes of finite deterministic graphs for which the bias is non-negative, and provide examples of finite deterministic graphs for which it is not. For certain classes of sparse and dense random graphs, we compute the scaling of the bias in the limit as the number of vertices tends to infinity.

Figures

Figures reproduced from arXiv: 2507.02627 by the authors.

**Figure 1.** An example of a finite undirected simple connected graph. An important question is whether the FP can be quantified, and whether ∆[n] can be analysed for key examples of large random graphs (which are relevant because social networks are typically vast in size and complex in shape). To do so, the key object to look at is the friendship-bias empirical distribution µn = 1 n X i∈[n] δ∆i,n . Only few papers have address… view at source ↗

**Figure 2.** A counter example for the TFP: n = 11, ∆x i,n = 0 for i = 1, 2, 3, 6, 9, 10, 11, ∆x i,n = 1 4 for i = 4, 8, ∆x i,n = − 1 3 for i = 5, 7, ∆x [n] = − 1 66 . In this example, the degree and the number of triangles do not positively correlate with each other: Vertices 4 and 8 have the largest degrees but are in no triangle. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Two examples where adding a triangle (coloured red) to the first graph reduces the average friendship-bias. The bias drops from 13 21 to 7 18 when the triangle is added at the vertex, as in the second graph. The bias drops from 13 21 to 7 24 when the triangle is added at the edge, as in the third graph. Our two main theorems below focus on partially completed star-graphs, which turn out to have a tractable structure… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: A partially completed star-graph G(2, 2, {2, 5}) with 15 vertices (including the center), two bands of adjacent triangles of width 2 and 4, two isolated triangles, and two tadpoles. This graph has ∆T [15] = 158 45 . We are now ready to state our first main result on pa…

**Figure 5.** Figure 5: Illustration of Theorem 1.7. Two partially completed star-graphs G1 and G2 of size 10 each are glued together at a vertex that is not the center. It is natural to wonder whether simultaneous gluing of more than two partially completed star-graphs also preserves the TFP…

**Figure 6.** Figure 6: for an illustration [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: shows a numerical plot λ 7→ ζ(λ), which is strictly increasing (a proof of this property is given in Remark 3.1). Thus, the expected value of TFB is positive for large enough sparse Erd˝os-R´enyi random graphs, and their limit is a strictly increasing function of the e…

**Figure 8.** Figure 8: Pairing algorithm for the Configuration Model with n = 6 and degree sequence (1, 3, 1, 3, 2, 4). The vertices are labelled clockwise from the right-top. Vertex i is assigned di halfedges. The half-edges are paired uniformly at random to become edges. Suppose that ck =…

**Figure 9.** Figure 9: A two-block graphon. From (1.10), the degree density is D(x) = Z [0,1] κ(x, y) dy = ( αp + γ(1 − p), if x ∈ [0, p], γp + β(1 − p), if x ∈ [p, 1], and the triangle density is T (x) = 1 2 Z [0,1]2 dy dz κ(x, y)κ(y, z)κ(z, x) = ( α 3p 2 + 2αγ2p(1 − p) + βγ2 (1 − p) 2 , if…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Generalized Friendship Paradox for Eigenvectors
math.PR 2026-07 conditional novelty 6.0 of 10

For inhomogeneous Erdős–Rényi graphs generated by a continuous graphon, the empirical distribution of eigenvector friendship bias converges to the law of (ρ/K(U)−1)φ(U), where ρ and φ are the principal eigenvalue and ...

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [10]

Meerpoel

V. Meerpoel. A study of the Generalized Friendship Paradox. Bachelor Thesis, Leiden University, 21 July 2024

work page 2024

[2] [1]

S.L. Feld. Why your friends have more friends than you do. Am. J. Sociol. , 96(6):1464– 1477, 1991. 28

work page 1991

[3] [2]

Christakis and J.H

N.A. Christakis and J.H. Fowler. Social network sensors for early detection of contagious outbreaks. PloS one , 5(9):e12948, 2010

work page 2010

[4] [3]

what do your friends think?

B. Nettasinghe and V. Krishnamurthy. “what do your friends think?”: Efficient polling methods for networks using friendship paradox. IEEE Transactions on Knowledge and Data Engineering, 33(3):1291–1305, 2021

work page 2021

[5] [4]

Cantwell, A

G.T. Cantwell, A. Kirkley, and M.E.J. Newman. The friendship paradox in real and model networks. J. Complex Netw. , 9(2):Paper No. cnab011, 2021

work page 2021

[6] [5]

S. Pal, F. Yu, Y. Novick, A. Swami, and A. Bar-Noy. A study on the friendship paradox– quantitative analysis and relationship with assortative mixing. Appl. Netw. Sci. , 4(1):1– 26, 2019

work page 2019

[7] [6]

Cao and S.M

Y. Cao and S.M. Ross. The friendship paradox. Math. Sci. , 41(1):61–64, 2016

work page 2016

[8] [7]

Hazra, F

R.S. Hazra, F. den Hollander, and A. Parvaneh. The friendship paradox for sparse random graphs. Probab. Theory Relat. Fields, 2025. https://doi.org/10.1007/s00440-025-01365-w

doi:10.1007/s00440-025-01365-w 2025