Title: Optimal community detection in dense bipartite graphs
Venue: NeurIPS 2025
Accepted: yes
Keywords: bipartite random graphs

minimax hypothesis testing

signal detection
Abstract: We consider the problem of detecting a community of densely connected vertices in a high-dimensional bipartite graph of size $n_1 \times n_2$. Under the null hypothesis, the observed graph is drawn from a bipartite Erdos-Renyi distribution with connection probability $p_0$. Under the alternative hypothesis, there exists an unknown bipartite subgraph of size $k_1 \times k_2$ in which edges appear with probability $p_1 = p_0 + \delta$ for some $\delta > 0$, while all other edges outside the subgraph appear with probability $p_0$. Specifically, we provide non-asymptotic upper and lower bounds on the smallest signal strength $\delta^*$ that is both necessary and sufficient to ensure the existence of a test with small enough type one and type two errors. We also derive novel minimax-optimal tests achieving these fundamental limits when the underlying graph is sufficiently dense. Our proposed tests involve a combination of hard-thresholded nonlinear statistics of the adjacency matrix, the analysis of which may be of independent interest. In contrast with previous work, our non-asymptotic upper and lower bounds match for any configuration of $n_1,n_2, k_1,k_2$.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider the problem of detection between on the one hand: a bipartite Erdos-Rényi graph with bipartition sizes n_1, n_2 and probability of an edge p_0, and on the other hand a bipartite graph where the edge presence probability is, between two sets of respective sizes k_1 and k_2, elevated from p_0 to being larger than p_0+delta. Their objective is to determine, for a target risk level \\eta (defined in Eq(2)), necessary and sufficient conditions on the magnitude of the elevation parameter delta, for the existence of a test Delta that discriminates between the two hypotheses with risk at most \\eta.\n\nThey then provide a lower bound on the critical value of delta, given in theorem 1, below which no such test exists. Under additional Assumption 1 (which guarantees that the graph be sufficiently dense), they give in Theorem 2 an upper bound on the critical value of delta above which a sophisticated test succeeds. This upper bound is within a constant factor of the lower bound, so that under Assumption 1 the two theorems characterize the magnitude of the critical value of delta. \n\nFinally the authors discuss implications of their results. They show that these generalize previously known thresholds in a balanced setting, and that their result reveals a novel phase transition specific to the unbalanced bipartite case (proposition 3). \n\nA discussion of limitations and future directions concludes the paper.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:The test setup considered is challenging, as the authors consider for alternative hypothesis a whole class of edge probabilities, specified by the elevation parameter delta. The result obtained is difficult, and it is interesting that it reveals a new phase transition for community detection specific to the unbalanced bipartite case. The underlying proof techniques seem powerful and delicate to use. The writing is good.\n\nQUESTIONS:\n-At the bottom of page 3, \\eta is first mentioned while not being defined yet, you may want to fiw that.\n\n-Could you comment on the fact that the constant C discussed at the bottom of page 4 depends on \\eta? What would happen in a regime where eta goes to zero?\n\n-In statement of theorem 1, notation c_delta for the constant is a bit awkward, as it might lead the reader to believe that this constant does depend on delta. However this would not make sense. Please use an alternative notation.\n-On page 6, line 106, under what probability distribution is expectation taken? P_0?\n\n-In Eq(14), is the maximum over sets J_1?\n\n-Can you comment on the choice of thresholds h_i used in Theorem 2?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

comments: SUMMARY:
The authors present new statistical tests for community detection in random bipartite graphs and prove that they are minimax optimal.

STRENGTHS AND WEAKNESSES:
Strengths:
- the problem studied in the paper is interesting
- the paper is well written and organized

Weaknesses:
- a statistics conference or journal might be a better fit
- it is difficult to assess the novel contribution w.r.t. to the state of the art (see questions below)

QUESTIONS:
- *theorem 2 of [2]  deals with G(n,p,Q) where p_i’s is the probability of a node belonging to community i, and pi’s are in (0,1). This does not seem to be consistent  with the claim that they “are confined to the restricted case of two equally sized communities”. Please clarify. 

- “the two sets of vertices may be of very different size, meaning that the results tailored for square adjacency matrices … do not capture the subtleties of community detection in bipartite graphs.” This seems to suggest that using a square adjacency matrix to represent a bipartite graph might be restrictive. Please clarify whether this is what you meant.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors presented a rigorous study of the community detection problem in bipartite graphs. In particular, the paper presented a lower bound on the minimax rate of separation, and described a novel optimal testing strategy that achieves the minimax rate when the observed graph is dense. Their non-asymptotic results hold for all possible dimensions of the observed graph and the planted subgraph, which reveals new behavior in the detection boundary.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The paper presents non-asymptotic, matching upper and lower bounds for $\\delta^*$ valid for all parameter configurations ($n_1,n_2,k_1,k_2$). This universality is a major advance over prior work constrained to asymptotic regimes, square matrices and symmetric communities. Besides, the introduction of the truncated degree test and max truncated degree test is significant. These adapt the truncated $\\mathcal{X}^2$ concept to the challenges of binomial data and the matrix setting, requiring careful function design and handling of moderate vs. large deviations.\n\nWeaknesses: (1) The upper bound and optimality of the proposed tests crucially depend on Assumption 1, requiring the graph to be sufficiently dense. This excludes the practically important regime of sparse bipartite graphs. (2) The max truncated degree test requires scanning over all subsets of $\\mathcal{V}_1$ or $\\mathcal{V}_2$, making it computationally infeasible for moderate-sized graphs. This paper doesn't offer any computationally efficient alternative achieving the rate, even heuristically. (3) This paper lacks empirical validation of the derived rates or the performance of the proposed tests.\n\nQUESTIONS:\n(1) Given the reliance of the upper bound on Assumption 1, what are the key technical obstacles preventing the extension of your optimal testing strategies to the sparse graph setting? Do you anticipate the minimax rate $\\delta^*$ itself changing fundamentally under sparsity?\n\n(2) For truncated degree tests, how to choose the thresholds $\\tau,h$?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of detecting a dense bipartite subgraph. More specifically, this is a hypothesis testing problem between a null distribution of bipartite Erd\\H{o}s–R\\´enyi graphs (of size $n_1 \\times n_2$) against an alternative distribution of bipartite random graph with a planted denser subgraph (of size $k_1 \\times k_2$). The result establishes the minimax rate of the edge density separation $\\delta^*$, for any values of $n_1, n_2, k_1$, and $k_2$. The optimal test consists of three parts: a total degree test, a truncated degree test, and a max truncated degree test. Each of them is optimal up to multiplicative constants under some conditions of the size parameters. A careful combination makes it optimal down to the complete regime, under a graph density constraint. It worth notice that the lower bound is not restricted on the graph density constraint.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. This work gives a lower bound of the minimax rate of density separation $\\delta^*$ under any $n_1,n_2,k_1,k_2$.\n2. Although the upper bound result relies on a density constraint, it is very interesting to be independent of balanced community or balanced planted bipartite graph assumptions.\n3. The proposed truncated degree test and max truncated degree test carefully extend the idea of the truncated $\\chi^2$-test by coming up a statistic of the normalized (under null) average degree to control the tail probability.\n\n**Weaknesses**\n1. Many parts of the paper are well-written, but I find Section 3 a bit lack of intuitions. It would be nice to explain more on what would each regime look like when each of $\\psi_{12}+\\psi_{21}$, $\\phi_{12}$, or $\\phi_{21}$ becomes the smallest among those three quantities. This would largely help the readers grab the idea of when would each test become more natural or useful.\n2. The constructed test potentially runs in exponential time, depending on the size of the planted dense bipartite subgraph. The potential of an information-computational gap does exist so we are not sure about whether this is a weakness of this constructed test or not. It would also be interesting to see whether there are some regimes where you can actually construct efficient tests.\n\nQUESTIONS:\n1. Could you clarify which specific regime in Assumption 1 is associated with Proposition 2?\n2. Could you elaborate on the intuition for when the max-truncated degree test is preferable to the truncated degree test, and conversely, in which regimes the truncated degree test would be more appropriate? The construction of $\\Delta_{a}^{h_1}$ and $\\Delta_{a}^{h_2}$ have a simple threshold, but the other thresholds in terms of $\\psi_{12}, \\psi_{21}, \\phi_{12},$ and $\\phi_{21}$ are a bit hard to grab the idea.\n3. Would you expect the lower bound to be tight when Assumption 1 fails?\n4. A minor point I noticed that is the notation of $\\Theta(n_1,n_2,k_1,k_2,\\delta)$ in equation (1) is inconsistent with other places.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}