Graph Quasirandomness for Hypothesis Testing of Stochastic Block Models
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arXiv
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| Format: | Preprint |
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2025
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| author | Bangachev, Kiril Bresler, Guy |
| author_facet | Bangachev, Kiril Bresler, Guy |
| contents | The celebrated theorem of Chung, Graham, and Wilson on quasirandom graphs implies that if the 4-cycle and edge counts in a graph $G$ are both close to their typical number in $\mathbb{G}(n,1/2),$ then this also holds for the counts of subgraphs isomorphic to $H$ for any $H$ of constant size. We aim to prove a similar statement where the notion of close is whether the given (signed) subgraph count can be used as a test between $\mathbb{G}(n,1/2)$ and a stochastic block model $\mathbb{SBM}.$
Quantitatively, this is related to approximately maximizing $H \longrightarrow |Φ(H)|^{\frac{1}{|\mathsf{V}(H)|}},$ where $Φ(H)$ is the Fourier coefficient of $\mathbb{SBM}$, indexed by subgraph $H.$ This formulation turns out to be equivalent to approximately maximizing the partition function of a spin model over alphabet equal to the community labels in $\mathbb{SBM}.$
We resolve the approximate maximization when $\mathbb{SBM}$ satisfies one of four conditions: 1) the probability of an edge between any two vertices in different communities is exactly $1/2$; 2) the probability of an edge between two vertices from any two communities is at least $1/2$ (this case is also covered in a recent work of Yu, Zadik, and Zhang); 3) the probability of belonging to any given community is at least $c$ for some universal constant $c>0$; 4) $\mathbb{SBM}$ has two communities. In each of these cases, we show that there is an approximate maximizer of $|Φ(H)|^{\frac{1}{|\mathsf{V}(H)|}}$ in the set $\mathsf{A} = \{\text{stars, 4-cycle}\}.$ This implies that if there exists a constant-degree polynomial test distinguishing $\mathbb{G}(n,1/2)$ and $\mathbb{SBM},$ then the two distributions can also be distinguished via the signed count of some graph in $\mathsf{A}.$ We conjecture that the same holds true for distinguishing $\mathbb{G}(n,1/2)$ and any graphon if we also add triangles to $\mathsf{A}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graph Quasirandomness for Hypothesis Testing of Stochastic Block Models Bangachev, Kiril Bresler, Guy Statistics Theory Combinatorics Probability The celebrated theorem of Chung, Graham, and Wilson on quasirandom graphs implies that if the 4-cycle and edge counts in a graph $G$ are both close to their typical number in $\mathbb{G}(n,1/2),$ then this also holds for the counts of subgraphs isomorphic to $H$ for any $H$ of constant size. We aim to prove a similar statement where the notion of close is whether the given (signed) subgraph count can be used as a test between $\mathbb{G}(n,1/2)$ and a stochastic block model $\mathbb{SBM}.$ Quantitatively, this is related to approximately maximizing $H \longrightarrow |Φ(H)|^{\frac{1}{|\mathsf{V}(H)|}},$ where $Φ(H)$ is the Fourier coefficient of $\mathbb{SBM}$, indexed by subgraph $H.$ This formulation turns out to be equivalent to approximately maximizing the partition function of a spin model over alphabet equal to the community labels in $\mathbb{SBM}.$ We resolve the approximate maximization when $\mathbb{SBM}$ satisfies one of four conditions: 1) the probability of an edge between any two vertices in different communities is exactly $1/2$; 2) the probability of an edge between two vertices from any two communities is at least $1/2$ (this case is also covered in a recent work of Yu, Zadik, and Zhang); 3) the probability of belonging to any given community is at least $c$ for some universal constant $c>0$; 4) $\mathbb{SBM}$ has two communities. In each of these cases, we show that there is an approximate maximizer of $|Φ(H)|^{\frac{1}{|\mathsf{V}(H)|}}$ in the set $\mathsf{A} = \{\text{stars, 4-cycle}\}.$ This implies that if there exists a constant-degree polynomial test distinguishing $\mathbb{G}(n,1/2)$ and $\mathbb{SBM},$ then the two distributions can also be distinguished via the signed count of some graph in $\mathsf{A}.$ We conjecture that the same holds true for distinguishing $\mathbb{G}(n,1/2)$ and any graphon if we also add triangles to $\mathsf{A}.$ |
| title | Graph Quasirandomness for Hypothesis Testing of Stochastic Block Models |
| topic | Statistics Theory Combinatorics Probability |
| url | https://arxiv.org/abs/2504.17202 |