## Abstract

A network may have weak signals and severe degree heterogeneity, and may be very sparse in one occurrence but very dense in another. SCORE (Ann. Statist. **43**, 57–89, 2015) is a recent approach to network community detection. It accommodates severe degree heterogeneity and is adaptive to different levels of sparsity, but its performance for networks with weak signals is unclear. In this paper, we show that in a broad class of network settings where we allow for weak signals, severe degree heterogeneity, and a wide range of network sparsity, SCORE achieves prefect clustering and has the so-called “exponential rate” in Hamming clustering errors. The proof uses the most recent advancement on entry-wise bounds for the leading eigenvectors of the network adjacency matrix. The theoretical analysis assures us that SCORE continues to work well in the weak signal settings, but it does not rule out the possibility that SCORE may be further improved to have better performance in real applications, especially for networks with weak signals. As a second contribution of the paper, we propose SCORE+ as an improved version of SCORE. We investigate SCORE+ with 8 network data sets and found that it outperforms several representative approaches. In particular, for the 6 data sets with relatively strong signals, SCORE+ has similar performance as that of SCORE, but for the 2 data sets (Simmons, Caltech) with possibly weak signals, SCORE+ has much lower error rates. SCORE+ proposes several changes to SCORE. We carefully explain the rationale underlying each of these changes, using a mixture of theoretical and numerical study.

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## Notes

- 1.
We model \(\mathbb {E}[A]\) by Ω −diag(Ω) instead of Ω because the diagonals of \(\mathbb {E}[A]\) are all 0. Here, “main signal”, “secondary signal”, and “noise” refers to Ω, −diag(Ω) and

*W*respectively. - 2.
For SBM, the diagonal entries of

*P*can be unequal. DCBM has more free parameters, so we have to assume that*P*has unit diagonal entries to maintain identifiability. - 3.
A multi-\(\log (n)\) term is a term

*L*_{n}> 0 that satisfies ”*L*_{n}*n*^{−δ}→ 0 and \(L_n n^{\delta }\to \infty \) for any fixed constant*δ*> 0 - 4.
For example, \(\frac {\hat {\xi }_{2}}{\hat {\xi }_{1}}\) is the

*n*-dimensional vector \((\frac {\hat {\xi }_{2}(1)}{\hat {\xi }_{1}(1)}, \frac {\hat {\xi }_{2}(2)}{\hat {\xi }_{1}(2)}, \ldots , \frac {\hat {\xi }_{2}(n)}{\hat {\xi }_{1}(n)})^{\prime }\). Note that we may choose to threshold all entries of the*n*× (*K*− 1) matrix by \(\pm \log (n)\) from top and bottom (Jin, 2015), but this is not always necessary. For all data sets in this paper, thresholding or not only has a negligible difference. - 5.
When translating the bound in Gao et al. (2018), we notice that

*𝜃*_{i}there have been normalized, so that their*𝜃*_{i}corresponds to our \((\theta _{i}/\bar {\theta })\). - 6.
This is analogous to the Students’

*t*-test, where for*n*samples from an unknown distribution, the*t*-test uses a normalization for the mean and a normalization for the variance.

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Jin, J., Ke, Z.T. & Luo, S. Improvements on SCORE, Especially for Weak Signals.
*Sankhya A* (2021). https://doi.org/10.1007/s13171-020-00240-1

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- Primary: 62H30
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- Secondary: 62P25