__ Summary and Contributions__: In this paper, the authors extend via a straight forward derivation the conjecture of detectability threshold of community detection problem, for k=2 (number of clusters), and for a network generated using a dynamic degree-corrected stochastic blockmodel with link persistency. One contribution of the paper is an extension of the results previously conjectured in Ref. [14] for dynamic SBM to dynamic DC-SBM using the results from [12,13]. Also the main contribution of the paper is the extension of Bethe Hessian spectral clustering as a dynamic spectral clustering on temporal networks.
-------post author response ----------
Thanks to the authors for their clarifications. After reading the authors' feedback, I have still serious concerns regarding the misunderstanding that can be created by this paper regarding Ref. [14]. If the authors have access to the codes by Ref. [14] that seems they have because of the footnote [8], then why they didn't provide the corresponding simulation with dynamical non-backtracking of [14]. To be fair to the readers and also to authors in Ref. [14], it would be correct that authors make sure to update their figure 3 with the regularized algorithm in Ref. [14] that seems have similar results with Alg. 1, instead of mentioning provided results is just for the second eigenvector, when it is obvious this eigenvector is not informative.
Regarding the novelty of the paper I still think this paper is not in the level of a NeurIPS paper since the contributions are partial and more importantly if the authors have been provided the results of Ref. [14] in Fig 3 in the first place, then the reviewers could judge the performance of their proposed spectral algorithm compared to existing methods. Although as I said before, the paper is really well-written and then I decided to increase my score.
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__ Strengths__: The paper is well written and technically sounds.

__ Weaknesses__: Although the paper is well written and explained well, however, I think the novelty of the paper is not enough. More importantly, a serious concern regarding the issues with some of the results previously shown in Ref. [14] that is not consistent with the results in this paper. I have enumerated some of the issues I found in the publication that I have explained below.
Some comments:
1. In page 3, line 98, the authors explain a method in the main text regarding how to derive the detectability limits for finite T, however, this is explained in Ref. [14]. I think the authors need to explain it in the "main text" that this is an explicit rederivation of what the authors in [14] explained before and not just in the appendix.
2. Line 152-153: The authors say "Letting \omega_{ij}=\xi if there exists a time instant t such that nodes i, j belong to Vt , and \omega_{ij} = h otherwise, ...". I think the authors meant \omega_{ij} = h for temporal edges, but it is needed to be written more clearly unless it means for all non-edges. Please make a correction.
3. The main concern is regarding the simulation of dynamical non-backtracking. The main contribution of the paper is that they show that the spectral method they propose is the one that detect communties as soon as theoretically possible and no spectral algorithm can do that, however, previously this has been shown in Ref. [14], where the authors show detectability is possible all the way down to the detectability threshold. Ref. [14] says "We then give two algorithms that are optimal in the sense that they succeed all the way down to this threshold. The first uses belief propagation, which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the belief propagation equations". It seems the authors's simulation for the proposed algorithm in Ref. [14] is problematic for simulation of dyn B method and it seems they have chosen an uninformative eigenvector for reconstruction (overlap is 0 !). See Fig. 3 in Ref. [14].
4. The results are limited for K=2 and I am not sure why the authors that provided the algorithm specifically for dynamic DC-SBM didn't provide more results for K>2, different values of \Phi, and larger T (T=4 is really small for a dynamic network).
some typos:
1. line 147: "whose theoretical support in given" --> "... is given"
2. line 519: "2.2" --> "3.2"

__ Correctness__: The algorithms seems valid for K=2, however, the results are not provided for K>2. Also one of the algorithms that the authors compared with i.e. the dynamical non-backtracking Ref. [14] is wrongly simulated. See Fig. 3 in Ref. [14].

__ Clarity__: Yes, well-written

__ Relation to Prior Work__: Yes, it is clearly explained

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The main contribution of this paper is the the development and analysis of a dynamic version of the Bethe-Hessian matrix-based algorithm for community detection, which also performs well in the sparse regime.

__ Strengths__: The topic of this paper pertains to temporal networks, an area of growing interest in the recent years.
The paper is clearly structured, and the supplementary material provides comprehensive additional details on the main steps of the proposed pipeline.
The proposed model is a sensible one, where a fraction of the nodes maintain their respective cluster membership, while the remaining ones receive random cluster re-assignments, and at each time step the adjacency matrix follows a degree corrected stochastic block model. The authors perform the sanity check that higher label persistence lead to being able to solve harder problems (sparser and more noise). The experiments in Figure 3 show superior performance to other methods from the literature, slightly under than if Belief Propagation algorithm, which however, is 2-3 orders of magnitude more computationally expensive for the problem instance considered.

__ Weaknesses__: The authors could make it more clear from the outset under what sparsity regime specifically they are able to operate in (in terms of average degree or density).
Figure 3 Left is not clear and is missing titles for the subplots and the quantity being heatmapped. The Figure caption is also missing k (=2 I presume).
Authors should clarify at the beginning of Section what is meant by overlap performance, perhaps less of a conventional name. Why not stick to something ubiquitous like Adjusted Rank Index?
For the numerical simulation in the appendix, concerning k, it might be interesting to see what empirically happens for unequal cluster sizes (preferably unequal in expectation), especially as a function of the ratio between the smallest and largest cluster (in other related settings of SBM models, the position of the informative eigenvalues may change).
Can the authors comment on the extension of this approach to directed graphs (where the adjacency matrix may or may not be assumed skew-symmetric)?
Minor typos:
Line 51/255: run â€” > ran?

__ Correctness__: The paper appears to be technically sound.

__ Clarity__: Yes, the paper is clearly written and well structured.

__ Relation to Prior Work__: Yes, the authors explain how this work relates to existing literature on clustering temporal graphs, and what gaps it aims to fill.

__ Reproducibility__: Yes

__ Additional Feedback__: ----- Post Rebuttal -----
After reading the rebuttal and the other reviewersâ€™ comments, I still believe this is solid submission for NeurIPS. I have lowered my score in light of the issues pointed out in relation to Ref [14].

__ Summary and Contributions__: The paper studies community detection in sparse dynamical graphs with the dynamical degree-corrected stochastic block model. It aims to address both sparsity and the so-called small label persistence issues by proposing a new spectral algorithm. Specifically, a dynamical Bethe-Hessian matrix is introduced. Authors claim that it can retrieves non-trivial communities for the case that only two communities exist (they also give an algorithm that can handle the case of more than two communities). Theoretical analysis and experimental studies are provided.
UPDATE: I have read the rebuttal.

__ Strengths__: The paper seems to make some theoretical contribution to tackling the sparsity and small label persistence issues in dynamical graphs.

__ Weaknesses__: There is no case study on real dynamical graphs. The proposed algorithms seem far away from being deployed in practice.
I feel the paper is difficult to read, probably because I am not familiar with the context of statistical physics favor.

__ Correctness__: seem correct

__ Clarity__: yes, but it could be improved for broader readers.

__ Relation to Prior Work__: yes

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper introduces a spectral clustering algorithm for a generalization of the degree-corrected stochastic blockmodel to a dynamic setting in which the groups labels evolve over time (are kept) according to a particular "persistence" dynamics in which the group label is either kept or randomly reassigned.
The authors generalize the ideas of spectral clustering via the Bethe Hessian to this dynamical setting and provide some theoretical discussion and analysis of their algorithms, in particular by relating their techniques to the (dynamic variant) of the backtracking matrix.
Overall, I find this study to be quite interesting and a welcome addition to the current literature on spectral clustering.

__ Strengths__: - The presented method has a clear grounding in spectral clustering schemes that have been developed for static graphs, and the authors do a good job in revisiting these foundations and appropriate extensions.
- spectral clustering is a very widely used technique, and I think there is a clear significance of extending these results to the dynamic setting

__ Weaknesses__: - As this work heavily draws on previous research on the Bethe Hessian for static graphs, the novelty is somewhat limited; in particular as the author can not give (due to the inherent additional complexity of the dynamical setting) not always give definitive algorithms and analysis, and have to make additional assumptions (e.g., knowing the persistence rate \eta)
-The empirical validation is somehwat limited, e.g., some application to real data would have strongly increase the impact of this work and there is a large number of temporal netwok data available by now (the auhtors provide some citations).

__ Correctness__: The claims made by the authors are by and large supported by theoretical analysis, even though the arguments used are not always fully rigorous but appeal to physcially motivated conjectures and/or previous literature on the subject.

__ Clarity__: The paper is written in a clear and cohesive manner

__ Relation to Prior Work__: The relations to previous work are appropriately highlighted in my opinion.
However, the novelty of their approach w.r.t to previous works could be discussed more clearly.

__ Reproducibility__: Yes

__ Additional Feedback__: p4. line 125 : \sigma should be s ?
p5. line 132: needless -> useless / non informative?
p.7, line 237: a closing parenthesis is missing
p.8, line 277: "we hinted at a greedy line-search -> maybe I misunderstood some of the remarks in section 4, but it appears there is (if anything) only a very subtle hint at a line search procedure in section 4?!
Following discussions and author response, I still have some reservations here, but see the paper slightly above the threshold.