Venue: ICLR 2022 Conference

Reviews:

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: The paper studies the problem of controlling the dynamics of a networked dynamical system,
under partial observations. Such systems arise in a number of applications, and therefore the
control problem is important and well motivated.
The authors consider a reduced order system from coarse data,
and derive bounds on the convergence and approximation error for the original dynamical systems model.
These are evaluated empirically on a small synthetic dataset
main_review: What is the motivation of studying the controlability problem on SBMs?

page 3: It would be useful to explain Gramian and other background on dynamical systems, since typical ML reviewers
might not be completely familiar with the literature

Definition 1, 3rd line: should be v\in V'_k. Also in the line right after Definition 1

page 4: what does "c-nodes can cover the fine graph" mean? What is the notion of covering here?

The paper initially discusses partial observability, but it is not clear how that fits into the problem
formulation

The use of the term synchronization here seems odd, since this is usually considered in the context of
dynamical systems. Here it is about coverage of the coarse communities.

What is the justification for assumptions 2 and 3?

The problem statement in section 4 is not at all clear.

Remark 2: "This assumption is a very weaker condition than asking..." (needs to be rephrased)

The experiments section is quite limited. It would be useful to consider other networks as well.

The use of "learning" is not consistent with learning theory, and should be explained better

page 6: "Broadly, our analysis highlight the role of" (should be highlights)
summary_of_the_review: The paper is written very poorly, and is hard to follow. Even the problem statement is not clear. The authors study the controlability problem for SBMs. It is not clear how well this is motivated from the applications they mention. The technical contribution is not very clear, and the experimental section is very weak. The fit for ICLR is not very clear
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper studies network controllability with partial knowledge of the network structure. It provides insight into how the controllability can be approximated from the topology. 
main_review: Strengths: It provides a clue about how to infer the global property of network controllability with partial knowledge of the connectome. As interdisciplinary research linking control theory with network science, this work is inspiring with relatively rigorous mathematical support. 

Weakness: 
1. The scope is a bit limited for the computer science community. The framework can be described in a more general sense. 
2. Since the network controllability has its application in practice, it would be better if the authors can provide illustrative examples on when such kind of approximation is needed. 
3. Is there any statistical significance for the results shown in Figure 2?
4. The assumptions on the diagonal elements are not consistent throughout the article. In the case of brain networks, the self-loop does not exist. In the case of the SBMs, the self-loop does exist from the probabilistic assumption. Indeed, the stability of Eqn (11) only requires a parameter to shrink the largest eigenvalues of $\tilde{A}$, which is unnecessary to be related to $tr(\tilde{A})$. 
5. Is the result a corollary from the inference on the hidden stochastic block structure?  The two steps of estimating controllability and block structures seem to be isolated with each other. 
summary_of_the_review: This paper established a framework for inferring the global controllability measurement based on the community information. The scope might be narrow for the computer science group. But the topic itself is very interesting and lays on the frontier of control theory and network neuroscience fields.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The paper studies the problem of quantifying certain controllability metrics, based on coarse scale observations of a network, which is assume to have community structure. 
The main contributions are some theoretical characterizations on when and how well this is possible, given the chosen setup.
main_review: In general, I think this is an interesting topic, which has so far received not a lot of attention in the literature. Learning network properties from aggregate descriptions is clearly of relevance in a number of settings.

However, I have several reservations about some of the results presented here. 

The motivation in terms of brain networks and linear control system is questionable, I think. Yes, there is a large literature now that uses linear control theory to study brain networks; but the brain is clearly a nonlinear system.

It seems the authors have neglected that there exists a branch in the literature concerned with "blind estimation" of community structure (such as SBMs) and other network properties, which has a very similar flavor to the work presented here. I think this should be discussed in more detail. See, e.g.,
Wai, Hoi-To, et al. "Blind community detection from low-rank excitations of a graph filter." IEEE Transactions on signal processing 68 (2019): 436-451.
Roddenberry, T. Mitchell, et al. "Exact blind community detection from signals on multiple graphs." IEEE Transactions on Signal Processing 68 (2020): 5016-5030.
Schaub, Michael T., Santiago Segarra, and John N. Tsitsiklis. "Blind identification of stochastic block models from dynamical observations." SIAM Journal on Mathematics of Data Science 2.2 (2020): 335-367.

Further, while the choice of the SBM is understandable from a theoretical perspective, from a practical point of view this is highly questionable: SBMs (and graphons) more generally, lead to dense graphs, while in practice we encounter highly sparse graphs. One could at least consider the degree corrected variant of the SBM.
Moreover, SBMs will not concentrate well in spectral norm in the "sparse regime" (node degrees being constant as the number of nodes increases), which is problematic for the setup considered here. 
I think the authors should at least discuss these issues in more detail.

Assumption 4 would merit more discussion -- it seems to a pretty strong assumption.


There are also some flaws in the technical presentation.

The adjacency matrix of a network has zero diagonal, hence normalizing by the trace is a bad idea -- the authors effectively assume the existence of self-loops here (presumably from the low-rank SBM assumption). For the coarse grained version (which will have self-loops) this is not a problem.

Similarly, the expression for the expectation of the SBM corresponds to a version in which self-loops are allowed -- though this is not unusual in the literature this should at least be commented upon.

I am not sure that the statement about the controllability for T vs n steps is correct. This is true for continuous time systems, but for discrete time systems it it seems choosing a small T could prevent a system from being fully controllable, which is controllable for T=n.

summary_of_the_review: The paper makes progress in an interesting direction: learning of network properties from aggregate observations.

What should be improved in a revised version  (see above)
a) the relations to previous work in the literature should be explained more clearly (ideas from concentration of measure, blind community detection and MOR)
b) a more careful discussion about the SBM and the weaknesses of this modelling assumption should be given.
c) a number of technical mistakes should be fixed
review_title: null

Meta-review: The paper considers the problem of controlling the dynamics of a networked dynamical system, under partial observations, considering a reduced order system from coarse data, and providing approximation bounds and an empirical evaluation.  Reviewers agree it is a borderline paper.  Technical results are nontrivial, and it introduces new questions, but the main contribution is rather narrow and it could be better written.
Decision: Reject