In the following you can find the list of my comments/remarks.
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1. Introduction I would suggest the authors to give a specific definition of the term "relay node", which is not commonly used in the literature, unless for experts. The meaning of a relay node can be inferred from the context, but it is better if it is explicitly given.
We have now made explicit that these are the nodes supporting synchronization, but remaining unsynchronized.
It is explicitly mentioned the role played by remote synchronization in designing strategies for the control of synchronization between groups of nodes in complex networks, but no further information is given on this point. Since this argument is really interesting, I would suggest the authors to add a short paragraph where they discuss this point in more details
We have added a new paragraph discussing this matter at the end of the section on remote synchronization in networks of homogeneous oscillators.
2. Remote synchronization in networks of inhomogeneous oscillators Which is the role played by the relay node, being a hub? The results would have been different if the presented topological configuration would not have been centered on the presence of a hub? Would have been still possible a remote synchronization without a single hub?
The role played by the hub-relay node is through intricate modulation effects of the amplitude fluctuations to allow synchronzation between the leaves. Its presence is fundamental to relay the information for remote synchronization which is not obtained in its absence. We have added a sentence explaining this fact.
Please correct "in an general sense" at the end of the paragraph.
3. Remote synchronization in networks of homogeneous oscillators For which kind of topologies are valid these results? Globally coupled networks? Tree or star networks?
The results hold for networks with symmetries. This has been now clarified at the beginning of the paragraph.
4. Remote synchronization of amplitude fluctuations At a certain point the authors write "A different, and pervasive, situation is when parametric mismatches are relatively small and unrelated to the topological features". Could the authors better explain the relationship between the parameters and the topological features? What does it happen when the parameter mismatches are not small? What would the authors expect when parameters are distributed according to a distribution with fat tails, thus allowing extreme events?
In this scenario, unlike the first one discussed (i.e., RS in networks of inhomogeneous oscillators) the characteristics is that there is no relationship between the parameter mismatches and the topological features of the nodes. When, on the contrary, parameter mismatches are not small, then the case of inhomogeneous oscillators is recovered and remote synchronization with the characteristics of that scenario may be observed. The case of distribution of parameter mismatches with fat tails has not yet been investigated in the current literature and is probably a very interesting, yet open question.
Please correct "at the interface the two".
5. I would add a final paragraph where discussing open problems and perspectives. What does the authors expect for more complicated networks? For example, what does it happen in a rich club graph, where a random network is enriched with some hub nodes? Some results are presented for the Kuramoto-Sakaguchi in sparse networks. Are you aware of results for other models?
Is it possible to relate remote synchronization with topological measures in case of inhomogeneous oscillators? For example, it is possible to find a correlation between remote synchronization and betweeness (or efficiency or clustering)? On the other hand, it is possible to explain remote synchronization in terms of dynamical properties? In other words, is there a relationship between remote synchronization and, for example, information transmission on shortest path lengths?)
We have added the following. Open problems and perspectives. From a theoretical point of view, a comprehensive understanding of the microscopic mechanisms underlying remote synchronization remains an open issue. It is also unsolved whether a single mechanism common to the different forms of remote synchronization here discussed is ultimately to be expected, or different causes should be ascribed to the distinct manifestations of the phenomenon and, if so, whether these can be grouped in any kind of classes. In the case of networks of homogeneous oscillators, the presence of symmetries is the primary element allowing for the onset of remote synchronization, but in networks of inhomogeneous oscillators, symmetries are not fundamental for observing the phenomenon which, on the contrary, seems to require an intricate interplay between structure and dynamics of the unit. In this latter scenario, the phenomenon consistently appears in random topologies with different densities of hubs, but the relationship of the extent of the phenomenon with the topological features of the network is still under investigation. Although significant results linking remote synchronization and correlated neuronal activities in the brain have been already obtained (Hövel et al., 2018; Vuksanović & Hövel, 2014), further investigation of this aspect is still required and has a high potential. Possible engineering applications of remote synchronization are also beginning to be considered.