General comments:

This article introduces the concept of simplical sets, and argues for their use in future network-based disease ecology research. I find that the introduction provides an excellent written description of simplical sets and the rationale for their use in disease ecology. However, I found the definition of simplical sets to be a little convoluted. For example, the first paragraph is quite clear, but after this, their definition is based on highlighting differences between simplical sets and hypergraphs (which was not formally defined), and by some assumptions of simplical sets (i.e. downward closure) which are also not clearly defined. I strongly suggest that the definition section is re-written to be more direct, and be sure to define jargon or network-theory specific terms when you first use them.

When discussing applications of simplical sets, the authors provide detailed examples for social networks and habitat overlap, but I find their examples at the largest scale of organization (global scale / macroscale host-parasite networks) is quite sparse in comparison, and does not go into as much depth or attention to detail as for the previous examples.

Finally, I find that the figures do not do justice to the manuscript. There is great opportunity to include visualizations of their definitions and worked examples (see line-specific comments below). I feel this would greatly improve the understanding of readers, and make the manuscript more impactful as a guide for those interested in adoption of simplical sets in their research.

Line specific comments:

Line 102: You say simplical sets are mathematically similar to a hypergraph representation, but have not yet defined what a hypergraph is. This will be important for accessibility to readers with a more general background. While a sense of what a hypergraph is can be ascertained from the rest of the paragraph, it might be best to clearly define it to begin with.

Lines 105-109: I had to re-read this sentence a few times to fully parse it’s meaning. Maybe you could either split it into two, or simplify it somehow to make it more of a direct statement?

Line 111: “Downward closure” is not directly defined upon first mention. Perhaps you could re-write this and the following sentence to define the property of downward closure, then say that simplical complexes must have this property.

Lines 118-127: It seems that some weighting of the interactions within a simplex would solve this. You might assume that all dyadic interactions could possibly exist, but some are always weighted to be 0 (no interaction). Is this possible, or are simplexes not usually defined with quantitative weights? This might be something good to clarify early on.

Line 149: In this line it would be nice to hint at the scales of host-parasite/pathogen networks you envision. Do you mean individual level networks or species level networks? And at what scale? Communities? Across landscapes? Networks of regional assemblages? Aspatial networks (networks where any evidence of a species level interaction is sufficient to identify a link, regardless or timing or location)?

Lines 173-183: This seems to be a nice example of where simplical sets may be useful, but I fear that for some there may not be enough information presented here. This might be a good opportunity for a conceptual diagram. You could display the two cases (traditional network versus simplical set), and include visualizations of the networks, and how the sigmoid functions would be calculated, and how this might change our conclusions about transmission in social networks.

Lines 196-199: Again, it is unclear how you arrive at the logic of dividing the signal by the order of the simplex. As with my last comment, I think some of these examples could be better demonstrated with a visualization paired with additional equations / worked calculations.

Line 216-222: The authors raise interesting properties of simplical sets not previously discussed: individuals participating in higher-order simplical sets, the stability of sets, and aspects of embedded topologies within simplical sets. I think these are quite interesting extensions of simplical sets, and offer the opportunity to characterize networks, as well as the roles of particular notes within higher order simplical sets. I again strongly urge the authors to use visualizations to explain these extensions and expand on the potential for simplical sets to more easily impart information about network structures compared to more commonly used properties of dyadic networks.

Lines 321-349: This section encourages the use of simplical sets in large-scale species interaction networks. While I can see their application as potentially useful, I find this section to be lacking in detail compared to the previous sections. It would be good to provide some more concrete examples of where simplical sets will be useful, and base this on previous research in this field.

For example, this section seems to have overlooked previous research that incorporates higher-order network structures when analyzing these networks. For example, see the use of motifs by Wardeh et al. (2021) in Nature Communications, and the use of network embedding by Poisot et al. (2022) on arXiv. To better argue for the use of simplical sets, it would be nice to see some discussion of their use in relation to other approaches, as you have done with the previous examples.

Wardeh, M., Blagrove, M.S.C., Sharkey, K.J. et al. Divide-and-conquer: machine-learning integrates mammalian and viral traits with network features to predict virus-mammal associations. Nat Commun 12, 3954 (2021). https://doi.org/10.1038/s41467-021-24085-w

Poisot, T., et al. (2022) Network embedding unveils the hidden interactions in the mammalian virome, arXiv preprint doi.org/10.48550/arXiv.2105.14973

Lines 367-432: This section is titled “How to use simplical sets in disease ecology?”, but does not make any direct reference to disease ecology. While these seem to constitute recent examples of metric calculation with simplical sets, their connection to disease ecology is not clearly made, and they do not offer any guidance on how to use them or implement them. If intended to reflect the title, this section should be re-framed within the context of disease ecology, or else perhaps it could be reframed as “recent examples of network metric calculations with simplical sets” or something similar.

The authors have done a fine job responding to my original concerns and suggestions. I am particularly satisfied with their efforts to increase clarity and understanding through expanded and novel figures.