Evolving scattering networks for engineering disorder

Network science provides a powerful tool for unraveling the complexities of social, technological and biological systems. Constructing networks using wave phenomena is also of great interest in devising advanced hardware for machine learning, as shown in optical neural networks. Although most wave-based networks have employed static network models, the impact of evolving models in network science provides strong motivation to apply dynamical network modeling to wave physics. Here the concept of evolving scattering networks for scattering phenomena is developed. The network is defined by links, node degrees and their evolution processes modeling multi-particle interferences, which directly determine scattering from disordered materials. I demonstrate the concept by examining network-based material classification, microstructure screening and preferential attachment in evolutions, which are applied to stealthy hyperuniformity. The results enable independent control of scattering from different length scales, revealing superdense material phases in short-range order. The proposed concept provides a bridge between wave physics and network science to resolve multiscale material complexities and open-system material design.

The referees' reports seem to be quite clear.Naturally, we will need you to address all of the points raised.
While we ask you to address all of the points raised, the following points need to be substantially worked on: -Compare (qualitatively *and* quantitatively if possible) your proposed method against existing methods for hyperuniform materials, scattering methods, quantum graphs, etc., as requested by Referees #2 and #3.
-Better clarify the novelty of your proposed method when compared to other methods.
-Better clarify and define the terms used in the paper, such as "wave", "wave network", and "evolving network", to avoid confusion.
-Add benchmark results beyond the Born approximation region to address point 2 from Referee #1.
Please use the following link to submit your revised manuscript and a point-by-point response to the referees' comments (which should be in a separate document to any cover letter):

[REDACTED]
scattering properties, including stealthy hyperuniform disordered structures, open-system material design and "super-dense" material phases for scattering.
The study presented is very comprehensive with many numerical experiments performed and discussed in great detail; the manuscript is also generally well-written (although the presentation is fragmented at times) and, in my view, is well-suited for publication in Nature Computational Science.
There are a couple of important issues that need addressing: 1. Throughout the manuscript (including the title) is mentioned the syntagm "wave networks".At face value (as mentioned above and noted in the manuscript), what the author does is to develop a "network-based" exploration/design/classification of disordered materials.While I fully agree that the network concepts are essential to the whole formalism, I find it misleading here as is can be easily confused with physical networks which are of relevance of the many applications mentioned in the manuscript (e.g.photonic band gap materials).Not to mention that for materials displaying physical networks (e.g.2D TE polarization or 3D photonic band gap materials), the current structure factorbased analysis is not appropriate.The author will need to address this aspect in the manuscript and make sure there is no confusion possible.
2. In the same note, the other word in the syntagm, "wave" is also a source of possible confusion.Throughout the manuscript, the author works in the Born approximation, a rather restrictive regime of scattering.While there are scattering physical phenomena where the Born approximation is well justified, many of the applications which seem to be the focus of the discussion in the manuscript (again band gap materials, be they electronic, acoustic/photonic or photonic) strongly rely upon multiple scattering and resonance scattering where the formalism developed here fails.The author does present a comparison between results under the Born approximation and "full-wave" simulations, but the agreement is unsurprising since the parameters were chosen to set the system in the Bornapproximation regime.To make my argument clearer, current study cannot predict if a conventional band gap opens in the structures designed, at what frequency it may open or what its size would be.Indeed, the gap eventually occurs due to backscattering states, but their nature is much more complex than presented here and resonance scattering, and multiple scattering play an essential role impossible to be captured in the simple model of scattering presented here.Again, the author would need to clarify this aspect and I wouldsuggest replacing the word "wave" with "scatteting" or similar.
Reviewer #2 (Remarks to the Author): The manuscript entitled "Evolving wave networks for engineering disorder" describes theoretical evolving wave networks, which address networks whose connections and number of nodes change in time, and which the author uses to describes complex and open physical systems.Moreover, a specific design properties is looked for, i.e. a special correlation, the so called hyper uniformity.
The manuscript is well-written, detailed, and address the growing field of wave network, as the author calls it.Network theory is transforming the way we model complex phenomena, and its application to physical systems, from complex materials to nonlinear interaction is growing.
The key novelty here is the addition of "evolution", i.e. time dependence.I had therefore high expectation of seeing some dynamical properties, while instead the manuscript describes algorithms to developed large networks adding a particle at a time.This is not new, it is similar to molecular dynamics approaches, and to the way hyper uniform materials have been designed so far (eg https://doi.org/10.1073/pnas.1705130114, or https://doi.org/10.1364/OPTICA.3.000763).I also find the manuscript full of text books results, eg where the structure factor is derived in Results, or figure 3, which are not new and should go to supplementary material.The network approach used in the introduction is soon lost in the manuscript, and I am not sure what new science I am learning from it, and further more which computational science.Therefore I cannot recommend it for publication in Nature Computer Science and instead I would recommend it for a more specialised journal.
More in details 1.The author never mentions quantum graphs, which are very close to wave networks.I believe they should be compared and introduced.
2. Evolving networks is the key concept, but it is never defined.
3. what is the difference of this network method and just a traditional scattering method?Which network property is used here, eg in fig 3 ?4. can the proposed method outperform current ones?
Reviewer #3 (Remarks to the Author): In this paper, the author developed a novel wave-network scheme for the design and optimization of photonic network materials, which was demonstrated in stealthy hyperuniform material systems.In particular, the author formulated the material optimization problem, in which the materials are composted of a varying number of isotropic scatters, into an evolving wave network in the context of scattering.One key novelty of the work is the proposed mapping between network science and wave physics, through the definition of nodes, weighted links, degree distributions, and evolution processes based on scattering theory.The utility of the framework was demonstrated by designing and achieving extraordinary material states such as effectively denser or sparser particle distributions for shortrange order while preserving long-range order of crystals or Poisson processes.This approach provides the network-based interpretation of wave phenomena, extending the candidate platforms for wave neural networks.
The paper is overall very well written and of great topical interest.The mapping between network and wave physics, as well as the concept using evolving network for photonic material design are very novel and inspiring.A key advantage compared to previously work on photonic material optimization and hyperuniform material design is that the current work allows varying particle numbers, enabling multi-scale open material system design.I'm happy to recommend its publication once the author addresses the following comments and suggestions for minor revisions: Although viewing material scattering as a wave network is very inspiring, the key quantity of interest that characterizes the scattering behavior is the statistic structure factor, which was the objective function employed in the optimization.In this regard, the current approach, especially on the fundamental mathematical level, bears similarity with the collective coordinate approach developed by Torquato and co-workers, which were discussed in many of the references cited in the paper.The key differences between the two include: (i) the collective coordinate approach does not explicitly consider an underlying "network" for the wave physics (ii) it does not allow varying particle numbers in its current implementation.It might be useful for the author to elaborate these points a little in the paper.
Some minor points: P3, L65, "…suggested network modeling…" should be "… suggested network model…"?In Fig 1, the network modeling for scattering process was not very apparent, as the links were not shown?

Reply to Editorial Requests
-Compare (qualitatively *and* quantitatively if possible) your proposed method against existing methods for hyperuniform materials, scattering methods, quantum graphs, etc., as requested by Referees #2 and #3.
The novelty of my work is based on "network model" and "evolution process" applied to wave physics.Both properties are essential for realizing evolving scattering networks.I included the following discussion and Table 1 to compare my proposal with several traditional methods.

(i) Collective coordinate method
The collective coordinate method developed by Torquato and co-workers [refs 37, 60] allows for designing hyperuniform materials by minimizing the potential energy obtained with the simplified modelling of scattering.The key novelty of my work against this method is (i-1) the realization of the "network model" characterized by scattering and (i-2) the application of the evolution with "varying" particle numbers.The potential energy in the collective coordinate method has been defined for a fixed number of particles.

(ii) Molecular dynamics
Molecular dynamics for designing hyperuniformity is the extension of the collective coordinate method, which is based on the potential energy minimization achieved by molecular dynamics with the equation of motion.Therefore, the key novelty of my work from this method is the same as those in (i) because the potential energy is again defined for a fixed number of particles.

(iii) Quantum graphs
The key novelty of my work against quantum graphs is (i) the extraction of the network kernel and (ii) the evolution process.
First, quantum graphs correspond to the network-based modelling of rigorous scattering matrix methods based on full interactions between wave nodes, which hinders the extraction of the kernel part of wave networks when participating elements are numerous.The solution for this challenging issue is to utilize the traditional approach in scattering problems: using the "approximated" Born series.Although the Born series and the differential wave equation are mathematically identical, the Born series enables the systematic approximation of scattering, allowing for extracting the network kernel using the first-order Born approximation.In the revised manuscript, I clarified such a novelty of my work against quantum graph theory.
(Lines 48-53) Notably, quantum graphs provide an analytical tool to fully describe the interactions between wave nodes [13][14][15][16] .However, extracting the kernel of a wave network requires platform-specific simplification, as shown in the reflectionless and single-channel assumptions for the network modelling of guided 19,27,33 and diffractive 20 systems.In this context, the efficient network modelling of scattering phenomena having complex interferences from multiple particles is still an open question.
(Lines 147-153) For example, quantum graphs enable the network modelling of scattering through the graph edges defined by the metric graph between particles and the governing Hamiltonian, and the graph vertices for field boundary conditions [13][14][15][16] .Although the quantum graph model for scattering corresponds to the network interpretation of a rigorous scattering matrix, this rigorous modelling, at the same time, hinders the extraction of the kernel part of wave networks, especially when participating elements are numerous, such as scattering from disordered materials.
(Lines 157-159) From the analysis of wave scattering using the Lippmann-Schwinger equation that allows for extracting kernel parts of scattering with the Born series 51 , I develop an evolution model for … Second, another novelty of my work is the "evolution", which has not been studied in quantum (Lines 154-156) I also note that all the previous network structures applied to wave phenomena [19][20][21]27,33 , including quantum graphs [13][14][15][16] , have employed static or generative models with time-independent … (iv) Wave neural networks Wave neural networks have been implemented with different platforms: the cascaded unitary and diagonal operators realized with phase delay lines and beam splitters [ref. 19], r multiple diffractive layers [ref. 20]. The concept of wave neural networks has not been applied to both the evolution process and the design of hyperuniformity.
Table 1.The comparison of traditional design methods and evolving scattering networks.The properties of interest denote the necessary conditions for applying evolving networks to wave physics, while preserving the advantages of network science: simplified and systematic understanding of complex phenomena.Here, "No" does not mean "impossible".It may be close to the answer to the question about the explicit implementation of the property.For example, although quantum graphs may be implemented with the evolution process, such an approach is still absent.

Yes
No No Yes No

Evolving scattering network
Yes Yes Yes Yes Yes Based on the listed comparisons, I carefully revised the manuscript, as follows: (Lines 338-345) The traditional strategy to achieve hyperuniform point patterns is the collective coordinate method 37,58,60 or its extension to molecular dynamics 38,61 , which minimizes the potential energy defined by particle positions and the wavevector k.Compared with the evolving scattering network model, the collective coordinate method does not explicitly consider an underlying network structure for wave physics.Furthermore, because the potential energy in the collective coordinate method is defined for the system of a fixed number of particles, the method does not allow varying particle numbers in its current implementation, as similar to generative models in network science 53 , … (Lines 392-395) … the preferential attachment is a dynamical process, in sharp contrast to the static or generative methodologies with preassigned rules, such as the collective coordinate method 37,58,60 or its extension to molecular dynamics 38,61 .
-Better clarify the novelty of your proposed method when compared to other methods: what does your method allow, more broadly speaking, that others do not?
First, compared with a traditional scattering method (e.g.collective coordinate approach) or evolutionary algorithms, the critical difference is the network concept based on wave interferences.
Although traditional approaches have been studied for designing wave scattering, these methods include neither the network viewpoint nor open-system natures-dynamically changing the system size.
Therefore, it is difficult to examine the impact of an individual network element in the traditional methods, lacking the bridge between scattering phenomena and wave neural networks that require individual elements, i.e. "neurons".I revised the manuscript to clarify the above discussion.
Second, the advantage of the proposed method is also clear when handling dynamically-varying scattering systems, as shown in Fig. 4-the first demonstration of material screening with SHU.For open-system problems, such as the alteration of a given system with the additional inclusion of elements (e.g., Fig. 4), the evolution process is a more superior strategy, as shown in the SHU conservation with evolving network models in Fig. 4.Although the final states of a material (or a network) obtained from those two approaches may not always be different, the evolution process can also reveal unexplored states, as shown in the discovery of super-dense material phases in shortrange order in Figs 5 and 6 by introducing the "preference" for material design for the first time.
-Better clarify and define the terms used in the paper, such as "wave", "wave network", and "evolving network", to avoid confusion.
I replaced the term "wave networks" with "scattering networks" in the revised manuscript (including the title) to clarify the impact of my work.I also included the definitions of "evolving network models" and "evolving scattering networks": (Lines 23-24) Evolving network models-the models that characterize the mechanisms and natures of timevarying networks-have stimulated significant advances in network science and related disciplines.
(Lines 65-67) Here, I propose the concept of evolving scattering networks-the open-system wave network models with a dynamically changing number of particles inside a system-which provides a novel tool for multiscale material design with target scattering responses.
-Add benchmark results beyond the Born approximation region to address point 2 from Referee #1.
As Referee #1 commented, my work is suitable to engineer weak scattering rather than reproducing multiple-scattering-based results, such as bandgaps.The engineering of hyperuniform patterns using evolving scattering networks is close to the necessary but not sufficient condition for bandgap physics.The validity of the first-order Born approximation compared to the full-wave analysis has been thoroughly studied in previous literature ([ref.51; J. Comput.Chem.23, 1297 (2002); Prog.
I agree that further generalization of evolving networks to higher-order Born series and fullvectorial wave equations is essential.I revised the manuscript following Referee #1's comment.
(Lines 498-499) To extend such intriguing features to strong scattering conditions beyond the first-order Born approximation, the concept of evolving wave networks needs to be generalized to higher-order Born series to cover multiple and resonant scattering and to full-vectorial wave equations for 3D structures.
(Lines 504-507) Because realizing hyperuniform patterns defined by structure factors is one of the necessary conditions for the complete bandgap by guaranteeing the unique existence …

Reply to Reviewer 1's report
Dear Editor, Please find enclosed my comments on the manuscript "Evolving wave networks for engineering disorder" by Yu, submitted for publication in Nature Computational Science.The study presented is very comprehensive with many numerical experiments performed and discussed in great detail; the manuscript is also generally well-written (although the presentation is fragmented at times) and, in my view, is well-suited for publication in Nature Computational Science.
I sincerely appreciate the reviewer's professional and positive comments on my manuscript.Following the reviewer's suggestions, I carefully revised my manuscript and now believe that my manuscript was greatly improved.
There are a couple of important issues that need addressing: 1. Throughout the manuscript (including the title) is mentioned the syntagm "wave networks".
At face value (as mentioned above and noted in the manuscript), what the author does is to develop a "network-based" exploration/design/classification of disordered materials.While I fully agree that the network concepts are essential to the whole formalism, I find it misleading here as is can be easily confused with physical networks which are of relevance of the many applications mentioned in the manuscript (e.g.photonic band gap materials).Not to mention that for materials displaying physical networks (e.g.2D TE polarization or 3D photonic band gap materials), the current structure factorbased analysis is not appropriate.The author will need to address this aspect in the manuscript and make sure there is no confusion possible.
Thank you very much for your comment.In the revised manuscript, I clarified that the concept of "wave networks" is distinct from "physical networks", which correspond to the networks defined by the composition of material or structural design.
(Lines 40-41) In addition to understanding physics in material or structural networks, realizing networks defined by wave-matter interactions has received considerable attention.
The limitation of the current structure-factor-based analysis was also clarified in the Discussion section (shown in the reply to Point 2).Also, as shown in the reply to Point 2, I replaced the term "wave networks" with "scattering networks" to accurately describe my results.

2.
In the same note, the other word in the syntagm, "wave" is also a source of possible confusion.Throughout the manuscript, the author works in the Born approximation, a rather restrictive regime of scattering.While there are scattering physical phenomena where the Born approximation is well justified, many of the applications which seem to be the focus of the discussion in the manuscript (again band gap materials, be they electronic, acoustic/photonic or photonic) strongly Indeed, the gap eventually occurs due to backscattering states, but their nature is much more complex than presented here and resonance scattering, and multiple scattering play an essential role impossible to be captured in the simple model of scattering presented here.Again, the author would need to clarify this aspect and I would suggest replacing the word "wave" with "scattering" or similar.
I completely agree with the reviewer's opinion on the terminology "wave networks".As the reviewer stated, my work is based on the Born approximation for scattering problems, and therefore, it is suitable to engineer weak scattering rather than reproducing multiple-scattering-based results, such as bandgaps.In this context, the engineering of hyperuniform patterns using evolving scattering networks is close to the necessary but not sufficient condition for bandgap physics.To handle bandgap phenomena, further generalization of the concept of evolving networks to higher-order Born series and full-vectorial wave equations is essential, which will be a future research topic.
Following the reviewer's comments, I replaced the term "wave networks" with "scattering networks" in the revised manuscript (including the title) to clarify the impact of my work.I also carefully revised the manuscript following the reviewer's suggestion.
(Lines 498-499) To extend such intriguing features to strong scattering conditions beyond the first-order Born approximation, the concept of evolving wave networks needs to be generalized to higher-order Born series to cover multiple and resonant scattering and to full-vectorial wave equations for 3D structures.
(Lines 504-507) Because realizing hyperuniform patterns defined by structure factors is one of the necessary conditions for the complete bandgap by guaranteeing the unique existence …

Reply to Reviewer 2's report
The manuscript entitled "Evolving wave networks for engineering disorder" describes theoretical evolving wave networks, which address networks whose connections and number of nodes change in time, and which the author uses to describes complex and open physical systems.Moreover, a specific design properties is looked for, i.e. a special correlation, the so called hyper uniformity.
The manuscript is well-written, detailed, and address the growing field of wave network, as the author calls it.Network theory is transforming the way we model complex phenomena, and its application to physical systems, from complex materials to nonlinear interaction is growing.
I sincerely appreciate the reviewer's positive comments on the topic and presentation of my manuscript.
In the revision, I have focused on clarifying the novelty of my work following the reviewer's comments.
The key novelty here is the addition of "evolution", i.e. time dependence.I had therefore high expectation of seeing some dynamical properties, while instead the manuscript describes algorithms to developed large networks adding a particle at a time.This is not new, it is similar to molecular dynamics approaches, and to the way hyper uniform materials have been designed so far (eg https://doi.org/10.1073/pnas.1705130114, or https://doi.org/10.1364/OPTICA.3.000763).
Thank you very much for your comment, which is certainly helpful in clarifying the novelty of my work.
The novelty of my work is based on the following two features: "network model" and "evolution process" for wave physics.Both properties are essential for realizing evolving networks of scattering phenomena.In this context, I included the discussion and Table 1 in this reply letter to compare my proposal with traditional design methods or other related topics.

(i) Collective coordinate method
The Table 1.The comparison of traditional design methods and evolving scattering networks.The properties of interest denote the necessary conditions for applying evolving networks to wave physics, while preserving the advantages of network science: simplified and systematic understanding of complex phenomena.Here, "No" does not mean "impossible".It may be close to the answer to the question about the explicit implementation of the property.For example, although quantum graphs may be implemented with the evolution process, such an approach is still absent.Based on the listed comparisons, I carefully revised the manuscript, as follows:

Network modelling
(Lines 338-345) The traditional strategy to achieve hyperuniform point patterns is the collective coordinate method 37,58,60 or its extension to molecular dynamics 38,61 , which minimizes the potential energy defined by particle positions and the wavevector k.Compared with the evolving scattering network model, the collective coordinate method does not explicitly consider an underlying network structure for wave physics.Furthermore, because the potential energy in the collective coordinate method is defined for the system of a fixed number of particles, the method does not allow varying particle numbers in its current implementation, as similar to generative models in network science 53 , … (Lines 392-395) … the preferential attachment is a dynamical process, in sharp contrast to the static or generative methodologies with preassigned rules, such as the collective coordinate method 37,58,60 or its extension to molecular dynamics 38,61 .
I also cordially emphasize that the dynamical features of the proposed scattering networks were studied in the original manuscript.The key dynamics of evolving networks are the time-varying network structures and properties obtained with the evolution process, e.g., the inclusion or annihilation of network nodes.After bridging wave parameters and network parameters in the section of "Evolving scattering networks", the results in Fig. 4g, (Lines 38-42) The use of network science is widespread throughout physics, as shown in the network modelling of material states 3 , potential landscapes 8 , and interacting quantum processors 10 , and in the field of quantum graph theory [13][14][15][16] .
(Lines 63-64) … will provide an insight into complex materials and artificial neural networks in photonics, acoustics, quantum graphs, and other wave mechanics.
In the revised manuscript, I emphasized the following aspects of the novelty of my work against quantum graphs.

I. Extracting the kernel network from scattering
A quantum graph is built with (i) the metric graph quantifying the inter-particle distances, (ii) the In the revised manuscript, I clarified such a novelty of my work against quantum graph theory.As Reviewer 1 also pointed out a similar issue-the restriction of my work in describing general wave phenomena-I changed the terminology "wave network" to "scattering network".
(Lines 48-53) Notably, quantum graphs provide an analytical tool to fully describe the interactions between wave nodes [13][14][15][16] .However, extracting the kernel of a wave network requires platform-specific simplification, as shown in the reflectionless and single-channel assumptions for the network modelling of guided 19,27,33 and diffractive 20 systems.In this context, the efficient network modelling of scattering phenomena having complex interferences from multiple particles is still an open question.
(Lines 147-153) For example, quantum graphs enable the network modelling of scattering through the graph edges defined by the metric graph between particles and the governing Hamiltonian, and the graph vertices for field boundary conditions [13][14][15][16] .Although the quantum graph model for scattering corresponds to the network interpretation of a rigorous scattering matrix, this rigorous modelling, at the same time, hinders the extraction of the kernel part of wave networks, especially when participating elements are numerous, such as scattering from disordered materials.
(Lines 157-159) From the analysis of wave scattering using the Lippmann-Schwinger equation that allows for extracting kernel parts of scattering with the Born series 51 , I develop an evolution model for …

II. Introducing the evolution process in network-based material design
As the reviewer commented, the key novelty of my manuscript is the "evolution" of a scattering network, which has not been studied in quantum graphs.(Lines 154-156) I also note that all the previous network structures applied to wave phenomena [19][20][21]27,33 , including quantum graphs [13][14][15][16] , have employed static or generative models with time-independent network … 2. Evolving networks is the key concept, but it is never defined.
Thank you very much for your comment.I included the definition of evolving network models: (Lines 23-24) Evolving network models-the models that characterize the mechanisms and natures of timevarying networks-have stimulated significant advances in network science and related disciplines.
I also defined evolving scattering networks more rigorously to clarify the novelty of my work against previous efforts.
(Lines 65-67) Here, I propose the concept of evolving scattering networks-the open-system wave network models with a dynamically changing number of particles inside a system-which provides a novel tool for multiscale material design with target scattering responses.
3. what is the difference of this network method and just a traditional scattering method?Which network property is used here, eg in fig 3?
Compared with a traditional scattering method (e.g.collective coordinate approach) or evolutionary algorithms, the critical difference is the network concept based on wave interferences.
The impact of the network interpretation is to decompose complex phenomena into simple subphenomena systematically.Although traditional approaches have been studied for designing wave scattering, these methods include neither the network viewpoint nor open-system naturesdynamically changing the system size (as stated in the reply to Point 1, quantum graphs have not been employed to evolving networks).Therefore, it is difficult to examine the impact of an individual network element in the traditional methods, lacking the bridge between scattering phenomena and wave neural networks that require individual elements, i.e. "neurons".
In this resubmission, I revised the manuscript to clarify the above discussion.
In this context, Fig. 3d-l demonstrates the length-scale-dependent network properties through the node degree distributions.As the reviewer stated, the nature of SHU materials-crystal-like longrange order and Poisson-like short-range order-is well known; it may be textbook results.However, its quantification using network parameters-node degrees-is a new result and demonstrates such an intermediate property using the network analysis for the first time.Also, the visualization of node degrees (Fig. 3j-l) not only shows the statistical homogeneity of hyperuniformity but also presents a sharp distinction when compared with the evolving network results in Fig. 4a-f and 5a,d,g.
Therefore, I'd like to maintain the current form of Fig. 3, while clarifying that the results themselves have been well-known.
(Lines 289-293) With this reciprocal-space design process, I revisit the comparison among the uncorrelated Poisson disorder (Fig. 3a), evolving SHU material (Fig. 3b), and … to interpret the length-scale natures of each material state with the network concept.
4. can the proposed method outperform current ones?
The optimization time consumption and computing resources are not the topic of this manuscript.
However, the advantage of the proposed method is clear when handling dynamically-varying scattering systems, as shown in Fig. 4-the first demonstration of material screening with SHU.
The difference of my approach compared to previous ones corresponds to the difference between evolving networks (changing node number) and static networks (fixed node number), or more generally, the difference between "open" systems and "closed" systems.The proposed method is in sharp contrast to the conventional energy minimization with a fixed number of particles, which satisfies the definition of "closed" systems.
Therefore, for open-system problems, such as the alteration of a given system with the additional inclusion of elements (e.g., Fig. 4), the evolution process is a more proper and natural strategy, as shown in the SHU conservation with evolving network models in Fig. 4.Although the final states of a material (or a network) obtained from those two approaches may not always be different, as analogous to the deterministic realization of scale-free networks [Physica A, 299, 559 (2001)], the application of (Lines 340-344) Compared with the evolving scattering network model, the collective coordinate method does not explicitly consider an underlying network structure for wave physics.Furthermore, because the potential energy in the collective coordinate method is defined for the system of a fixed number of particles, the method does not allow varying particle numbers in its current implementation, as similar to ... Some minor points: P3, L65, "…suggested network modeling…" should be "… suggested network model…"?
Thank you very much for this careful comment.I have corrected this part in the revised manuscript. In

Decision Letter, first revision:
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graphs.There exist only a few examples of handling time-varying quantum graphs, which are not about the networks with varying network sizes but about time-varying unitary evolution [Наносистемы: физика, химия, математика 6, 173 (2015)].I clarified this point.
The paper presents a comprehensive exploration of network-based classification of microstructured material and preferential attachment during material-design evolution.There are a number of interesting results and ideas put forward including a novel way to generate materials with certain scattering properties, including stealthy hyperuniform disordered structures, open-system material design and "super-dense" material phases for scattering.
rely upon multiple scattering and resonance scattering where the formalism developed here fails.The author does present a comparison between results under the Born approximation and "full-wave" simulations, but the agreement is unsurprising since the parameters were chosen to set the system in the Born-approximation regime.To make my argument clearer, current study cannot predict if a conventional band gap opens in the structures designed, at what frequency it may open or what its size would be.
suggested Optica paper[Optica 3, 763-767 (2016): ref. 58] employed the collective coordinate method developed by Torquato and co-workers [Phys.Rev.E 70, 046122 (2004); J.Appl.Phys.104,   033504 (2008): refs 37, 60].This method leads to the design of hyperuniform materials by minimizing the potential energy that is obtained with the simplified modelling of scattering.The key novelty of my work against this method is (i-1) the realization of the network modelling defined by scattering properties and (i-2) the application of the evolution process with varying particle numbers.Notably, the potential energy in the collective coordinate method has been defined for a fixed number of particles.(ii) Molecular dynamics Molecular dynamics employed in the suggested PNAS paper [Proc.Natl.Acad.Sci.114, 95709574 (2017): ref. 61] is the extension of the collective coordinate method for hyperuniformity, which was also proposed by Torquato and co-workers [Phys.Rev. X 5, 021020 (2015): ref. 38].This method is based on the potential energy minimization achieved by molecular dynamics with the equation of motion.Therefore, the key novelty of my work from this method is the same as those in (i) because the potential energy is again defined for a fixed number of particles.(iii) Quantum graphs : The novelty against quantum graphs is discussed in the reply to Point 1 in below.(iv) Wave neural networks Wave neural networks have attracted significant attention for machine learning or quantum computing.The networks have been implemented with different platforms: the cascaded unitary and diagonal operators realized with phase delay lines and beam splitters [Nat.Photon.11, 441 (2017): ref. 19], or multiple diffractive layers [Science 361, 1004-1008 (2018); ref. 20].Wave phenomena in these platforms are usually simplified for network modelling, for example, by neglecting reflections to achieve the unitary condition.The concept of wave neural networks has not been applied to both the evolution process and the design of hyperuniformity.
Fig. 3g-i denote the network parameters-node degree distributions-which are the conventional quantities for characterizing network structures [Barabási, A.-L. Network science (Cambridge university press, 2016)].The traditional material classification and its network-based interpretation are apparently not the same things, as similar to the novelty and insight of quantum graphs that interpret known quantum phenomena.Therefore, I cordially claim that these node degree distributions are not the textbook results, as shown in the original manuscript ("(Lines 296- 299) Although Fig. 3a-c shows a well-known structure factor of each material phase, Fig. 3d-i demonstrates that node degree distributions of evolving scattering networks operate as a useful tool for characterizing microstructures at the length scale of interest, bridging network analysis and material statistics.").Furthermore, Fig. 3j-l-the visualization of node degrees-are essential to understand the results of Fig. 4a-f and Fig. 5a,d,g, because Fig. 3j-l shows which elements are important in the network modelling, and this property is critical in understanding the effect of preferential attachments in the evolution process.Therefore, I decided to maintain the current form of Fig. 3.
Hamiltonian characterizing the physically defined graph edges with the metric graph, and (iii) the graph vertices reproducing the boundary condition between waves.Using quantum graphs, scattering phenomena are modelled by the scattering matrix that connects the incident and scattered fields (at the graph leads) through the internal waves (at the graph edges) where the internal waves are defined by the directions of graph edges and differential wave equations[PRL 85, 968 (2000): ref. 15].This method therefore corresponds to the network-based modelling of rigorous scattering matrix methods, fully describing the interactions between wave nodes.Although such a description allows for extracting important features in complicated systems composed of a few particles, the rigorous description of conventional quantum graph theory, at the same time, hinders the extraction of the kernel part of wave networks, especially when participating elements are numerous: e.g., examining scattering from disordered materials.Therefore, when using quantum graph models, it is not straightforward to obtain the simplified and systematic network interpretation of scattering from disordered materials while maintaining physical validity, which is one of the goals of my work.The solution for this challenging issue is to utilize the traditional approach in scattering problems: using the recursive form of the Lippman-Schwinger equation and the following Born series, and its approximation, which is applied to evolving scattering network model in the form of the firstorder Born approximation.As described in Supplementary Notes S1 and S4, I start from the structure factor originating from the approximation of the Lippman-Schwinger equation-the widely-used integral form of the governing equation to simplify scattering phenomena.Although the LippmanSchwinger equation and the differential wave equation are mathematically identical, the recursive form of the Lippman-Schwinger equation enables the systematic approximation of scattering waves, allowing for extracting the network kernel in my work.
In this work, I introduced the concept of evolving scattering networks, which is fundamentally different from previous wave networks or quantum graphs in terms of handling "open systems": allowing for the alteration of matter (i.e., an increasing particle number) and energy (i.e., minimizing the cost function) inside the system.There exist only a few examples of handling time-varying quantum graphs, and these works are not about the evolving network structures with varying network sizes or topology but about time-varying unitary quantum evolution [Наносистемы: физика, химия, математика 6, 173 (2015)], which may correspond to generative network models.I clarified this point in the revised manuscript.
the evolution process can reveal unexplored system states, as shown in the seminal finding of scalefree networks [Science 286, 509 (1999)], and the discovery of super-dense material phases in shortrange order in Figs 5 and 6 by introducing the concept of "preference" for material design for the first time.I sincerely appreciate the reviewer for this insightful comment, clarifying the novelty of my work when compared with the collective coordinate approach.Following the reviewer's suggestion, I included the extended discussion for this point in the revised manuscript.
Fig 1, the network modeling for scattering process was not very apparent, as the links were not shown?I appreciate the reviewer's helpful suggestion.To emphasize the network modelling, I included the links in the revised Fig. 1f, while I assume the plotting of a part of link weights (significant values of |wp,q K |) because the network is fully connected.(Lines 133-139) Red and blue solid lines represent the positive and negative signs of existing link weights defined by Eq. (5), respectively.Red and blue arrows also represent the positive and negative signs of newly included link weights after adding the (n+1) th particle, respectively.Only the links with significant values of |wp,q K | are assumed to be plotted because a scattering network is fully-connected.The black arrow describes the k-impulse component cos[k•(rp -rq)] of the link weight between the p th and q th particles.The transparency of the solid lines and arrows denotes the magnitude of the weights.

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