Navigability, Walkability, and Perspicacity Associated with Canonical Ensembles of Walks in Finite Connected Undirected Graphs—Toward Information Graph Theory
Round 1
Reviewer 1 Report
I really liked the article under review. It is well organized, uses an understandable mathematical apparatus, has scientific and practical significance. Separately, I want to note the presence of a practical example, which is a rarity in articles with a mathematical bias.
Notes:
1. The introduction is very pretentious. The connection between the James Webb Space Telescope and the topic of the article is difficult to assess.
2. Neither the introduction, nor the key words, nor the conclusions in any way tell the reader that the entire study is based on the practical problems of routing in the city. But this is an important advantage of this work.
3. One sentence from a new line (line 85) looks like a typo.
4. In general, the authors lose sight of the fact that it would be nice to at least superficially describe in what problems the proposed theory can be used. Finite Connected Undirected Graphs are used in chemistry, machine learning, databases, social graphs, transportation, etc.
For example, there is such an application area of networks-on-chip design. In them, a very important problem is the choice of the topology of the communication system [https://doi.org/10.5772/intechopen.97262], which is exactly described by finite connected undirected graphs. An important problem is the evaluation of the properties of such graphs, incl. their routability [https://doi.org/10.12694/scpe.v22i3.1896], [https://doi.org/10.1007/978-981-15-6048-4_40], and the development of routing algorithms for them [https ://doi.org/10.1016/j.sysarc.2016.04.011], [https://doi.org/10.3390/a16010010]. And, for example, topology graphs in application specific NoCs [https://doi.org/10.1016/j.sysarc.2018.10.001], [https://doi.org/10.1109/IECON48115.2021.9589829], [https:/ /doi.org/10.1016/j.comnet.2019.03.014], [https://doi.org/10.1145/3520241], are generally very similar in nature to graphs considered in the context of walkability analysis in urban neighborhoods.
5. Correction of remark 4 could also improve the impression of the References section, where new works are mainly the works of the author himself (8 self-citations for 44 references, in my opinion, is too much). Justify the relevance of references [11], [21], [38] or remove them.
Described above.
Author Response
Response to Reviewer 1
We profoundly thank our reviewer for helping us to improve the quality of our manuscript. Reviewer's suggestions really helped us to formulate a coherent logical concept (i.e., Information Graph Theory) of our article. We have revised our manuscript thoroughly, and the changes made to the text in response to accordingly these suggestions highlighted in blue are summarized in the table given below.
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1. The introduction is very pretentious. The connection between the James Webb Space Telescope and the topic of the article is difficult to assess. |
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We thank our reviewer for this important comment. We have removed the pretentious beginning of the Introduction section and the first two references in the list. The new beginning of Introduction sounds as follows: Graph models are useful for the computer networking, sociology, urban morphology, smart city networks, chemistry, engineering, communication, logistic and management, security applications, etc., as helping us to answer the fundamental question about relations between the \textit{local} and \textit{global} properties of a networking system. In our work, we assess the quality of these relations by some \textit{information} quantities at different graph connectivity scales, ranging from the connectivity between the nearest neighbors to the connectivity with respect to infinitely long paths available in the graph. The proposed information approach to the description of graph structures (that can be named as \textit{information graph theory}) may be especially useful for the assessment of \textit{communication efficiency} between {\color{blue} individual system's units} at different time and connectivity scales”, etc. |
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2. Neither the introduction, nor the key words, nor the conclusions in any way tell the reader that the entire study is based on the practical problems of routing in the city. But this is an important advantage of this work. |
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Following reviewer’s suggestions, we have added the following paragraphs to the Introduction section: “One possible application of graph information theory is the design of \textit{Network-on-Chip} (NoC), the network-based communications subsystems on an integrated circuit, most typically between semiconductor intellectual property cores schematizing various functions of the computer system in a system on a single chip\footnote{\color{blue} We profoundly thank our reviewer for pointing us at the possible NoC related application of graph information theory.} \cite{Venkataraman:2019}. NoCs provide the advantage of customized architecture, increased scalability, and bandwidth and come in many network topologies, many of which are still experimental \cite{Misbah:2021}. The interconnections between multiple cores on a chip present a considerable influence on the performance and communication of the chip design regarding the throughput, end-to-end delay, and packets loss ratio. Based on the huge amount of supported heterogeneous cores, efficient communication between the associated processors has to be considered at all levels of the system design to ensure global interconnection \cite{Alimi:2022}. Although hierarchical architectures have addressed the majority of the associated challenges of the traditional interconnection techniques, the main limiting factor is scalability, and NoC is proved to be a promising solution. As communicating nodes require routing algorithm for successful transmission of packets, it is important to design a NoC routing algorithm capable of providing less congested paths, better energy efficiency, and high scalability \cite{Kaleem:2021}. It is important to mention that the same choices of routing algorithms when considered for more than one performance parameter of the network, do not yield expected results since different applications exhibit different network performances for the same routing technique \cite{Gogoi:2022}. Fault-tolerant mapping and routing techniques at different levels of a NoC was surveyed in \cite{Kadri:2019}. In \cite{Romanov:2021}, a method for choosing hierarchical coordinates and a greedy forwarding algorithm for path finding was presented. Nevertheless, reliability is becoming a major concern in NoC design \cite{Kadri:2019}. In particular, the accurate predictions of network packet latency, contention delay, and the static and dynamic energy consumption of the network are in demand \cite{Trajkovic:2022}.
Another evident application field of graph information theory is the study of urban morphology, i.e. urban forms and structures affecting sustainability and urban growth. Along with the spreading of new technologies and the availability of big data, cities are increasingly viewed not simply as places in space but as systems of networks and flows \cite{aAcci:2020}. This became particularly evident in the concept of a \textit{smart city}, as synergizing of architecture, technology, and the \textit{Internet of Things} that enables the collection and data exchange of objects embedded with electronics, software, sensors, and network connectivity \cite{Rassia:2017}.
All above mentioned research problems involves a variety of information quantities resolving uncertainty about evolution of some diffusion processes at different connectivity scales in a graph, and therefore the entire approach to graphs summarized in the present paper may be named as \textit{information graph theory}.”
The following sentence has been added at the end of the Abstract of the manuscript: “As evaluating communication efficiency between individual system's units at different time and connectivity scales, information graph theory is in demand for a wide range of applications, such as designing the Network-on-Chip architecture and engineering urban morphology within the concept of smart city.”
The Keywords of the manuscript have been changed, as well: “\keyword{ Statistical Ensembles of Walks; {\color{blue} Information Graph Theory}; Navigability, Walkability, and Perspicacity {\color{blue} in Networking Systems}}”
We thank our reviewer for this valuable suggestion. |
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3. One sentence from a new line (line 85) looks like a typo. |
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Thank you, the line has been changed to: |
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4. In general, the authors lose sight of the fact that it would be nice to at least superficially describe in what problems the proposed theory can be used. Finite Connected Undirected Graphs are used in chemistry, machine learning, databases, social graphs, transportation, etc. We thank our reviewer for this valuable suggestion. As we mentioned above, the following texts have been added to the Introduction section: “One possible application of graph information theory is the design of \textit{Network-on-Chip} (NoC), the network-based communications subsystems on an integrated circuit, most typically between semiconductor intellectual property cores schematizing various functions of the computer system in a system on a single chip\footnote{\color{blue} We profoundly thank our reviewer for pointing us at the possible NoC related application of graph information theory.} \cite{Venkataraman:2019}. NoCs provide the advantage of customized architecture, increased scalability, and bandwidth and come in many network topologies, many of which are still experimental \cite{Misbah:2021}. The interconnections between multiple cores on a chip present a considerable influence on the performance and communication of the chip design regarding the throughput, end-to-end delay, and packets loss ratio. Based on the huge amount of supported heterogeneous cores, efficient communication between the associated processors has to be considered at all levels of the system design to ensure global interconnection \cite{Alimi:2022}. Although hierarchical architectures have addressed the majority of the associated challenges of the traditional interconnection techniques, the main limiting factor is scalability, and NoC is proved to be a promising solution. As communicating nodes require routing algorithm for successful transmission of packets, it is important to design a NoC routing algorithm capable of providing less congested paths, better energy efficiency, and high scalability \cite{Kaleem:2021}. It is important to mention that the same choices of routing algorithms when considered for more than one performance parameter of the network, do not yield expected results since different applications exhibit different network performances for the same routing technique \cite{Gogoi:2022}. Fault-tolerant mapping and routing techniques at different levels of a NoC was surveyed in \cite{Kadri:2019}. In \cite{Romanov:2021}, a method for choosing hierarchical coordinates and a greedy forwarding algorithm for path finding was presented. Nevertheless, reliability is becoming a major concern in NoC design \cite{Kadri:2019}. In particular, the accurate predictions of network packet latency, contention delay, and the static and dynamic energy consumption of the network are in demand \cite{Trajkovic:2022}.
Another evident application field of graph information theory is the study of urban morphology, i.e. urban forms and structures affecting sustainability and urban growth. Along with the spreading of new technologies and the availability of big data, cities are increasingly viewed not simply as places in space but as systems of networks and flows \cite{aAcci:2020}. This became particularly evident in the concept of a \textit{smart city}, as synergizing of architecture, technology, and the \textit{Internet of Things} that enables the collection and data exchange of objects embedded with electronics, software, sensors, and network connectivity \cite{Rassia:2017}.
All above mentioned research problems involves a variety of information quantities resolving uncertainty about evolution of some diffusion processes at different connectivity scales in a graph, and therefore the entire approach to graphs summarized in the present paper may be named as \textit{information graph theory}.”
Moreover, the Reference list have been enriched by 11 new references, and the most of them are relatively resent, belonging to 2019-2022: {\color{blue} \bibitem{Johnson:2014} Johnson, J., \textit{Hypernetworks in the Science of Complex Systems}, Vol. \textbf{3} World Scientific Series "\textit{Complexity Science}", Imperial College Press, 2014.
\bibitem{Venkataraman:2019} Venkataraman, N.L., Kumar, R., "Design and analysis of application specific network on chip for reliable custom topology". \textit{Comput. Netw}. \textbf{158}, C 69–76, 2019.
\bibitem{Misbah:2021} Misbah Manzoor, Roohie Naaz Mir, Najeeb-Ud-Din Hakim, " A Review Of Design Approaches For Enhancing The Performance Of Nocs At Communication Centric Level", \textit{Scalable Computing: Practice and Experience}, Vol. \textbf{22}, Issue 3, pp. 347–364, 2021.
\bibitem{Alimi:2022} Alimi I, K. Patel R, Aboderin O, \textit{et al.} "Network-on-Chip Topologies: Potentials, Technical Challenges, Recent Advances and Research Direction", in \textit{Network-on-Chip - Architecture, Optimization, and Design Explorations}. IntechOpen. DOI: 10.5772/intechopen.97262, 2022.
\bibitem{Kaleem:2021} Kaleem, M., Isnin, I.F.B., "A Survey on Network on Chip Routing Algorithms Criteria". In: Saeed, F., Al-Hadhrami, T., Mohammed, F., Mohammed, E. (eds) \textit{Advances on Smart and Soft Computing. Advances in Intelligent Systems and Computing}, vol \textbf{1188}. Springer, Singapore, 2021.
\bibitem{Gogoi:2022} Gogoi, A., Ghoshal, B., Sachan, A., Kumar, R., Manna, K., "Application driven routing for mesh based Network-on-Chip architectures", \textit{Integration, the VLSI Journal}. \textbf{84}:C, pp 26-36. Online publication date: 1-May-2022 at \url{https://doi.org/10.1016/j.vlsi.2021.12.008}, 2022.
\bibitem{Kadri:2019} Nassima Kadri, Mouloud Koudil, "A survey on fault-tolerant application mapping techniques for Network-on-Chip", \textit{Journal of Systems Architecture}, Vol. \textbf{92}, P. 39-52, 2019.
\bibitem{Romanov:2021} Romanov, N., Myachin, Sukhov, A., "Fault-Tolerant Routing in Networks-on-Chip Using Self-Organizing Routing Algorithms," \textit{IECON 2021 – 47th Annual Conference of the IEEE Industrial Electronics Society}, Toronto, ON, Canada, pp. 1-6, 2021.
\bibitem{Trajkovic:2022} Trajkovic, J., Karimi, S., Hangsan, S., Zhang, W., "Prediction Modeling for Application-Specific Communication Architecture Design of Optical NoC". \textit{ACM Trans. Embed. Comput. Syst.} \textbf{21}, 4, Article 35, 2022.
\bibitem{aAcci:2020} Luca D'Acci., L., (Editor), Michael Batty (Foreword by) \textit{The Mathematics of Urban Morphology} in Birkhauser Series: \textit{Modeling and Simulation in Science}. ISBN: 9783030123833, 2020.
\bibitem{Rassia:2017} Rassia, S.Th., Pardalos, P., (Eds.) “Smart City Networks: Through the Internet of Things”, In Springer Series Optimization and Its Applications. ISBN: 978-3-319-61312-3, 2017. } |
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5. Correction of remark 4 could also improve the impression of the References section, where new works are mainly the works of the author himself (8 self-citations for 44 references, in my opinion, is too much). Justify the relevance of references [11], [21], [38] or remove them. |
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Following reviewer’s suggestions, we removed the reference [38] from the list and added the following justification to for the references [11], [21] to the text: In addition, as we mentioned above, 11 new recent references have been added to the Reference list of the manuscript. |
The entire manuscript has been edited, and multiple corrections have been made throughout the text. We profoundly thank our reviewer again for his multiple very important suggestions and advises. The special thanks to the reviewer have been added as a footnote at the frontpage of the manuscript:
“\footnote{\color{blue} We profoundly thank our reviewer for pointing us at the possible NoC related application of graph information theory.}”
Author Response File: 
Author Response.docx
Reviewer 2 Report
This is a very interesting paper. The author is proposing to apply information theory to graph-theory problems. In a way, the other direction is address, too: graph theory may also be used to solve problems in information theory. The author proposes, as he writes in the title, to go towards graph information theory, where graph theory and information theory use each other's techniques.
The study leads to the definition of the canonical ensemble of walks, which is shown to be related to an already known topological index called the eigenvector centrality. Other indices, the navigability, strive and perspicacity are defined by marrying techniques in information theory with results in spectral graph theory.
As was already said, this paper is very compelling. It is recommended to publish this paper in this journal as is.
Author Response
Response to Reviewer 2
We profoundly thank our reviewer for his kind words and suggestion to accept the manuscript.
In order to further improve the quality of our manuscript, we have revised our manuscript thoroughly, and the changes made to the text in response to accordingly these suggestions highlighted in blue are summarized in the table given below.
|
We have removed the pretentious beginning of the Introduction section and the first two references in the list. The new beginning of Introduction sounds as follows: Graph models are useful for the computer networking, sociology, urban morphology, smart city networks, chemistry, engineering, communication, logistic and management, security applications, etc., as helping us to answer the fundamental question about relations between the \textit{local} and \textit{global} properties of a networking system. In our work, we assess the quality of these relations by some \textit{information} quantities at different graph connectivity scales, ranging from the connectivity between the nearest neighbors to the connectivity with respect to infinitely long paths available in the graph. The proposed information approach to the description of graph structures (that can be named as \textit{information graph theory}) may be especially useful for the assessment of \textit{communication efficiency} between {\color{blue} individual system's units} at different time and connectivity scales”, etc. |
|
We have added the following paragraphs to the Introduction section: “One possible application of graph information theory is the design of \textit{Network-on-Chip} (NoC), the network-based communications subsystems on an integrated circuit, most typically between semiconductor intellectual property cores schematizing various functions of the computer system in a system on a single chip\footnote{\color{blue} We profoundly thank our reviewer for pointing us at the possible NoC related application of graph information theory.} \cite{Venkataraman:2019}. NoCs provide the advantage of customized architecture, increased scalability, and bandwidth and come in many network topologies, many of which are still experimental \cite{Misbah:2021}. The interconnections between multiple cores on a chip present a considerable influence on the performance and communication of the chip design regarding the throughput, end-to-end delay, and packets loss ratio. Based on the huge amount of supported heterogeneous cores, efficient communication between the associated processors has to be considered at all levels of the system design to ensure global interconnection \cite{Alimi:2022}. Although hierarchical architectures have addressed the majority of the associated challenges of the traditional interconnection techniques, the main limiting factor is scalability, and NoC is proved to be a promising solution. As communicating nodes require routing algorithm for successful transmission of packets, it is important to design a NoC routing algorithm capable of providing less congested paths, better energy efficiency, and high scalability \cite{Kaleem:2021}. It is important to mention that the same choices of routing algorithms when considered for more than one performance parameter of the network, do not yield expected results since different applications exhibit different network performances for the same routing technique \cite{Gogoi:2022}. Fault-tolerant mapping and routing techniques at different levels of a NoC was surveyed in \cite{Kadri:2019}. In \cite{Romanov:2021}, a method for choosing hierarchical coordinates and a greedy forwarding algorithm for path finding was presented. Nevertheless, reliability is becoming a major concern in NoC design \cite{Kadri:2019}. In particular, the accurate predictions of network packet latency, contention delay, and the static and dynamic energy consumption of the network are in demand \cite{Trajkovic:2022}.
Another evident application field of graph information theory is the study of urban morphology, i.e. urban forms and structures affecting sustainability and urban growth. Along with the spreading of new technologies and the availability of big data, cities are increasingly viewed not simply as places in space but as systems of networks and flows \cite{aAcci:2020}. This became particularly evident in the concept of a \textit{smart city}, as synergizing of architecture, technology, and the \textit{Internet of Things} that enables the collection and data exchange of objects embedded with electronics, software, sensors, and network connectivity \cite{Rassia:2017}.
All above mentioned research problems involves a variety of information quantities resolving uncertainty about evolution of some diffusion processes at different connectivity scales in a graph, and therefore the entire approach to graphs summarized in the present paper may be named as \textit{information graph theory}.”
The following sentence has been added at the end of the Abstract of the manuscript: “As evaluating communication efficiency between individual system's units at different time and connectivity scales, information graph theory is in demand for a wide range of applications, such as designing the Network-on-Chip architecture and engineering urban morphology within the concept of smart city.”
The Keywords of the manuscript have been changed, as well: “\keyword{ Statistical Ensembles of Walks; {\color{blue} Information Graph Theory}; Navigability, Walkability, and Perspicacity {\color{blue} in Networking Systems}}”
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The line 85 has been changed to: |
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The Reference list have been enriched by 11 new references, and the most of them are relatively resent, belonging to 2019-2022: {\color{blue} \bibitem{Johnson:2014} Johnson, J., \textit{Hypernetworks in the Science of Complex Systems}, Vol. \textbf{3} World Scientific Series "\textit{Complexity Science}", Imperial College Press, 2014.
\bibitem{Venkataraman:2019} Venkataraman, N.L., Kumar, R., "Design and analysis of application specific network on chip for reliable custom topology". \textit{Comput. Netw}. \textbf{158}, C 69–76, 2019.
\bibitem{Misbah:2021} Misbah Manzoor, Roohie Naaz Mir, Najeeb-Ud-Din Hakim, " A Review Of Design Approaches For Enhancing The Performance Of Nocs At Communication Centric Level", \textit{Scalable Computing: Practice and Experience}, Vol. \textbf{22}, Issue 3, pp. 347–364, 2021.
\bibitem{Alimi:2022} Alimi I, K. Patel R, Aboderin O, \textit{et al.} "Network-on-Chip Topologies: Potentials, Technical Challenges, Recent Advances and Research Direction", in \textit{Network-on-Chip - Architecture, Optimization, and Design Explorations}. IntechOpen. DOI: 10.5772/intechopen.97262, 2022.
\bibitem{Kaleem:2021} Kaleem, M., Isnin, I.F.B., "A Survey on Network on Chip Routing Algorithms Criteria". In: Saeed, F., Al-Hadhrami, T., Mohammed, F., Mohammed, E. (eds) \textit{Advances on Smart and Soft Computing. Advances in Intelligent Systems and Computing}, vol \textbf{1188}. Springer, Singapore, 2021.
\bibitem{Gogoi:2022} Gogoi, A., Ghoshal, B., Sachan, A., Kumar, R., Manna, K., "Application driven routing for mesh based Network-on-Chip architectures", \textit{Integration, the VLSI Journal}. \textbf{84}:C, pp 26-36. Online publication date: 1-May-2022 at \url{https://doi.org/10.1016/j.vlsi.2021.12.008}, 2022.
\bibitem{Kadri:2019} Nassima Kadri, Mouloud Koudil, "A survey on fault-tolerant application mapping techniques for Network-on-Chip", \textit{Journal of Systems Architecture}, Vol. \textbf{92}, P. 39-52, 2019.
\bibitem{Romanov:2021} Romanov, N., Myachin, Sukhov, A., "Fault-Tolerant Routing in Networks-on-Chip Using Self-Organizing Routing Algorithms," \textit{IECON 2021 – 47th Annual Conference of the IEEE Industrial Electronics Society}, Toronto, ON, Canada, pp. 1-6, 2021.
\bibitem{Trajkovic:2022} Trajkovic, J., Karimi, S., Hangsan, S., Zhang, W., "Prediction Modeling for Application-Specific Communication Architecture Design of Optical NoC". \textit{ACM Trans. Embed. Comput. Syst.} \textbf{21}, 4, Article 35, 2022.
\bibitem{aAcci:2020} Luca D'Acci., L., (Editor), Michael Batty (Foreword by) \textit{The Mathematics of Urban Morphology} in Birkhauser Series: \textit{Modeling and Simulation in Science}. ISBN: 9783030123833, 2020.
\bibitem{Rassia:2017} Rassia, S.Th., Pardalos, P., (Eds.) “Smart City Networks: Through the Internet of Things”, In Springer Series Optimization and Its Applications. ISBN: 978-3-319-61312-3, 2017. } |
|
We removed the reference [38] from the list and added the following justification to for the references [11], [21] to the text: |
The entire manuscript has been edited, and multiple corrections have been made throughout the text.
Author Response File: Author Response.docx
