## Periodic Reporting for period 2 - Scale-FreeBack (Scale-Free Control for Complex Physical Network Systems)

Reporting period: 2018-03-01 to 2019-08-31

• What is the problem/issue being addressed?

Technology achievements were typically built upon fundamental theoretical findings, but nowadays technology seems to be evolving faster than our ability to develop new concepts and theories. For example intelligent traffic systems benefit from many technical innovations; mobile phones, radars, cameras and magnetometers can be used to measure traffic flow and provide large sets of valuable data. However, these huge technological advances have not been used to the full so far: traffic lights are far from functioning optimally and traffic management systems do not always prevent the occurrence of congestions.

So what is missing? Such systems affect our daily lives; why aren’t they on pace with technological advances? This is perhaps because they have become far more complex than the analytical tools available for managing them. Systems have many components, communicate with each other, have self-decision-making mechanisms, share an enormous amount of information, and form networks. Research in control systems has challenged some of these features, but not in a very concerted manner.

Scale-FreeBack proposes a new paradigm by approaching the problem with a new holistic vision, and novel design methods ensuring the scalability of the whole chain (modelling, observation, and control), and not just that of single components. For this purpose, it is first proposed to investigate appropriate scale-free dynamic modeling approaches breaking down the network system complexity, and then to devise control algorithms specifically tailored for such models. A fundamental innovation in the control approach adopted here is that it will inherit the “scale-free” property of the models, and thus be scale-free by design. Scale-free control refers here to the ability of a control algorithm to be easily scaled to system networks of arbitrary dimensions.

• Why is it important for society?

Scale-FreeBack proposes to take advantage of the new opportunities presented by the latest large-scale sensing technologies, while dealing with the demanding control issues which arise because of the need for cyber-physical security specific to large-scale interconnected networks. In addition to investigating the purely theoretical aspects, the Scale-FreeBack team also expects to come up with some innovative control solutions for improving traffic management systems, but theory in applicable in many other complex large-scale systems where control systems are essential (Smart grids, intelligent cities, power networks, etc.)

• What are the overall objectives?

The overall objective of Scale-FreeBack is to develop holistic scale-free control methodology addressing complex network systems in the widest sense. Scale-FreeBack is intended to go beyond traditional control approaches dealing with complexity on the sole basis of numerical optimization methods, by setting the foundations for a new scale-free network control theory dealing with complex physical networks of an arbitrary dimension which are of major societal interest.

The project will focus first on developing appropriate mathematical scale-free dynamic modeling approaches which can be used to break down the complexity of network systems, and then on building estimation and control algorithms which will be specifically tailored to these models. It is also planned to apply, test and validate the findings obtained in the field of road traffic networks throughout the duration of the project. The project intent to meet the following challenges;

1. When and how can a complex large-scale (homogenous) network system be represented by a scale-free model having the requisite controllability/observability properties?

2. How can the internal states of a scale-free network system be monitored and estimated by using information originating from sources of various kinds ?

3. How will it be possible to design scale-free control algorithms and make them robust/resili

Technology achievements were typically built upon fundamental theoretical findings, but nowadays technology seems to be evolving faster than our ability to develop new concepts and theories. For example intelligent traffic systems benefit from many technical innovations; mobile phones, radars, cameras and magnetometers can be used to measure traffic flow and provide large sets of valuable data. However, these huge technological advances have not been used to the full so far: traffic lights are far from functioning optimally and traffic management systems do not always prevent the occurrence of congestions.

So what is missing? Such systems affect our daily lives; why aren’t they on pace with technological advances? This is perhaps because they have become far more complex than the analytical tools available for managing them. Systems have many components, communicate with each other, have self-decision-making mechanisms, share an enormous amount of information, and form networks. Research in control systems has challenged some of these features, but not in a very concerted manner.

Scale-FreeBack proposes a new paradigm by approaching the problem with a new holistic vision, and novel design methods ensuring the scalability of the whole chain (modelling, observation, and control), and not just that of single components. For this purpose, it is first proposed to investigate appropriate scale-free dynamic modeling approaches breaking down the network system complexity, and then to devise control algorithms specifically tailored for such models. A fundamental innovation in the control approach adopted here is that it will inherit the “scale-free” property of the models, and thus be scale-free by design. Scale-free control refers here to the ability of a control algorithm to be easily scaled to system networks of arbitrary dimensions.

• Why is it important for society?

Scale-FreeBack proposes to take advantage of the new opportunities presented by the latest large-scale sensing technologies, while dealing with the demanding control issues which arise because of the need for cyber-physical security specific to large-scale interconnected networks. In addition to investigating the purely theoretical aspects, the Scale-FreeBack team also expects to come up with some innovative control solutions for improving traffic management systems, but theory in applicable in many other complex large-scale systems where control systems are essential (Smart grids, intelligent cities, power networks, etc.)

• What are the overall objectives?

The overall objective of Scale-FreeBack is to develop holistic scale-free control methodology addressing complex network systems in the widest sense. Scale-FreeBack is intended to go beyond traditional control approaches dealing with complexity on the sole basis of numerical optimization methods, by setting the foundations for a new scale-free network control theory dealing with complex physical networks of an arbitrary dimension which are of major societal interest.

The project will focus first on developing appropriate mathematical scale-free dynamic modeling approaches which can be used to break down the complexity of network systems, and then on building estimation and control algorithms which will be specifically tailored to these models. It is also planned to apply, test and validate the findings obtained in the field of road traffic networks throughout the duration of the project. The project intent to meet the following challenges;

1. When and how can a complex large-scale (homogenous) network system be represented by a scale-free model having the requisite controllability/observability properties?

2. How can the internal states of a scale-free network system be monitored and estimated by using information originating from sources of various kinds ?

3. How will it be possible to design scale-free control algorithms and make them robust/resili

The work produced during this period has been in connection with the PhD and Post-doc students:

• Nicolas Martin (PhD)

• Stéphane Mollier (PhD)

• Muhammad Umar Niazi (PhD)

• Martin Rodriguez-Vega (PhD)

• Liudmila Tumash (PhD)

• Denis Nikitin (PhD)

• Ujjwal Pratap (PhD)

• Giocomo Casadei (Post-doc)

• Vadim Bertrand (Research Engineer)

WP1. Scale-free modeling and aggregation.

WP1.-T1.1-2. On-line partitioning algorithms for evolutionary scale-free Networks. This work is done in connection with the ERC PhD Student Nicolas Martin (Dec.16-Dec19) in co-direction with one of my team colleague Paolo Frasca from the CNRS. Here we deal with the problem of how to find a scale-free abstraction out of an arbitrary network and how to use this abstraction to benefit of the scale-free properties in term of control.

We devise an algorithm, called MergeToScaleFree, and allows finding a partition of an arbitrary network resulting in a scale-free abstraction while preserving two properties of the initial network: the equilibrium point of the dynamic equation (up to a projection) and the flow network property. This algorithm has been applied on large-scale networks such as the urban traffic network of Grenoble (20 000 nodes). This work has been presented at the ERC workshop in September 2017, at the 6th International Conference on Complex Networks and Their Applications in Lyon in December 2017, and at the ECC conference in Cyprus in June 2018. Finally, a journal version of this work was published at the IEEE Transactions on Network Science and Engineering. We, now, work on a direct application of this algorithm in the context of epidemic spreading. In particular, we search the best subset of nodes to remove in the network to reduce the spreading in a homogeneous network using the scale-free abstraction. The strategy that we propose is called MergeToCure. This work was presented at the NecSys conference in Groningen in August 2018.

Publications:

• Nicolas Martin, Paolo Frasca, Carlos Canudas-de-Wit. Network reduction towards a scale-free structure preserving physical properties. COMPLEX NETWORKS 2017 - 6th International Conference on Complex Networks and Their Applications, Nov 2017, Lyon, France. pp.1-3 2017

• Nicolas Martin, Paolo Frasca, Carlos Canudas-de-Wit. A network reduction method inducing scale-free degree distribution. European Control Conference 2018, Jun 2018, Limassol, Cyprus.

• Nicolas Martin, Paolo Frasca, Carlos Canudas-de-Wit. Large-scale network reduction towards scale-free structure. IEEE Transactions on Network Science and Engineering, IEEE, 2018, pp.1-12.

• Nicolas Martin, Paolo Frasca, Carlos Canudas de Wit. MergeToCure: a New Strategy to Allocate Cure in an Epidemic over a Grid-like network Using a Scale-Free Abstraction. NecSys, Aug 2018, Groningen, Netherlands. 51 (23), pp.34-39 2018, IFAC-PapersOnLine

WP1.-T1.3. Aggregated Scale-Free Models for large-scale traffic system.

Large scale traffic networks are a popular topic nowadays due to the impact traffic has in our everyday life, both economically and health-wise. City management are interested in understanding the evolution of traffic and its patterns over the city in order to take decision on potential changes and to design new and more functional infrastructure. Within the ERC, we have explored several edges of this problem: 2-D macroscopic models for traffic fluid models, and aggregation of velocity data to predict traveling times between main points of interest.

Novel Macroscopic 2-D Models for Urbain Traffic Networks. This work is done in connection with the ERC PhD Student Stéphane Mollier (Oct.16-Oct19) in co-direction with one of my team colleague Maria Laura Delle Monache from INRIA and Benjamin Seibold, Associate Professor of Mathematics at the Applied Mathematics and Scientific Computing Group, Department of Mathematics Temple University, Philadelphia USA. Here we deal with the problem of developing new 2-D flu

• Nicolas Martin (PhD)

• Stéphane Mollier (PhD)

• Muhammad Umar Niazi (PhD)

• Martin Rodriguez-Vega (PhD)

• Liudmila Tumash (PhD)

• Denis Nikitin (PhD)

• Ujjwal Pratap (PhD)

• Giocomo Casadei (Post-doc)

• Vadim Bertrand (Research Engineer)

WP1. Scale-free modeling and aggregation.

WP1.-T1.1-2. On-line partitioning algorithms for evolutionary scale-free Networks. This work is done in connection with the ERC PhD Student Nicolas Martin (Dec.16-Dec19) in co-direction with one of my team colleague Paolo Frasca from the CNRS. Here we deal with the problem of how to find a scale-free abstraction out of an arbitrary network and how to use this abstraction to benefit of the scale-free properties in term of control.

We devise an algorithm, called MergeToScaleFree, and allows finding a partition of an arbitrary network resulting in a scale-free abstraction while preserving two properties of the initial network: the equilibrium point of the dynamic equation (up to a projection) and the flow network property. This algorithm has been applied on large-scale networks such as the urban traffic network of Grenoble (20 000 nodes). This work has been presented at the ERC workshop in September 2017, at the 6th International Conference on Complex Networks and Their Applications in Lyon in December 2017, and at the ECC conference in Cyprus in June 2018. Finally, a journal version of this work was published at the IEEE Transactions on Network Science and Engineering. We, now, work on a direct application of this algorithm in the context of epidemic spreading. In particular, we search the best subset of nodes to remove in the network to reduce the spreading in a homogeneous network using the scale-free abstraction. The strategy that we propose is called MergeToCure. This work was presented at the NecSys conference in Groningen in August 2018.

Publications:

• Nicolas Martin, Paolo Frasca, Carlos Canudas-de-Wit. Network reduction towards a scale-free structure preserving physical properties. COMPLEX NETWORKS 2017 - 6th International Conference on Complex Networks and Their Applications, Nov 2017, Lyon, France. pp.1-3 2017

• Nicolas Martin, Paolo Frasca, Carlos Canudas-de-Wit. A network reduction method inducing scale-free degree distribution. European Control Conference 2018, Jun 2018, Limassol, Cyprus.

• Nicolas Martin, Paolo Frasca, Carlos Canudas-de-Wit. Large-scale network reduction towards scale-free structure. IEEE Transactions on Network Science and Engineering, IEEE, 2018, pp.1-12.

• Nicolas Martin, Paolo Frasca, Carlos Canudas de Wit. MergeToCure: a New Strategy to Allocate Cure in an Epidemic over a Grid-like network Using a Scale-Free Abstraction. NecSys, Aug 2018, Groningen, Netherlands. 51 (23), pp.34-39 2018, IFAC-PapersOnLine

WP1.-T1.3. Aggregated Scale-Free Models for large-scale traffic system.

Large scale traffic networks are a popular topic nowadays due to the impact traffic has in our everyday life, both economically and health-wise. City management are interested in understanding the evolution of traffic and its patterns over the city in order to take decision on potential changes and to design new and more functional infrastructure. Within the ERC, we have explored several edges of this problem: 2-D macroscopic models for traffic fluid models, and aggregation of velocity data to predict traveling times between main points of interest.

Novel Macroscopic 2-D Models for Urbain Traffic Networks. This work is done in connection with the ERC PhD Student Stéphane Mollier (Oct.16-Oct19) in co-direction with one of my team colleague Maria Laura Delle Monache from INRIA and Benjamin Seibold, Associate Professor of Mathematics at the Applied Mathematics and Scientific Computing Group, Department of Mathematics Temple University, Philadelphia USA. Here we deal with the problem of developing new 2-D flu

Progress beyond the state of the art on Scale-free dynamic network modelling & analysis.

The work on model aggregation toward scale-free structures and its related application has been done in connection with the ERC PhD Student Nicolas Martin. We have for the first time in the control arena proposed an algorithm for such model aggregation with the suited scale-free structure. In addition to seek for such structure, we also propose to make this aggregation by preserving some original physical system properties (such as mass conservation), but also other relevant control properties (such as equilibria values). Next steps will be to include more complex control properties such as controllability and/or observability, and to consider evolutionary networks, which structure change (slowly) in time. The new control properties to preserve during the network modeling aggregation will be in connection with the mathematical properties fund already in the results in WP2.

During the summer 2018 (June 11 to August 20) Nicolas Martin will visit Prof Jun-ichi Imura, from the School of Engineering at the Tokyo Institute of Technology laboratory in Tokyo, Japan. Prof Imura participate to our workshop las year and a collaboration with his group is envisioned along the visit of Nicolas Martin. This visit will be in the frame of the JSPS summer program. We expected that this collaboration will allow us to strength our current work on network reduction and network control at large scale systems.

The work on Aggregated Scale-Free Models for large-scale traffic system is done in connection with the ERC PhD Student Stéphane Mollier. The main progress beyond the state of the art is the derivation for the first time of a two-dimensional PDE describing how the flow along the network is propagate via a continuum distribution. The result can be seen as the generalization of the well-known 1-D version of the LWR model, and its corresponding space-time discretization as the generalization of the Cell Transmission Model. In addition, we have studied several extension to make the model even more representative. This includes making the associated fundamental diagram space dependent allowing for highest capacity in areas with high road density, and devising a new method to validate the results from a microscopic simulation. This type of comparison is of high value as it is the closest possible simulation comparison that can be achieved with respect the reality. To be more precise, each vehicle motion from the microscopic simulation, is associate to a local density function, then all density functions of each vehicle in the network are added to from a special (and time varying) density distribution which is then compared with the density distribution predicted by our 2-D LWR model. The model then is able to predict shock and advection waves in the 2-D plane. Future work include: the model extension for networks with two-ways roads, aggregation in a SF graph structure.

Progress beyond the state of the art on Scale-free dynamic network modelling & analysis.

The work on model aggregation toward scale-free structures and its related application has been done in connection with the ERC PhD Student Nicolas Martin. We have for the first time in the control arena proposed an algorithm for such model aggregation with the suited scale-free structure. In addition to seek for such structure, we also propose to make this aggregation by preserving some physical system original properties (such as amass conservation), but also other relevant control properties (such as equilibria values). Next steps will be to include more complex control properties such as controllability and/or observability, and to consider evolutionary networks, which structure change (slowly) in time. The new control properties to preserve during the network modeling aggregation will be in connection with the mathematical properties fund already in the results in WP2.

During the summer

The work on model aggregation toward scale-free structures and its related application has been done in connection with the ERC PhD Student Nicolas Martin. We have for the first time in the control arena proposed an algorithm for such model aggregation with the suited scale-free structure. In addition to seek for such structure, we also propose to make this aggregation by preserving some original physical system properties (such as mass conservation), but also other relevant control properties (such as equilibria values). Next steps will be to include more complex control properties such as controllability and/or observability, and to consider evolutionary networks, which structure change (slowly) in time. The new control properties to preserve during the network modeling aggregation will be in connection with the mathematical properties fund already in the results in WP2.

During the summer 2018 (June 11 to August 20) Nicolas Martin will visit Prof Jun-ichi Imura, from the School of Engineering at the Tokyo Institute of Technology laboratory in Tokyo, Japan. Prof Imura participate to our workshop las year and a collaboration with his group is envisioned along the visit of Nicolas Martin. This visit will be in the frame of the JSPS summer program. We expected that this collaboration will allow us to strength our current work on network reduction and network control at large scale systems.

The work on Aggregated Scale-Free Models for large-scale traffic system is done in connection with the ERC PhD Student Stéphane Mollier. The main progress beyond the state of the art is the derivation for the first time of a two-dimensional PDE describing how the flow along the network is propagate via a continuum distribution. The result can be seen as the generalization of the well-known 1-D version of the LWR model, and its corresponding space-time discretization as the generalization of the Cell Transmission Model. In addition, we have studied several extension to make the model even more representative. This includes making the associated fundamental diagram space dependent allowing for highest capacity in areas with high road density, and devising a new method to validate the results from a microscopic simulation. This type of comparison is of high value as it is the closest possible simulation comparison that can be achieved with respect the reality. To be more precise, each vehicle motion from the microscopic simulation, is associate to a local density function, then all density functions of each vehicle in the network are added to from a special (and time varying) density distribution which is then compared with the density distribution predicted by our 2-D LWR model. The model then is able to predict shock and advection waves in the 2-D plane. Future work include: the model extension for networks with two-ways roads, aggregation in a SF graph structure.

Progress beyond the state of the art on Scale-free dynamic network modelling & analysis.

The work on model aggregation toward scale-free structures and its related application has been done in connection with the ERC PhD Student Nicolas Martin. We have for the first time in the control arena proposed an algorithm for such model aggregation with the suited scale-free structure. In addition to seek for such structure, we also propose to make this aggregation by preserving some physical system original properties (such as amass conservation), but also other relevant control properties (such as equilibria values). Next steps will be to include more complex control properties such as controllability and/or observability, and to consider evolutionary networks, which structure change (slowly) in time. The new control properties to preserve during the network modeling aggregation will be in connection with the mathematical properties fund already in the results in WP2.

During the summer