Soufiane Zhejiayu Ma
email: ma.zhejiayu@easybroadcast.fr

Roubia Easybroadcast

Frédéric Giroire
email: frederic.giroire@univ-cotedazur.fr

Guillaume Urvoy-Keller
email: guillaume.urvoy-keller@univ-cotedazur.fr

Q-learning for Waiting Time Control in CDN/V2V Live streaming

Keywords: hybrid P2P, live streaming, q-learning, machine learning

HTTP-based streaming has become the dominant technology for streaming due to the widespread adoption of the HTTP protocol. Many streaming providers use a combination of Content Delivery Network (CDN) and Viewer-to-Viewer (V2V) technology, known as Hybrid CDN/V2V live streaming, for both efficiency and cost-effectiveness. V2V technology allows for offloading streaming traffic from the CDN and reducing operational costs, and WebRTC technology facilitates direct V2V transfer, as it is natively supported by all browsers. In a WebRTC-based V2V network, some viewers cache the video chunks on their devices, while others wait and fetch chunks from their neighbors. A common strategy used to determine when a viewer should stop waiting for chunk delivery and revert to the CDN is called Random Waiting Time Control (RWC). However, due to the complex dynamics in the V2V system, RWC is far from optimal. In this work, we have formulated the Waiting Time Control determination problem as a reinforcement learning problem and proposed a Q-learning-based Waiting Time Control (QWC) solution. We conducted offline experiments in the Grid5000 [1] testbed and validated our results through a 14-day A/B testing in the wild. Our findings showed that QWC improves overall streaming Qualityof-Experience (QoE) in rebuffering (-29% fewer events), video quality (+17% higher), and buffer length (+5% longer), with a slightly improved V2V ratio (+5% more).

I. INTRODUCTION

The HTTP-based streaming has become a standard way to effortlessly provide live streaming and VoD service thanks to the universality of the HTTP protocol. Popular protocols are Apple HLS (HTTP Live Streaming) and Google MPEG-DASH (Dynamic Adaptive Streaming over HTTP). The Content Delivery Network (CDN) providers, such as Akamai [2], Cloudfront [START_REF]Cloudfront[END_REF], and Cloudflare [START_REF]Clouflare[END_REF] provide robust streaming delivery over multiple regions. However, the downside of CDNs is their cost which surges as the audience increases. As a solution, the V2V (viewer-to-viewer) technology based on WebRTC (Web Real-Time Communication) is used to reduce the operational cost of streaming services. It enables the browsers to open data channels with each other when they are watching the same stream so that the audience forms an active edge-device resource cache. This translates into fewer requests to the CDN. This work has been supported by the French government through the UCA JEDI (ANR-15-IDEX-01) and EUR DS4H (ANR-17-EURE-004) Investments in the Future projects, by the European Project dAIEDGE, and by Inria associated team EfDyNet.

Video players, such as HLS, can be easily used with an open-source JavaScript library. This simplicity facilitates the deployment of Web-based streaming services. Technically, the HLS player controls how the video chunks are fetched. The default behavior is straightforward in HLS players: they issue an HTTP request to get the current manifest listing all the available video chunks, and they start fetching the first one. The manifest file is updated periodically in live streaming, and leads the player to continue playing forward.

The HLS player that has been enhanced with V2V technology operates in the following manner: Initially, the viewers form a V2V overlay network that comprises a maximum of 10 other viewers. Subsequently, the manifest file is retrieved in the same manner as before. An external JavaScript library manages the task of chunk fetching, where a viewer with a new chunk notifies its neighbors about the availability of this chunk, which they can then request. In this scenario, a crucial question is: "How long should a viewer wait for a chunk to be accessible in its neighborhood before resorting to the CDN?" The answer to this question is complex in the context of live streaming, where new chunks must be swiftly distributed to maintain all clients in sync. Additionally, there are challenges associated with peer dynamics, such as peer churn [START_REF] Liu | Opportunities and challenges of peer-to-peer internet video broadcast[END_REF] (the departure of peers, which disrupts the V2V networks) and flash crowds [START_REF] Stavrou | A lightweight, robust p2p system to handle flash crowds[END_REF] (a sudden surge of audience joining the live stream). A random waiting time is often employed to prevent synchronization among viewers, as reported in [START_REF] Barbosa | Bemtv: Hybrid cdn/peerto-peer architecture for live video distribution over the internet[END_REF]. However, its influence on the V2V ratio 1 is difficult to predict. We advocate an adaptive strategy to adjust the waiting time.

In this work, we make the following contributions: [START_REF] Balouek | Adding virtualization capabilities to the Grid'5000 testbed[END_REF] We formulate the problem of selecting the waiting time as a reinforcement learning task and use a Q-learning algorithm for the agent to learn to make optimal waiting time decisions. The agent is rewarded for a short V2V download time and penalized for an inappropriate waiting time that leads to a rebuffering event (such as when the buffer is less than the average chunk transfer time). (2) Through an A/B testing deployment in a real-world scenario, we demonstrate that our Q-learning agent can enhance the waiting time decisions and increase various QoE metrics for the users without sacrificing the V2V ratio.

The paper is organized as follows: Section II frames the waiting control problem for a Hybrid CDN/V2V live streaming system and discusses the related work. In Section III, we introduce the Q-learning reinforcement learning algorithm and our design of state space, action space, and reward functions. We validate the design with an offline experiment in a large experimental platform, Grid5000 [START_REF] Balouek | Adding virtualization capabilities to the Grid'5000 testbed[END_REF]. Finally, in Section IV, we analyze the statistics collected from an A/B testing campaign in order to compare the Random Waiting Time Control (RWC) and the Q-learning-based Waiting Time Control (QWC).

II. BACKGROUND

To understand the Waiting Time Control problem in Hybrid CDN/V2V live streaming, we first present the player module of our HTTP-based hybrid CDN/V2V system. Second, we formulate the Waiting Time Control problem. Lastly, we present the related work by including other P2P or hybrid CDN/P2P systems.

Our hybrid CDN/V2V live streaming system relies on We-bRTC for building the data channels. Our system architecture involves a tracker for matching the viewers and a JavaScript library which enables the connections among Web users. Similar approach is used in many other hybrid CDN/V2V systems [2, 8]- [START_REF] Ma | When locality is not enough: Boosting peer selection of hybrid cdn-p2p live streaming systems using machine learning[END_REF].

A. Hybrid CDN/V2V chunk fetching module

Before diving into the hybrid CDN/V2V chunk fetching module (CFM), we first present the two important modules of a pure-CDN HTTP streaming player. A typical pure CDN HTTP streaming player, such as the one provided by HLS.js [START_REF]Hls.js[END_REF] and Dash.js [START_REF]Dash-Industry-Forum[END_REF] has a CFM. In HLS.js, the module is called Fragment Loader (fLoader) while in Dash.js the module is named as XHRLoader (XMLHttpRequest Loader). The function of this module is relatively straightforward: it makes HTTP requests and returns the fetched resource to the player. Note that the player module is a built-in module that controls the fetching of manifest files, quality changes, playing, pausing, stopping, seeking, and downloading of the video chunks. The player module and the CFM are in a master-slave relationship where the CFM takes orders from the player module, as shown in Figure 1.

The process of incorporating V2V capability into the pure-CDN HLS.js and Dash.js involves a re-implementation of the CFM. This implementation entails: (1) Whenever the V2V neighbors have cached the chunks, one should be obtain the chunks from its neighbors; (2) If there is a cache miss or if the V2V fetching operation times out, traditional HTTP requests are sent to retrieve the chunks from the CDN.

In our platform, the player module remains unaltered, and we integrate a V2V-enabled CFM. This "plug-and-play" feature enables the V2V service to be seamlessly integrated without the risk of disrupting other built-in functionalities of HLS or DASH, such as ABR (Adaptive Bitrate Streaming) algorithms.

B. Performance Metrics

To evaluate the system performance, we use different Quality-of-Service (QoS) and Quality-of-Experience (QoE) metrics: Fig. 1: The sequence diagram shows the player module, the chunk fetching module (CFM), and the CDN/V2V data fetching scheme. The total chunk fetching time in V2V mode comprises availability waiting time and data transfer time.

• The V2V ratio measures the percentage of data downloaded in V2V mode.

C. Waiting Time Control Problem

As mentioned earlier, the player module in the platform calls the CFM sequentially to fetch chunks. The CFM of the hybrid CDN/V2V system then decides whether to fetch the chunks from other neighbors or from the CDN. It's important to note that there is a waiting time that can be added between the moment the player requests a chunk and the moment the chunk is cached, as shown in Figure 1. This waiting time, known as the availability waiting time T av , provides a window for the chunk to be downloaded by a neighboring viewer and advertised within the network, allowing the actual V2V chunk fetching to occur after T av has passed.

In this section, we address the question of how the availability waiting time T av influences the system performance. We do so from both the perspective of individual viewers and of the performance of the entire system. For individual viewers, we examine how the availability waiting time affects their QoE. For the performance of the entire system, we examine the impact of T av on the V2V ratio.

1) Viewers having higher latency can achieve higher V2V ratio: If a peer selects a longer waiting time before fetching 2a, we see that the peers that fetch chunks late enough experience a higher chunk availability from neighbors (given by the number of providers) during their session. Furthermore, the availability is often translated into higher V2V ratio. The late peers retrieve their chunks more frequently from other peers. As demonstrated in Figure 2b, the wider the range of the random waiting time, the higher the overall V2V ratio becomes. a chunk, it will increase its latency in relation to the latest chunk of the stream. This extra waiting time provides an opportunity for other neighbors to download the next needed chunk and become providers. As a result, the peer will benefit from increased chunk availability and be able to download the chunk using V2V mode. This phenomenon is demonstrated in Figure 2a which reports results from an experiment in the Grid5000 platform involving hundreds of simulated viewers. We observe that the peers with a larger delay (larger overall streaming latency) from the live streaming playback head have more chunk providers (top plot) and, thus, are more likely to fetch the chunks from other users (bottom plot). However, the selection of waiting time is a trade-off and must balance various factors. If the waiting time is set too high, it may negatively impact the viewing experience, particularly for live events such as sports where users want to watch the content as soon as it is produced.

2) Desynchronization of users promotes V2V ratio.: From a system-wide viewpoint, availability is strongly tied to the level of desynchronization among the users. As the viewers become more desynchronized, the availability of chunks increases. The reasoning behind this is that if there is minimal desynchronization, the viewers are likely to be watching the same chunk, leaving little room for V2V exchanges. Another experiment was conducted on the Grid5000 platform where different groups of viewers were configured to take a random waiting time from different ranges. As shown in Figure 2b, the group with the widest range of waiting time (100ms -1s) achieved the highest V2V ratio.

To conclude, the waiting time problem reveals a trade-off between a higher V2V ratio and the Quality of Experience.

D. Related work

In this section, we examine previous studies that have explored the chunk diffusion strategy in both pure P2P and hybrid CDN/P2P2 networks. To emphasize the involvement of actual viewers, we use the term V2V throughout this work. One such study [START_REF] Yousef | Enabling adaptive bitrate algorithms in hybrid cdn/p2p networks[END_REF] models the system with a large window of chunks that can be downloaded between the playback head and the latest chunk. The system makes a decision every w seconds to schedule the chunks for downloading and to determine the seeders to which the requests are sent. This approach leads to a significant latency between the source of the streaming and the playback head, which is generally not ideal for live streaming. In contrast, in our work, the latest chunk is requested as soon as it is listed in the manifest file, reducing the latency between the source and the playback head.

While previous studies, such as LiveSky [START_REF] Yin | Design and deployment of a hybrid cdn-p2p system for live video streaming: experiences with livesky[END_REF], have focused on the architecture of the V2V network and its impact on performance, our approach prioritizes QoE metrics, such as low latency and fewer rebuffering events. The LiveSky system proposed a hybrid mesh-pull and tree-push multiple substream approach, in which a parent peer can continuously push content to its children, as a solution to the long delays observed in pure pull approached in P2P live streaming networks [START_REF] Zhang | A peer-to-peer network for live media streaming using a push-pull approach[END_REF]. However, this complex tree-based and mesh-based structure still involves at least three round-trip times (RTTs) in the download process, making it less efficient.

In contrast, our work aims to achieve better QoE metrics by relying on a CDN to reduce the complexity of the V2V network and remove the drawback of pure-pull in P2P systems with respect to delay. By utilizing a CDN, we can ensure that the latest chunk is requested soon after it is listed in the manifest file, minimizing latency and improving the overall streaming experience for the users. This approach aligns with the current goals of streaming service providers, which prioritize delivering high-quality, low-latency streams to their users.

PRIME [START_REF] Magharei | Prime: Peer-to-peer receiver-driven meshbased streaming[END_REF] organizes the peers in different levels, implicitly assuming that the source has low uploading capacity. Peers on the first level fetch the chunks available at the source, and then the peers on the second level pull from their parents and so on. Additional random links are used to form a mesh network, improving the resilience to churn. While they put significant effort in adjusting the number of children with the uplink capacity and content rate, such multi-level architecture creates latency which increases linearly with the level. Consequently, the users on the leaf nodes can experience a significant latency which degrades users' viewing experiences, especially in realtime events such as a sport event. In addition, the evaluation is done through simulations only. In contrast, we rely both on controlled experiments in a testbed and experiment in the wild to evaluate our proposal.

A recent closely related work is [START_REF] Yousef | Enabling adaptive bitrate algorithms in hybrid cdn/p2p networks[END_REF], where the authors consider also a hybrid CDN/P2P network for video live streaming. Their focus is on the interplay between the ABR algorithm and the CFM module since the bandwidth estimation done at the ABR level is difficult due the difference of downloading rate between a CDN server and a regular peer. In our work, we address a different problem, namely the trade-off between the V2V ratio and QoE metrics, using reinforcement learning to decide how long to wait for the neighbors before falling back to the CDN.

III. DESIGN AND OFFLINE EXPERIMENTS

This section details the Reinforcement Learning (RL) agent and the Q-learning [START_REF] Sutton | Reinforcement learning: An introduction[END_REF] algorithm we use. Furthermore, we explain the challenges encountered during an offline experiment in Grid5000 to motivate the discussion on reward shaping and an interpretation of the learned policies.

A. Reinforcement Learning Formulation

The training setup follows the standard Reinforcement Learning framework to train a Q-learning model. In Q-learning, a tabular structure known as the Q-table is maintained, which stores the Q-values Q(s, a) for all s ∈ S and a ∈ A, where S represents the set of discrete states and A represents the set of discrete actions. The Q-value Q(s, a) estimates the expected reward for the agent if it is in state s and takes action a, and then continues to interact with the environment according to some policy π until the end of the episode. A policy π maps the current state s to an action a. The of the algorithm is a Bellman equation as a simple value iteration update, using the weighted average of the current value and the new information:

Q(s t , a t ) = Q(s t , a t ) + α[r t + γ max a∈A Q(s t+1 , a) -Q(s t , a)].
(1) More details about Q-learning can be found in [START_REF] Sutton | Reinforcement learning: An introduction[END_REF].

In our waiting time control problem, the Q-learning algorithm is part of the CFM as follows. Prior to each download request from the player module at each step t, the RL agent observes the current state s t = (b t , n t , p t ), encoding Low-onbuffer, Low-on-neighbor, and the previous role, respectively. The agent takes an action a t which is to choose a range of waiting time. As a feedback for the action a t taken, the agent receives a reward r t based on the agent's contribution to V2V transfers and a penalty received if an undesired rebuffering event occurs.

1) State s t = (b t , n t , p t ): Low-on-buffer is determined by a threshold b low (in seconds) so that b t = 1 when the buffer is below b low , otherwise b t = 0. The low-on-buffer state is crucial in helping the agent to learn that a low buffer level could cause a rebuffering event. Similarly, Low-on-neighbor is determined by a threshold n low . The rationale is that an agent might be able to identify the correct waiting time according to the number of connected neighbors, e.g., wait shorter when abundant resources exist. Previous role captures the previous behavior of an agent and the dynamics within the neighborhood.The possible values of the previous role include:

(1) CDN-only, when the previous chunk was only downloaded from the CDN and no further exchange took place; (2) CDNseeder, when the previous chunk was downloaded from the CDN and subsequently sent to neighbors; (3) leecher-only, when the previous chunk was obtained from a neighbor and no uploading occurred; (4) leecher-seeder, when the previous chunk was obtained from a neighbor and then sent to another peer.

2) Action a t ∈ {1, 2, ..., k} : The Random Waiting Time Control (RWC) takes a random waiting time (milliseconds) value in the range [0, 4000). This is the legacy behavior of our library in our operational system. In the Q-learning-based Waiting Time Control (QWC), we subdivide the bounding ranges into k ranges. For instance, suppose k = 4: we get four different actions {0, 1, 2, 3} with the corresponding waiting time range [0, 1000), [1000,2000), [2000,3000), and [3000, 4000). When the agent takes action a t = 0, it samples a value from the range [0, 1000) as its waiting time.

3) Reward r t ∈ R: The reward function provides feedback to the agent to shape its behavior in fulfilling three distinct requirements: (1) promoting more V2V chunk exchanges, (2) reducing the waiting time, and (3) preventing rebuffering events. Thus, the reward function is composed of three distinct parts, as illustrated in Equation (2). The first two requirements are satisfied by the first line of the equation, where v t+1 is measured at time t + 1 and indicates whether the chunk was downloaded from V2V after the agent took action a t at time t. Similarly, u t+1 represents the number of times the chunk at time t was sent to the agent's neighbors. The discount factor η < 1 discounts the reward for longer wait, i.e., a longer wait results in a smaller reward. Note that the reward for chunk uploads (u t+1 ) is not discounted in the reward function because uploading is a desirable behavior, even when the waiting time is long. The second line of the equation fulfills the last requirement by giving the agent a negative reward -p in case the buffer is too low.

r t = v t+1 × η at + u t+1 , if buffer t+1 > b danger -p, otherwise (2)
This reward function rewards both uploading and downloading from V2V. Similarly, we are also interested in testing different reward functions such as download which rewards only downloading from V2V, and upload which rewards only uploading V2V chunk. We will detail the impact of choosing download-upload, download and upload as the reward function in Section III-B1. 4) Policy π(s t ) = a t : We employ the ϵ-greedy policy where the agent takes a random action with probability ϵ and a greedy action arg max a Q(s, a) with probability 1 -ϵ. ϵ-greedy policy helps to balance the exploration and exploitation in case the agent gets stuck on a sub-optimal policy [17, Ch. 2.3]. In a stationary environment, in the beginning, the agent starts interacting with the environment with a relatively high ϵ. As time progresses, the agent is supposed to switch from exploring to exploiting by shrinking ϵ. A common way to shrink ϵ is by using a decay parameter: after each action, ϵ := max(ϵ × δ, ϵ min ).

B. Offline experiments on Grid5000

We conducted an offline experiment campaign on Grid5000 [START_REF] Balouek | Adding virtualization capabilities to the Grid'5000 testbed[END_REF], a large-scale academic grid testbed. We set up a hybrid CDN/V2V live streaming system using a Kubernetes cluster with containerized headless Chrome browsers, which serve as viewers. Traffic control, using tc [START_REF] Hubert | tc: Manipulate traffic control settings[END_REF], can be added to limit the bandwidth of WebRTC. The configuration of the containerized Chrome browser and traffic control settings were done through a Docker file.

The experiments allow us to gain insight into (1) the influence of the reward function and (2) the advantages of using a blacklisting algorithm.

1) Reward function: The choice of reward function has a significant effect on the definition of roles within a neighborhood. As demonstrated in Figure 3a, three reward functions were tested: download, download-upload, and upload. The results show that download-upload and upload functions lead to improved performance over time, while the download function results in a gradual decline in the V2V ratio. The reason why a download-only reward fails is that by rewarding only V2V downloads, viewers are incentivized to make riskier decisions, such as waiting longer (see Figure 3b), which increases the chances of reverting to CDN.

2) Blacklisting: The previous experiments on the reward function were conducted under the assumption of equal download and upload bandwidths for all agents. However, in reality, agents have varying bandwidth capacities, particularly in terms of uplink. Therefore, we considered a scenario with 60 agents divided into three different bandwidth categories (4, 8, and 12 Mbps), each category representing 1, 2, and 3 times the streaming rate, each with 20 agents.

As shown in Figure 4a, the sliding average index over time for actions taken by agents in the three categories was analyzed. The shortest waiting time corresponds to action 0 while the longest waiting time corresponds to action 3. We expected that high-bandwidth agents would act as seeders and low-bandwidth ones as leechers. However, this was not the case, even with the use of a "download-upload" reward function, as agents across all three bandwidth categories took similar actions.

The reason for this behavior is a constraint in WebRTC. We-bRTC uses UDP for low latency, but connections are unreliable and senders are not informed when a chunk is lost. As a result, low-bandwidth peers send chunks and receive rewards even if the chunk is lost, leading them to also choose a short waiting time like high-bandwidth peers.

To mitigate this, we designed a blacklisting algorithm. This prevents a peer from requesting chunks from neighbors who have sent lost chunks, and instead requests chunks from other neighbors. This reduces future chunk losses and reduces rewards for uploading among low-bandwidth peers, who will then wait longer to receive rewards from downloading in V2V instead. In contrast, high-bandwidth peers are not affected and continue acting as seeders.

As demonstrated in Figure 4b, with the blacklisting algorithm activated, we observe a distinct average action time series compared to before. There is improved separation between the bandwidth categories. Low-bandwidth peers tend to take larger actions and wait longer to download from V2V, while highbandwidth peers take smaller actions and act as seeders.

IV. EXPERIMENT AND RESULTS

We compare QWC and RWC in production to address the following inquiries: (1) Does QWC enhance the QoE metrics compared to RWC? (2) How does QWC affect various user subgroups, particularly those with longer and shorter sessions? (3) How QWC performs during peak or non-peak hours?

A. Deployment in the wild

We used A/B testing campaigns for comparing QWC and RWC. The reason is that audience varies daily, so comparing two algorithms simultaneously removes the time-related variance. In A/B testing, the peers are assigned uniformly at random to one bucket. Then, they can only find and connect to other ones within the same bucket, and each bucket uses a different algorithm (RWC or QWC).

RWC and QWC are compared during a 14-day A/B testing deployment in the wild. RWC randomly chooses a waiting time between two bounding values; QWC learns through interactions how to choose a more specific range as discussed in Section III-A2.

For each client, statistical data is sent by the JavaScript library every 30 seconds. The library aggregates the data downloaded from the CDN and V2V, the number of rebuffering events, and the video quality. We analyzed the metrics for each one-hour interval, resulting in 336 (14 x 24) data points for the online A/B testing. Each day typically has one audience peak for a given channel, and we further analyze the performance during peak and non-peak hours.

We tested QWC on two Moroccan Over-The-Top (OTT) TV channels, which are HLS channels. Each chunk (.ts file) consists of 6 seconds of video with three different video qualities: 500KB, 900KB, and 1.3MB per chunk. The default HLS Adaptive Bitrate algorithm controls the quality switches. The majority of users are from Morocco, and most use a PC with Windows or a mobile phone to watch the live stream.

As shown in Figure 5, during the fourteen days of A/B testing, we observed a strong periodicity of the number of viewers at different hours of the day. On some specific days, (a) Upload and download-upload groups do not suffer from performance drop as in download group (ϵmin = 0.1). These experiments are carried out in a homogeneous 8Mbps environment.

(b) The download reward function drives the users to taken a larger waiting time to get rewards whereas the upload and download-upload encourage the users to be a seeder. we saw larger peaks corresponding to highly popular events. The largest peak was on January 9, 2023, when each bucket had more than 4,000 viewers during the peak.

B. Reward function and Hyperparameters

For our A/B testing campaign, we use the download-upload reward function combined with the blacklisting algorithm (see

Section III-B).
The values of the hyperparameters of the Q-learning algorithm are displayed in Table I. They were chosen based on the offline experiments. We set ϵ min = 0.1 to keep a minimum of 10% of exploration rate throughout the agent's lifetime. We use α = 0.7 as the learning rate. Because of the high churn of such a live streaming system, a very large number of peers stay a very short time in the system, less than 10 minutes. These peers only take 100 actions (1 chunk per action, 10 chunks per minute). Thus, they should learn as fast as possible good moves as they do not have the time to learn optimal actions. For this reason, we chose a high value of the learning rate to speed up the learning.

The hyperparameter γ, taken from the range [0, 1], represents the discount factor for future rewards. When γ = 0, the agent only considers short-term rewards, since future rewards are zeroed out. Conversely, when γ = 1, the agent values future rewards equally to current rewards and may be willing to delay them. However, in our problem, we do not want rewards to TABLE I: Hyperparameters settings be delayed, as the sessions are short and agents will only experience a limited number of rewards. Lastly, δ is the decay rate of ϵ. We set δ to be 0.935 so that after approximately 33 chunks (3.3 minutes), the agents will reach their minimum exploration rate. This is because we desire the users to spend one-third of their time learning and two-thirds of their time exploiting, as the average viewing time of viewers is 10 minutes.

C. Overall streaming performance

Fig. 6: Timeseries of the one-week long A/B testing. The top and bottom figures display the average chunk size and the rebuffering count per user. The average chunk size in the QWC bucket is significantly higher than viewers in the RWC bucket most of the time. Furthermore, the rebuffering is significantly reduced in QWC.

The main performance metrics are reported in Table II. They are all improved with the use of our Q-learning algorithm. Indeed, during the 14-day A/B testing comparing RWC and QWC, we see a 5% increase in the V2V ratio in QWC. Furthermore, there are 29% fewer rebuffering events in QWC than in RWC. QWC reduced the number of rebuffering per user from 1.399 to 0.981. The average duration of the rebuffering events is reduced by 8%. In terms of buffer length, the average buffer level increases from 126.73 to 133.01 seconds, a 5% of improvement. And finally, QWC offers a 17% improvement to the average chunk size. Figure 6 shows the timeseries of the two metrics which are significantly improved by QWC.

1) QWC encourages a higher bitrate: Adaptive HTTP Live Streaming is widely adopted and the quality is adapted according to the user's bandwidth and buffer occupation. The player issues an up-switch when the ABR (Adaptive Bitrate Algorithm) determines that a higher quality is more appropriate. There are several reasons for quality up-switches: [START_REF] Balouek | Adding virtualization capabilities to the Grid'5000 testbed[END_REF] an abundance of buffer, (2) a shorter chunk fetching time, which leads to a higher bandwidth estimation, and (3) reduced rebuffering events, resulting in fewer quality down-switches. Fig. 7: Improvement of QWC over RWC grouped by Day of Week. Each instance represents the improvement for a onehour duration. Note that, the median rebuffer count per user is smaller than 1.0 indicating a smaller number of rebuffering events when using our Q-learning algorithm. We now detail the analysis in terms of buffer length, chunk fetching time, and rebufferings Firstly, the buffer length is larger in QWC. Figure 7 reports the above metrics over the 14 days of the experiments. The blue boxplots show that the buffer is more abundant in QWC, as the median improvement is mostly above 1, except on Saturday. However, this could be due to the high variance of non-peak hours. To prove this point, we show in Figure 8 that the variance of the buffer length improvement is significantly higher during non-peak hours. During peak hours, we mostly observe improvements in terms of buffer length in the QWC bucket.

Secondly, the chunk fetching time is shorter in QWC. As shown in Figure 1, the chunk fetching time is a composition of waiting time and data transfer time. The chunk fetching time should be reduced in QWC. As previously mentioned in Section III-A, QWC rewards the agent to minimize the waiting time for chunks. The reward function specifically rewards the agent for waiting a shorter period of time to score a V2V chunk, while penalizing it for waiting longer. This leads to the agent learning the strategies: (1) when it lags behind its neighbors and a longer waiting time is not required, it waits for a shorter period to obtain a V2V chunk, (2) when it is ahead of its neighbors, it acts as a seeder for the next chunk to increase its rewards, and (3) when there are no active cached chunks from other viewers and waiting would result in the risk of rebuffering, the agent selects the CDN as its streaming source. Thirdly, in terms of rebuffering count, QWC has shown a significant improvement as depicted in Figure 7. The highest improvement can be seen on Wednesday and Thursday, with a median improvement of 48% and 40% respectively. Nevertheless, Figure 8 reveals that the improvement in rebuffering count is less pronounced during peak hours, with a median decrease of 25%, compared to non-peak hours where the median decrease is 40%.

2) QWC slightly increases V2V ratio: One of the key factors to consider in determining the suitability of using QWC in a production environment is whether the reduced waiting time results in a negative impact on the V2V ratio. The results of our experiments, as depicted in Figure 7, show that the V2V ratio is improved on all days except Thursday, where a slight degradation is observed. However, on Wednesday, Saturday, and Sunday, the improvement is significant. It is important to note that the improvement occurs both during peak and non-peak hours, although the variance is higher during nonpeak hours, leading to some periods of time with a slight degradation. Nevertheless, the V2V ratio is increased by 5% globally.

D. Per-viewer QoE analysis

The overall performance of the streaming system is improved with the use of the QWC method, as evidenced by the improvement of Quality of Experience (QoE) metrics. However, in a V2V streaming system, there is a high rate of churn, with a significant number of users entering and exiting the system. To better understand the impact of churn on the user's QoE, we examine the performance metrics based on the duration of the user's viewing session. Specifically, we analyze the performance metrics for users who stay in the system for more than 10 minutes and those who have an enter-and-leave behavior.

The results of this analysis are shown in Figure 9, which presents empirical Cumulative Distribution Function (CDF) plots comparing QWC and RWC for different metrics. The results for long sessions are shown at the bottom of the figure, while the results for short sessions are shown at the top. From this comparison, we can make several key observations. For example, the comparison of the V2V ratio in the first column of Figure 9 reveals that 40% of the short sessions do not download any V2V chunks. Additionally, there is a slight difference between the V2V ratios of most agents in the short sessions, with the top 25% of users having a slightly higher V2V ratio. However, the V2V ratio becomes more polarized for the long sessions, with more mass accumulating at two bounding values. This indicates that the blacklisting algorithm effectively assigns roles to agents based on their uploading capacity.

Rebuffering events are not common in short viewing sessions, with approximately 75% of these sessions being rebuffering-free. In comparison, only about 50% of the long viewing sessions are rebuffering-free. The QWC method leads to fewer rebuffering events than the RWC method, with only around 25% of users experiencing more than three rebuffering events in QWC compared to more than four rebuffering events in RWC. This improvement is particularly evident for long sessions, which correspond to users who are more engaged with the content.

The last column of the results shows that QWC also leads to an improvement in the chunk size for both short and long sessions. In QWC, more than 50% of sessions have a chunk size of 1.2MB, compared to only around 30% in RWC. As expected, the improvement in chunk size is more pronounced for long sessions, which represent the actual viewers, than for short sessions.

V. CONCLUSION AND FUTURE RESEARCH

This work addresses the waiting time control problem using a Q-learning algorithm in a commercial hybrid CDN/V2V live streaming system. We validated our design with a 14-day A/B testing campaign that demonstrated that our proposed solution can reduce the waiting time and improve buffer length without affecting the V2V ratio, significantly enhancing the viewers' QoE metrics, such as video quality and rebuffering events.

In our work, we utilized the Q-learning algorithm as the method of choice for solving the problem at hand. However, there are other approaches that could potentially be used for this problem as well. One such alternative is Deep Q-Networks (DQN) [START_REF] Mnih | Playing atari with deep reinforcement learning[END_REF], which utilizes a neural network to approximate the Q-values. This allows for the system to consider a larger number of states, making it a more robust solution. Another approach that could be considered is the Deep Deterministic Policy Gradient (DDPG) [START_REF] Lillicrap | Continuous control with deep reinforcement learning[END_REF]. This method allows for the agents to operate in a continuous action space, which eliminates the need for manually dividing the random waiting time range. Fig. 9: The Empirical Cumulative Distribution Function plots of different metrics per session. The blue line represents QWC, and the orange line represents RWC. The first (second, resp.) row represents the short (long, resp.) sessions that last less (more, resp.) than 10 minutes. Improvements due to the Q-learning algorithm are more pronounced for users really watching the stream (long sessions).

In addition to modeling independent agents in our Multiagent Reinforcement Learning approach [START_REF] Tan | Multi-agent reinforcement learning: Independent vs. cooperative agents[END_REF], it is important to consider the risk of these agents adopting a self-serving behavior and locking into specific patterns of action. To mitigate this risk, a potential avenue for future research is to explore the use of cooperative agents that share information with one another. The idea behind this is that by working together, the individual actions of the agents can be combined to form a joint action, which can lead to faster convergence and a more optimal solution. This cooperative Multi-agent Reinforcement Learning approach aligns all agents towards a shared goal of achieving a higher V2V ratio, promoting collaboration and coordination among the agents. In this way, the use of cooperative agents can result in a more efficient and effective solution, improving the overall performance of the system.

  Fig.2: In Figure2a, we see that the peers that fetch chunks late enough experience a higher chunk availability from neighbors (given by the number of providers) during their session. Furthermore, the availability is often translated into higher V2V ratio. The late peers retrieve their chunks more frequently from other peers. As demonstrated in Figure2b, the wider the range of the random waiting time, the higher the overall V2V ratio becomes.

Fig. 3 :

 3 Fig.3: Reward functions and their impact on the V2V ratio (ϵ min = 0.1)

Fig. 4 :

 4 Fig.4: Time-series of the average index of actions per peer bandwidth category without (a) or with (b) blacklisting. The figures demonstrate that the high-bandwidth peers automatically take the role of seeder when using the blacklisting algorithm.

Fig. 5 :

 5 Fig. 5: Number of viewers per bucket demonstrates a strong periodicity with the peak time coming at night. The maximum reaches more than 4 000 viewers per bucket.

Fig. 8 :

 8 Fig. 8: Improvement of QWC over RWC grouped by peak (IsPeak=True) vs non peak (IsPeak=False) hours. This figure shares the same legend as Figure 7. Each point is labeled as Peak or Non-peak based on the number of users. When the number of users is over 200, we label the hour as a peak hour.

•

  The Rebuffering Count (RebC) measures the number of rebuffering events due to buffer starvation.• The Rebuffering Time (RebT) measures the duration of the rebufferings.

• The Buffer Length measures, in seconds, how long the players' buffer is filled. If the player's buffer is low, there is a higher risk of rebuffering.

• The Average Chunk Size measures the stream quality in terms of the chunk size whose unit is bytes. For instance, in a multi-quality HLS streaming, each quality has a separate manifest file that lists the corresponding links for .ts objects. Those .ts objects have different file sizes. It is thus convenient to use the average chunk size to indicate the overall streaming quality.

TABLE II :

 II Summary of the A/B testing.

The percent of data downloaded from V2V instead of CDN. It is one of the KPIs (Key Performance Index) for Hybrid CDN/V2V Live Streaming systems

In this work, we use the term V2V to emphasize the participation of actual viewers.


Dr Jeremy Salaam

Dr Thibault Fogeron

Dr Guillaume Pilet

Dr Radu Bolbos

Dr Christophe Bucher

Dr Lhoussain Khrouz

Prof. Dr Jens Hasserodt
email: jens.hasserodt@ens-lyon.fr

Unprecedented Relaxivity Gap in pH-Responsive Fe III -Based MRI Probes

come   L'archive ouverte pluridisciplinaire

Converting MRI from a classic imaging modality into a molecular one is a technical as well as a scientific challenge that continues to require tremendous efforts. [1] A chemical solution in the form of a molecular probe able to respond to a biomolecular analyte with a sufficient signal change would have the power to redefine our current view on medical imaging and diagnosis. Academic proposals of probe designs based on the paramagnetic ions Gd(III) or Mn (II) do not yet show enough sensitivity for practical in vivo detection of analytes of biomedical interest. In fact, obtaining a satisfactory signalto-background ratio is not easy to achieve for these metal ions due to the significant relaxivity these probes already generate in their intact state, that is, prior to the encounter of the biomolecular analyte. [2] [3] [4] [5] [6] In addition, the use of gadolinium-based contrast agents (GBCAs) has raised significant health concerns since the early 2000s. [7] [8] [9] Transition metal complexes, and particularly iron complexes, are now finally beginning to arouse interest as MRI probes [10] [11] for the detection of biomarkers of tissue states such as changing redox potentials, enzyme activity, or changing pH. Prior results from our lab are in line with the observation that these chelates show relaxivity gaps between their un-activated and activated forms that habitually beat the competition. [12] [13] Previously, we had already reported iron-based designs that operate in an irreversible manner as these promise the highest free-energy differences between the un-activated and activated forms and thus a complete response to the bio-analyte. [14] An increase by a factor of 15 (FePyC3A) appears to be the highest gap reported to date, [15] an order of magnitude higher than those reported for probes based on other paramagnetic metal ions.

Here, we describe two pH-responsive, iron(III)-based, molecular probes, 1 and 2, whose un-activated and activated forms exhibit an unprecedented relaxivity gap. Previously, two pioneering designs of a pH-responsive MRI probe had already been reported: a Gadolinium-based probe designed to serve the classic proton-based MRI modality (1H channel) and showing a relaxivity change of factors of 2 to 4 over a pH range from 5 to 9 [16] , and a ratiometric lanthanide-based probe operating via a CEST modality. [17] A more recent report describes a pH-responsive FeII-based DOTA complex for 19F and CEST imaging. [18] Scheme 1. Activation mechanism of complexes 1 and 2 upon protonation of the coordinated oxazolone type moieties giving rise to an MRI signal in water.

Chelates 1 and 2 were originally intended to be assessed as iron(II) complexes, but they spontaneously oxidize already during synthesis to become ferric chelates. Even though ferric systems cannot adopt a formally spin-silent state (diamagnetism), we will show below that 1 and 2 are capable of unprecedented relaxivity increases upon activation. As desired, the pyrrolidone-based coordinating pendant arms within 1 and 2 reversibly coordinate or de-coordinate the metal center in water in response to changing pH. While the pendant arm in 1 represents a cyclic carbamate, that of 2 constitutes a lactam (Scheme 1).

The cyclic nature of these coordinating sites has been chosen deliberately because this denies the system the ability to coordinate the metal center with any heteroatom on the ring other than nitrogen. As such, either nitrogen coordinates, or water. Free oxazolidinone and pyrrolidone exhibit pKa's in DMSO of 20.8 [19] and 24.2 [20] , respectively. Importantly, their tautomeric equilibria (iminol vs. carboxamide) are also largely in favor of the carboxamide version. [21] We thus hypothesized that iron chelates incorporating these heterocycles would only be able to enforce their coordination upon deprotonation to the amidate form. Any protonation to the electroneutral carboxamide form would then automatically result in the heterocycles' de-coordination and thus to a relaxivity increase, hopefully close to physiological pH.

Ferric complexes 1 and 2 were synthesized by alkylation of DPTACN with two tosylated reagents easily accessible from commercial material, followed by complexation with FeCl3 (see SI). Both chelates are purified by RP-HPLC. The last synthetic step, complexation, yielded 1 and 2 in 32% and 23%, respectively. The UV spectra of 1 and 2 in water exhibit an intense d-d band at 475 nm and 500 nm, respectively, which is characteristic for ferric complexes (Figures S1 andS2). Their recrystallization in a MeCN/Et2O mixture furnished purple-colored monocrystals suitable for X-ray diffraction analysis. ORTEP representation of X-ray structure analyses [22] of complex 1 (left) and 2 (right). Hydrogens are omitted for clarity.

The resulting structures (Figure 1) [22] showed hexa-coordinate complexes with iron-nitrogen bonds congruent with a lowspin state (ca. 2.00 Å, tables S1 and S2). For complex 1 the C-O bond length of the carbamate group is 1.210 Å, and for complex 2 the same bond for the lactam group is 1.22 Å. These values are in agreement with a C-O double bond as found in the free forms of 2-oxazolidinone and 2-pyrrolidone (1.215 Å [23] and 1.238 Å [24] , respectively) while the length of single C-O bonds of the corresponding tautomeric forms should be closer to 1.3 Å. Also, the Fe-N bond of these two moieties is 0.1 Å shorter than the Fe-N bonds of the picolyl pendant arms. It thus has to be concluded that the coordination does not impose an iminolate/imidate form, but rather an amidate, with the negative charge located close to the nitrogen-iron bond. The structure of 1 shows some level of distortion due to the methyl group that requires significant space.

Magnetic moments for both complexes were measured by the Evans method using an NMR spectrometer (Figure S4) in neutral and acidic media. Magnetic moments of 2.12 µB (Bohr Magnetons) at pH 7.2, and 2.35 µB at pH 0.8 for complex 1, and 1.87 µB at pH 7.0 and 1.94 µB at pH 1.2 for complex 2 were determined. The absence of a switch from a low-to a high-spin state due to de-coordination of the pendant arm was also checked by EPR spectroscopy for complex 2 that displayed no signature for a high-spin version of Fe(III), neither in acidic nor in neutral water (Figure S5).

Complex 1 and 2 were submitted to detailed electrochemical analysis in aqueous electrolytes (0.1 M KNO3) The cyclic voltammogram (CV) of 2 recorded at a glassy carbon working electrode (100 mV/s) displays one reversible wave at E1/2 = -0.11 V (vs. ECS) attributed to a one-electron reduction of the iron(III) center (Figure S6). The reversible character of this wave is consistent with the conclusion that the coordination scheme of the iron center does not change significantly over the time scale of this measurement. Conversely, similar measurements with 1 yielded an irreversible reduction wave at Ep = + 0.10 V associated with a re-oxidation signal observed for the reverse scan at 0.42 V (Figure S6). In this case, the irreversible nature of the iron-centered reduction wave is attributed to the poor stability of the electro-generated iron(II) complex which evolves at the CV time scale into a different complex. The observation of a re-oxidation wave at a more positive potential also suggests that the Fe(II) ion is harder to oxidize in the EC product than in the initial complex. Taken together, these data thus lead us to propose that this complex is the reduced version of compound 1'.

The preferred tautomeric forms of the heterocyclic pendant arms turned out to be those displaying a negative charge on the nitrogen, i.e., the amidate versions. Accordingly, the iron center is stabilized in its ferric state. Hence, our attempt to obtain an iron(II) complex by use of a ferrous salt during synthesis directly led to the ferric complex. Our TACN-based ligands exercise a significant ligand field meant to impose the low-spin state on the corresponding ferrous complexes, and so it does not come as a surprise that the ferric complex is also found in the low-spin state. Indeed, high-spin iron(III) complexes usually exhibit coordination spheres with hard, oxygen-based coordination sites. [25] It is worth noting that the result of a reaction between the ligand and a ferrous salt shows the exact same characteristics as the one with a ferric salt. No de-complexation was observed after spontaneous oxidation, nor any potential loss of stability. This demonstrates the equally good fit of the DPTACN system for iron centers in both oxidation states.

Relaxation time measurements (T1) of the complexes' aqueous solutions (4 mM) were conducted with a standard NMR spectrometer at 9.4 T (Figure 2). Acid titration from neutral pH (HCl addition) shows progressive reduction of T1 indicating the de-coordination of the heterocycle-bearing pendant arm and concurrent access of water to the first coordination sphere. The T1 values thus reached (0.1 to 0.2 sec) are congruent with a relaxation mechanism involving inner sphere water. As stated above, a non-coordinated lactam (pyrrolidone) exhibits a pKa of 24.2 in DMSO, that of a cyclic carbamate (oxazolidinone) is found at 12 in water and 20.8 in DMSO. When coordinated to the iron(III) center and considering the strength of the 5-membred chelate ring involved, the pKa value decreases for both moieties to approximately the same value of 3.5. The initial T1 plateau at neutral pH was different for both complexes. We investigated the temperature dependency of the activation for one of the complexes (Figure S7), and the T1 difference is more pronounced at higher temperature (37 °C). This is largely because the T1 values of the initial, coordinatively closed states differ substantially. Indeed, the temperature dependence of proton relaxation times has been reported previously. [26] This data also illustrates that the pKa (3.5) is not temperature-dependent.

The kinetics of this pH-dependent response were obtained my measuring T1 every 10 minutes after an initial acidification. The results show a significant decrease in T1 during the first 10 minutes (83% and 68% decrease for complex 1 and 2, respectively), and a further, slower decrease until stabilization at approximately one hour. These kinetics were further investigated with complex 2 by preparing different solutions at different pH and monitoring the T1 values at larger time points (Figure S8). No further evolution can be detected for the longer waiting times.

That the T1 values of 1 and 2 at pH 7 do not reach the value for pure water (3.05 sec) should not come as a surprise. We cannot expect the acid-base equilibrium to be of a completely binary nature, i.e., that at pH extremes only one coordination state is populated to a 100 percent. Rather, small proportions of water-coordinated complexes at pH 7 are likely responsible for the observed T1 values, of course in conjunction with an additional contribution from the outer-sphere influence of the low-spin but paramagnetic metal center. Other contributing factors might be the high concentration used (4 mM, chosen to observe a significant gap in T1 by titration) and the non-zero spin value of the low-spin ferric center.

The switching capacity of 1 and 2 is remarkably robust. It can endure several cycles of de-coordination/re-coordination without showing any sign of fatigue (Figure 3). The population of the de-coordinated state 1' and 2' (reflected by the observed T1 values) depends on proton concentration and is consistent for every pH value tested. Subsequently, we determined the relaxivities (at 20°C) for both 1 and 2, as well as for their activated forms 1' and 2', at the pH values where their respective proportions are the highest in solution, i.e., at pH 7 and 2, respectively. At 7 Tesla (MRI), r1 for 1 rose from 0.033 to 2.34 mM -1 s -1 when moving from pH 7 to pH 2. On the other hand, the relaxivity of chelate 2 increased from r1 = 0.06 to 1.60 mM -1 s -1 . Thus, at 7 T, their relaxivities soar by factors of 71 (1) and 27 (2) as a response to a pH change from 7 to 2 (Table 1). This behavior is reflected in the hypersignal differences generated by an MRI instrument (7 Tesla) for aqueous solutions of 1 and 2 at pH 2 vs. pH 7 (Figure 4). The absence of any notable relaxivity at pH 7 for both coordinatively closed complexes is congruent with the same observations in a previous report on low-spin iron(III) complexes in aqueous solution whose first coordination sphere is fully occupied by their ligand. [27] On the opposite side, the relaxivity for the activated form is in the high region of values for iron-based contrast agents and does well compete with values reported for GBCAs. Both of these two features are equally important, as they ensure a low background signal for the un-activated form of the probe and a sizeable MRI signal after activation, i.e., the desired significant signal gap for a responsive probe as emphasized in the introduction.

Two key design features are likely responsible for the unprecedented performance described above: (1) use of the DPTACN macrocycle that ensures high water stability, the adoption of resolutely low-spin states for both forms of the probes and a facile approach of water molecules, and (2) heterocycle-bearing, pendant arms which ensure the adoption of a single coordination state at opposite sides of a given pH interval. Also, these heterocycles might contribute to the high observed relaxivities for the activated forms.

As described above, the low-spin state of the ferric centers does not change with a water molecule replacing the coordinated heterocycle, a phenomenon not observed with ferrous centers. It is notable that this does not compromise the observation of a high signal gap upon activation of the ferric versions. Finally, we also measured the relaxivity of the activated version with an NMR spectrometer (9.4 T). The observed values rose by a factor of 1.3 for both complexes when going from 7 T to 9.4 T (Figures S9, S11 and table S4). This agrees with a previous report where an iron-based chelate shows increased relaxivity with increasing magnetic field [28] , while Gadolinium-and Mn-based contrast agents see a systematic decrease. [START_REF] Hasserodt | Contrast Agents for Magnetic Resonance Imaging[END_REF] Past reports of analyte-responsive Gadolinium-based small-molecule contrast agents spoke of relaxivity changes of 20, 40, 75, even 100 percent. [30] In all these cases, their unreacted versions were decent contrast agents, sporting relaxivities anywhere between 2 and 4 mM -1 s -1 , in other words also giving off a sizeable signal in the absence of the analyte. The same holds true for more recent materials-based, pH-responsive contrast agents still employing Gadolinium, even though a relaxivity change by a factor of 4 was achieved. [31] However, a satisfying performance solely regarding the criterion of relaxivity change is not sufficient for a viable solution. Many other parameters have to be taken into account, above all stability (robustness). [32] Recent work reported a relaxivity increase by a factor of 15 using an iron-based approach. [15] Such reports currently arouse increasing interest in the field. Our proposal to overcome the sensitivity issue with the concept of an iron-based molecular probe date back to 2005 [33] and 2008. [12] The system described here exhibits heretofore unknown relaxivity increases by factors of 27 and 71, respectively. The unactivated version is essentially MRI-silent. This unprecedented signal gap competes well with the performance of 19 F MRI or CEST-MRI probes but outperforms them in view of the vitally important advantages associated with the use of 1 H MRI. The system's robust performance encourages further development into a probe responding to the presence of biomolecular analytes at physiological pH.

Figure 1 .

 1 Figure 1. ORTEP representation of X-ray structure analyses[22] of complex 1 (left) and 2 (right). Hydrogens are omitted for clarity.

Figure 2 .

 2 Figure 2. A: T1 vs pH titration of 1 (grey line) and 2 (dark line) (4 mM) in water monitored by NMR (9.4T); B: Time-dependent response of 1 (top) and 2 (bottom) to acidification.

Figure 3 .

 3 Figure 3. Switching capacity of complex 1 (grey line) and 2 (dark line).

Figure 4 .

 4 Figure 4. MRI phantom images of aqueous solutions of 1 (left panel) and 2 (right) at pH 7 (upper sample tubes) and 2 (lower).

Table 1 .

 1 r1 values (mM -1 s -1 ) obtained at 20°C in pure water at pH 7 (1 and 2) and pH 2 (1' and 2'), and at 7 T MRI field strength (extracted from phantom images).

	Compound	1	1'	2	2'
	r1 (mM -1 s -1 )	0.033 ± 0.001	2.34 ± 0.04	0.06 ± 0.0005	1.60 ± 0.28

Acknowledgements

We thank Dr. Pascal Fries, CEA Grenoble, for the measurement of the NMRD profiles.

Keywords: Molecular MRI • pH-responsive molecular probes • iron(III) chelates • relaxivity • detection sensitivity


Luca Maricchiolo¹
email: luca.maricchiolo@u-pec.fr

Parametric analysis of urban form, from geometrical to topological

Keywords: urban form, urban design, morphometrics, parametric design, city information modelling Conference urban form, urban design, morphometrics, parametric design, city information modelling

This paper presents a methodology to read urban space and to display it through parametric representation. It is based on quantitative analysis of geometrical properties of different urban sequences, parameterized on a sample trajectory representing observer path. A labelled t(x) parameter allows to evaluate such path, systemizing on it visual, spatial, and morphological variations, through some indicators: a. Dimensions D(t): urban space large, measured orthogonally to the observer trajectory tangent at t(x); b. Focal distance Fd(t): visual depth tangent to observer trajectory at t(x); c. Sequences variations: number of dimensions and focal distance deviations greater than respective average value, representing major changes in visual perception along the path, so that they produce the transition from one scene to another one in an urban sequence. Quantitative data depending on a unique parameter t(x) lead to display graphical and statistical representation, as computing of average value, range, and standard deviation. This methodology has been tested on few case studies in European cities: Rome, Palermo, Naples and Barcelona, comparing historical urban fabrics to XIX century planned ones, highlighting constants, differences and variations. The parametrical analysis of sequences indicators allows to shift from geometrical information to a topological model, abstracting results from shape specificity. The configuration pattern linking metric and relational information is experimented as a design pattern. This puts urban analysis in the perspective of design practice, leading to the script of an input matrix for an informed parametric urban modelling.

Introduction.

The research presented in this paper addresses the issues of data collection in urban morphology and their use into computational urban design. Particularly, it proposes a quantitative methodology to read urban space, with the aim of discussing links between urban morphology analysis and urban design. Studies of urban morphology are traditionally aimed at understanding the generation of voids and massesurban morphogenesisrelying on qualitative approaches to identify figures, elements and syntactic structures as well as their forming processes The disciplinary evolution has proposed a great variety of approaches, mainly addressing two dimensions: the planimetric one, based on the building footprint and street network; the three-dimensional one, based on the articulation between spaces and buildings. Among these, different schools addressed specific mechanisms. Michael Conzen [START_REF] Conzen | The use of town plans in the study of urban history[END_REF], with the Anglo-Saxon one, identify three key elements and syntaxes: roads and their composition in road systems; plots and their aggregation in urban blocks; buildings and their arrangement in plots and blocks. Saverio Muratori [START_REF] Muratori | Studi per una operante storia di Venezia[END_REF], Gianfranco Caniggia and Gian Luigi Maffei [START_REF] Caniggia | 1: Lettura dell'edilizia di base[END_REF] focused on local syntaxes, linking building typologies with the assembly rules along paths in the conformation of street and block shapes. The introduction of mathematical models, computer aided design tools for urban analysis [START_REF] Martin | Urban Spaces and Structures[END_REF], and graph theories in network analysis [START_REF] Steadman | Architectural Morphology: An Introduction to the Geometry of Building Plans[END_REF], moves towards approaches based on measurable characters. Within the framework of quantitative knowledge in the field, two main methods can be distinguished: configurational analysis and spatial analysis (Erin et al., 2017). Configurational analysis aims at quantifying the ability of street network to structure movements within the fabric, basing on the structural features of the network and of the relationships established with other elements in the urban space. The theoretical reference of Space Syntax [START_REF] Hillier | Space is the machine[END_REF] addresses the network of public spaces modelling through graphs. Graphs can be built either as a system of axes intersected in nodes, or as a system of adjacent convex spaces (Fusco and Tirico, 2006). Configurational analysis introduced an innovation in the sense of a quantitative approach in urban morphology, based on the "network" component, leading to methods to address accessibility (Urban Newtork Analysis. Sevtsuk and Mekonnen, 2012), perspective continuity (Axial analysis. [START_REF] Figueiredo | Continuity lines in the axial system[END_REF] minimal paths (Angular analysis. [START_REF] Turner | Angular analysis: a method for the quantification of space[END_REF] and centrality analysis (Multiple Centrality Assessment. [START_REF] Porta | The network analysis of urban streets: a primal approach[END_REF] Spatial analysis, based on Geographic Information Systems (GIS), concerns all the elements present in the urban fabric and focuses on the spatial relationships and configurations between them. It addresses categorization of elements and relations, as building types, neighbourhoods typologies, land use, mobility, density, building-plot relationship, referring to different models to analyse their arrangement in the urban form. Despite most of GIS based analyses rely on « point and click » method, whose reproducibility is affected by the sequence of operations, some computational tools have been recently implemented, in both Python and R, as well as plug-ins in GIS software, to systematize spatial analysis [START_REF] Fleischmann | Evolution of Urban Patterns: Urban Morphology as an Open Reproducible Data Science[END_REF]. Morphological tessellation using Thiessen polygons on building footprints is a reproductible method to identify an analytical spatial unit for individual buildings surroundings [START_REF] Fleischmann | Measuring Urban Form: Overcoming Terminological Inconsistencies for a Quantitative and Comprehensive Morphologic Analysis of Cities[END_REF]. Furthermore, tessellation using Thissen polygons around a path identifies a new analytic urban unit, the proximity band around a street segment, that is particularly interesting as it adopts a perceptive point of view, different from traditional planimetric spatial analysis. [START_REF] Araldi | From the Street to the Metropolitan Region: Pedestrian Perspective in Urban Fabric Analysis[END_REF] The development of Geographic Data Sciences is still primarily linked to functional and mobility issues. The issues related to shape, geometry and proportions of space seem to be of less interest, although these represent a major success factor of an urban form. Analysis scale matters. GIS is oriented to territorial development and urban planning, while the scale of urban design is often underrepresented due to lack of detail and loss of three-dimensional approach and internal focus. Urban design measures address urban form at the highest scale of definition, and they request internal observation, as external data (e.g. satellite observation) can hardly be exploited, due to lack of definition. Basic units are urban frontage, streets or squares, whose measurement can be used as a support to explain or govern a whole range of urban phenomena, as vitality, safety, accessibility. In this perspective, quantitative approach provides objective measures to be compared with urban performances [START_REF] Lynch | A theory of good city form[END_REF] to empower multidisciplinary analysis [START_REF] Clifton | Quantitative analysis of urban form: a multidisciplinary review[END_REF] and modelling (e.g. City Information Modelling. [START_REF] Gil | City Information Modelling: A Conceptual Framework for Research and Practice in Digital Urban Planning[END_REF]. The analysis methodology hereby presented addresses a quantitative measurement of urban space adopting internal focalization, basing on pedestrian's point of view to question perceptive experience. If perception is subjective, however some physical features of space are objectively measurable, as dimensions, spacing, proportions [START_REF] Ewing | Identifying and measuring urban design qualities related to walkability[END_REF]. The investigative categories adopted in the present research to collect countable data from several urban sequences in a comparative approach follows the theoretical approach proposed by Kevin Lynch (Lynch, 1961). This will be discussed to better understand the combinatorial criterion of few urban sequences in Mediterranean historic cities, whose common features can be explained by the limitation of combinatorial possibilitiesas grammar and syntaxis limit to a finite number the possible combinations of letters in language [START_REF] Hillier | Space is the machine[END_REF]. Thus, analysis aims at recognizing recurring geometrical and topological features as to produce similar perceptive situations.

. This paper presents the indicators, the methodology and the collected data of the analysis of six urban sequences of historic cities in Mediterranean Europe, followed by the results discussion oriented towards design. The general objective is to develop a methodology to read urban space, to highlight recurring characters and to explore results potential in a design algorithm. This is based on consecutive steps, whose results are presented during the discussion:

-the parametric measurement of six sequences of public space in six historical fabrics; -the evaluation of the results on the basis of some indicators aimed at identifying the sequences; -the construction of a topological model based on the identified sequences; -the testing of the topological model in a design algorithm.

Measurement adopts an internal focus and is returned in parametric representation along a sample path through urban fabric. The transition from Cartesian to parametric representation aims at abstracting analytic metric data from local shape specificities: the unique and reproducible independent variable is the walking time along an urban path, which is adopted as a fundamental parameter. Metric data are, thus, represented as function of t parameter on a sample trajectory. This allows getting away from geometric features, ensuring a stronger results comparability and absolute values fit for design algorithm. Dependent variables and indicators are based on the normative theory of Kevin Lynch [START_REF] Lynch | Site planning[END_REF], which offers a reference benchmark. The analysis is carried on in a CAD environment (Rhinoceros3D) through a visual programming tool (Grasshopper), based on building footprint from GIS information. On the basis of qualitative results, identification of local topologies and sequences configuration on a graph shapes a design pattern.

Measurement. Quantitative analysis of urban sequences.

On local indicators framework.

Several approaches to urban studies based on human perception suggest the transition to a representation of urban space from an external focus (plan) to an internal focus, based on inner perception or cognitive spaces. Perceptive role of space in shaping historical memory and assuming adequate performances has widely been raised in literature (from Sitte, 1889[START_REF] Jacobs | The death and life of great American cities (Vintage books[END_REF] but it is Kevin Lynch, in Site Planning [START_REF] Lynch | Site planning[END_REF], who operates an analytical, dimensional and normative description of the space of the city, which he will subsequently relate to a performance and multidimensional approach [START_REF] Lynch | A theory of good city form[END_REF].

In Site Planning, Lynch focuses on sensorial function of space, often ignored in site arrangement plans in favour of functional or technical needs. As argued by Lynch, urban space is, the most if not all, experienced through a sequence of views, driven by site directions and rhythms, whose element parallax movement can affect global perception. Thus, proportioning and articulation are not necessarily conceived in absolute terms, but relating to the perception and the movement of people in space. Basing on sequences structure, inner focalisation and parallax, Lynch addresses a few tentative quantities that can be assigned to the size and the proportion of external spaces: (a) Space dimensions. Basing on human view capacity, urban space can be dimensioned moving from its perception. Humans can detect a man about 1,2 km away, recognize him at 25 m, see his face as a clear portrait at 15 m, feel him to be in direct relation to us, whether pleasant or intrusive, from 1 to 3 m. Consequently, about 15 m large space appear intimate, until 25 m it is human scaled; smaller, it appears too small and oppressive; for instance, most of the successful enclosed squares of the past not exceeded 130 m in the smaller dimension.

(b) Parallax. Urban perception is shaped by the limits to the angle of clear vision and to the rapidity of scanning. Thus, focal distance from the observer to the object affects urban perception: an element whose height equals distance is seen in detail but not as a whole; while distance is two times height, it would be seen clearly and entirely; from three times and more, visual field becomes predominant on the element, till this disappear to be part of the scene. (c) Proportions. Most comfortable spaces are height a third of large; below, space ceases to be enclosed; over, it seems a pit. (d) Sequences. Made of both spaces and paths, they should be measured by successive balances at each point of view. Evaluation results from cumulative effect, as a lack of formal balance at a moment can be less important than a qualitative connected chain of spaces. Sequences size on congenial physical characteristics: continuity and closure of form; differentiation, dominance or contrast of a figure on a ground; symmetry, order, repetition or simplicity of forms. Movement is as much important: the road suggests direction, eyes follow it as a thread that ties the whole together.

(e) Rhythmical continuity. Sequences should be linked harmoniously to create consistent variations alternating open to close, simple to intricate, brilliant to subdued. "The space should be considered as a total patternnot seen flatwise from the air, but as a progression through which one moves" (Lynch, 1962, p. 81) Although empirically thought, "these statements seem to derive from the optical characteristics of the human eye, and from the size of the objects which are generally of greatest interest to it, i.e., other human being" (Lynch, 1962, p, 60).

Urban sequence specimens.

The analysis methodology was tested on different urban fabrics. The demonstrative purpose of the methodology is independent of local quality of urban specimens, aiming at highlighting some tools to link spatial quantitative analysis to design.

Four sequences in the historical urban fabrics of Rome, Naples, Palermo and Barcelona, have been compared with two paradigmatic XIX century urban plans: the first extension of Rome as the post-unification capital, Rione Prati, consisting of regular orthogonal grid of approximately 100 x 70 meters, and Barcelona eixample planned by Ildefons Cerdà, based on 110 meters squared blocks. For each, the analysis focuses on an arbitrary path along a quasi-linear trajectory, traced avoiding main axes to simulate a random walk experiencing the urban sequences. This, to grasp diffused urban features rather than extraordinary ones, as they appear to a grater number of observers.

Analysis parameter and variables

Sequences features are measured and represented as a function of an independent parameter, walking time t(x) along the path, ranging from t(0) at the start and t(1) at the end, equal to about 20 minutes walking on an approximate length of 2km (+/-100m) at 5km/h. Analysis variables depending on t are : (a) Dimensions D(t). The scale of space is mostly represented by large, measured orthogonally to the path, from a built front to the other on opposite sides. Intersections with transversal streets are assumed as gates, as they don't interrupt perceptive continuity of the fronts, while shaping convex urban voids. Parametric representation of D(t) shows the variation of urban space dimensions, highlighting discontinuities and rates of change. Lack of dependence from urban form geometry allows comparison. (b) Focal distance Fd(t). Urban form diversity is expressed by measuring the variation of axial viewpoints, on a direction tangent to the path. The more frequent are the viewpoints changes, and the shorter are their focal length, so space experience and perception are richest. Focal distance measures, hence, length from the observer to the first urban front or other perspective obstacle on an axial direction, namely tangent to the path at t(x). Parametric restitution shows space depth: lower values express a tight succession of scenes, so a strong articulation of the path, while higher values express long perspectives and more linear paths. (c) Sequences S(t). The passage from a spatial sequence to the successive is identified by discontinuities in space dimensions and focal depths variations, or by the rate of change of such variables. A variation equal to or greater than, positive or negative, identifies the passage from one sequence to the next; in the same way, a variation (always positive) of the focal depth equal to the previous one identifies a sequence passage. The sequences are therefore described by both sudden variations of expansion or compression (e.g.: the way in or the way out of a square), and sudden variations of visual depth (e.g.: a turn along a path). Graphical comparison between cases study directly shows compressions, dilations, and focal progressions differences, as to explain how urban sequences and their ratio change in relation of urban space shape. Some considerations can be drawn on four main topics: 1. Dimensions and scale. As expected, average dimensions are lower for historical fabrics rather than planned ones. In all Rome, Palermo, Barcelona, and Naples historical centres average values are around 10 m large, fitting with human scale and recognizability of boundaries. Furthermore, dimensions don't exceed 200 m, keeping space scale not to far from Lynch's reference for enclosed squares. 2. Diversity. The gaps between minimum, maximum and average values express the variation rate; therefore, the standard deviation of data qualifies space in its capacity to provide compression and release produced when the observer passes from a narrow to a broader space. 3. Articulation. Focal distances express axial variations along the path, as turns, curves, visual obstacles. Intensity of gaps shows less linearity in space, whose articulation provides a higher number of perspectives per path unity. In historical fabrics (i, ii, iii, iv) visual depth varies from about 3 to 950m, associated with the long straight of Naples decumano inferiore. In planned fabric (v, vi) infinite length refers to focal depth exceeding sample limits, not finding any visual obstacle within the path. 4. Rhythms. The number of sequences per length unity reveals sequence ratio, that is the frequency of variations along the path, thus urban space rhythm The following tables display key values of the collected data and their statistical comparison.

Discussion. The topology of urban sequences.

General topology of open spaces consists of a continuous surface space with as many holes as built elements (both isolated buildings and blocks). All squares and streets are a connected bidimensional surface, as any point can be reached by travelling on this surface. It is bi-dimensional and orientable, as it is easily recognizable an above and a below and there are not ambiguities between the two faces, although few overlaps are possible. It is bordered by all urban space internal boundaries, like urban frontages and enclosures, and by a hypothetical external city limit. Hence, as a general topology, urban space is a surface topologically equivalent to a disk with n perforations and an external boundary, which are independent of geometry [START_REF] Brelsford | The topology of cities (Santa Fe Institute[END_REF]. Furthermore, accessible urban space is isomorphic to its access network, which can be described by graphs identifying nodes and connections [START_REF] Agryzkov | Different types of graphs to model a city[END_REF]. Access network graph provides a configurational scheme, whose nodes are access points as well as intersections or aggregation spaces. Such urban representation is highly reproducible and generalizable, providing information about integration, connectivity and depth, as relational features, independent from element own qualities [START_REF] Brelsford | The topology of cities (Santa Fe Institute[END_REF]. As a development of network analysis, several criteria can identify nodes and connections, to make graphs more consistent to the real or to the perceived urban form: these could be based, for instance, on isovists, street names, or axial analysis including only the fewest and longest lines [START_REF] Hillier | Space is the machine[END_REF]. Such normalizations of graphs aim at linking continuous paths that are not, actually, a system of segments, to better clarify relations between the parts of access network [START_REF] D'autilia | Linee, graphs e contrazioni. Un approccio rigoroso alla space syntax[END_REF].

Despite their effectiveness about cities configuration understanding, graphs representations do not provide any information about real quality of nodes and connections. Furthermore, they lead to some inconsistencies and ambiguities, for instance in the representation of a square with a fountain, that could be seen as a node in itself, or as network of small segments with as any intersections as the streets going to the square [START_REF] D'autilia | Linee, graphs e contrazioni. Un approccio rigoroso alla space syntax[END_REF]. Another weakness concerns the base support, focusing just on relations and removing metric or geometric features data. This leads to some ambiguities in integration computation, including loss of urban scale and over-reliance on connections axiality [START_REF] Ratti | Urban texture and space syntax. Some inconsistencies[END_REF]. Topology, as a non-metric science, could be very effective in the study of relations, especially at a global scale. However, at a local scale such representation suffers from a lack of information about spatial qualities; moreover, it is hard to dissociate pedestrian and human-scaled perceptions from metric information, as walkability relies more on distances and spatial comfort rather than on connections depth [START_REF] Ratti | Urban texture and space syntax. Some inconsistencies[END_REF]. Urban sequences, as defined in Site planning [START_REF] Lynch | Site planning[END_REF], consist of a linear configuration (or chain) of urban voids, which quantitative description requires both metric and non-metric relational data. Abstraction from geometric information allows to focus on relational properties and to highlight configuration. Nevertheless, within this research, some metric-derived data are supposed to be kept, as the position and the scale of each sequence.

The hypothesis hereby explored consists of introduce few nodes features within a topological representation of the analysed urban sequences. The topology of urban sequences is restricted to the analysed path, not including general city topology. This is made of a chain of cognitive spaces, topologically identified by surface, boundaries (e.g. urban frontages) and gates [START_REF] Stojanovski | City information modeling (CIM) and urbanism: blocks, connections, territories, people and situations[END_REF]. Identifying as gates the connections between each convex void space, it allows to define a local topology of public spaces as a bi-dimensional continuous surface bordered by both enclosures and gates. In an enclosed square, delimitated by compact urban fronts, access streets do not break unity of perceptiveor cognitiveenclosure. This is consistent with the perception of urban sequences as a chain of different, unitary, spaces, interconnected one to each other, although the whole path is actually just one topological surface.

Local topologies lead to identify a range of urban objects enriching the basic urban form ontology made of streets, buildings, and blocks [START_REF] Kropf | Ambiguity in the definition of built form[END_REF]. This concerns elementary syntactic units whose internal and external relational properties are recurrent in urban forms, as to be defined as building blocks for an informed and generalizable city modelling. Perceptive and relational features, defining basic spaces as frontages, streets, squares, are at the basis of urban performances and could be very useful if integrated in base grammar for urban models [START_REF] Stojanovski | City Information Modelling (CIM) and Urban Design -Morphological Structure, Design Elements and Programming Classes in CIM', Computing for a Better Tomorrow[END_REF]. Finally, representation of urban sequences as a chain of local topologies, interconnected by gates, fulfils internal focalization of urban space. Urban sequences configuration consists of two syntactic principles: a. The chain itself, as a configurational path referring to gates and interconnections b. Local topologies, referring to cognitive spaces and represented as nodes The parametric analysis is used as a criterion to qualify both principles: the parameterization of the path identifies the position (and the depth) of every space on the chain; the quantitative data define scale of each space. This leads to define three syntactic units, basing on the spread of analytic values highlighted by parametric analysis, represented through an informed dual graph of urban sequences (in which nodes refer to each convex space and connections refer to gates): a. Expanded units (e.g. squares), consisting of spaces whose dimensions exceed the average values of the whole sequence, represented as nodes; b. Compressed units (e.g. streets), consisting of spaces whose dimensions are less then average values, represented as nodes as well; c. Gates, referring to links between units, equivalent to discontinuities of both dimensions and focal distances indicators. The identification of these three syntactic elements is essentially based on local spatial qualities, with the aim of including in city modelling qualitative features, as shape, human-scale, perceptive experience, through quantitative analysis and reproducible methods.

Conclusion. Towards urban modelling and design

Urban morphology and urban design are theoretically linked, as morphology focuses on how and why urban elements are composed. Design takes advantage of morphology categories, both in terms of theoretical approaches (normative theories rooted in the results of morphology studies), and of operational tools (methods to analyse local contexts as basis for design). Nevertheless, a mainly qualitative approach in morphology makes it difficult to have a highly generalizable systematisation of results and a direct application in design guidelines. Quantitative analysis in the field of urban morphology can fill this gap, providing precise and intelligible results to be used in the project. Parametric urbanism can offer a great support, relying on algorithms and rules on the basis of local parameters: "the challenge will be the capacity of quantitative urban morphology to serve as input and/or reference framework to parametric urbanism, in order to avoid the divergence between morphological studies and design proposals" (Erin et al. 2017). With this regard, the development of a configurationally pattern, following quantitative analysis of an existing urban form, aims at exploring in parametric design a few principles recognized to be successful, namely human scale and articulation of public spaces sequences. This approach, whose experimentation is restricted to few indicators on sample case studies, leads to the definition of a simple algorithm to shape urban voids along a matrix path. Geometry of the path and the boundaries are independent variables, defined a priori in the parametric algorithm, while distances and dimensions of spatial units depend on path parameterisation. Hence, the algorithm defines occurrence and scale of space units, as to produce homeomorphic sequences reproducing similar space progressions. This pattern outlines a way to link design with analysis, operating within evidence-based knowledge to feed a parametric algorithm. The definition of ontological categories from quantitative morphology can improve object-based urban modelling, offering a support for multidimensional data [START_REF] Duarte | City induction: a model for formulating, generating, and evaluating urban designs[END_REF]; on the other hand, it suggests further research development to increase spatial and performance-based data usability in city information modelling.

Systematic analysis of urban space and sequences, thus, addresses multidimensional analysis by providing a data-driven semantic framework for city modelling, aiming at enhancing spatial and perceptive features in the whole process of analysing, designing and managing cities.

Illustrations and tables

Figure 1 .

 1 Figure 1. From cartesian to parametric representation. (a) Kevin Lynch, urban sequences in Venice (Lynch, 1962); (b) sample path in Rome case study; (c) the sample path in cartesian representation (above) and unfolded along the parametrized trajectory (below)

  (i) Rome, Centro Storico. The analysed path goes through the zona urbanistica Centro Storico, crossing the medieval and the Renaissance fabrics, and avoiding main straight axes resulting of later planification. It begins in Rione Ponte, crosses Parione, Sant'Eustachio and Pigna districts, touching nodal points such as Piazza Navona, Piazza della Rotonda, to reach Trevi district and end in Piazza del Quirinale. (ii) Palermo, Albergheria and Kalsa. The analysed path runs through the two mandamenti on the south side of the Cassaro, presently via Vittorio Emanuele. It moves from Piazza della Vittoria in front of Normanni Palace toward the sea in north-east direction, crossing Ballarò marketplace, Piazza Bologni, Piazza Bellini, and reaching the waterfront through via Alloro. (iii) Barcelona, Barrio Gotico. The path crosses medieval district parallel to the coastline, from the eixample on south-west to the eixample on north-east. It starts in Raval neighbourhood at Sant Antoni marketplace proceeding along Carrer de Sant Antoni Abat and Carrer del Carme, to cross La Rambla and enter Barrio Gotico in Carrer de la Portaferrissa until the cathedral. It runs through plan Cerdà's demolitions in Plaça Nova and Avenida de Francesc Cambò, facing Santa Catarina market to finally reach the example on the north-east side of Bairro Gotico in Placeta del Commerç. (iv) Naples, Centro Storico. The path crosses the still visible Roman structure of the historical fabric. Moving from piazza Carità along via Toledo, it reaches the castrum at piazza del Gesù Nuovo and runs along the decumano inferiore (via San Biagio dei Librai), crosses the cardus of via Duomo, and it ends in Piazza Garibaldi, a vast XIX century square that became bigger after modern central station relocation. (v) Rome, Prati. On the orthogonal XIX century grid, the analysed path is one of the axes, namely Via Cola di Rienzo from Piazza Risorgimento to Piazza della Libertà. (vi) Barcelona, Eixample. The analysis concerns a sample path within Cerda's fabric, particularly Carrer del Conseil de Cent, from Carrer del Comte d'Urgell to Carrer de Bailèn.

Figure 2 .

 2 Figure 2. Analysis of a sample path in Rome, Centro Storico (a), along a parametric trajectory (b): (c) dimensions, sequences, and focal distances in parametric representation on basis t(0,1) ...............................................................................................

Figure 3 .

 3 Figure 3. Graphical representation of the comparative analysis: (i) Rome, Centro Storico, (ii) Palermo, (iii) Barcelona, Barrio gotico, (iv) Naples, (v) Rome, Prati, (vi) Barcelona, Eixample

Figure 4 .

 4 Figure 4. Data visualisation per type for every case study: (a) dimensions; (b) focal distances; (c) sequences per 100mDimensions and focal distances provide a measurement of urban space scale, diversification, articulation and rhythms: Analytic quantitative data allow a comparation and an objective evaluation of the case studies, on the basis of related indicators. Obviously, such evaluation is restricted to the analysed path, not giving any information about the identified fabric as a whole. Moreover, it is arbitrary, due to its dependency from the chosen path and the travel direction. However, for demonstrative purpose, it shows a methodology fit to be reproduced in every possible path within a neighbourhood, to offer more significant statistical data about the whole urban fabric. Evaluation data are not suitable for developing a design algorithm, due to strict compliance with the specific case-study: values strictly depend on shape, while a broad generalisation of the relational principle, fit for design, requires stronger abstraction. Hence, the elaboration of a topological model addresses the script of configurational rules to feed a morphogenetic algorithm. ................................................................................................

Figure 5 .

 5 Figure 5. Urban sequence (i) represented as a chain of local topologies with proportional data based on dimensions D(t). ................................................................................................

Figure 6 .

 6 Figure 6. Urban sequences design through a parametric algorithm based on urban sequence (i) configuration, adopting three different sample paths with the related geometric languages. ..................................................................................................

Figure 1 .

 1 Figure 1. From cartesian to parametric representation: (a) Urban sequences in Venice (Lynch, 1962); (b) sample path in Rome case study (Author, 2022); (c) the sample path in cartesian representation (above) and unfolded along the parametrized trajectory (below) (Author, 2022); Figure 2. Analysis of a sample path in Rome, Centro Storico (a), along a parametric trajectory (b): (c) dimensions, sequences, and focal distances in parametric representation on basis t(0,1) (Author, 2022); Figure 3. Graphical representation of the comparative analyses: (i) Rome, Centro Storico, (ii) Palermo, (iii) Barcelona, Barrio gotico, (iv) Naples, (v) Rome, Prati, (vi) Barcelona, Eixample (Author, 2022); Figure 4. Data visualisation per type for every case study: (a) dimensions; (b) focal distances; (c) sequences per 100 m (Author, 2022); Figure 5. Urban sequence (i) represented as a chain of local topologies with proportional data based on dimensions D(t) (Author, 2022); Figure 6. Urban sequences design through a parametric algorithm based on urban sequence (i) configuration, adopting three different sample paths with the related geometric languages (Author, 2022); Table 1. Key values of parametric analysis, per indicator, per case study (Author, 2022). ..................................................................................................................

Methodology. A parametric approach for the pedestrian point of view.

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Table 1 .

 1 Key values of parametric analysis, per indicator, per case study

	indicator	u (i) Rome	(ii) Palermo	(iii) Barcelona	(iv) Naples	(v) Rome	(vi) Barcelona
			Centro Storico		Bairro Gotico		Prati	Eixample
	path length m	2042.0	1789.0	1972	2068	1127	1887
	D(max)	m	175.0	186.5	146.0	163.6	116.5	60.5
	D(average) m	10.4	8.2	9.5	14.4	24.8	20.0
	D(min)	m	2.6	1.9	4.2	4.2	22.3	16.8
	Fd(max)	m	235.5	300.5	302.5	955,9	Inf	Inf
	Fd(average) m	50.6	59.5	99.4	242,6	n.d.	n.d.
	Fd(min)	m	2.9	8.6	3.0	6.5	Inf	Inf
	S	val	37	37	33	19	4	1
	S(ratio)	val	1.8	2	1.7	0.9	0.3	0.05