## 1 Introduction  
The traffic performance of a network, based on traffic metrics that describe the whole network, is investigated via the macroscopic fundamental diagram (MFD). Proof of its existence (independently of the demand) is described in [1]. Some important findings of references [1; 2; 3] are that the MFD is a property of the network itself (i.e., its control and infrastructure) and not of the demand, and that the maximum flow is associated with a density value (the critical density) independently of the origin-destination table. In [3], the network MFD presents less scatter than the one that focuses on individual links. Also, the MFD reveals a well-defined, smoothly declining curve, considering the speeds and densities observed at different times of day and between multiple days. In [4], it was observed that when the flow vs. speed data of individual fixed detectors were aggregated, the scatter nearly disappeared in the resultant plot (a well-defined curve). Also, regularity conditions were proposed (though they were determined to be not necessary) in relation to the existence of the MFD. These included (i) slow-varying and distributed demand, (ii) a redundant network (with several route options), (iii) a homogeneous network (with similar links), and (iv) links with an approximate fundamental diagram. The relationship between production (average flow) with accumulation (average density) and the spatial inhomogeneity of density is presented in [5]. In this paper, the authors explain the nucleation effect (in which congestion concentrates in the bottlenecks that first appear and then propagates from these bottlenecks) and presents the relation of the standard deviation of density to production and with accumulation. If simulations start with homogeneous distribution of vehicles, uniform demand, and the same control for traffic lights, different spatial distribution of congestion can be observed due to the randomized turning of vehicles at intersections [6]. A sufficient condition for the existence of low-scatter MFD in arterial networks is described in [7]: if the spatial distribution of link density is the same for two time intervals with the same number of vehicles in the network, then the average flows should also be the same; nevertheless, no well-defined MFD was observed for the freeway network study case. Freeway networks do not have well-defined MFDs (density vs. flow) [8]. Freeway networks are different from arterial networks because they do not have traffic signals, are not redundant, and because of their transient states. The identified causes for freeways being hysteretic and path-dependent were different spatial heterogeneity in vehicle density in the onset and offset of congestion and capacity drop phenomena during the congestion offset (with lower flows at occupancy values near the critical point). The MFD on freeways was investigated in [9], the relationship between vehicle hours traveled (VHT) and vehicle kilometers traveled (VKT) presents a well-defined shape (with no scatter) if all the links, and all the lanes of each link, are in the same regime (congested or uncongested); as the best performance (the higher VHT) on a freeway happens when all links (and all lanes) are under the same regime, policies to homogeneously spread congestion can be useful for maximizing the number of completed trips. The existence of well-defined MFDs was observed through clustering the zone under study in [10]. The partitioning mechanism consisted of applying three algorithms: segmenting, merging, and boundary adjustment. The resultant clusters had low density variation in the links and high spatial compactness. The time-varying nature of travel demands can induce scattering in the MFD, and appropriate boundary conditions (adjustments in the rationing of travel demand) are required to obtain a well-defined MDF, as demonstrated in [11].