## 1 Introduction  
The unprecedented advancements in modern technology have caused the fields of electricity, electronics, and computer systems to merge in such a way that they are no longer disjointed sets. Particularly, to make electric power systems more automated, secure, and resilient, the application of machine learning, artificial intelligence, big data, and deep learning in the modern utility grids and smart grids is accelerating at an exponential rate [1, 2]. As such, the importance of a high computational power is undeniable to ensure that the widely encompassing systems are operating as desired. Due to its high computational power, massively parallel computation (MPC) is required in all aspects of modern life, including disaster prevention and mitigation, healthcare, drug design, personalized medicine, etc. Moreover, due to the enormous shift of professional activities into a full virtual mode at the onset of the COVID-19 pandemic, the world has truly realized the importance of fast computation at a low power consumption. Computational speeds as high as exa-flops or even zetta-flops are not only good-to-have in the next-generation supercomputers for MPCs, but rather a must-have criterion in order to keep marching forward in the computer network industry [3, 4, 5, 6]. Although low dimensional networks are better in terms of their lesser delay and higher throughputs [7; 8], they are not suitable for interconnecting millions of nodes for a hierarchical interconnection network (HIN) [9; 10; 11]. For next-generation systems of MPC, the interconnection of more than a million nodes, a combination of low-dimensional networks and the topology is required [12]. HINs have been adequately described in the literature as a revolutionary milestone for massively parallel computers and nest-generation parallel computing. For instance, the extended hypercube [13], Tori-connected mesh (TESH) [14], hierarchical torus network (HTN) [15], rectangular twisted torus meshes (RTTM) [16], midimew connected mesh network (MMN) [17; 18], hierarchical Tori connected mesh network (HTM) [19], shifted completely connected network (SCCN) [20], 3-dimensional Tori connected torus network (3D-TTN) [21], and many more networks have been proposed and assessed in recent years as a means to promote MPC systems. The numerous works pertinent to the HIN is testament to its numerous benefits in terms of network performance and computational speed. The prime advantage of HIN lies in the reduced link cost and the inherent coordination and communication among the many nodes within the network [22]. The concept of a novel HIN named Tori-connected flattened butterfly network (TFBN) had been posited by the authors in an earlier work [23] along with a statistical analysis of its performance. The network contains several interconnected basic modules (BMs) for forming the networks that indicate higher level in the hierarchy. The BMs are actually a 2D flattened butterfly network [24] and each higher level is a 2D torus network; hence the name of the new network. The specialty of the TFBN is that it reduces network congestion and augments the throughput in the BMs. Although the very first study on the proposed TFBN shows good static network performance compared to other networks, further investigation is substantial to establish TFBN as a reliable HIN topology. Since the practical implementation is quite expensive, a static evaluation and then experimentation by simulation is a better alternative to assess the superiority of any network. In this research, we explore the density parameters of the network statically. The main objectives of this paper are to statically assess the packing density, message traffic density, and a new parameter called traffic congestion versus packing density trade-off factor (TCPDTF) to show the eminence of the proposed TFBN. The rudimentary study on the packing density of TFBN is presented earlier, and only the packing density of a Level-2 TFBN with only 256 nodes is presented in that article [25].