# Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers
## 2 Network Architecture of MTNs  
In this section, the architecture of MTNs is illustrated. With regards to our previous paper [31], we only updated the layout of the horizontal links as seen in Figure 1. This difference provides a low diameter and average distances in the network. The main goal of HINs is to achieve a low diameter and average distance values, as described in Section 4. These two properties were not evaluated in our previous paper. Furthermore, this study also evaluated a new modified network to evaluate its static performance and cost effectiveness.  
Figure 1: Hierarchy of MTNs (2,2,0).  
The basic building block of a midimew-connected torus network is the basic module (BM), which was \(2^{m}\times 2^{m}\) torus, where \(m\) is a positive integer. The hierarchy connection which was implemented at higher-level connections with the \(2^{m}\times 2^{m}\) torus BM used midimew links. The 2D torus block represented level 1, the lowest level. Higher levels were the result of midimew connections in level 1. The packet flow in the network links was different at the lowest level (level 1), as compared to that at higher levels. All of the levels used bidirectional links for flexible connections. The following definitions illustrated the network topology:  
**Definition 1**.: _The 2D torus (level 1) was a \(2^{m}\times 2^{m}\) BM containing \(2^{2m}\) nodes created by \(2^{m}\) rows and \(2^{m}\) columns._  
**Definition 2**.: _An MTN (\(m\), \(L\), \(q\)) had a \(2^{m}\times 2^{m}\) BM, where \(L\) is the level of hierarchy and \(q\) is the inter-level connectivity. The connections that formed the higher levels used \(4\times 2^{q}\) of each of the BM's free ports, \(2(2^{q})\) for the horizontal connections and \(2(2^{q})\) for the diagonal connections, where \(q\in\{0,1,\ldots,m\}\). The minimal level of inter-connectivity was when \(q=0\), while maximum was when \(q=m\). In this paper, \(m=2\) for better granularity._  
**Definition 3**.: _The highest level that could be obtained by the MTNs (\(m\), \(L\), \(q\)) was \(L_{max}=2^{m-q}+1\) level._  
**Definition 4**.: _The wire complexity, \(W_{L}\), was the total number of wires used to link the network elements. Therefore, the \(W_{L}\) of MTNs was found using Equation (1), where \(W_{BM}\) is the number of links in a single BM._  
\[W_{L}= W_{BM}\times 2^{m\times 2(L-1)}+\sum_{x=2}^{L}2(2^{q})\times 2^{m\times 2 (L-1)} \tag{1}\]  
The following figures are provided to illustrate these definitions. Figure 1 represents an MTN (2,2,0). Figure 2 shows the overall network connections forming level 2. Further details of each level are shown in Table 1.  
Figure 2: Complete level 2 connection of MTNs (2,2,0).  
The modification to the network, as shown in Figure 1 and based on the previous version [31], did not change the total number of nodes, free ports, or wire complexity, as shown in Table 1. This indicated that both versions supported the same number of nodes for supercomputers. However, different connections were needed to achieve better values for diameter and average distance, as shown in Section 4.