# TFBN: A Cost Effective High Performance Hierarchical Interconnection Network
## 4 Static Network Performance Evaluation of a TFBN
### Arc Connectivity and Bisection Width  
The MPC system consisting of many nodes and they are connected by many links. It is quite normal that the node or the links may have a fault [36], or it may fail to operate properly. This fault or failure of links may cause the disjoint of the underlying interconnection network deployed to construct the MPC system. Two disjoint parts of the network might be just two parts or it might be two equal halves of the network. The minimal number of links that need to be cut or to be faulty to disconnect the network into two separate parts is known as arc connectivity. In addition, the minimum number of links that need to be cut or to be faulty to disconnect the network into two equal halves is called bisection width.  
Arc connectivity is statically used to indicate the fault-tolerant capability of any interconnection network. The actual fault tolerance depends on the reconfiguration of the faulty nodes or links by redundant wires or links or by rerouting the packet by any alternative path. However, the ratio between arc connectivity and degree of node statically indicates the fault tolerance capability of any interconnection network. We have evaluated the arc connectivity of the proposed TFBN along with the TTN, TESH, torus, and mesh networks, and the evaluated result is charted in Table 1. It is portrayed that the arc connectivity of TFBN is equal to that of toroidal networks such as conventional torus and hierarchical TTN. In addition, it is double that of conventional mesh and hierarchical TESH network. It is noted from Table 1 that the high node degree of the TFBN results in a low ratio of arc connectivity and node degree. Therefore, static fault tolerance performance of the proposed TFBN is lower than conventional torus and TTN, and it is more than conventional mesh and hierarchical TESH networks.  
The moderate value of bisection width is preferable. If it is low, then the merging of the yielded result of the computation outcome by the individual node will be slowed down because the bisection links will be congested due to heavy traffic flow between two halves of the interconnection network. This congestion will be severe for a bit complement traffic pattern where all the packets cross the bisection of the network. Thus, the bisection links will be heavily congested. Therefore, to reduce this congestion and contention, high bisection width is better. On the other hand, high bisection width will create the complexity in Very large-scale integration (VLSI) or wafer-stack implementation or the Network-on-chip (NoC) design of an interconnection network. Therefore, a kind of optimum and moderate value is expected for any interconnection network to be considered for the next generation MPC system that is neither too high nor too low. We have evaluated the bisection width of TFBN along with other networks considered in this paper. It is depicted in Table 1 that the TFBN has moderate bisection width like hierarchical TTN and TESH networks, and it is quite a bit lower than mesh and torus networks.  
The question may arise as to why we have considered Level-2 (256 nodes) and Level-3 (4096 nodes) for the evaluation and comparison of performance. The proposed TFBN is the updated work of TTN, whereby the basic module of TTN is replaced by a flattened butterfly network. It is shown in our another study that TTN results in good performance for more than 1 million nodes compared to other networks [32; 33]. For both the Level-2 and Level-3 networks, the proposed TFBN yielded superior performance to that of TTN. Therefore, it is believed that the TFBN will result in superior performance for Level 4 and 5 networks compared to other networks.