Constructing Large-scale Low-latency Network from Small Optimal Networks
  The construction of large-scale, low-latency networks becomes difficult as the number of nodes increases. In general, the way to construct a theoretically optimal solution is unknown. However, it is known that some methods can construct suboptimal networks with low-latency. One such method is to construct large-scale networks from optimal or suboptimal small networks, using the product of graphs. There are two major advantages to this method. One is that we can reuse small, already known networks to construct large-scale networks. The other is that the networks obtained by this method have graph-theoretical symmetry, which reduces the overhead of communication between nodes. A network can be viewed as a graph, which is a mathematical term from combinatorics. The design of low-latency networks can be treated as a mathematical problem of finding small diameter graphs with a given number of nodes ( called order ) and a given number of connections between each node ( called degree ). In this paper, we overview how to construct large graphs from optimal or suboptimal small graphs by using graph-theoretical products. We focus on the case of diameter 2 in particular. As an example, we introduce a graph of order 256, degree 22 and diameter 2, which granted us the Deepest Improvement Award at the Graph Golf competition. Moreover, the average shortest path length of the graph is the smallest in graphs of order 256 and degree 22. 
Ryosuke Mizuno, Yawara Ishida
math.CO