  In this paper, we discuss how to design the graph topology to reduce the
communication complexity of certain algorithms for decentralized optimization.
Our goal is to minimize the total communication needed to achieve a prescribed
accuracy. We discover that the so-called expander graphs are near-optimal
choices. We propose three approaches to construct expander graphs for different
numbers of nodes and node degrees. Our numerical results show that the
performance of decentralized optimization is significantly better on expander
graphs than other regular graphs.
