  Decentralized optimization is an emerging paradigm in distributed learning in
which agents achieve network-wide solutions by peer-to-peer communication
without the central server. Since communication tends to be slower than
computation, when each agent communicates with only a few neighboring agents
per iteration, they can complete iterations faster than with more agents or a
central server. However, the total number of iterations to reach a network-wide
solution is affected by the speed at which the agents' information is ``mixed''
by communication. We found that popular communication topologies either have
large maximum degrees (such as stars and complete graphs) or are ineffective at
mixing information (such as rings and grids). To address this problem, we
propose a new family of topologies, EquiTopo, which has an (almost) constant
degree and a network-size-independent consensus rate that is used to measure
the mixing efficiency.
  In the proposed family, EquiStatic has a degree of $\Theta(\ln(n))$, where
$n$ is the network size, and a series of time-dependent one-peer topologies,
EquiDyn, has a constant degree of 1. We generate EquiDyn through a certain
random sampling procedure. Both of them achieve an $n$-independent consensus
rate. We apply them to decentralized SGD and decentralized gradient tracking
and obtain faster communication and better convergence, theoretically and
empirically. Our code is implemented through BlueFog and available at
\url{https://github.com/kexinjinnn/EquiTopo}
