  We show that, in many settings, the worst-case performance of a distributed
optimization algorithm is independent of the number of agents in the system,
and can thus be computed in the fundamental case with just two agents. This
result relies on a novel approach that systematically exploits symmetries in
worst-case performance computation, framed as Semidefinite Programming (SDP)
via the Performance Estimation Problem (PEP) framework. Harnessing agent
symmetries in the PEP yields compact problems whose size is independent of the
number of agents in the system. When all agents are equivalent in the problem,
we establish the explicit conditions under which the resulting worst-case
performance is independent of the number of agents and is therefore equivalent
to the basic case with two agents. Our compact PEP formulation also allows the
consideration of multiple equivalence classes of agents, and its size only
depends on the number of equivalence classes. This enables practical and
automated performance analysis of distributed algorithms in numerous complex
and realistic settings, such as the analysis of the worst agent performance. We
leverage this new tool to analyze the performance of the EXTRA algorithm in
advanced settings and its scalability with the number of agents, providing a
tighter analysis and deeper understanding of the algorithm performance.
