  Symmetries are fundamental to dynamical processes in complex networks such as
cluster synchronization, which have attracted a great deal of current research.
Finding symmetric nodes in large complex networks, however, has relied on
automorphism groups in algebraic group theory, which are solvable in
quasipolynomial time. We articulate a conceptually appealing and
computationally extremely efficient approach to finding and characterizing all
symmetric nodes by introducing a structural position vector (SPV) for each and
every node in the network. We prove mathematically that nodes with the
identical SPV are symmetrical to each other. Utilizing six representative
complex networks from the real world, we demonstrate that all symmetric nodes
can be found in linear time, and the SPVs can not only characterize the
similarity of nodes but also quantify the nodal influences in spreading
dynamics on the network. Our SPV-based framework, in additional to being
rigorously justified, provides a physically intuitive way to uncover,
understand and exploit symmetric structures in complex networks.
