  Synchronization is critical for system level behaviour in physical, chemical,
biological and social systems. Empirical evidence has shown that the network
topology strongly impacts the synchronizablity of the system, and the analysis
of their relationship remains an open challenge. We know that the eigenvalue
distribution determines a network's synchronizability, but analytical
expressions that connect network topology and all relevant eigenvalues (e.g.,
the extreme values) remain elusive.
  Here, we accurately determine its synchronizability by proposing an
analytical method to estimate the extreme eigenvalues using perturbation
theory. Our analytical method exposes the role global and local topology
combine to influence synchronizability. We show that the smallest non-zero
eigenvalue which determines synchronizability is estimated by the smallest
degree augmented by the inverse degree difference in the least connected nodes.
From this, we can conclude that there exists a clear negative relationship
between the smallest non-zero eigenvalue and the local assortativity of nodes
with smallest degree values. We validate the accuracy of our framework within
the setting of a Scale-free (SF) network and can be driven by commonly used
ODEs (e.g., 3-dimensional Rosler or Lorenz dynamics). From the results, we
demonstrate that the synchronizability of the network can be tuned by rewiring
the connections of these particular nodes while maintaining the general degree
profile of the network.
