  Complex networks often possess communities defined based on network
connectivity. When dynamics undergo in a network, one can also consider
dynamical communities; i.e., a group of nodes displaying a similar dynamical
process. We have investigated both analytically and numerically the development
of dynamical community structure, where the community is referred to as a group
of nodes synchronized in frequency, in networks of phase oscillators. We first
demonstrate that using a few example networks, the community structure changes
when network connectivity or interaction strength is varied. In particular, we
found that community switching, i.e., a portion of oscillators change the group
to which they synchronize, occurs for a range of parameters. We then propose a
three-oscillator model: a bidirectional, weighted chain of three Kuramoto phase
oscillators, as a theoretical framework for understanding the community
formation and its variation. Our analysis demonstrates that the model shows a
variety of partially synchronized patterns: oscillators with similar natural
frequencies tend to synchronize for weak coupling, while tightly connected
oscillators tend to synchronize for strong coupling. We obtain approximate
expressions for the critical coupling strengths by employing a perturbative
approach in a weak coupling regime and a geometric approach in a strong
coupling regimes. Moreover, we elucidate the bifurcation types of transitions
between different patterns. Our theory might be useful for understanding the
development of partially synchronized patterns in a wider class of complex
networks than community structured networks.
