  The purpose of fingerprinting is to compare long messages with low
communication complexity. Compared with its classical version, the quantum
fingerprinting can realize exponential reduction in communication complexity.
Recently, the multi-party quantum fingerprinting is studied on whether the
messages from many parties are the same. However, sometimes it is not enough
just to know whether these messages are the same, we usually need to know the
relationship among them. We provide a general model of quantum fingerprinting
network, defining the relationship function $f^R$ and giving the corresponding
decision rules. In this work, we take the four-party quantum fingerprinting
protocol as an example for detailed analysis. We also choose the optimal
parameters to minimize communication complexity in the case of asymmetric
channels. Furthermore, we compare the multi-party quantum fingerprinting with
the protocol based on the two-party quantum fingerprinting and find that the
multi-party protocol has obvious advantages, especially in terms of
communication time. Finally, the method of encoding more than one bit on each
coherent state is used to further improve the performance of the protocol.
