We propose describing a kind of cooperation–competition systems by node-weighted bipartite graphs. In such systems the nodes can be divided into two types. One expresses a kind of activities, organizations or events, named “acts”; and another expresses the “actors” participating in some acts. In each act the actors basically show collaboration relationship with each other, however they play different roles in the cooperation. Usually every actor tries to play the most important role; therefore the role difference can be regarded as a kind of competition. The node weight usually signifies the role or “importance degree” of each actor. We propose using some statistical properties for the description of such kind of systems. The properties without considering node weight can describe the cooperation situation and configuration. The properties with node weight may describe the competition results. Two example systems are reported to show the description method and the corresponding empirical investigation results. One is the undergraduate elective network, and another the library network of Yangzhou University. The description method may be effective for the similar systems in different fields.
