Here we present an agent-based model where agents interact with other agents by playing a hybrid of dictator and ultimatum games in a co-evolving social network. The basic assumption about the behaviour of the agents in both games is that they try to attain superior socioeconomic positions relative to other agents. As the model parameters we have chosen the relative proportions of the dictator and ultimatum game strategies being played between a pair of agents in a single social transaction and a parameter depicting the living costs of the agents. The motivation of the study is to examine how different types of social interactions affect the formation of social structures and networks, when the agents have a tendency to maximize their socioeconomic standing. We find that such social networks of agents invariably undergo a community formation process from simple chain-like structure to more complex networks as the living cost parameter is increased. The point where this occurs, depends also on the relative proportion of the dictator and ultimatum games being played. We find that it is harder for complex social structures to form when the dictator game strategy in social transactions of agents becomes more dominant over that of the ultimatum game.