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The notion of w-density for the graphs with positive weights on vertices and nonnegative weights on edges is introduced.A weighted graph is called w-balanced if its w-density is no less than the w-density of any subgraph of it.In this paper,a good characterization of w-balanced weighted graphs is given.Applying this characterization,many large w-balanced weighted graphs are formed by combining smaller ones.In the case where a graph is not w-balanced,a polynomial-time algorithm to find a subgraph of maximum w-density is proposed.It is shown that the w-density theory is closely related to the study of SEW(G,w) games.