Generalized strength of weighted scale-free networks.

An analysis to describe statistical properties of weighted complex networks is proposed. Effective structures of weighted networks depend on how strongly weights w are paid attention or which weights are relevant to the network problem. Defining the metaweight w;{q} with a real parameter q , we characterize systematically weighted complex networks depending on the level of importance of weights. It is found that power-law distribution functions R_{q}[s(q)] of metastrengths s(q) defined by s_{i}(q)= summation operator_{j}w_{ij};{q} , where i and j denote node indices for any q are characterized by only three exponents if the weight distribution is independent of network topology. We also examine the validity of our analytical arguments and the meaning of power-law forms of R_{q}[s(q)] for different q values by illustrating some examples.