Network systems evolving by preferential attachment of new vertices and by link loading are considered. The probability of node attaching is assumed to depend linearly on degrees of existing nodes to which new ones are connected. The rule of the network loading allows both deterministic and random processes of distributing weights among links. Specifically, the rule is determined by assigning to each link a weight dependent on randomly modified intervals between times at which linked vertices are connected to the system. This enables the investigation of the influence of local and global loading processes on statistical properties of weighted networks. It is demonstrated that, for different values of parameters controlling these processes, the distribution of weights concentrated at vertices of such networks displays a large variety of scale-free and truncated scale-free behaviors. It is also shown that statistical properties of the studied weighted networks and statistical properties of some complex evolving spatial systems exhibit a close relationship.
