  We introduce the weighted random graph (WRG) model, which represents the
weighted counterpart of the Erdos-Renyi random graph and provides fundamental
insights into more complicated weighted networks. We find analytically that the
WRG is characterized by a geometric weight distribution, a binomial degree
distribution and a negative binomial strength distribution. We also
characterize exactly the percolation phase transitions associated with edge
removal and with the appearance of weighted subgraphs of any order and
intensity. We find that even this completely null model displays a percolation
behavior similar to what observed in real weighted networks, implying that edge
removal cannot be used to detect community structure empirically. By contrast,
the analysis of clustering successfully reveals different patterns between the
WRG and real networks.
