Mathematical modelling of real complex networks aims to characterize their architecture and decipher their underlying principles. Self-repeating patterns and multifractality exist in many real-world complex systems such as brain, genetic, geoscience, and social networks. To better comprehend the multifractal behavior in the real networks, we propose the weighted multifractal graph model to characterize the spatiotemporal complexity and heterogeneity encoded in the interaction weights. We provide analytical tools to verify the multifractal properties of the proposed model. By varying the parameters in the initial unit square, the model can reproduce a diverse range of multifractal spectrums with different degrees of symmetry, locations, support and shapes. We estimate and investigate the weighted multifractal graph model corresponding to two real-world complex systems, namely (i) the chromosome interactions of yeast cells in quiescence and in exponential growth, and (ii) the brain networks of cognitively healthy people and patients exhibiting late mild cognitive impairment leading to Alzheimer disease. The analysis of recovered models show that the proposed random graph model provides a novel way to understand the self-similar structure of complex networks and to discriminate different network structures. Additionally, by mapping real complex networks onto multifractal generating measures, it allows us to develop new network design and control strategies, such as the minimal control of multifractal measures of real systems under different functioning conditions or states.