The paper explores the evolution of networks where nodes compete for links based on their "fitness," leading to multiscaling in connectivity. In traditional models, all nodes grow at the same rate, but this study introduces a fitness parameter that determines how quickly a node acquires links. Nodes with higher fitness can outcompete others, even if they are younger, leading to a power-law distribution of connectivity with a dynamic exponent dependent on fitness. The model incorporates a generalized preferential attachment mechanism where the probability of a new node connecting to an existing one depends on both its connectivity and fitness. This leads to a continuum theory that predicts the connectivity distribution as a weighted sum of power-laws, with a correction term for logarithmic scaling. The study demonstrates that when nodes have different fitness values, the system exhibits multiscaling, where the dynamic exponent varies with fitness. This allows fitter nodes to grow faster, even if they are added later. The model is validated through numerical simulations showing that the predicted connectivity distribution matches the observed data, with a few highly connected nodes appearing as "super hubs" in log-log plots. The results highlight the importance of fitness in competitive systems, showing that even with varying fitness, nodes follow a power-law growth in connectivity. The model provides a framework for understanding the evolution of complex networks, including social, web, and biological systems, by incorporating both growth and fitness-based competition. The findings suggest that the fitter nodes dominate in terms of connectivity, following a power-law time dependence with a higher exponent than less fit peers. The model's predictions can be tested on real-world networks, offering insights into the underlying mechanisms of network evolution.The paper explores the evolution of networks where nodes compete for links based on their "fitness," leading to multiscaling in connectivity. In traditional models, all nodes grow at the same rate, but this study introduces a fitness parameter that determines how quickly a node acquires links. Nodes with higher fitness can outcompete others, even if they are younger, leading to a power-law distribution of connectivity with a dynamic exponent dependent on fitness. The model incorporates a generalized preferential attachment mechanism where the probability of a new node connecting to an existing one depends on both its connectivity and fitness. This leads to a continuum theory that predicts the connectivity distribution as a weighted sum of power-laws, with a correction term for logarithmic scaling. The study demonstrates that when nodes have different fitness values, the system exhibits multiscaling, where the dynamic exponent varies with fitness. This allows fitter nodes to grow faster, even if they are added later. The model is validated through numerical simulations showing that the predicted connectivity distribution matches the observed data, with a few highly connected nodes appearing as "super hubs" in log-log plots. The results highlight the importance of fitness in competitive systems, showing that even with varying fitness, nodes follow a power-law growth in connectivity. The model provides a framework for understanding the evolution of complex networks, including social, web, and biological systems, by incorporating both growth and fitness-based competition. The findings suggest that the fitter nodes dominate in terms of connectivity, following a power-law time dependence with a higher exponent than less fit peers. The model's predictions can be tested on real-world networks, offering insights into the underlying mechanisms of network evolution.