The paper presents a solution for the time- and age-dependent connectivity distribution of a growing random network (GRN). The network is constructed by adding new sites that connect to existing sites with a probability \( A_k \) that depends on the number of pre-existing links \( k \). For homogeneous connection kernels \( A_k \sim k^\gamma \), different behaviors are observed depending on the value of \( \gamma \): 1. **For \( \gamma < 1 \)**: The number of sites with \( k \) links, \( N_k \), varies as a stretched exponential. 2. **For \( \gamma > 1 \)**: A single site connects to nearly all other sites, leading to a "winner-take-all" phenomenon. 3. **For \( \gamma = 1 \)**: A power law distribution \( N_k \sim k^{-\nu} \) is found, where \( \nu \) can be tuned to any value in the range \( 2 < \nu < \infty \). The authors use a rate equation approach to solve the GRN model, which is simpler than standard probabilistic or generating function techniques. They focus on the connectivity distribution \( \hat{N}_k(t) \) and solve the rate equations for different values of \( \gamma \). For \( \gamma < 1 \), the connectivity distribution decreases exponentially. For \( \gamma > 1 \), a single "gel" site connects to almost every other site. For \( \gamma = 1 \), the power law distribution is observed, and the exponent \( \nu \) can be tuned to any value in the range \( 2 < \nu < \infty \). The paper also discusses the age-dependent structure of the network, showing that older sites are more highly connected. The age distribution of sites with \( k \) links is peaked around a characteristic age \( a_k \), which depends on \( \gamma \). Overall, the study provides a comprehensive understanding of the connectivity and age distributions in growing random networks, with implications for various real-world networks such as citation networks and information networks.The paper presents a solution for the time- and age-dependent connectivity distribution of a growing random network (GRN). The network is constructed by adding new sites that connect to existing sites with a probability \( A_k \) that depends on the number of pre-existing links \( k \). For homogeneous connection kernels \( A_k \sim k^\gamma \), different behaviors are observed depending on the value of \( \gamma \): 1. **For \( \gamma < 1 \)**: The number of sites with \( k \) links, \( N_k \), varies as a stretched exponential. 2. **For \( \gamma > 1 \)**: A single site connects to nearly all other sites, leading to a "winner-take-all" phenomenon. 3. **For \( \gamma = 1 \)**: A power law distribution \( N_k \sim k^{-\nu} \) is found, where \( \nu \) can be tuned to any value in the range \( 2 < \nu < \infty \). The authors use a rate equation approach to solve the GRN model, which is simpler than standard probabilistic or generating function techniques. They focus on the connectivity distribution \( \hat{N}_k(t) \) and solve the rate equations for different values of \( \gamma \). For \( \gamma < 1 \), the connectivity distribution decreases exponentially. For \( \gamma > 1 \), a single "gel" site connects to almost every other site. For \( \gamma = 1 \), the power law distribution is observed, and the exponent \( \nu \) can be tuned to any value in the range \( 2 < \nu < \infty \). The paper also discusses the age-dependent structure of the network, showing that older sites are more highly connected. The age distribution of sites with \( k \) links is peaked around a characteristic age \( a_k \), which depends on \( \gamma \). Overall, the study provides a comprehensive understanding of the connectivity and age distributions in growing random networks, with implications for various real-world networks such as citation networks and information networks.