The resilience of the Internet to random breakdowns is studied using percolation theory. The Internet's node connectivity follows a power-law distribution, $ P(k) = c k^{-\alpha} $. The study shows that for $ \alpha \leq 3 $, the Internet is highly robust to random node removal, with a critical fraction $ p_c > 0.99 $ needed to disintegrate the network. This is because the second moment of the connectivity distribution, $ \kappa_0 = \langle k^2 \rangle / \langle k \rangle $, determines the critical threshold. For $ \alpha < 3 $, $ \kappa_0 $ diverges, leading to a critical threshold approaching 100% node removal. For the Internet, with $ \alpha \approx 2.5 $, the spanning cluster remains connected even at nearly 100% breakdown. This is due to the power-law distribution's properties, which allow the network to maintain connectivity despite high node removal. The study also shows that for networks with power-law connectivity distributions, the critical threshold depends on $ \alpha $, with $ \alpha > 3 $ leading to a finite critical threshold, while $ \alpha < 3 $ results in a high threshold. Computer simulations confirm these findings, showing that for $ \alpha = 3.5 $, the network disintegrates at $ p_c \approx 0.5 $, while for $ \alpha = 2.5 $, the network remains connected even at near 100% breakdown. The Internet's resilience is attributed to its power-law connectivity distribution and large size, making it highly robust to random failures.The resilience of the Internet to random breakdowns is studied using percolation theory. The Internet's node connectivity follows a power-law distribution, $ P(k) = c k^{-\alpha} $. The study shows that for $ \alpha \leq 3 $, the Internet is highly robust to random node removal, with a critical fraction $ p_c > 0.99 $ needed to disintegrate the network. This is because the second moment of the connectivity distribution, $ \kappa_0 = \langle k^2 \rangle / \langle k \rangle $, determines the critical threshold. For $ \alpha < 3 $, $ \kappa_0 $ diverges, leading to a critical threshold approaching 100% node removal. For the Internet, with $ \alpha \approx 2.5 $, the spanning cluster remains connected even at nearly 100% breakdown. This is due to the power-law distribution's properties, which allow the network to maintain connectivity despite high node removal. The study also shows that for networks with power-law connectivity distributions, the critical threshold depends on $ \alpha $, with $ \alpha > 3 $ leading to a finite critical threshold, while $ \alpha < 3 $ results in a high threshold. Computer simulations confirm these findings, showing that for $ \alpha = 3.5 $, the network disintegrates at $ p_c \approx 0.5 $, while for $ \alpha = 2.5 $, the network remains connected even at near 100% breakdown. The Internet's resilience is attributed to its power-law connectivity distribution and large size, making it highly robust to random failures.