This paper explores the emergence of modularity in complex networks, showing that it can arise from fluctuations in the establishment of links. The authors demonstrate that modularity in networks is analogous to the ground-state energy of a spin system. They show that both random graphs and scale-free networks have high modularity due to fluctuations in link formation. This finding suggests that modularity should be evaluated against a random graph null model to determine statistical significance. Modularity is defined as the difference between the actual number of links within modules and the expected number in a random network. The authors show that modularity in low-dimensional regular lattices, ER random graphs, and scale-free networks can be high due to the large number of possible partitions and fluctuations in link formation. They derive analytical expressions for modularity in these network types, showing that it depends on network size and other parameters. The authors also show that modularity in random graphs can be as high as that of graphs with modular structures imposed at the onset. This suggests that modularity is not necessarily a result of evolutionary pressures but can emerge from the statistical properties of random networks. The results imply that modularity should be compared to a random graph null model to determine statistical significance. The authors conclude that modularity in complex networks can arise from fluctuations in link formation and that this should be taken into account when defining statistically significant modularity.This paper explores the emergence of modularity in complex networks, showing that it can arise from fluctuations in the establishment of links. The authors demonstrate that modularity in networks is analogous to the ground-state energy of a spin system. They show that both random graphs and scale-free networks have high modularity due to fluctuations in link formation. This finding suggests that modularity should be evaluated against a random graph null model to determine statistical significance. Modularity is defined as the difference between the actual number of links within modules and the expected number in a random network. The authors show that modularity in low-dimensional regular lattices, ER random graphs, and scale-free networks can be high due to the large number of possible partitions and fluctuations in link formation. They derive analytical expressions for modularity in these network types, showing that it depends on network size and other parameters. The authors also show that modularity in random graphs can be as high as that of graphs with modular structures imposed at the onset. This suggests that modularity is not necessarily a result of evolutionary pressures but can emerge from the statistical properties of random networks. The results imply that modularity should be compared to a random graph null model to determine statistical significance. The authors conclude that modularity in complex networks can arise from fluctuations in link formation and that this should be taken into account when defining statistically significant modularity.