The Louvain algorithm, a popular method for community detection in networks, has a major flaw: it may produce arbitrarily badly connected communities. In some cases, communities may even be disconnected, especially when the algorithm is run iteratively. This issue was not widely recognized until now. In contrast, the Leiden algorithm, introduced in this paper, guarantees that communities are connected. It also converges to a partition where all subsets of communities are locally optimally assigned. Additionally, the Leiden algorithm is faster than the Louvain algorithm and provides explicit guarantees for the quality of the partitions. The Louvain algorithm works in two phases: local moving of nodes and aggregation of the network. It starts from a singleton partition and iteratively moves nodes to improve the quality function. However, this process can lead to disconnected communities. The Leiden algorithm improves upon the Louvain algorithm by incorporating a fast local move approach, random neighbour move, and refinement of the partition. This allows it to find better partitions and guarantees that communities are connected. The Leiden algorithm is tested on several benchmark and real-world networks. It is found to be faster than the Louvain algorithm and produces better partitions. The algorithm is also guaranteed to produce partitions where all communities are connected and locally optimally assigned. The Leiden algorithm is particularly effective in handling complex networks where the Louvain algorithm may produce disconnected communities. The paper also discusses the guarantees provided by the Leiden algorithm. After each iteration, it is guaranteed that all communities are γ-separated and γ-connected. In stable iterations, all nodes are locally optimally assigned, and all communities are subpartition γ-dense. The Leiden algorithm converges to a partition where all communities are uniformly γ-dense and subset optimal. These guarantees make the Leiden algorithm a stronger and more reliable method for community detection compared to the Louvain algorithm.The Louvain algorithm, a popular method for community detection in networks, has a major flaw: it may produce arbitrarily badly connected communities. In some cases, communities may even be disconnected, especially when the algorithm is run iteratively. This issue was not widely recognized until now. In contrast, the Leiden algorithm, introduced in this paper, guarantees that communities are connected. It also converges to a partition where all subsets of communities are locally optimally assigned. Additionally, the Leiden algorithm is faster than the Louvain algorithm and provides explicit guarantees for the quality of the partitions. The Louvain algorithm works in two phases: local moving of nodes and aggregation of the network. It starts from a singleton partition and iteratively moves nodes to improve the quality function. However, this process can lead to disconnected communities. The Leiden algorithm improves upon the Louvain algorithm by incorporating a fast local move approach, random neighbour move, and refinement of the partition. This allows it to find better partitions and guarantees that communities are connected. The Leiden algorithm is tested on several benchmark and real-world networks. It is found to be faster than the Louvain algorithm and produces better partitions. The algorithm is also guaranteed to produce partitions where all communities are connected and locally optimally assigned. The Leiden algorithm is particularly effective in handling complex networks where the Louvain algorithm may produce disconnected communities. The paper also discusses the guarantees provided by the Leiden algorithm. After each iteration, it is guaranteed that all communities are γ-separated and γ-connected. In stable iterations, all nodes are locally optimally assigned, and all communities are subpartition γ-dense. The Leiden algorithm converges to a partition where all communities are uniformly γ-dense and subset optimal. These guarantees make the Leiden algorithm a stronger and more reliable method for community detection compared to the Louvain algorithm.