Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. MATH This will compute the Leiden clusters and add them to the Seurat Object Class. Contrary to what might be expected, iterating the Louvain algorithm aggravates the problem of badly connected communities, as we will also see in our experimental analysis. It is a directed graph if the adjacency matrix is not symmetric. It is good at identifying small clusters. It states that there are no communities that can be merged. The property of -separation is also guaranteed by the Louvain algorithm. In particular, it yields communities that are guaranteed to be connected. CPM does not suffer from this issue13. The Louvain algorithm is illustrated in Fig. Faster unfolding of communities: Speeding up the Louvain algorithm. First iteration runtime for empirical networks. CAS Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. Performance of modularity maximization in practical contexts. A partition of clusters as a vector of integers Examples Louvain has two phases: local moving and aggregation. One of the best-known methods for community detection is called modularity3. The Louvain algorithm starts from a singleton partition in which each node is in its own community (a). Neurosci. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. Newman, M. E. J. As such, we scored leiden-clustering popularity level to be Limited. Phys. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . The current state of the art when it comes to graph-based community detection is Leiden, which incorporates about 10 years of algorithmic improvements to the original Louvain method. Phys. Moreover, Louvain has no mechanism for fixing these communities. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. Rev. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Cite this article. Phys. Leiden algorithm. Google Scholar. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. First calculate k-nearest neighbors and construct the SNN graph. Sci. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. We denote by ec the actual number of edges in community c. The expected number of edges can be expressed as \(\frac{{K}_{c}^{2}}{2m}\), where Kc is the sum of the degrees of the nodes in community c and m is the total number of edges in the network. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). to use Codespaces. Biological sequence clustering is a complicated data clustering problem owing to the high computation costs incurred for pairwise sequence distance calculations through sequence alignments, as well as difficulties in determining parameters for deriving robust clusters. Data Eng. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. 2016. In fact, although it may seem that the Louvain algorithm does a good job at finding high quality partitions, in its standard form the algorithm provides only one guarantee: the algorithm yields partitions for which it is guaranteed that no communities can be merged. CPM has the advantage that it is not subject to the resolution limit. In general, Leiden is both faster than Louvain and finds better partitions. This is very similar to what the smart local moving algorithm does. Nodes 13 should form a community and nodes 46 should form another community. These nodes are therefore optimally assigned to their current community. Optimising modularity is NP-hard5, and consequentially many heuristic algorithms have been proposed, such as hierarchical agglomeration6, extremal optimisation7, simulated annealing4,8 and spectral9 algorithms. MathSciNet In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. 2. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. ADS A smart local moving algorithm for large-scale modularity-based community detection. 4, in the first iteration of the Louvain algorithm, the percentage of badly connected communities can be quite high. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. J. A structure that is more informative than the unstructured set of clusters returned by flat clustering. Rev. In this post, I will cover one of the common approaches which is hierarchical clustering. We study the problem of badly connected communities when using the Louvain algorithm for several empirical networks. An overview of the various guarantees is presented in Table1. Powered by DataCamp DataCamp The fast local move procedure can be summarised as follows. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). Sci. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. If nothing happens, download GitHub Desktop and try again. They identified an inefficiency in the Louvain algorithm: computes modularity gain for all neighbouring nodes per loop in local moving phase, even though many of these nodes will not have moved. The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. Then the Leiden algorithm can be run on the adjacency matrix. As can be seen in Fig. The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. Disconnected community. Community detection is an important task in the analysis of complex networks. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. Reichardt, J. Besides being pervasive, the problem is also sizeable. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). The smart local moving algorithm (Waltman and Eck 2013) identified another limitation in the original Louvain method: it isnt able to split communities once theyre merged, even when it may be very beneficial to do so. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. To do this we just sum all the edge weights between nodes of the corresponding communities to get a single weighted edge between them, and collapse each community down to a single new node. Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. Narrow scope for resolution-limit-free community detection. https://leidenalg.readthedocs.io/en/latest/reference.html. In other words, communities are guaranteed to be well separated. 20, 172188, https://doi.org/10.1109/TKDE.2007.190689 (2008). https://doi.org/10.1038/s41598-019-41695-z. Good, B. H., De Montjoye, Y. IEEE Trans. The algorithm then locally merges nodes in \({{\mathscr{P}}}_{{\rm{refined}}}\): nodes that are on their own in a community in \({{\mathscr{P}}}_{{\rm{refined}}}\) can be merged with a different community. sign in In other words, modularity may hide smaller communities and may yield communities containing significant substructure. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. Proc. ISSN 2045-2322 (online). The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. In the Louvain algorithm, a node may be moved to a different community while it may have acted as a bridge between different components of its old community. & Moore, C. Finding community structure in very large networks. Traag, V. A. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). We consider these ideas to represent the most promising directions in which the Louvain algorithm can be improved, even though we recognise that other improvements have been suggested as well22. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. Brandes, U. et al. For a full specification of the fast local move procedure, we refer to the pseudo-code of the Leiden algorithm in AlgorithmA.2 in SectionA of the Supplementary Information. In practical applications, the Leiden algorithm convincingly outperforms the Louvain algorithm, both in terms of speed and in terms of quality of the results, as shown by the experimental analysis presented in this paper. However, this is not necessarily the case, as the other nodes may still be sufficiently strongly connected to their community, despite the fact that the community has become disconnected. In the previous section, we showed that the Leiden algorithm guarantees a number of properties of the partitions uncovered at different stages of the algorithm. Provided by the Springer Nature SharedIt content-sharing initiative. Below we offer an intuitive explanation of these properties. As can be seen in Fig. Furthermore, if all communities in a partition are uniformly -dense, the quality of the partition is not too far from optimal, as shown in SectionE of the Supplementary Information. Figure4 shows how well it does compared to the Louvain algorithm. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. Phys. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. A tag already exists with the provided branch name. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. Soft Matter Phys. Phys. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). To study the scaling of the Louvain and the Leiden algorithm, we rely on a variant of a well-known approach for constructing benchmark networks28. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. From Louvain to Leiden: guaranteeing well-connected communities, $$ {\mathcal H} =\frac{1}{2m}\,{\sum }_{c}({e}_{c}-{\rm{\gamma }}\frac{{K}_{c}^{2}}{2m}),$$, $$ {\mathcal H} ={\sum }_{c}[{e}_{c}-\gamma (\begin{array}{c}{n}_{c}\\ 2\end{array})],$$, https://doi.org/10.1038/s41598-019-41695-z. Somewhat stronger guarantees can be obtained by iterating the algorithm, using the partition obtained in one iteration of the algorithm as starting point for the next iteration. Both conda and PyPI have leiden clustering in Python which operates via iGraph. (2) and m is the number of edges. Google Scholar. The count of badly connected communities also included disconnected communities. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. Once no further increase in modularity is possible by moving any node to its neighboring community, we move to the second phase of the algorithm: aggregation. Then, in order . For each community, modularity measures the number of edges within the community and the number of edges going outside the community, and gives a value between -1 and +1. That is, no subset can be moved to a different community. For higher values of , Leiden finds better partitions than Louvain. The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. 2015. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. Instead, a node may be merged with any community for which the quality function increases. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. S3. 1 I am using the leiden algorithm implementation in iGraph, and noticed that when I repeat clustering using the same resolution parameter, I get different results. The Leiden algorithm also takes advantage of the idea of speeding up the local moving of nodes16,17 and the idea of moving nodes to random neighbours18. A Smart Local Moving Algorithm for Large-Scale Modularity-Based Community Detection. Eur. Article Correspondence to An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. It partitions the data space and identifies the sub-spaces using the Apriori principle. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. The Louvain method for community detection is a popular way to discover communities from single-cell data. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). Requirements Developed using: scanpy v1.7.2 sklearn v0.23.2 umap v0.4.6 numpy v1.19.2 leidenalg Installation pip pip install leiden_clustering local This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. Sci. To use Leiden with the Seurat pipeline for a Seurat Object object that has an SNN computed (for example with Seurat::FindClusters with save.SNN = TRUE ). http://arxiv.org/abs/1810.08473. Electr. modularity) increases. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. The solution proposed in smart local moving is to alter how the local moving step in Louvain works. This is because Louvain only moves individual nodes at a time. Rather than progress straight to the aggregation stage (as we would for the original Louvain), we next consider each community as a new sub-network and re-apply the local moving step within each community. The Leiden community detection algorithm outperforms other clustering methods. Google Scholar. Google Scholar. However, after all nodes have been visited once, Leiden visits only nodes whose neighbourhood has changed, whereas Louvain keeps visiting all nodes in the network. If nothing happens, download Xcode and try again. This makes sense, because after phase one the total size of the graph should be significantly reduced. 10, 186198, https://doi.org/10.1038/nrn2575 (2009). It implies uniform -density and all the other above-mentioned properties. Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. and JavaScript. Rev. One of the most widely used algorithms is the Louvain algorithm10, which is reported to be among the fastest and best performing community detection algorithms11,12. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). The two phases are repeated until the quality function cannot be increased further. 5, for lower values of the partition is well defined, and neither the Louvain nor the Leiden algorithm has a problem in determining the correct partition in only two iterations. Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). The Leiden algorithm is partly based on the previously introduced smart local move algorithm15, which itself can be seen as an improvement of the Louvain algorithm. Subpartition -density does not imply that individual nodes are locally optimally assigned. Based on project statistics from the GitHub repository for the PyPI package leiden-clustering, we found that it has been starred 1 times. Modularity optimization. The algorithm then moves individual nodes in the aggregate network (e). Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). The Louvain local moving phase consists of the following steps: This process is repeated for every node in the network until no further improvement in modularity is possible. Ph.D. thesis, (University of Oxford, 2016). 2 represent stronger connections, while the other edges represent weaker connections. By submitting a comment you agree to abide by our Terms and Community Guidelines. In this new situation, nodes 2, 3, 5 and 6 have only internal connections. Waltman, L. & van Eck, N. J. Scaling of benchmark results for difficulty of the partition. Sci. The parameter functions as a sort of threshold: communities should have a density of at least , while the density between communities should be lower than . 2(a). This can be a shared nearest neighbours matrix derived from a graph object. These nodes can be approximately identified based on whether neighbouring nodes have changed communities. Are you sure you want to create this branch? This phenomenon can be explained by the documented tendency KMeans has to identify equal-sized , combined with the significant class imbalance associated with the datasets having more than 8 clusters (Table 1). Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. There is an entire Leiden package in R-cran here Discov. Phys. Computer Syst. & Girvan, M. Finding and evaluating community structure in networks. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. Acad. ACM Trans. By moving these nodes, Louvain creates badly connected communities. Louvain pruning keeps track of a list of nodes that have the potential to change communities, and only revisits nodes in this list, which is much smaller than the total number of nodes. Communities in Networks. In addition, to analyse whether a community is badly connected, we ran the Leiden algorithm on the subnetwork consisting of all nodes belonging to the community. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. Louvain algorithm. Number of iterations before the Leiden algorithm has reached a stable iteration for six empirical networks. You signed in with another tab or window. The authors show that the total computational time for Louvain depends a lot on the number of phase one loops (loops during the first local moving stage).