More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. Rev. By moving these nodes, Louvain creates badly connected communities. Badly connected communities. Number of iterations until stability. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. The leidenalg package facilitates community detection of networks and builds on the package igraph. Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. In the first step of the next iteration, Louvain will again move individual nodes in the network. ADS Scientific Reports (Sci Rep) Rev. In subsequent iterations, the percentage of disconnected communities remains fairly stable. It starts clustering by treating the individual data points as a single cluster then it is merged continuously based on similarity until it forms one big cluster containing all objects. & Bornholdt, S. Statistical mechanics of community detection. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. In an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities. We name our algorithm the Leiden algorithm, after the location of its authors. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. The algorithm moves individual nodes from one community to another to find a partition (b). Communities in Networks. In later stages, most neighbors will belong to the same community, and its very likely that the best move for the node is to the community that most of its neighbors already belong to. 2013. Fortunato, S. Community detection in graphs. B 86, 471, https://doi.org/10.1140/epjb/e2013-40829-0 (2013). The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. 2010. where >0 is a resolution parameter4. Not. The value of the resolution parameter was determined based on the so-called mixing parameter 13. Based on this partition, an aggregate network is created (c). 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. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. Besides being pervasive, the problem is also sizeable. Rev. Clustering is a machine learning technique in which similar data points are grouped into the same cluster based on their attributes. A. Then the Leiden algorithm can be run on the adjacency matrix. Two ways of doing this are graph modularity (Newman and Girvan 2004) and the constant Potts model (Ronhovde and Nussinov 2010). In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. Data Eng. Therefore, by selecting a community based by choosing randomly from the neighbors, we choose the community to evaluate with probability proportional to the composition of the neighbors communities. Modularity optimization. CAS contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. It therefore does not guarantee -connectivity either. Newman, M. E. J. 2008. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. Phys. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. We use six empirical networks in our analysis. 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. Elect. For those wanting to read more, I highly recommend starting with the Leiden paper (Traag, Waltman, and Eck 2018) or the smart local moving paper (Waltman and Eck 2013). Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). Sci. Iterating the Louvain algorithm can therefore be seen as a double-edged sword: it improves the partition in some way, but degrades it in another way. Soft Matter Phys. Computer Syst. CPM does not suffer from this issue13. E Stat. Sci. Finding community structure in networks using the eigenvectors of matrices. Hence, in general, Louvain may find arbitrarily badly connected communities. Traag, V. A., Van Dooren, P. & Nesterov, Y. CPM has the advantage that it is not subject to the resolution limit. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. Article Basically, there are two types of hierarchical cluster analysis strategies - 1. The property of -separation is also guaranteed by the Louvain algorithm. Bae, S., Halperin, D., West, J. D., Rosvall, M. & Howe, B. Scalable and Efficient Flow-Based Community Detection for Large-Scale Graph Analysis. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. It is good at identifying small clusters. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. In particular, we show that Louvain may identify communities that are internally disconnected. The count of badly connected communities also included disconnected communities. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. An aggregate network (d) is created based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. 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). In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. Both conda and PyPI have leiden clustering in Python which operates via iGraph. The Beginner's Guide to Dimensionality Reduction. Importantly, the first iteration of the Leiden algorithm is the most computationally intensive one, and subsequent iterations are faster. The algorithm continues to move nodes in the rest of the network. These nodes are therefore optimally assigned to their current community. This way of defining the expected number of edges is based on the so-called configuration model. Large network community detection by fast label propagation, Representative community divisions of networks, Gausss law for networks directly reveals community boundaries, A Regularized Stochastic Block Model for the robust community detection in complex networks, Community Detection in Complex Networks via Clique Conductance, A generalised significance test for individual communities in networks, Community Detection on Networkswith Ricci Flow, https://github.com/CWTSLeiden/networkanalysis, https://doi.org/10.1016/j.physrep.2009.11.002, https://doi.org/10.1103/PhysRevE.69.026113, https://doi.org/10.1103/PhysRevE.74.016110, https://doi.org/10.1103/PhysRevE.70.066111, https://doi.org/10.1103/PhysRevE.72.027104, https://doi.org/10.1103/PhysRevE.74.036104, https://doi.org/10.1088/1742-5468/2008/10/P10008, https://doi.org/10.1103/PhysRevE.80.056117, https://doi.org/10.1103/PhysRevE.84.016114, https://doi.org/10.1140/epjb/e2013-40829-0, https://doi.org/10.17706/IJCEE.2016.8.3.207-218, https://doi.org/10.1103/PhysRevE.92.032801, https://doi.org/10.1103/PhysRevE.76.036106, https://doi.org/10.1103/PhysRevE.78.046110, https://doi.org/10.1103/PhysRevE.81.046106, http://creativecommons.org/licenses/by/4.0/, A robust and accurate single-cell data trajectory inference method using ensemble pseudotime, Batch alignment of single-cell transcriptomics data using deep metric learning, ViralCC retrieves complete viral genomes and virus-host pairs from metagenomic Hi-C data, Community detection in brain connectomes with hybrid quantum computing. Note that communities found by the Leiden algorithm are guaranteed to be connected. Clustering is the task of grouping a set of objects with similar characteristics into one bucket and differentiating them from the rest of the group. A Comparative Analysis of Community Detection Algorithms on Artificial Networks. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. This algorithm provides a number of explicit guarantees. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. A Smart Local Moving Algorithm for Large-Scale Modularity-Based Community Detection. Eur. In other words, communities are guaranteed to be well separated. and L.W. Rather than evaluating the modularity gain for moving a node to each neighboring communities, we choose a neighboring node at random and evaluate whether there is a gain in modularity if we were to move the node to that neighbors community. Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). In this section, we analyse and compare the performance of the two algorithms in practice. 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. Are you sure you want to create this branch? Phys. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. We now consider the guarantees provided by the Leiden algorithm. A structure that is more informative than the unstructured set of clusters returned by flat clustering. As we prove in SectionC1 of the Supplementary Information, even when node mergers that decrease the quality function are excluded, the optimal partition of a set of nodes can still be uncovered. Importantly, the problem of disconnected communities is not just a theoretical curiosity. The two phases are repeated until the quality function cannot be increased further. & Fortunato, S. Community detection algorithms: A comparative analysis. Moreover, when the algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are guaranteed to be locally optimally assigned. This aspect of the Louvain algorithm can be used to give information about the hierarchical relationships between communities by tracking at which stage the nodes in the communities were aggregated. After each iteration of the Leiden algorithm, it is guaranteed that: In these properties, refers to the resolution parameter in the quality function that is optimised, which can be either modularity or CPM. o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. To install the development version: The current release on CRAN can be installed with: First set up a compatible adjacency matrix: An adjacency matrix is any binary matrix representing links between nodes (column and row names). While smart local moving and multilevel refinement can improve the communities found, the next two improvements on Louvain that Ill discuss focus on the speed/efficiency of the algorithm. This represents the following graph structure. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. 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. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. Phys. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. However, as increases, the Leiden algorithm starts to outperform the Louvain algorithm. Nat. Leiden consists of the following steps: Local moving of nodes Partition refinement Network aggregation The refinement step allows badly connected communities to be split before creating the aggregate network. Uniform -density means that no matter how a community is partitioned into two parts, the two parts will always be well connected to each other. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. 2016. 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. Learn more. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. A community is subset optimal if all subsets of the community are locally optimally assigned. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). S3. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. Inf. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. Modularity is used most commonly, but is subject to the resolution limit. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. There is an entire Leiden package in R-cran here Note that the object for Seurat version 3 has changed. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. Article If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. These nodes can be approximately identified based on whether neighbouring nodes have changed communities. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. After running local moving, we end up with a set of communities where we cant increase the objective function (eg, modularity) by moving any node to any neighboring community. Communities were all of equal size. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). However, so far this problem has never been studied for the Louvain algorithm. IEEE Trans. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. E Stat. In the meantime, to ensure continued support, we are displaying the site without styles This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. * (2018). Work fast with our official CLI. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). 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 . However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). Porter, M. A., Onnela, J.-P. & Mucha, P. J. Bullmore, E. & Sporns, O. Traag, V. A. leidenalg 0.7.0. Article Wolf, F. A. et al. E Stat. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. For all networks, Leiden identifies substantially better partitions than Louvain. Waltman, L. & van Eck, N. J. Article 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Nodes 16 have connections only within this community, whereas node 0 also has many external connections. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. As such, we scored leiden-clustering popularity level to be Limited. Sci Rep 9, 5233 (2019). Randomness in the selection of a community allows the partition space to be explored more broadly. It is a directed graph if the adjacency matrix is not symmetric. For larger networks and higher values of , Louvain is much slower than Leiden. Any sub-networks that are found are treated as different communities in the next aggregation step. If nothing happens, download Xcode and try again. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. Phys. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. and JavaScript. Ayan Sinha, David F. Gleich & Karthik Ramani, Marinka Zitnik, Rok Sosi & Jure Leskovec, Zhenqi Lu, Johan Wahlstrm & Arye Nehorai, Natalie Stanley, Roland Kwitt, Peter J. Mucha, Scientific Reports Phys. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. Removing such a node from its old community disconnects the old community. You signed in with another tab or window. Rev. Moreover, the deeper significance of the problem was not recognised: disconnected communities are merely the most extreme manifestation of the problem of arbitrarily badly connected communities. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. We used modularity with a resolution parameter of =1 for the experiments. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. 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 starts from a singleton partition (a). As can be seen in Fig. Please The Leiden algorithm is considerably more complex than the Louvain algorithm. A community is subpartition -dense if it can be partitioned into two parts such that: (1) the two parts are well connected to each other; (2) neither part can be separated from its community; and (3) each part is also subpartition -dense itself. Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). Clauset, A., Newman, M. E. J. It means that there are no individual nodes that can be moved to a different community. CAS The algorithm then moves individual nodes in the aggregate network (d). How many iterations of the Leiden clustering algorithm to perform. Figure3 provides an illustration of the algorithm. The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. A new methodology for constructing a publication-level classification system of science. Finally, we demonstrate the excellent performance of the algorithm for several benchmark and real-world networks. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in A partition of clusters as a vector of integers Examples Get the most important science stories of the day, free in your inbox. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). conda install -c conda-forge leidenalg pip install leiden-clustering Used via. The nodes are added to the queue in a random order. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . For both algorithms, 10 iterations were performed. 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. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. Reichardt, J. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. Knowl. Phys. Phys. In addition, we prove that, when the Leiden algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are locally optimally assigned. As shown in Fig. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. . N.J.v.E. Louvain has two phases: local moving and aggregation. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. 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. An overview of the various guarantees is presented in Table1. The corresponding results are presented in the Supplementary Fig. In contrast, Leiden keeps finding better partitions in each iteration. The nodes that are more interconnected have been partitioned into separate clusters. ACM Trans. Runtime versus quality for empirical networks. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. It implies uniform -density and all the other above-mentioned properties. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. That is, no subset can be moved to a different community. 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. This will compute the Leiden clusters and add them to the Seurat Object Class. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). Acad. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. Cluster cells using Louvain/Leiden community detection Description. Percentage of communities found by the Louvain algorithm that are either disconnected or badly connected compared to percentage of badly connected communities found by the Leiden algorithm. Then optimize the modularity function to determine clusters. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. To elucidate the problem, we consider the example illustrated in Fig. Rev. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. Empirical networks show a much richer and more complex structure. Agglomerative clustering is a bottom-up approach. Phys. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. If you cant use Leiden, choosing Smart Local Moving will likely give very similar results, but might be a bit slower as it doesnt include some of the simple speedups to Louvain like random moving and Louvain pruning. This can be a shared nearest neighbours matrix derived from a graph object. In fact, when we keep iterating the Leiden algorithm, it will converge to a partition for which it is guaranteed that: A community is uniformly -dense if there are no subsets of the community that can be separated from the community. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis.