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. ISSN 2045-2322 (online). We used modularity with a resolution parameter of =1 for the experiments. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. 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. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. Article The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in a credit line to the material. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. Traag, V. A. leidenalg 0.7.0. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). This makes sense, because after phase one the total size of the graph should be significantly reduced. It therefore does not guarantee -connectivity either. Phys. Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. For each network, we repeated the experiment 10 times. Nodes 06 are in the same community. B 86, 471, https://doi.org/10.1140/epjb/e2013-40829-0 (2013). 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. 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. Ronhovde, Peter, and Zohar Nussinov. Phys. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Below we offer an intuitive explanation of these properties. Google Scholar. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. Am. The thick edges in Fig. 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. How many iterations of the Leiden clustering algorithm to perform. Soft Matter Phys. 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. Finding and Evaluating Community Structure in Networks. Phys. PubMed In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. J. Comput. 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. It means that there are no individual nodes that can be moved to a different community. Mech. The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. In the meantime, to ensure continued support, we are displaying the site without styles In terms of the percentage of badly connected communities in the first iteration, Leiden performs even worse than Louvain, as can be seen in Fig. http://dx.doi.org/10.1073/pnas.0605965104. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. V. A. Traag. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. If material is not included in the articles Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. You will not need much Python to use it. Complex brain networks: graph theoretical analysis of structural and functional systems. As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM). Based on this partition, an aggregate network is created (c). 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). The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. At this point, it is guaranteed that each individual node is optimally assigned. Slider with three articles shown per slide. Rev. We start by initialising a queue with all nodes in the network. The Leiden community detection algorithm outperforms other clustering methods. Clustering is a machine learning technique in which similar data points are grouped into the same cluster based on their attributes. 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. This function takes a cell_data_set as input, clusters the cells using . An aggregate. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. performed the experimental analysis. Leiden is faster than Louvain especially for larger networks. Second, to study the scaling of the Louvain and the Leiden algorithm, we use benchmark networks, allowing us to compare the algorithms in terms of both computational time and quality of the partitions. Clustering the neighborhood graph As with Seurat and many other frameworks, we recommend the Leiden graph-clustering method (community detection based on optimizing modularity) by Traag *et al. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. Ph.D. thesis, (University of Oxford, 2016). We now consider the guarantees provided by the Leiden algorithm. In this post, I will cover one of the common approaches which is hierarchical clustering. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. Hence, in general, Louvain may find arbitrarily badly connected communities. Louvain pruning is another improvement to Louvain proposed in 2016, and can reduce the computational time by as much as 90% while finding communities that are almost as good as Louvain (Ozaki, Tezuka, and Inaba 2016). In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . 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 ). On the other hand, after node 0 has been moved to a different community, nodes 1 and 4 have not only internal but also external connections. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. This amounts to a clustering problem, where we aim to learn an optimal set of groups (communities) from the observed data. You signed in with another tab or window. A tag already exists with the provided branch name. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. Obviously, this is a worst case example, showing that disconnected communities may be identified by the Louvain algorithm. The property of -separation is also guaranteed by the Louvain algorithm. 20, 172188, https://doi.org/10.1109/TKDE.2007.190689 (2008). The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . 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. The algorithm then moves individual nodes in the aggregate network (e). First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. Note that communities found by the Leiden algorithm are guaranteed to be connected. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. In this way, Leiden implements the local moving phase more efficiently than Louvain. 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. 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. 2007. We then remove the first node from the front of the queue and we determine whether the quality function can be increased by moving this node from its current community to a different one. Other networks show an almost tenfold increase in the percentage of disconnected communities. Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. MathSciNet (We ensured that modularity optimisation for the subnetwork was fully consistent with modularity optimisation for the whole network13) The Leiden algorithm was run until a stable iteration was obtained. Phys. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. 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. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. This is well illustrated by figure 2 in the Leiden paper: When a community becomes disconnected like this, there is no way for Louvain to easily split it into two separate communities. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . S3. https://doi.org/10.1038/s41598-019-41695-z. We use six empirical networks in our analysis. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. The steps for agglomerative clustering are as follows: Sci. V.A.T. E 69, 026113, https://doi.org/10.1103/PhysRevE.69.026113 (2004). In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. Waltman, Ludo, and Nees Jan van Eck. The solution provided by Leiden is based on the smart local moving algorithm. Cluster Determination Source: R/generics.R, R/clustering.R Identify clusters of cells by a shared nearest neighbor (SNN) modularity optimization based clustering algorithm. Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. Therefore, clustering algorithms look for similarities or dissimilarities among data points. Louvain has two phases: local moving and aggregation. Wolf, F. A. et al. Internet Explorer). The reasoning behind this is that the best community to join will usually be the one that most of the nodes neighbors already belong to. Article Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. 2013. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). However, it is also possible to start the algorithm from a different partition15. PubMed Central Subpartition -density is not guaranteed by the Louvain algorithm. In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. Scaling of benchmark results for difficulty of the partition. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. Phys. However, values of within a range of roughly [0.0005, 0.1] all provide reasonable results, thus allowing for some, but not too much randomness. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. A smart local moving algorithm for large-scale modularity-based community detection. All communities are subpartition -dense. 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). The Louvain algorithm is illustrated in Fig. We therefore require a more principled solution, which we will introduce in the next section. Theory Exp. Rev. Google Scholar. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. Porter, M. A., Onnela, J.-P. & Mucha, P. J. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. Modularity optimization. Directed Undirected Homogeneous Heterogeneous Weighted 1. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. There is an entire Leiden package in R-cran here It identifies the clusters by calculating the densities of the cells. CPM is defined as. Inf. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. We name our algorithm the Leiden algorithm, after the location of its authors. Badly connected communities. After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. 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. We keep removing nodes from the front of the queue, possibly moving these nodes to a different community. http://arxiv.org/abs/1810.08473. Cite this article. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Then the Leiden algorithm can be run on the adjacency matrix. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. Rev. Thank you for visiting nature.com. volume9, Articlenumber:5233 (2019) bioRxiv, https://doi.org/10.1101/208819 (2018). Both conda and PyPI have leiden clustering in Python which operates via iGraph. 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. Nat. In other words, communities are guaranteed to be well separated. Modularity is used most commonly, but is subject to the resolution limit. For each set of parameters, we repeated the experiment 10 times. Importantly, the number of communities discovered is related only to the difference in edge density, and not the total number of nodes in the community. N.J.v.E. (2) and m is the number of edges. When a sufficient number of neighbours of node 0 have formed a community in the rest of the network, it may be optimal to move node 0 to this community, thus creating the situation depicted in Fig. MathSciNet Eng. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. 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. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. The Leiden algorithm is considerably more complex than the Louvain algorithm. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. Clustering with the Leiden Algorithm in R This package allows calling the Leiden algorithm for clustering on an igraph object from R. See the Python and Java implementations for more details: https://github.com/CWTSLeiden/networkanalysis https://github.com/vtraag/leidenalg Install This is similar to what we have seen for benchmark networks. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. Eng. J. This is not too difficult to explain. The docs are here. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. Article It is a directed graph if the adjacency matrix is not symmetric. We used the CPM quality function. In all experiments reported here, we used a value of 0.01 for the parameter that determines the degree of randomness in the refinement phase of the Leiden algorithm. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in It implies uniform -density and all the other above-mentioned properties. They show that the original Louvain algorithm that can result in badly connected communities (even communities that are completely disconnected internally) and propose an alternative method, Leiden, that guarantees that communities are well connected. Note that nodes can be revisited several times within a single iteration of the local moving stage, as the possible increase in modularity will change as other nodes are moved to different communities. The Louvain method for community detection is a popular way to discover communities from single-cell data. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. 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. CAS In this section, we analyse and compare the performance of the two algorithms in practice. With one exception (=0.2 and n=107), all results in Fig. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. Blondel, V D, J L Guillaume, and R Lambiotte. After the first iteration of the Louvain algorithm, some partition has been obtained. Some of these nodes may very well act as bridges, similarly to node 0 in the above example. For example, the red community in (b) is refined into two subcommunities in (c), which after aggregation become two separate nodes in (d), both belonging to the same community. Rev. MATH Newman, M. E. J. to use Codespaces. Clearly, it would be better to split up the community. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. Rev. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. Trying to fix the problem by simply considering the connected components of communities19,20,21 is unsatisfactory because it addresses only the most extreme case and does not resolve the more fundamental problem. This should be the first preference when choosing an algorithm. This way of defining the expected number of edges is based on the so-called configuration model. First calculate k-nearest neighbors and construct the SNN graph. wrote the manuscript. 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. CPM does not suffer from this issue13. MathSciNet Rev. This represents the following graph structure. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). 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For higher values of , Leiden finds better partitions than Louvain. & Girvan, M. Finding and evaluating community structure in networks. modularity) increases. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. A partition of clusters as a vector of integers Examples E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). The Leiden algorithm is clearly faster than the Louvain algorithm. Nonetheless, some networks still show large differences. Basically, there are two types of hierarchical cluster analysis strategies - 1. 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. 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 https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. Technol. Traag, V. A. In single-cell biology we often use graph-based community detection methods to do this, as these methods are unsupervised, scale well, and usually give good results. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. 2015. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. If nothing happens, download GitHub Desktop and try again. 104 (1): 3641. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. In short, the problem of badly connected communities has important practical consequences. In subsequent iterations, the percentage of disconnected communities remains fairly stable. Disconnected community. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. This is very similar to what the smart local moving algorithm does. ADS 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. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. Sci Rep 9, 5233 (2019). Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. This contrasts with the Leiden algorithm. 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. The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. Besides the Louvain algorithm and the Leiden algorithm (see the "Methods" section), there are several widely-used network clustering algorithms, such as the Markov clustering algorithm [], Infomap algorithm [], and label propagation algorithm [].Markov clustering and Infomap algorithm are both based on flow .