Proc. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). U. S. A. 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. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. The degree of randomness in the selection of a community is determined by a parameter >0. We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). Inf. Modularity (networks) - Wikipedia Phys. CAS With one exception (=0.2 and n=107), all results in Fig. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. Because the percentage of disconnected communities in the first iteration of the Louvain algorithm usually seems to be relatively low, the problem may have escaped attention from users of the algorithm. Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. 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). The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. Four popular community detection algorithms are explained . Phys. PubMed Central cluster_leiden: Finding community structure of a graph using the Leiden If nothing happens, download GitHub Desktop and try again. Eng. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM). 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. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. As discussed earlier, the Louvain algorithm does not guarantee connectivity. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). Leiden algorithm. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. Waltman, Ludo, and Nees Jan van Eck. 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. Modularity is a scale value between 0.5 (non-modular clustering) and 1 (fully modular clustering) that measures the relative density of edges inside communities with respect to edges outside communities. Zenodo, https://doi.org/10.5281/zenodo.1469357 https://github.com/vtraag/leidenalg. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. Correspondence to The algorithm continues to move nodes in the rest of the network. In other words, communities are guaranteed to be well separated. Importantly, the problem of disconnected communities is not just a theoretical curiosity. Such a modular structure is usually not known beforehand. Then optimize the modularity function to determine clusters. 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. 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. 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. ADS 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. Sci. 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. http://arxiv.org/abs/1810.08473. Consider the partition shown in (a). The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. Soc. Louvain has two phases: local moving and aggregation. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. At some point, the Louvain algorithm may end up in the community structure shown in Fig. Slider with three articles shown per slide. We generated networks with n=103 to n=107 nodes. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). 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. 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. We now compare how the Leiden and the Louvain algorithm perform for the six empirical networks listed in Table2. Cite this article. 104 (1): 3641. Sci. For example: If you do not have root access, you can use pip install --user or pip install --prefix to install these in your user directory (which you have write permissions for) and ensure that this directory is in your PATH so that Python can find it. A major goal of single-cell analysis is to study the cell-state heterogeneity within a sample by discovering groups within the population of cells. Please The random component also makes the algorithm more explorative, which might help to find better community structures. We here introduce the Leiden algorithm, which guarantees that communities are well connected. However, so far this problem has never been studied for the Louvain algorithm. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. On Modularity Clustering. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. 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. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. For the Amazon and IMDB networks, the first iteration of the Leiden algorithm is only about 1.6 times faster than the first iteration of the Louvain algorithm. You are using a browser version with limited support for CSS. 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. Acad. Phys. igraph R manual pages Are you sure you want to create this branch? The constant Potts model tries to maximize the number of internal edges in a community, while simultaneously trying to keep community sizes small, and the constant parameter balances these two characteristics. For larger networks and higher values of , Louvain is much slower than Leiden. E Stat. It states that there are no communities that can be merged. After the refinement phase is concluded, communities in \({\mathscr{P}}\) often will have been split into multiple communities in \({{\mathscr{P}}}_{{\rm{refined}}}\), but not always. Hierarchical Clustering: Agglomerative + Divisive Explained | Built In Number of iterations before the Leiden algorithm has reached a stable iteration for six empirical networks. Subpartition -density is not guaranteed by the Louvain algorithm. The corresponding results are presented in the Supplementary Fig. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. Clustering with the Leiden Algorithm in R Google Scholar. Faster Unfolding of Communities: Speeding up the Louvain Algorithm. Phys. 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. For each set of parameters, we repeated the experiment 10 times. In particular, benchmark networks have a rather simple structure. Google Scholar. This will compute the Leiden clusters and add them to the Seurat Object Class. GitHub - MiqG/leiden_clustering: Cluster your data matrix with the Rev. At each iteration all clusters are guaranteed to be connected and well-separated. reviewed the manuscript. Rev. V.A.T. If we move the node to a different community, we add to the rear of the queue all neighbours of the node that do not belong to the nodes new community and that are not yet in the queue. 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. Number of iterations until stability. We generated benchmark networks in the following way. Note that this code is designed for Seurat version 2 releases. Phys. Segmentation & Clustering SPATA2 - GitHub Pages This contrasts to benchmark networks, for which Leiden often converges after a few iterations. We now show that the Louvain algorithm may find arbitrarily badly connected communities. Article However, it is also possible to start the algorithm from a different partition15. The numerical details of the example can be found in SectionB of the Supplementary Information. Complex brain networks: graph theoretical analysis of structural and functional systems. Nonlin. You signed in with another tab or window. MathSciNet However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. Newman, M. E. J. ADS From Louvain to Leiden: guaranteeing well-connected communities - Nature 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). Phys. Detecting communities in a network is therefore an important problem. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. The speed difference is especially large for larger networks. 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). 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. The Louvain algorithm10 is very simple and elegant. Source Code (2018). 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. ISSN 2045-2322 (online). First calculate k-nearest neighbors and construct the SNN graph. Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. Here is some small debugging code I wrote to find this. 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. Article Node mergers that cause the quality function to decrease are not considered. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. sign in In this new situation, nodes 2, 3, 5 and 6 have only internal connections. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. MathSciNet The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. Each community in this partition becomes a node in the aggregate network. The count of badly connected communities also included disconnected communities. A new methodology for constructing a publication-level classification system of science. Phys. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. 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. Community detection - Tim Stuart Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. Modularity is a popular objective function used with the Louvain method for community detection. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. Duch, J. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. A score of 0 would mean that the community has half its edges connecting nodes within the same community, and half connecting nodes outside the community. 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. An aggregate. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). Performance of modularity maximization in practical contexts. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. Unsupervised clustering of cells is a common step in many single-cell expression workflows. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. Traag, V. A. Theory Exp. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. 2(b). CAS 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. Hierarchical Clustering Explained - Towards Data Science In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. 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. Technol. Google Scholar. The two phases are repeated until the quality function cannot be increased further. The community with which a node is merged is selected randomly18. Here we can see partitions in the plotted results. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. Brandes, U. et al. A tag already exists with the provided branch name. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. 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. 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. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. E Stat. In the worst case, almost a quarter of the communities are badly connected. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). Use Git or checkout with SVN using the web URL. 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 . Package 'leiden' October 13, 2022 Type Package Title R Implementation of Leiden Clustering Algorithm Version 0.4.3 Date 2022-09-10 Description Implements the 'Python leidenalg' module to be called in R. Enables clustering using the leiden algorithm for partition a graph into communities. This way of defining the expected number of edges is based on the so-called configuration model. Note that the object for Seurat version 3 has changed. By moving these nodes, Louvain creates badly connected communities. 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. 2004. scanpy.tl.leiden Scanpy 1.9.3 documentation - Read the Docs 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.