Rev. Nonlin. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. Modularity optimization. If nothing happens, download Xcode and try again. It was found to be one of the fastest and best performing algorithms in comparative analyses11,12, and it is one of the most-cited works in the community detection literature. 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. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. It is a directed graph if the adjacency matrix is not symmetric. Complex brain networks: graph theoretical analysis of structural and functional systems. The Leiden algorithm starts from a singleton partition (a). 2.3. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. In addition, a node is merged with a community in \({{\mathscr{P}}}_{{\rm{refined}}}\) only if both are sufficiently well connected to their community in \({\mathscr{P}}\). Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). ADS Then optimize the modularity function to determine clusters. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. Bullmore, E. & Sporns, O. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. The degree of randomness in the selection of a community is determined by a parameter >0. Good, B. H., De Montjoye, Y. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. Cite this article. One of the best-known methods for community detection is called modularity3. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). 2 represent stronger connections, while the other edges represent weaker connections. The value of the resolution parameter was determined based on the so-called mixing parameter 13. An aggregate. For example an SNN can be generated: For Seurat version 3 objects, the Leiden algorithm has been implemented in the Seurat version 3 package with Seurat::FindClusters and algorithm = "leiden"). This is very similar to what the smart local moving algorithm does. to use Codespaces. Article Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). 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. AMS 56, 10821097 (2009). The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. Bae, S., Halperin, D., West, J. D., Rosvall, M. & Howe, B. Scalable and Efficient Flow-Based Community Detection for Large-Scale Graph Analysis. import leidenalg as la import igraph as ig Example output. 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. Runtime versus quality for benchmark networks. 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. 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. Leiden is faster than Louvain especially for larger networks. 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. Phys. Acad. 69 (2 Pt 2): 026113. http://dx.doi.org/10.1103/PhysRevE.69.026113. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. Scientific Reports (Sci Rep) IEEE Trans. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. This is not too difficult to explain. Yang, Z., Algesheimer, R. & Tessone, C. J. A partition of clusters as a vector of integers Examples In the initial stage of Louvain (when all nodes belong to their own community), nearly any move will result in a modularity gain, and it doesnt matter too much which move is chosen. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. However, so far this problem has never been studied for the Louvain algorithm. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . 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. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. 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. This will compute the Leiden clusters and add them to the Seurat Object Class. This is similar to what we have seen for benchmark networks. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). Edges were created in such a way that an edge fell between two communities with a probability and within a community with a probability 1. 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. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. Eur. 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). 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. Modularity is used most commonly, but is subject to the resolution limit. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. As can be seen in Fig. MathSciNet 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 here introduce the Leiden algorithm, which guarantees that communities are well connected. 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. All communities are subpartition -dense. performed the experimental analysis. You will not need much Python to use it. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. For the results reported below, the average degree was set to \(\langle k\rangle =10\). Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. There are many different approaches and algorithms to perform clustering tasks. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. In this way, Leiden implements the local moving phase more efficiently than Louvain. Computer Syst. An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. Such a modular structure is usually not known beforehand. MATH 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. Traag, V. A. volume9, Articlenumber:5233 (2019) In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. For higher values of , Leiden finds better partitions than Louvain. MathSciNet The percentage of disconnected communities even jumps to 16% for the DBLP network. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. Sci. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). Neurosci. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. reviewed the manuscript. Clearly, it would be better to split up the community. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). Article We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. The speed difference is especially large for larger networks. E Stat. 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 We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. Leiden is the most recent major development in this space, and highlighted a flaw in the original Louvain algorithm (Traag, Waltman, and Eck 2018). One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. We generated networks with n=103 to n=107 nodes. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. At this point, it is guaranteed that each individual node is optimally assigned. 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. The Louvain algorithm starts from a singleton partition in which each node is in its own community (a). Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. A score of -1 means that there are no edges connecting nodes within the community, and they instead all connect nodes outside the community. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? This algorithm provides a number of explicit guarantees. 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. J. Exp. In fact, for the Web of Science and Web UK networks, Fig. 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. PubMed To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. Waltman, Ludo, and Nees Jan van Eck. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. Blondel, V D, J L Guillaume, and R Lambiotte. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. 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. Eng. Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. Four popular community detection algorithms are explained . Louvain keeps visiting all nodes in a network until there are no more node movements that increase the 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. Traag, V A. PubMedGoogle Scholar. Here we can see partitions in the plotted results. Number of iterations until stability. Our analysis is based on modularity with resolution parameter =1. Google Scholar. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). 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. Google Scholar. Phys. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. 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. Therefore, clustering algorithms look for similarities or dissimilarities among data points. running Leiden clustering finished: found 16 clusters and added 'leiden_1.0', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 12 clusters and added 'leiden_0.6', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 9 clusters and added 'leiden_0.4', the Subpartition -density does not imply that individual nodes are locally optimally assigned. The constant Potts model (CPM), so called due to the use of a constant value in the Potts model, is an alternative objective function for community detection. As can be seen in Fig. Sci. On Modularity Clustering. This makes sense, because after phase one the total size of the graph should be significantly reduced. This can be a shared nearest neighbours matrix derived from a graph object. In short, the problem of badly connected communities has important practical consequences. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. Phys. As discussed earlier, the Louvain algorithm does not guarantee connectivity. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). U. S. A. Detecting communities in a network is therefore an important problem. The Beginner's Guide to Dimensionality Reduction. Article At some point, node 0 is considered for moving. CAS There is an entire Leiden package in R-cran here Rev. That is, no subset can be moved to a different community. Sci. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). Acad. Each community in this partition becomes a node in the aggregate network. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. Resolution Limit in Community Detection. Proc. Porter, M. A., Onnela, J.-P. & Mucha, P. J. Phys. This way of defining the expected number of edges is based on the so-called configuration model. When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. Article A Comparative Analysis of Community Detection Algorithms on Artificial Networks. 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). Performance of modularity maximization in practical contexts. Google Scholar. In particular, we show that Louvain may identify communities that are internally disconnected. Agglomerative clustering is a bottom-up approach. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. Rev. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. & Bornholdt, S. Statistical mechanics of community detection. PubMed Central Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. 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. Importantly, the problem of disconnected communities is not just a theoretical curiosity. To obtain CAS 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. V. A. Traag. It partitions the data space and identifies the sub-spaces using the Apriori principle. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). Besides being pervasive, the problem is also sizeable. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. 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. For this network, Leiden requires over 750 iterations on average to reach a stable iteration. Inf. Soft Matter Phys. We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. Nonetheless, some networks still show large differences. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. CAS The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. Correspondence to First iteration runtime for empirical networks. DBSCAN Clustering Explained Detailed theorotical explanation and scikit-learn implementation Clustering is a way to group a set of data points in a way that similar data points are grouped together. Communities in Networks. & Arenas, A. 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. Zenodo, https://doi.org/10.5281/zenodo.1469357 https://github.com/vtraag/leidenalg. This contrasts to benchmark networks, for which Leiden often converges after a few iterations. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. The count of badly connected communities also included disconnected communities. conda install -c conda-forge leidenalg pip install leiden-clustering Used via. Rev. Empirical networks show a much richer and more complex structure. The algorithm then moves individual nodes in the aggregate network (d). After the first iteration of the Louvain algorithm, some partition has been obtained. Newman, M. E. J. Leiden is both faster than Louvain and finds better partitions. Phys. Louvain algorithm. Work fast with our official CLI. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). 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. The steps for agglomerative clustering are as follows: Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). N.J.v.E. It identifies the clusters by calculating the densities of the cells. The solution provided by Leiden is based on the smart local moving algorithm. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. ADS The Leiden algorithm starts from a singleton partition (a). See the documentation for these functions. https://doi.org/10.1038/s41598-019-41695-z. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. In the worst case, almost a quarter of the communities are badly connected. Google Scholar. The property of -separation is also guaranteed by the Louvain algorithm. By creating the aggregate network based on \({{\mathscr{P}}}_{{\rm{refined}}}\) rather than P, the Leiden algorithm has more room for identifying high-quality partitions. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. There was a problem preparing your codespace, please try again. 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. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. 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. The random component also makes the algorithm more explorative, which might help to find better community structures. 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. We used the CPM quality function. Powered by DataCamp DataCamp Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. The nodes are added to the queue in a random order. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). 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 . 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. Directed Undirected Homogeneous Heterogeneous Weighted 1. Rev. Wolf, F. A. et al. Figure3 provides an illustration of the algorithm. Higher resolutions lead to more communities and lower resolutions lead to fewer communities, similarly to the resolution parameter for modularity. Int. Learn more. 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. Technol. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). Arguments can be passed to the leidenalg implementation in Python: In particular, the resolution parameter can fine-tune the number of clusters to be detected. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. Waltman, L. & van Eck, N. J. The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. Newman, M E J, and M Girvan. 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. Are you sure you want to create this branch? All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. 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 solution proposed in smart local moving is to alter how the local moving step in Louvain works. Google Scholar. Moreover, Louvain has no mechanism for fixing these communities. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). The differences are not very large, which is probably because both algorithms find partitions for which the quality is close to optimal, related to the issue of the degeneracy of quality functions29. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. The nodes that are more interconnected have been partitioned into separate clusters. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. 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. Article Brandes, U. et al. Nodes 16 have connections only within this community, whereas node 0 also has many external connections. . CPM does not suffer from this issue13. 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. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. CAS We now show that the Louvain algorithm may find arbitrarily badly connected communities. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. (2) and m is the number of edges. The refined partition \({{\mathscr{P}}}_{{\rm{refined}}}\) is obtained as follows. We thank Lovro Subelj for his comments on an earlier version of this paper. In other words, communities are guaranteed to be well separated. Not. Waltman, L. & van Eck, N. J. J. Comput. 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. Below we offer an intuitive explanation of these properties. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward.
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