Each community in this partition becomes a node in the aggregate network. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. Disconnected community. Phys. For larger networks and higher values of , Louvain is much slower than Leiden. Modularity is given by. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. 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. Clustering algorithms look for similarities or dissimilarities among data points so that similar ones can be grouped together. As can be seen in Fig. Traag, V A. In other words, communities are guaranteed to be well separated. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. However, so far this problem has never been studied for the Louvain algorithm. Acad. 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. 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 . Nodes 16 have connections only within this community, whereas node 0 also has many external connections. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. ADS Elect. A score of -1 means that there are no edges connecting nodes within the community, and they instead all connect nodes outside the community. Phys. As shown in Fig. wrote the manuscript. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). 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. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees. We here introduce the Leiden algorithm, which guarantees that communities are well connected. Learn more. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. 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. B 86, 471, https://doi.org/10.1140/epjb/e2013-40829-0 (2013). 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). S3. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. Traag, V. A. 2.3. Besides the relative flexibility of the implementation, it also scales well, and can be run on graphs of millions of nodes (as long as they can fit in memory). The DBLP network is somewhat more challenging, requiring almost 80 iterations on average to reach a stable iteration. Source Code (2018). Modularity is a popular objective function used with the Louvain method for community detection. Community detection in complex networks using extremal optimization. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Therefore, clustering algorithms look for similarities or dissimilarities among data points. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. The Beginner's Guide to Dimensionality Reduction. In contrast, Leiden keeps finding better partitions in each iteration. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. Badly connected communities. J. 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. 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"). At each iteration all clusters are guaranteed to be connected and well-separated. ADS Electr. Communities may even be disconnected. After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. 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. 10, 186198, https://doi.org/10.1038/nrn2575 (2009). It is a directed graph if the adjacency matrix is not symmetric. First, we created a specified number of nodes and we assigned each node to a community. Speed and quality of the Louvain and the Leiden algorithm for benchmark networks of increasing size (two iterations). Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. Nat. Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? The speed difference is especially large for larger networks. Traag, V. A., Van Dooren, P. & Nesterov, Y. The fast local move procedure can be summarised as follows. Rev. Provided by the Springer Nature SharedIt content-sharing initiative. However, after all nodes have been visited once, Leiden visits only nodes whose neighbourhood has changed, whereas Louvain keeps visiting all nodes in the network. We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. It states that there are no communities that can be merged. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). This makes sense, because after phase one the total size of the graph should be significantly reduced. ACM Trans. Figure3 provides an illustration of the algorithm. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. Sci. Besides being pervasive, the problem is also sizeable. Technol. We thank Lovro Subelj for his comments on an earlier version of this paper. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). PubMed Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. The nodes are added to the queue in a random order. CPM has the advantage that it is not subject to the resolution limit. 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. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. Data 11, 130, https://doi.org/10.1145/2992785 (2017). Fortunato, S. Community detection in graphs. Then the Leiden algorithm can be run on the adjacency matrix. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. 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. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. A new methodology for constructing a publication-level classification system of science. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. 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 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). Here we can see partitions in the plotted results. We study the problem of badly connected communities when using the Louvain algorithm for several empirical networks. J. Comput. 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. 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. Nonlin. 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. CAS Inf. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). Phys. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. 2004. By moving these nodes, Louvain creates badly connected communities. 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. J. We name our algorithm the Leiden algorithm, after the location of its authors. Soft Matter Phys. 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 practice, this means that small clusters can hide inside larger clusters, making their identification difficult. Phys. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . 68, 984998, https://doi.org/10.1002/asi.23734 (2017). However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. Modularity optimization. Scientific Reports (Sci Rep) Below we offer an intuitive explanation of these properties. In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. 2(b). ADS o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. 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. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. 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. 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. We used the CPM quality function. Google Scholar. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. 20, 172188, https://doi.org/10.1109/TKDE.2007.190689 (2008). Communities may even be internally disconnected. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. CAS A partition of clusters as a vector of integers Examples & Fortunato, S. Community detection algorithms: A comparative analysis. Discov. Instead, a node may be merged with any community for which the quality function increases. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). A Comparative Analysis of Community Detection Algorithms on Artificial Networks. Google Scholar. Performance of modularity maximization in practical contexts. Soft Matter Phys. Louvain has two phases: local moving and aggregation. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. Rev. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. In particular, we show that Louvain may identify communities that are internally disconnected. 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. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). conda install -c conda-forge leidenalg pip install leiden-clustering Used via. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. Eur. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. 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. Google Scholar. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in 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). Computer Syst. Blondel, V D, J L Guillaume, and R Lambiotte. Google Scholar. 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. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. When node 0 is moved to a different community, the red community becomes internally disconnected, as shown in (b). The percentage of disconnected communities is more limited, usually around 1%. In the worst case, almost a quarter of the communities are badly connected. and JavaScript. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. For a full specification of the fast local move procedure, we refer to the pseudo-code of the Leiden algorithm in AlgorithmA.2 in SectionA of the Supplementary Information. Faster Unfolding of Communities: Speeding up the Louvain Algorithm. Phys. 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. 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. (2) and m is the number of edges. The Leiden algorithm starts from a singleton partition (a). Rev. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. The solution provided by Leiden is based on the smart local moving algorithm. Note that this code is . In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. 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. For higher values of , Leiden finds better partitions than Louvain. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. This should be the first preference when choosing an algorithm. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. 2008. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. http://dx.doi.org/10.1073/pnas.0605965104. If nothing happens, download Xcode and try again. Again, if communities are badly connected, this may lead to incorrect inferences of topics, which will affect bibliometric analyses relying on the inferred topics. 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 example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. In the meantime, to ensure continued support, we are displaying the site without styles Phys. For each set of parameters, we repeated the experiment 10 times. Value. Directed Undirected Homogeneous Heterogeneous Weighted 1. Randomness in the selection of a community allows the partition space to be explored more broadly. E Stat. In the first iteration, Leiden is roughly 220 times faster than Louvain. Waltman, L. & van Eck, N. J. 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. Porter, M. A., Onnela, J.-P. & Mucha, P. J. The property of -separation is also guaranteed by the Louvain algorithm. AMS 56, 10821097 (2009). The second iteration of Louvain shows a large increase in the percentage of disconnected communities. 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. 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. Nodes 06 are in the same community. Rev. It implies uniform -density and all the other above-mentioned properties. This may have serious consequences for analyses based on the resulting partitions. This is very similar to what the smart local moving algorithm does. SPATA2 currently offers the functions findSeuratClusters (), findMonocleClusters () and findNearestNeighbourClusters () which are wrapper around widely used clustering algorithms. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. Article In particular, it yields communities that are guaranteed to be connected. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. Clauset, A., Newman, M. E. J. PubMed Central Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. Other networks show an almost tenfold increase in the percentage of disconnected communities. An aggregate. As can be seen in Fig. 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. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). Hence, by counting the number of communities that have been split up, we obtained a lower bound on the number of communities that are badly connected. In fact, for the Web of Science and Web UK networks, Fig. It does not guarantee that modularity cant be increased by moving nodes between communities. The percentage of disconnected communities even jumps to 16% for the DBLP network. Community detection is an important task in the analysis of complex networks. Both conda and PyPI have leiden clustering in Python which operates via iGraph. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). V.A.T. Communities in Networks. I tracked the number of clusters post-clustering at each step. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. The larger the increase in the quality function, the more likely a community is to be selected. Mech. The refined partition \({{\mathscr{P}}}_{{\rm{refined}}}\) is obtained as follows. This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. There is an entire Leiden package in R-cran here Knowl. Importantly, the first iteration of the Leiden algorithm is the most computationally intensive one, and subsequent iterations are faster. We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. This contrasts with the Leiden algorithm. Our analysis is based on modularity with resolution parameter =1. This step will involve reducing the dimensionality of our data into two dimensions using uniform manifold approximation (UMAP), allowing us to visualize our cell populations as they are binned into discrete populations using Leiden clustering. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. Sci. Use Git or checkout with SVN using the web URL. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. The algorithm continues to move nodes in the rest of the network. Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. Louvain algorithm. However, Leiden is more than 7 times faster for the Live Journal network, more than 11 times faster for the Web of Science network and more than 20 times faster for the Web UK network. 2 represent stronger connections, while the other edges represent weaker connections. We now consider the guarantees provided by the Leiden algorithm. As discussed earlier, the Louvain algorithm does not guarantee connectivity. leidenalg. Traag, Vincent, Ludo Waltman, and Nees Jan van Eck. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . Phys. Community detection is often used to understand the structure of large and complex networks. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. We now compare how the Leiden and the Louvain algorithm perform for the six empirical networks listed in Table2. The numerical details of the example can be found in SectionB of the Supplementary Information. All communities are subpartition -dense. & Moore, C. Finding community structure in very large networks. However, this is not necessarily the case, as the other nodes may still be sufficiently strongly connected to their community, despite the fact that the community has become disconnected. All authors conceived the algorithm and contributed to the source code. In this case we know the answer is exactly 10. A. and L.W. 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly 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. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. N.J.v.E. 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