Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. Eur. 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. We use six empirical networks in our analysis. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. This problem is different from the well-known issue of the resolution limit of modularity14. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. The algorithm then moves individual nodes in the aggregate network (d). Porter, M. A., Onnela, J.-P. & Mucha, P. J. This is very similar to what the smart local moving algorithm does. How many iterations of the Leiden clustering algorithm to perform. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. Subpartition -density is not guaranteed by the Louvain algorithm. Therefore, clustering algorithms look for similarities or dissimilarities among data points. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Rev. The larger the increase in the quality function, the more likely a community is to be selected. The percentage of disconnected communities is more limited, usually around 1%. We generated networks with n=103 to n=107 nodes. In contrast, Leiden keeps finding better partitions in each iteration. Number of iterations until stability. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). Fortunato, Santo, and Marc Barthlemy. ADS 4, in the first iteration of the Louvain algorithm, the percentage of badly connected communities can be quite high. Computer Syst. Waltman, Ludo, and Nees Jan van Eck. 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. Figure3 provides an illustration of the algorithm. Rev. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. IEEE Trans. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. We here introduce the Leiden algorithm, which guarantees that communities are well connected. The nodes are added to the queue in a random order. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. A new methodology for constructing a publication-level classification system of science. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. Complex brain networks: graph theoretical analysis of structural and functional systems. Lancichinetti, A. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. Node mergers that cause the quality function to decrease are not considered. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. PubMed Central (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. 2007. This will compute the Leiden clusters and add them to the Seurat Object Class. The two phases are repeated until the quality function cannot be increased further. In the Louvain algorithm, a node may be moved to a different community while it may have acted as a bridge between different components of its old community. Nat. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). Here we can see partitions in the plotted results. Finding and Evaluating Community Structure in Networks. Phys. Runtime versus quality for benchmark networks. Google Scholar. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Here is some small debugging code I wrote to find this. The Leiden community detection algorithm outperforms other clustering methods. Rev. Removing such a node from its old community disconnects the old community. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. 2 represent stronger connections, while the other edges represent weaker connections. The property of -separation is also guaranteed by the Louvain algorithm. A tag already exists with the provided branch name. Other networks show an almost tenfold increase in the percentage of disconnected communities. 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. The Leiden algorithm provides several guarantees. 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. Please Basically, there are two types of hierarchical cluster analysis strategies - 1. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. Duch, J. In particular, benchmark networks have a rather simple structure. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. 2. At this point, it is guaranteed that each individual node is optimally assigned. MATH At some point, the Louvain algorithm may end up in the community structure shown in Fig. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). It implies uniform -density and all the other above-mentioned properties. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. The percentage of disconnected communities even jumps to 16% for the DBLP network. 68, 984998, https://doi.org/10.1002/asi.23734 (2017). The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. (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. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. Phys. CAS Sci. Faster unfolding of communities: Speeding up the Louvain algorithm. Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. Sci. 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. Google Scholar. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. Nodes 13 should form a community and nodes 46 should form another community. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. Based on project statistics from the GitHub repository for the PyPI package leiden-clustering, we found that it has been starred 1 times. 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. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). PubMed This may have serious consequences for analyses based on the resulting partitions. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. Randomness in the selection of a community allows the partition space to be explored more broadly. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. This enables us to find cases where its beneficial to split a community. 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. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. Hence, for lower values of , the difference in quality is negligible. Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. 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. We start by initialising a queue with all nodes in the network. The algorithm continues to move nodes in the rest of the network. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). E Stat. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Communities in Networks. Unsupervised clustering of cells is a common step in many single-cell expression workflows. The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. where >0 is a resolution parameter4. Cite this article. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. We first applied the Scanpy pipeline, including its clustering method (Leiden clustering), on the PBMC dataset. Nonetheless, some networks still show large differences. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). PubMed Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). V.A.T. 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). Source Code (2018). This contrasts with the Leiden algorithm. and L.W. Run the code above in your browser using DataCamp Workspace. Then, in order . We name our algorithm the Leiden algorithm, after the location of its authors. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Communities may even be disconnected. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. 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). Article Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. 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. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. The corresponding results are presented in the Supplementary Fig. 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). 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. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. Rev. See the documentation for these functions. S3. Nodes 06 are in the same community. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Phys. This is very similar to what the smart local moving algorithm does. Besides being pervasive, the problem is also sizeable. ADS volume9, Articlenumber:5233 (2019) A Simple Acceleration Method for the Louvain Algorithm. Int. Finding community structure in networks using the eigenvectors of matrices. ADS Performance of modularity maximization in practical contexts. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. The Web of Science network is the most difficult one. 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. In other words, communities are guaranteed to be well separated. Good, B. H., De Montjoye, Y. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. Article Article You will not need much Python to use it. As we prove in SectionC1 of the Supplementary Information, even when node mergers that decrease the quality function are excluded, the optimal partition of a set of nodes can still be uncovered. An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. Figure4 shows how well it does compared to the Louvain algorithm. Newman, M. E. J. It partitions the data space and identifies the sub-spaces using the Apriori principle. 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. http://dx.doi.org/10.1073/pnas.0605965104. We keep removing nodes from the front of the queue, possibly moving these nodes to a different community. 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. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. 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. 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. In this case, refinement does not change the partition (f). Get the most important science stories of the day, free in your inbox. You are using a browser version with limited support for CSS. Nodes 16 have connections only within this community, whereas node 0 also has many external connections. Modules smaller than the minimum size may not be resolved through modularity optimization, even in the extreme case where they are only connected to the rest of the network through a single edge. At each iteration all clusters are guaranteed to be connected and well-separated. 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). In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. 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. GitHub on Feb 15, 2020 Do you think the performance improvements will also be implemented in leidenalg? However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. 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. Acad. 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. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. 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. Traag, Vincent, Ludo Waltman, and Nees Jan van Eck. It is good at identifying small clusters. 104 (1): 3641. We used modularity with a resolution parameter of =1 for the experiments. Work fast with our official CLI. For larger networks and higher values of , Louvain is much slower than Leiden. 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. Community detection is often used to understand the structure of large and complex networks. Data 11, 130, https://doi.org/10.1145/2992785 (2017). Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. Four popular community detection algorithms are explained . The Louvain algorithm starts from a singleton partition in which each node is in its own community (a). Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Knowl. Ph.D. thesis, (University of Oxford, 2016). Scaling of benchmark results for difficulty of the partition. Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? An overview of the various guarantees is presented in Table1. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. 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. 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. Based on this partition, an aggregate network is created (c). 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. Article 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. 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. The leidenalg package facilitates community detection of networks and builds on the package igraph. As such, we scored leiden-clustering popularity level to be Limited. 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 ). Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. This algorithm provides a number of explicit guarantees. Knowl. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta, http://dx.doi.org/10.1073/pnas.0605965104, http://dx.doi.org/10.1103/PhysRevE.69.026113, https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf, http://dx.doi.org/10.1103/PhysRevE.81.046114, http://dx.doi.org/10.1103/PhysRevE.92.032801, https://doi.org/10.1140/epjb/e2013-40829-0, Assign each node to a different community. 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 count of badly connected communities also included disconnected communities. In short, the problem of badly connected communities has important practical consequences. Badly connected communities. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. Phys. 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 . On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. However, so far this problem has never been studied for the Louvain algorithm. Phys. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. Slider with three articles shown per slide.
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