puzzlef/louvain-adjust-tolerance: Effect of adjusting start tolerance and tolerance-norm of the Louvain algorithm for community detection

Effect of adjusting start tolerance and tolerance-norm of the Louvain algorithm for community detection.

Louvain is an algorithm for detecting communities in graphs. Community detection helps us understand the natural divisions in a network in an unsupervised manner. It is used in e-commerce for customer segmentation and advertising, in communication networks for multicast routing and setting up of mobile networks, and in healthcare for epidemic causality, setting up health programmes, and fraud detection is hospitals. Community detection is an NP-hard problem, but heuristics exist to solve it (such as this). Louvain algorithm is an agglomerative-hierarchical community detection method that greedily optimizes for modularity (iteratively).

Modularity is a score that measures relative density of edges inside vs outside communities. Its value lies between −0.5 (non-modular clustering) and 1.0 (fully modular clustering). Optimizing for modularity theoretically results in the best possible grouping of nodes in a graph.

Given an undirected weighted graph, all vertices are first considered to be their own communities. In the first phase, each vertex greedily decides to move to the community of one of its neighbors which gives greatest increase in modularity. If moving to no neighbor's community leads to an increase in modularity, the vertex chooses to stay with its own community. This is done sequentially for all the vertices. If the total change in modularity is more than a certain threshold, this phase is repeated. Once this local-moving phase is complete, all vertices have formed their first hierarchy of communities. The next phase is called the aggregation phase, where all the vertices belonging to a community are collapsed into a single super-vertex, such that edges between communities are represented as edges between respective super-vertices (edge weights are combined), and edges within each community are represented as self-loops in respective super-vertices (again, edge weights are combined). Together, the local-moving and the aggregation phases constitute a pass. This super-vertex graph is then used as input for the next pass. This process continues until the increase in modularity is below a certain threshold. As a result from each pass, we have a hierarchy of community memberships for each vertex as a dendrogram. We generally consider the top-level hierarchy as the final result of community detection process.

Louvain algorithm is a hierarchical algorithm, and thus has two different tolerance parameters: tolerance and passTolerance. tolerance defines the minimum amount of increase in modularity expected, until the local-moving phase of the algorithm is considered to have converged. We compare the increase in modularity in each iteration of the local-moving phase to see if it is below tolerance. passTolerance defines the minimum amount of increase in modularity expected, until the entire algorithm is considered to have converged. We compare the increase in modularity across all iterations of the local-moving phase in the current pass to see if it is below passTolerance. passTolerance is normally set to 0 (we want to maximize our modularity gain), but the same thing does not apply for tolerance. Adjusting values of tolerance between each pass have been observed to impact the runtime of the algorithm, without significantly affecting the modularity of obtained communities.

In this experiment we change the initial value of tolerance (for local-moving phase) from 1e-00 to 1e-6 in steps of 10. For each initial value of tolerance, we use a tolerance-norm of L1, L2, or L∞. We compare the results, both in terms of quality (modularity) of communities obtained, and performance. We choose the remaining Louvain parameters as resolution = 1.0, toleranceDeclineFactor = 10 (the rate at which we recude tolerance after every pass), and passTolerance = 0.0. In addition we limit the maximum number of iterations in a single local-moving phase with maxIterations = 500, and limit the maximum number of passes with maxPasses = 500. We run the Louvain algorithm until convergence (or until the maximum limits are exceeded), and measure the time taken for the computation (performed 5 times for averaging), the modularity score, the total number of iterations (in the local-moving phase), and the number of passes. This is repeated for seventeen different graphs.

From the results, we make the following observations. A tolerance-norm of L1 converges the fastest, followed by L∞, and then L2 (except for initial tolerance below 10^-2). This could be due to delta-modularity for any vertex being small, so that squiring it (as with L2-norm) reduces the net error significantly. In general we observe that the modularity obtained with all tolerance-norms is the same, but for some graphs, best modularity is achieved with an initial tolerance of 10^-5 with L∞-norm, and an initial tolerance of > 10^-6 with L2-norm. As L∞-norm considers only the max error, we believe it would be a suitable tolerance-norm of choice for dynamic Louvain algorithm. Hence, we ask you to consider an initial tolerance of 10^-5 with L∞-norm to be the suitable choice in the general case.

All outputs are saved in a gist and a small part of the output is listed here. Some charts are also included below, generated from sheets. The input data used for this experiment is available from the SuiteSparse Matrix Collection. This experiment was done with guidance from Prof. Kishore Kothapalli and Prof. Dip Sankar Banerjee.

<br>

$ g++ -std=c++17 -O3 main.cxx
$ ./a.out ~/data/web-Stanford.mtx
$ ./a.out ~/data/web-BerkStan.mtx
$ ...

# Loading graph /home/subhajit/data/web-Stanford.mtx ...
# order: 281903 size: 2312497 [directed] {}
# order: 281903 size: 3985272 [directed] {} (symmetricize)
# [-0.000497 modularity] noop
# [0e+00 batch_size; 0 batch_count; 00792.589 ms; 0025 iters.; 009 passes; 0.923382580 modularity] louvainSeqLast
# [0e+00 batch_size; 0 batch_count; 00573.409 ms; 0004 iters.; 001 passes; 0.766543329 modularity] louvainSeqFirst
# [5e+02 batch_size; 1 batch_count; 00224.703 ms; 0004 iters.; 004 passes; 0.914939582 modularity] louvainSeqDynamicLast
# [5e+02 batch_size; 1 batch_count; 00480.665 ms; 0025 iters.; 009 passes; 0.923243225 modularity] louvainSeqDynamicFirst
# [5e+02 batch_size; 2 batch_count; 00227.230 ms; 0004 iters.; 004 passes; 0.914955676 modularity] louvainSeqDynamicLast
# ...
# [-1e+05 batch_size; 4 batch_count; 00397.172 ms; 0011 iters.; 006 passes; 0.876155496 modularity] louvainSeqDynamicFirst
# [-1e+05 batch_size; 5 batch_count; 00206.570 ms; 0004 iters.; 004 passes; 0.869377553 modularity] louvainSeqDynamicLast
# [-1e+05 batch_size; 5 batch_count; 00406.253 ms; 0012 iters.; 006 passes; 0.876216054 modularity] louvainSeqDynamicFirst
#
# Loading graph /home/subhajit/data/web-BerkStan.mtx ...
# order: 685230 size: 7600595 [directed] {}
# order: 685230 size: 13298940 [directed] {} (symmetricize)
# [-0.000316 modularity] noop
# [0e+00 batch_size; 0 batch_count; 01248.049 ms; 0028 iters.; 009 passes; 0.935839474 modularity] louvainSeqLast
# [0e+00 batch_size; 0 batch_count; 00926.964 ms; 0005 iters.; 001 passes; 0.798873782 modularity] louvainSeqFirst
# [5e+02 batch_size; 1 batch_count; 00527.935 ms; 0003 iters.; 003 passes; 0.932618558 modularity] louvainSeqDynamicLast
# [5e+02 batch_size; 1 batch_count; 00884.497 ms; 0025 iters.; 009 passes; 0.935987055 modularity] louvainSeqDynamicFirst
# [5e+02 batch_size; 2 batch_count; 00529.566 ms; 0003 iters.; 003 passes; 0.932615817 modularity] louvainSeqDynamicLast
# ...

<br> <br>

• Fast unfolding of communities in large networks; Vincent D. Blondel et al. (2008)
• Community Detection on the GPU; Md. Naim et al. (2017)
• Scalable Static and Dynamic Community Detection Using Grappolo; Mahantesh Halappanavar et al. (2017)
• From Louvain to Leiden: guaranteeing well-connected communities; V.A. Traag et al. (2019)
• CS224W: Machine Learning with Graphs | Louvain Algorithm; Jure Leskovec (2021)
• The University of Florida Sparse Matrix Collection; Timothy A. Davis et al. (2011)

<br> <br>

<br>