puzzlef/louvain-openmp-adjust-schedule: Effect of adjusting schedule in OpenMP-based Louvain algorithm for community detection

Effect of adjusting schedule in OpenMP-based 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.

There exist two possible approaches of vertex processing with the Louvain algorithm: ordered and unordered. With the ordered approach (original paper's approach), the local-moving phase is performed sequentially upon each vertex such that the moving of a previous vertex in the graph affects the decision of the current vertex being processed. On the other hand, with the unordered approach the moving of a previous vertex in the graph does not affect the decision of movement for the current vertex. This unordered approach (aka relaxed approach) is made possible by maintaining the previous and the current community membership status of each vertex (along with associated community information), and is the approach followed by parallel Louvain implementation on the CPU as well as the GPU. We are interested in looking at performance/modularity penalty (if any) associated with the unordered approach.

In this experiment we attempt performing the OpenMP-based Louvain algorithm using the ordered approach with all different schedule kinds (static, dynamic, guided, and auto), which adjusting the chunk size from 1 to 65536 in multiples of 2. In all cases, we use a total of 12 threads. We choose the Louvain parameters as resolution = 1.0, tolerance = 1e-2 (for local-moving phase) with tolerance decreasing after every pass by a factor of toleranceDeclineFactor = 10, and a passTolerance = 0.0 (when passes stop). 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 observe that little to no benefit is provided with a multi-threaded OpenMP-based ordered Louvain algorithm, compared to the sequential ordered approach. This is somewhat surprising (not surprising to me) and indicates that when multiple reader threads and a writer thread access common memory locations (here it is the community membership of each vertex) perfomance is degraded and would tend to approach the performance of a sequential algorithm if there is just too much overlap. What should you do in that case? A simple approach would be to just go ahead with the sequential (ordered) Louvain algorithm, or partition the graph is such a way that each partition can be run independently, and then combined back together. We could also try this same experiment with the unordered approach, and see if it performs better than the ordered sequential approach (the ordered approach is an order of magnitude slower than the unordered approach, but is easily parallelizable). Partially ordered approaches for vertex processing may also be explored. Vertex ordering via graph coloring has been explored by Halappanavar et al.

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.

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$ 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)
# OMP_NUM_THREADS=12
# [-0.000497 modularity] noop
# [00636.194 ms; 0025 iterations; 009 passes; 0.923383 modularity] louvainSeq
# [00698.879 ms; 0026 iterations; 009 passes; 0.923119 modularity] louvainOmp {sch_kind: static, chunk_size: 1}
# [00655.290 ms; 0025 iterations; 009 passes; 0.922996 modularity] louvainOmp {sch_kind: static, chunk_size: 2}
# [00661.412 ms; 0026 iterations; 009 passes; 0.923447 modularity] louvainOmp {sch_kind: static, chunk_size: 4}
# ...
# [00617.610 ms; 0025 iterations; 009 passes; 0.927243 modularity] louvainOmp {sch_kind: auto, chunk_size: 16384}
# [00620.235 ms; 0025 iterations; 009 passes; 0.923699 modularity] louvainOmp {sch_kind: auto, chunk_size: 32768}
# [00614.724 ms; 0023 iterations; 009 passes; 0.923273 modularity] louvainOmp {sch_kind: auto, chunk_size: 65536}
#
# Loading graph /home/subhajit/data/web-BerkStan.mtx ...
# order: 685230 size: 7600595 [directed] {}
# order: 685230 size: 13298940 [directed] {} (symmetricize)
# OMP_NUM_THREADS=12
# [-0.000316 modularity] noop
# [01208.700 ms; 0028 iterations; 009 passes; 0.935839 modularity] louvainSeq
# [01747.514 ms; 0027 iterations; 009 passes; 0.938295 modularity] louvainOmp {sch_kind: static, chunk_size: 1}
# [01569.095 ms; 0025 iterations; 009 passes; 0.933566 modularity] louvainOmp {sch_kind: static, chunk_size: 2}
# [01541.691 ms; 0025 iterations; 009 passes; 0.938013 modularity] louvainOmp {sch_kind: static, chunk_size: 4}
# ...

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• 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)

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