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PPT overview of paper accepted for 2019 Southeastern International Conference on Combinatorics, Graph Theory & Computing. Details a persistence approach to community detection and a new quantum persistence-based algorithm based on the coloring problem.

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- 1. Colleen Farrelly and Uchenna Chukwu
- 2. Discerning community structure and community strength within network data is a major focus in network analytics. Connected brain regions related to different functions Social ties Subscale identification within a psychometric measure Many approaches exist to either quantify strength of communities or cluster network data: Louvain clustering and Louvain modularity Random walk clustering However, existing methods don’t provide evolving slices of community structure, which can delineate the strength of ties within communities and the point at which individuals join and drop out of those communities. Our approach, which blends filtration methods, quantum computing, and k-core algorithms, can provide those evolving slices.
- 3. There is a well-known link between the graph coloring problem and k-core algorithms. The graph coloring problem requires finding the minimum number of colors required to color each vertex of the graph such that no two vertices have the same color. This solution gives the chromatic number, or the minimum number of colors needed to solve the problem for a given graph. Call the chromatic number, k. If the vertices of a given graph that necessitate the use of k colors, rather than k-1 colors, are removed, the chromatic number can drop to k-1 for the graph, referred to as degeneracy of the graph. Decomposing a graph in this manner gives the shells of a graph that require a given number of colors. This produces a series of connected graphs based on chromatic number degeneracy of the graph, called the k-cores of the graph. K-core algorithms are common tools in community detection problems, as well as visualization problems and the coloring problem itself.
- 4. Graph filtrations have become relatively commonplace in the topological analysis of network data, particularly within brain imaging and connectomics. Given a weighted graph, iteratively set higher and higher edge weights to 0 and analyze each “slice” of the graph for local or global network properties. This is a lower-dimensional analogue of filtrations used in persistent homology; rather than filter on distance to build simplicial complexes, filter links directly within a graph (1-skeleton of potentially larger simplicial complex). This gives evolving slices of graph metrics. Filtration set to 0.2 edge weight
- 5. Two solutions exist to the graph coloring problem on qubit-based quantum computers: Simulated annealing solutions (mainly for D-Wave’s quantum computer) Gate-based solutions (mainly for Rigetti’s and IBM’s quantum computers) Use the D-Wave extension of networkx Python package, which relies on QUBO formulation of the coloring algorithm on a given graph QUBO formulation expands Ising model into linear and quadratic terms for optimization. Tools from the Ocean package’s nextworkx extension to do this automatically, like the min_vertex_coloring function. The DWaveSampler simulates the D-Wave quantum computing set up, allowing the code to run on a simulated quantum computer based on simulated annealing. Post-processing of the raw results yields the minimum coloring solution.
- 6. Trial graphs: 3 toy graphs with 6 nodes and varying connectivity patterns and link weights The KONECT Windsurfer network, which is a weighted network with 43 vertices Community-finding algorithms compared with quantum persistent k-cores: Louvain clustering (igraph R package) Walktrap clustering (igraph R package) Spinglass clustering (igraph R package) Hierarchical clustering (hclust function in R) Graph filtration: Comparison algorithms were run with and without filtration to compare to standard use of algorithms in community-finding algorithms and their persistence-based analogues. Results were also compared to persistent k-core results for benchmarking of quantum persistent k-core algorithm. Filtration values were set to 0.2, 0.4, 0.6, 0.8, and 1, respectively.
- 8. Clustering methods generally agree across methods for all 4 weighted graphs. Graph 1 presented some issues to Walktrap community detection. Others were consistent across algorithms. This suggests that the true clustering on these graphs is the solution found by the classical community detection algorithm.
- 9. Graph filtration yielded some interesting results related to the decomposition of the network, particularly within the Windsurfer dataset. Left figure is unfiltered result with Louvain clustering. With a filtration value of 0.6 (anything below this set to 0; right figure), ties begin to separate into isolated groups and smaller, interconnected clusters.
- 10. Filtration yielded a series of k- core values for each vertex in the 4 networks, tracking changes in degeneracy across each graph. Table 1 (above) shows the filtration for graph 1; vertices 1, 3, and 4 form links between the two cluster groups found, with vertex 4 showing a strong connection to distal nodes in its own group. Table 3 (below) shows the filtration for graph 3; vertices 1, 2, and 3 have strong intergroup connections. Filtration Value V1 V2 V3 V4 V5 V6 0.2 2 1 2 2 1 1 0.4 2 1 2 2 1 1 0.6 1 1 1 1 1 1 0.8 0 1 1 1 1 0 1 0 0 0 0 0 0 Filtration Value V1 V2 V3 V4 V5 V6 0.2 3 3 3 3 3 3 0.4 2 2 2 2 1 2 0.6 2 2 2 0 1 1 0.8 1 1 1 0 0 1 1 0 0 0 0 0 0
- 11. The quantum persistent k-cores/coloring algorithm was expected to need k+1 colors to find an optimal coloring solution to each filtration of each graph. For the 3 toy graphs, this matched the persistent k-cores results exactly. Each coloring resulted in a graph with most likely solution conforming to the coloring problem solution with k+1 colors needed for the best solution. For graphs with only isolated vertices, the algorithm correctly identified that no nodes were connected in the graphs. This confirms that the quantum persistent k-cores algorithm can both find the optimal coloring and distinguish how many colors are needed within the connected communities, which corresponded correctly in all toy graph cases. The surfer network posed computational problems for the algorithm, suggesting that this large of a coloring/k-cores problem is not feasible for the current systems.
- 12. These experiments demonstrate the usefulness of graph filtration in community detection problems. Common community detection algorithms, such as Louvain clustering or walktrap clustering, can find structure and subgroups within the graph in a more nuanced fashion when combined with graph filtration, providing an evolution of subgroup separation. Filtrations also illuminate how graph degeneracy and, hence, communities structures evolve by edge weight and strength of vertex connections. Quantum coloring algorithms probabilistically converge to known solutions from k-core algorithms, suggesting that quantum algorithms are viable for these problems and do not degrade in performance under extreme filtrations. It is hoped that these tools will provide the network analytics and graph theory community with a starting point of persistent graph tools for community finding and subgroup mining.
- 13. Lee, H., Chung, M. K., Kang, H., Kim, B. N., & Lee, D. S. (2011, September). Computing the shape of brain networks using graph filtration and Gromov-Hausdorff metric. In International Conference on Medical Image Computing and Computer- Assisted Intervention (pp. 302-309). Springer, Berlin, Heidelberg. Farrelly, C. M., Schwartz, S. J., Amodeo, A. L., Feaster, D. J., Steinley, D. L., Meca, A., & Picariello, S. (2017). The analysis of bridging constructs with hierarchical clustering methods: An application to identity. Journal of Research in Personality, 70, 93-106. Hahm, J., Lee, H., Park, H., Kang, E., Kim, Y. K., Chung, C. K., ... & Lee, D. S. (2017). Gating of memory encoding of time- delayed cross-frequency MEG networks revealed by graph filtration based on persistent homology. Scientific reports, 7, 41592. Hübner, K. L. Coloring Complex Networks. Kageura, K., & Abekawa, T. (2007). Modelling and exploring the network structure of terminology using the Potts spin glass model. In Proceedings of the 10th Conference of the Pacific Association for the Computational Linguistics (PACLING) (pp. 236- 245). Radzikowski, J., Stefanidis, A., Jacobsen, K. H., Croitoru, A., Crooks, A., & Delamater, P. L. (2016). The measles vaccination narrative in Twitter: a quantitative analysis. JMIR public health and surveillance, 2(1). Rider, A. K., Siwo, G., Emrich, S. J., Ferdig, M. T., & Chawla, N. V. (2014). A supervised learning approach to the ensemble clustering of genes. International journal of data mining and bioinformatics, 9(2), 199-219. Rossi, R. A., & Ahmed, N. K. (2014). Coloring large complex networks. Social Network Analysis and Mining, 4(1), 228. Shaydulin, R., Ushijima-Mwesigwa, H., Safro, I., Mniszewski, S., & Alexeev, Y. (2018). Community Detection Across Emerging Quantum Architectures. arXiv preprint arXiv:1810.07765. Shepitsen, A., Gemmell, J., Mobasher, B., & Burke, R. (2008, October). Personalized recommendation in social tagging systems using hierarchical clustering. In Proceedings of the 2008 ACM conference on Recommender systems (pp. 259-266). ACM. Titiloye, O., & Crispin, A. (2011). Quantum annealing of the graph coloring problem. Discrete Optimization, 8(2), 376-384. Ushijima-Mwesigwa, H., Negre, C. F., & Mniszewski, S. M. (2017, November). Graph partitioning using quantum annealing on the D-Wave system. In Proceedings of the Second International Workshop on Post Moores Era Supercomputing (pp. 22-29). ACM. Wang, Y., & Perkowski, M. (2011, May). Improved complexity of quantum oracles for ternary grover algorithm for graph coloring. In Multiple-Valued Logic (ISMVL), 2011 41st IEEE International Symposium on (pp. 294-301). IEEE.