Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Soft happy colourings and community structure of networks (2405.15663v2)

Published 24 May 2024 in cs.DM

Abstract: For $0<\rho\leq 1$, a $\rho$-happy vertex $v$ in a coloured graph $G$ has at least $\rho\cdot \mathrm{deg}(v)$ same-colour neighbours, and a $\rho$-happy colouring (aka soft happy colouring) of $G$ is a vertex colouring that makes all the vertices $\rho$-happy. A community is a subgraph whose vertices are more adjacent to themselves than the rest of the vertices. Graphs with community structures can be modelled by random graph models such as the stochastic block model (SBM). In this paper, we present several theorems showing that both of these notions are related, with numerous real-world applications. We show that, with high probability, communities of graphs in the stochastic block model induce $\rho$-happy colouring on all vertices if certain conditions on the model parameters are satisfied. Moreover, a probabilistic threshold on $\rho$ is derived so that communities of a graph in the SBM induce a $\rho$-happy colouring. Furthermore, the asymptotic behaviour of $\rho$-happy colouring induced by the graph's communities is discussed when $\rho$ is less than a threshold. We develop heuristic polynomial-time algorithms for soft happy colouring that often correlate with the graphs' community structure. Finally, we present an experimental evaluation to compare the performance of the proposed algorithms thereby demonstrating the validity of the theoretical results.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (36)
  1. Emmanuel Abbe. Community detection and stochastic block models: Recent developments. Journal of Machine Learning Research, 18(177):1–86, 2018.
  2. Exact recovery in the stochastic block model. IEEE Transactions on Information Theory, 62(1):471–487, 2016.
  3. Community detection in general stochastic block models: Fundamental limits and efficient algorithms for recovery. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 670–688, 2015.
  4. Parameterized complexity of happy coloring problems. Theoretical Computer Science, 835:58–81, 2020.
  5. Linear time algorithms for happy vertex coloring problems for trees. In Veli Mäkinen, Simon J. Puglisi, and Leena Salmela, editors, Combinatorial Algorithms, pages 281–292, Cham, 2016. Springer International Publishing.
  6. Efficient and principled method for detecting communities in networks. Phys. Rev. E, 84:036103, Sep 2011.
  7. Béla Bollobás. Random Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2nd edition, 2001.
  8. A heuristic algorithm using tree decompositions for the maximum happy vertices problem. Journal of Heuristics, Nov 2023.
  9. Amin Coja-Oghlan. Graph partitioning via adaptive spectral techniques. Electronic Colloquium on Computational Complexity, 19(02):227–284, 2010.
  10. Algorithms for graph partitioning on the planted partition model. In Dorit S. Hochbaum, Klaus Jansen, José D. P. Rolim, and Alistair Sinclair, editors, Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques, pages 221–232, Berlin, Heidelberg, 1999. Springer Berlin Heidelberg.
  11. Algorithms for graph partitioning on the planted partition model. Random Structures & Algorithms, 18(2):116–140, 2001.
  12. Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E, 84:066106, Dec 2011.
  13. Reinhard Diestel. Graph theory. Graduate texts in mathematics 173. Springer, 5th edition, 2017.
  14. On random graphs i. Publicationes Mathematicae Debrecen, 6:290, 1959.
  15. Community detection in social networks. Encyclopedia with Semantic Computing and Robotic Intelligence, 01(01):1630001, 2017.
  16. A simple and effective algorithm for the maximum happy vertices problem. TOP, 30(1):181–193, April 2022.
  17. Stochastic blockmodels: First steps. Social Networks, 5(2):109–137, 1983.
  18. The metropolis algorithm for graph bisection. Discrete Applied Mathematics, 82(1):155–175, 1998.
  19. Stochastic blockmodels and community structure in networks. Phys. Rev. E, 83:016107, Jan 2011.
  20. A review of stochastic block models and extensions for graph clustering. Applied Network Science, 4(1):122, Dec 2019.
  21. Rhyd Lewis. Guide to Graph Colouring: Algorithms and Applications. Springer Cham, 2nd edition, 2021.
  22. Finding happiness: An analysis of the maximum happy vertices problem. Computers & Operations Research, 103:265–276, 2019.
  23. The maximum happy induced subgraph problem: Bounds and algorithms. Computers & Operations Research, 126:105114, 2021.
  24. Birds of a feather: Homophily in social networks. Annual Review of Sociology, 27(1):415–444, 2001.
  25. Reconstruction and estimation in the planted partition model. Probability Theory and Related Fields, 162(3):431–461, Aug 2015.
  26. Graph spectra and the detectability of community structure in networks. Phys. Rev. Lett., 108:188701, May 2012.
  27. M. E. J. Newman. Communities, modules and large-scale structure in networks. Nature Physics, 8(1):25–31, Jan 2012.
  28. Hierarchical organization in complex networks. Phys. Rev. E, 67:026112, Feb 2003.
  29. Understanding the psycho-sociological facets of homophily in social network communities. IEEE Computational Intelligence Magazine, 14(2):28–40, 2019.
  30. Recombinative approaches for the maximum happy vertices problem. Swarm and Evolutionary Computation, 75:101188, 2022.
  31. Tackling the Maximum Happy Vertices Problem in Large Networks. 4OR, 58(9):2696–2711, 2020.
  32. Collective dynamics of ‘small-world’ networks. Nature, 393(6684):440–442, Jun 1998.
  33. Homophily preserving community detection. IEEE Transactions on Neural Networks and Learning Systems, 31(8):2903–2915, 2020.
  34. Comparative study for inference of hidden classes in stochastic block models. Journal of Statistical Mechanics: Theory and Experiment, 2012(12):P12021, dec 2012.
  35. Algorithmic aspects of homophyly of networks. Theoretical Computer Science, 593:117–131, 2015.
  36. New algorithms for a simple measure of network partitioning. Theoretical Computer Science, 957:113846, 2023.

Summary

We haven't generated a summary for this paper yet.