Proportionally fair (PF) controller placement, introduced in [1] and formalised in [2] with Logistic Regression and Random Forest estimators of the pairwise coverage matrix Π on topological features, and extended in [3] to agricultural sensor coverage, is the framework this paper evaluates. This paper stress-tests that pipeline: we replace the topological-feature classifiers with a graph-native estimator (a two-layer GraphSAGE encoder with a pairwise readout) and probability-profile clustering with embedding-space clustering, keeping the linearised PF objective, the partition-matroid constraint, and the heap-based selection rule unchanged. The four resulting estimators are evaluated on Barabási-Albert topologies of 75, 300, and 500 nodes along four axes: worst-case placement quality, training wall-clock time, expected calibration error, and interpretability. Logistic Regression delivers the highest worst-case coverage at moderate-to-heavy attack intensities (run-mean margins up to 3.8 functional nodes at 75 and 20.0 at 300), trains roughly 24× faster than GraphSAGE at 500 nodes, gives raw probabilities better calibrated than the deep estimator (ECE 0.064 versus 0.237, both ≈ 0.07 after isotonic regression), and remains directly inspectable through its per-feature coefficients. GraphSAGE surpasses neither supervised baseline on any axis. The result is qualified and scale-dependent. One component of the deep pipeline does produce a measurable gain: embedding-space K-means surpasses probability-space K-means by 19.34 functional nodes at 300 and 4.33 at 500 while trailing by 1.40 at 75. Within the empirical scope of this study (synthetic Barabási-Albert topologies of 75, 300, and 500 nodes; failure model with 12% targeted removals of high-betweenness nodes plus 88% uniform random removals), the recommended deployment configuration retains Logistic Regression as the production estimator of Π and confines the deep encoder to the clustering step on topologies above the regime-transition size. Generalisation to operational SDN backbones, to graphs of substantially different degree distribution, or to adaptive-adversary attack models is outside the empirical scope and is listed as a limitation in Section 8.
References
[1]
Ballasheni, E., Nace, D., Tomaszewski, A. and Zyle, A. (2023) A Probabilistic Optimization Approach to the Equitable Controller Location Problem. IEEE Transactions on Network and Service Management, 22, 1812-1824.
[2]
Ballasheni, E., Shinko, I., Picorri, L., Kolici, V., Zyle, A. and Nace, D. (2025) Optimizim Probabilistik dhe Machine Learning per Vendosjen e Kontrolloreve ne Rrjetet SDN. Buletini i Shkencave Teknike, No. 1, 71-79. https://upt.edu.al/wp-content/uploads/2026/02/Buletini-i-Shkencave-Teknike-2025-Nr.-1-1.pdf
[3]
Ballasheni, E., Shinko, I., Picorri, L., Kolici, V., Nace, D. and Zyle, A. (2025) Sensor Placement for Agricultural Field Coverage. 2025 International Congress on Smart Agriculture and Sustainable Systems, Marrakech, 5-9 December 2025, 1-6.
[4]
Kreutz, D., Ramos, F.M.V., Esteves Verissimo, P., Esteve Rothenberg, C., Azodolmolky, S. and Uhlig, S. (2015) Software-Defined Networking: A Comprehensive Survey. Proceedings of the IEEE, 103, 14-76. https://doi.org/10.1109/jproc.2014.2371999
[5]
Bernardos, C., de la Oliva, A., Serrano, P., Banchs, A., Contreras, L., Jin, H., et al. (2014) An Architecture for Software Defined Wireless Networking. IEEE Wireless Communications, 21, 52-61. https://doi.org/10.1109/mwc.2014.6845049
[6]
Bari, M.F., Roy, A.R., Chowdhury, S.R., Zhang, Q., Zhani, M.F., Ahmed, R., et al. (2013) Dynamic Controller Provisioning in Software Defined Networks. Proceedings of the 9th International Conference on Network and Service Management (CNSM 2013), Zurich, 14-18 October 2013, 18-25. https://doi.org/10.1109/cnsm.2013.6727805
[7]
Dixit, A., Hao, F., Mukherjee, S., Lakshman, T.V. and Kompella, R. (2013) Towards an Elastic Distributed SDN Controller. ACM SIGCOMM Computer Communication Review, 43, 7-12. https://doi.org/10.1145/2534169.2491193
[8]
Breiman, L. (2001) Random Forests. Machine Learning, 45, 5-32. https://doi.org/10.1023/a:1010933404324
[9]
Kipf, T.N. and Welling, M. (2017) Semi-Supervised Classification with Graph Convolutional Networks. International Conference on Learning Representations (ICLR), Toulon, 24-26 April 2017, 1-14.
[10]
Hamilton, W.L., Ying, R. and Leskovec, J. (2017) Inductive Representation Learning on Large Graphs. Advances in Neural Information Processing Systems (Neu-rIPS), Long Beach, 4-9 December 2017, 1024-1034.
[11]
Veli?kovi?, P., Cucurull, G., Casanova, A., Romero, A., Liò, P. and Bengio, Y. (2018) Graph Attention Networks. International Conference on Learning Representations (ICLR), Vancouver, 30 April-3 May 2018, 1-12.
[12]
Shwartz-Ziv, R. and Armon, A. (2022) Tabular Data: Deep Learning Is Not All You Need. Information Fusion, 81, 84-90. https://doi.org/10.1016/j.inffus.2021.11.011
[13]
Grinsztajn, L., Oyallon, E. and Varoquaux, G. (2022) Why Do Tree-Based Models Still Outperform Deep Learning on Typical Tabular Data? Advances in Neural Information Processing Systems 35, New Orleans, 28 November-9 December 2022, 507-520. https://doi.org/10.52202/068431-0037
[14]
Heller, B., Sherwood, R. and McKeown, N. (2012) The Controller Placement Problem. Proceedings of the 1st Workshop on Hot Topics in Software Defined Networks, Helsinki, 13 August 2012, 7-12. https://doi.org/10.1145/2342441.2342444
[15]
Tanha, M., Sajjadi, D., Ruby, R. and Pan, J. (2018) Capacity-Aware and Delay-Guaranteed Resilient Controller Placement for Software-Defined WANs. IEEE Transactions on Network and Service Management, 15, 991-1005. https://doi.org/10.1109/tnsm.2018.2829661
[16]
Muller, L.F., Oliveira, R.R., Luizelli, M.C., Gaspary, L.P. and Barcellos, M.P. (2014) Survivor: An Enhanced Controller Placement Strategy for Improving SDN Survivability. 2014 IEEE Global Communications Conference, Austin, 8-12 December 2014, 1909-1915. https://doi.org/10.1109/glocom.2014.7037087
[17]
Kelly, F.P., Maulloo, A.K. and Tan, D.K.H. (1998) Rate Control for Communication Networks: Shadow Prices, Proportional Fairness and Stability. Journal of the Operational Research Society, 49, 237-252. https://doi.org/10.1057/palgrave.jors.2600523
[18]
Ogryczak, W., Luss, H., Pióro, M., Nace, D. and Tomaszewski, A. (2014) Fair Optimization and Networks: A Survey. Journal of Applied Mathematics, 2014, Article ID: 612018. https://doi.org/10.1155/2014/612018
[19]
Luss, H. (2012) Equitable Resource Allocation: Models, Algorithms, and Applications. Wiley. https://doi.org/10.1002/9781118449189
[20]
Khalil, E., Dai, H., Zhang, Y., Dilkina, B. and Song, L. (2017) Learning Combinatorial Optimization Algorithms over Graphs. Advances in Neural Information Processing Systems (NeurIPS), Long Beach, 4-9 December 2017, 6348-6358.
[21]
Kool, W., van Hoof, H. and Welling, M. (2019) Attention, Learn to Solve Routing Problems! International Conference on Learning Representations (ICLR), New Orleans, 6-9 May 2019, 1-25.
[22]
Li, Z., Chen, Q. and Koltun, V. (2018) Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search. Advances in Neural Information Processing Systems (NeurIPS), Montréal, 3-8 December 2018, 537-546.
[23]
Gasse, M., Chételat, D., Ferroni, N., Charlin, L. and Lodi, A. (2019) Exact Combinatorial Optimization with Graph Convolutional Neural Networks. Advances in Neural Information Processing Systems (NeurIPS), Vancouver, 8-14 December 2019, 15580-15592.
[24]
Almasan, P., Suárez-Varela, J., Rusek, K., Barlet-Ros, P. and Cabellos-Aparicio, A. (2022) Deep Reinforcement Learning Meets Graph Neural Networks: Exploring a Routing Optimization Use Case. Computer Communications, 196, 184-194. https://doi.org/10.1016/j.comcom.2022.09.029
[25]
Platt, J. (1999) Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods. In: Advances in Large Margin Classifiers, MIT Press, 61-74.
[26]
Zadrozny, B. and Elkan, C. (2002) Transforming Classifier Scores into Accurate Multiclass Probability Estimates. Proceedings of the 8th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Edmonton, 23-26 July 2002, 694-699. https://doi.org/10.1145/775047.775151
[27]
Niculescu-Mizil, A. and Caruana, R. (2005) Predicting Good Probabilities with Supervised Learning. Proceedings of the 22nd International Conference on Machine Learning-ICML‘05, Bonn, 7-11 August 2005, 625-632. https://doi.org/10.1145/1102351.1102430
[28]
Guo, C., Pleiss, G., Sun, Y. and Weinberger, K.Q. (2017) On Calibration of Modern Neural Networks. Proceedings of the 34th International Conference on Machine Learning (ICML), Sydney, 6-11 August 2017, 1321-1330.
[29]
Domingos, P. (2012) A Few Useful Things to Know about Machine Learning. Communications of the ACM, 55, 78-87. https://doi.org/10.1145/2347736.2347755
[30]
Kingma, D.P. and Ba, J. (2015) Adam: A Method for Stochastic Optimization. International Conference on Learning Representations (ICLR), San Diego, 7-9 May 2015, 1-15.
[31]
Rousseeuw, P.J. (1987) Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis. Journal of Computational and Applied Mathematics, 20, 53-65. https://doi.org/10.1016/0377-0427(87)90125-7
[32]
Ying, R., Bourgeois, D., You, J., Zitnik, M. and Leskovec, J. (2019) GNNExplainer: Generating Explanations for Graph Neural Networks. Advances in Neural Information Processing Systems (NeurIPS), Vancouver, 8-14 December 2019, 9240-9251.
[33]
Xie, S.M. and Ermon, S. (2019) Reparameterizable Subset Sampling via Continuous Relaxations. Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, Macao, 10-16 August 2019, 3919-3925. https://doi.org/10.24963/ijcai.2019/544
[34]
Ballasheni, E., Shinko, I., Nace, D., Zyle, A. and Kolici, V. (2025) Strategic Placement of Access Points for Maximizing Wi-Fi Quality. 2025 20th Annual System of Systems Engineering Conference (SoSE), Tirana, 8-11 June 2025, 1-6. https://doi.org/10.1109/sose66311.2025.11083808