Graph Neural Network for Minimum Dominating Set

The dominating set concept in graphs has been used in many applications. In large graphs finding the minimum dominating set is difficult. The minimum dominating set problem in graphs seek a set D of minimum number of vertices such that each vertex of the graph is either in D or adjacent to a vertex in D. In a graph on n nodes if there is a single node of degree n-1 then that single node forms a minimum dominating set. In the proposed work, a designed network called Graph Neural Network (GNN) is used to identify the node of degree N-1 in a graph having a single node of degree N-1 as it forms the minimum dominating set in the graph.. The network is simulated for graphs with nodes varying from 5 to 15. The state dimension of the input vectors are analyzed for better convergence. It has been found that the minimum dominating set was correctly identified from 80% to 90% of the graphs when the state dimension was 2 and 3. It has been observed that when the state dimension was 2, the convergence was fast as it requires minimum hidden neurons than other state dimensions. It has also been observed that when the state dimension was greater than 3, convergence requires hours of time and more number of hidden neurons. GNN was able to identify the minimum dominating set in a graph on n vertices which has a single node of degree n-1.

[1]  Erkki Mäkinen,et al.  A Neural Network Model to Minimize the Connected Dominating Set for Self-Configuration of Wireless Sensor Networks , 2009, IEEE Transactions on Neural Networks.

[2]  Alessandro Sperduti,et al.  A general framework for adaptive processing of data structures , 1998, IEEE Trans. Neural Networks.

[3]  Ah Chung Tsoi,et al.  Sentence Extraction by Graph Neural Networks , 2010, ICANN.

[4]  Alessandro Sperduti,et al.  Supervised neural networks for the classification of structures , 1997, IEEE Trans. Neural Networks.

[5]  Marco Furini,et al.  International Journal of Computer and Applications , 2010 .

[6]  Ah Chung Tsoi,et al.  Graph neural networks for ranking Web pages , 2005, The 2005 IEEE/WIC/ACM International Conference on Web Intelligence (WI'05).

[7]  Ah Chung Tsoi,et al.  Document Mining Using Graph Neural Network , 2006, INEX.

[8]  Ah Chung Tsoi,et al.  The Graph Neural Network Model , 2009, IEEE Transactions on Neural Networks.

[9]  Franco Scarselli,et al.  Neural networks for relational learning: an experimental comparison , 2011, Machine Learning.

[10]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[11]  Ah Chung Tsoi,et al.  Computational Capabilities of Graph Neural Networks , 2009, IEEE Transactions on Neural Networks.