We introduce the concepts of “gravitational” and “elastic” potential energy in the context of networks and investigate their implications on interpreting certain well-known graph-theoretic parameters and optimization problems. In the case of gravitational potential energy, we treat the nodes of a graph as “particles” with masses and consider the potential energy of gravitational interactions between them. We prove that the maximum clique in a graph represents the minimum gravitational potential energy structure. This result yields a natural physics-based interpretation of the well-known Motzkin-Strauss formulation for the maximum clique problem. In the case of elastic potential energy, we assume that graph nodes are located on an elastic surface, the graph edges represent “springs” connecting the nodes on the surface, and consider the potential energy corresponding to elastic deformations of springs. We show that under certain reasonable assumptions on surface deformation, the second-smallest and the largest eigenvalues of the graph Laplacian matrix correspond to the minimum and the maximum elastic potential energies of a graph, respectively. Moreover, the associated eigenvectors correspond to the node heights in the minimum and maximum energy configurations. Note that the second-smallest eigenvalue of the graph Laplacian is widely known as algebraic connectivity, and it turns out that this concept also has a simple and intuitive physics-based interpretation.
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This research was supported in part by the Air Force Research Laboratory (AFRL) and the Defense Threat Reduction Agency (DTRA). This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. The research was performed while the first author held a National Research Council Research Associateship Award at AFRL. The authors would like to thank the referees for the constructive comments that helped to improve the paper.
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Veremyev, A., Boginski, V. & Pasiliao, E.L. Potential energy principles in networked systems and their connections to optimization problems on graphs. Optim Lett 9, 585–600 (2015). https://doi.org/10.1007/s11590-014-0774-2
- Potential energy
- Graph theory
- Maximum clique
- Motzkin-strauss formulation
- Graph laplacian spectra
- Algebraic connectivity