Abstract
The scope of these lecture notes is to provide an introduction to modern statistical physics mean-field methods for the study of phase transitions and optimization problems over random structures. We first give a brief introduction to the field using as tutorial example the percolation problem in random graphs. Next we describe the so called cavity method and the related message-passing algorithms (Belief Propagation and variants) which can be used to analyze and solve optimization problems over random structures.
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Notes
- 1.
Throughout this chapter, we will omit for simplicity the Boltzmann’s constant k; this is always possible by choosing appropriate measurement units, such that k = 1.
- 2.
When the graph does not form a single connected component it is often called a forest, but this distinction is moot four our purposes.
- 3.
Using the most biased variable is a simple and reasonable heuristic which works well in practice, but other strategies may be considered.
- 4.
One step may correspond to one update of all messages (synchronous update scheme), or more often to the update of the messages associated to one randomly chosen variable or function node (asynchronous update scheme).
- 5.
This requires to extend the BP equations to models with continuous variables, which was omitted here for simplicity, but is rather straightforward.
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Baldassi, C., Braunstein, A., Ramezanpour, A., Zecchina, R. (2015). Statistical Physics and Network Optimization Problems. In: Fagnani, F., Fosson, S., Ravazzi, C. (eds) Mathematical Foundations of Complex Networked Information Systems. Lecture Notes in Mathematics(), vol 2141. Springer, Cham. https://doi.org/10.1007/978-3-319-16967-5_2
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