Graph Theory

Random graphs are of central importance in probability theory, combinatorics, and statistical physics. The purpose of these lectures is to review, in a non-rigorous manner, the typical properties of random graphs, with a strong emphasis on the Erdos-Renyi ensemble. We will illustrate those properties, and in particular the percolation transition, on the special case of random systems of Boolean equations. Statistical mechanics techniques will be introduced to study rare properties, that is, large deviations from the typical case. In the second lecture we will focus on dynamical processes modifying random graph structures, and analyze their evolution. Furthermore the replica method for 'dilute' systems will be presented. Finally, in the third lecture, random spatial maps will be considered, in relationship with the modeling of recent experiments in neurobiology. *Lecture 1: [[http://www.phys.ens.fr/~monasson/Netadis/lecture1.pdf|Typical and Rare Properties of Random Graphs]]\\ *Lecture 2: [[http://www.phys.ens.fr/~monasson/Netadis/lecture2.pdf|Dynamical Processes on Random Graphs]]\\ *Lecture 3: [[http://www.phys.ens.fr/~monasson/Netadis/lecture3.pdf|Storage of spatial maps in Hopfield-like models]]. For details about the statistical mechanics derivation of the phase diagram of the model, [[http://www.phys.ens.fr/~monasson/Articles/a83.pdf|see this paper]]. Suggestions for further readings:\\ *B. Bollobas, Random Graphs, Cambridge University Press (2001)\\ *N.C. Wormald, The differential equation method for random graph processes and greedy algorithms, in Lectures on Approximation and Randomized Algorithms (M. Karonski and H.J. Proemel, eds), pp. 73-155. PWN, Warsaw (1999)\\ *D.J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge University Press (1992)\\ *for more on statistical mechanics and random Boolean systems, see R. Monasson, [[http://www.phys.ens.fr/~monasson/Articles/a65.pdf|Introduction to Phase Transitions in Random Optimization Problems]], Lecture Notes of Les Houches Summer School, Elsevier (2006) and references therein.