Core percolation in random graphs: a critical phenomena analysis
Service de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France
Revised: 26 June 2001
Published online: 15 December 2001
We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at : below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs.
PACS: 02.10.-v – Logic, set theory, and algebra / 02.50.-r – Probability theory, stochastic processes, and statistics / 64.60.Ak – Renormalization-group, fractal, and percolation studies of phase transitions / 64.60.Fr – Equilibrium properties near critical points, critical exponents
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2001