The average number of distinct sites visited by a random walker on random graphs

Caterina De Bacco 1 Satya N. Majumdar 1 Peter Sollich 2

Journal of Physics A: Mathematical and Theoretical, Institute of Physics: Hybrid Open Access, 2015, 48, pp.205004

We study the linear large $n$ behavior of the average number of distinct sites $S(n)$ visited by a random walker after $n$ steps on a large random graph. An expression for the graph topology dependent prefactor $B$ in $S(n) = Bn$ is proposed. We use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message-passing equations to calculate its value. Numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs. Scaling with system size and average graph connectivity are also analysed.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. King's College London