Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

http://library.um.ac.id/free-contents/downloadpdf.php/buku/methods-of-modern-mathematical-physics-vol-i-functional-analysis-reed-michael-barry-simon-12066.pdf

Methods of modern mathematical physics (vol.) I : functional analysis / Reed Michael, Barry Simon)

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.

/pdf/unbounded-laplacians-on-graphs-basic-spectral-properties-and-1cbu2po6b3.pdf

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geometrical growing networks are present in a large set of real networks describing biological, social and technological systems.

/pdf/emergent-complex-network-geometry-z3xa9sijxl.pdf

Emergent complex network geometry.

Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are all intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale-invariance, self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments dealing with these approaches to network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.

Network Geometry

Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001

Combinatorial curvature for planar graphs

For a (d,f)-regular planar graph, which is an infinite planar graph embedded in the plane such that the degree of each vertex is d and the degree of each face is f, we determine two kinds of isoperimetric constants in concrete form.

Isoperimetric Constants of (d,f) Regular Planar Graphs

For a magnetic Schrodinger operator on a graph, which is a generalization of
					classical Harper operator, we study some spectral properties: the Bloch property
					and the behaviour of the bottom of the spectrum with respect to magnetic fields.
					We also show some examples which have interesting properties.

https://projecteuclid.org/download/pdf_1/euclid.nmj/1114631555

Weak Bloch property for discrete magnetic Schrodinger operators

In terms of the exponential growth of a non-compact Riemannian manifold, we give an upper bounds for the bottom of the essential spectrum of the Laplacian. This is an improvement of Brooks' result.

/pdf/a-remark-on-exponential-growth-and-the-spectrum-of-the-4avvngjqz5.pdf

A remark on exponential growth and the spectrum of the Laplacian

We introduce the boundary area growth as a new quantity for an infinite graph. Using this, we give some upper bounds for the bottom of the spectrum of the discrete Laplacian which relates closely to the transition operator. We also give some applications and examples.

Yusuke Higuchi

Papers

Combinatorial curvature for planar graphs

Isoperimetric Constants of (d,f) Regular Planar Graphs

Weak Bloch property for discrete magnetic Schrodinger operators

A remark on exponential growth and the spectrum of the Laplacian

Boundary Area Growth and the Spectrum of Discrete Laplacian