… 1.7 ONE-DIMENSIONAL GRAPHS The finite difference grid discussed previously in this
chapter is now regarded as a graph consisting of N nodes, also called vertices, connected by L …

… was used to prove deterministic properties of graphs. For example, if we can show that a
random graph has with a positive probability a certain property, then a graph must exist with this …

… -theoretical framework and discuss the main topological properties of power grids. In
Section 3, we introduce the Exponential Random Graph model along with the proposed …

… Based on the above mathematical foundation, we apply random walks for power grid anal…
[39] with this generic randomwalk method and present a hierarchical random-walk method. …

… of power grids and therefore, generated random graphs are more similar to power graphs in
… Also, due to the variety in the inputs of the proposed model, the created random graphs are …

… the random graph G we consider the two standard models G(n, p) and G(n, M). As usual, a
random G ∈ G(n, p) is a graph … of cubes or grids in G ∈ G(n, M), for M in a certain range. For a …

… In this paper, we propose using a synthetic grid procedure that samples random graphs
using the probability distribution specified by an ERG model of the form given in Eq. (8). Our …

… , and the power grid has addressed the resilience of these networks to either random or targeted
… on random graphs with vertex degrees distributed according to the truncated power law …

… , while networks like the power grid, the Internet backbone, … properties to those of the familiar
random graph. We do not claim … random graphs 20,21, but they differ from random graphs in …

… in a random graph on n vertices of a graph isomorphic to a given graph of order Cn and
average degree at least three are investigated. In particular it is proved that the random graph G(…