Intentional Risk Management through Complex Networks Analysis by Victor Chapela Regino Criado Santiago Moral & Miguel Romance

Author:Victor Chapela, Regino Criado, Santiago Moral & Miguel Romance
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

In the following sections, after the introduction of the notation we will use and the basic mathematical models (Sect. 4.2), we will see how to compute the PageRank of each edge from the PageRank of its nodes (Sect. 4.3), how to compute it by using the line-graph (Sect. 4.4) and the relationship between these two approaches. Finally, we establish a result that connects both approaches (Sect. 4.5).

4.2 Mathematical Formulation and Notation

Through this section, as usual, we consider a directed network G = (X, E), where and E ⊆ X × X is the set of edges, in this case, ordered pairs as (i, j) ∈ E where i, j ∈ X. In the following sections of this chapter, we will consider a directed and weighted network G = (X, E) joint to a function w : E ⟶ [0, +∞)in such a way that for each edge (i, j) ∈ E, the coefficient w(i, j) is called weight of (i,j) ∈ E. If we have a directed network G = (X, E) and this network does not have an associated weight-function, then we will say that G is a non weighted network.

Given a directed and weighted network G = (X, E) such that for each (i, j) ∈ E its weight is given by w(i, j), the (weighted) adjacency matrix of G is the matrix given by

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