[ot][spam][crazy][wrong] Reading a Book from Library Genesis

[Karl]

Library genesis can be found at https://libgen.is/ right now, it can
move.  It has books for download, for example that textbook you paid
$300 for, there is a very good chance some revision of it, or a book
with the same content, is on library genesis.

I've downloaded this book, "Deep Learning on Graphs", by Yao Ma and
Jiliang Tang, from library genesis.  You could download it for example
from http://library.lol/main/1BEBBF5A91CE94A5F2D5C50935BBAC9E .

Let's read it!  Even better, let's skim through it.

I'll start with section 2.4: Spectral Graph Theory

----

Spectral graph theory studies the properties of a graph through
analyzing the eigenvalues and eigenvectors of its Laplacian matrix.
In this section, we first introduce the Laplacian matrix of a graph
and then discuss the key properties of the Laplacian matrix and its
eigenvalues and eigenvectors.

2.4.1 Laplacian Matrix

In this subsection, we introduce the Laplacian matrix of a graph,
which is another matrix representation for graphs in addition to the
adjacency matrix.

Defintion 2.28 (Laplacian Matrix)   For a given graph G = {V,E} with A
as its adjacency matrix, its Laplacian matrix is defined as
    L = D - A, (2.18)
where D is a diagonal degree matrix D = diag(d(v_1),...,d(v_|V|))

Another definition of the Laplacian matrix is a normalized version of
Eq. (2.18).

Definition 2.29 (Normalized Laplacian Matrix)    For a given graph G=
{V,E} with A as its adjacency matrix, its normalized Laplacian matrix
is defined as
    L = D^(-1/2)(D - A)D^(-1/2) = I - D^(-1/2)AD^(-1/2). (2.19)

Next, we focus on the discussion of the (unnormalized) Laplacian
matrix as defined in Definition 2.28.  However, in some later chapters
of this book, the normalized Laplacian matrix will also be utilized.
Unless specifically mentioned, we refer to the Laplacian matrix as the
unnormalized one defined in Definition 2.28.
----
That was the top of doc page 27, pdf page 46.

The Laplacian matrix is defined in terms of the adjacency matrix, so
let's search the document for the first occurences of the phrase
"adjacency matrix", to see what that is.

Now I'm in section 2.2 Graph Representations, document page 18, pdf page 37.

----
2.2 Graph Representations

In this section, we introduce the definition of graphs.  We focus on
simple unweighted graphs and will discuss more complex graphs in the
following sections.

Definition 2.1 (Graph)    A graph can be denoted as G = {V,E}, where
V= {v_1,...,v_N} is a set of N = |V| nodes and E = {e_1,...,e_M} is a
set of M edges.

Nodes are essential entities in a graph.  In social graphs, users are
viewed as nodes, whereas in chemical compound graphs, chemical atoms
are treated as nodes.  The size of a given graph G is defined by its
number of nodes; i.e., N = |V|.  The set of edges E describes the
connections between nodes.  An edge e_i connects two nodes v^1_ei and
v^2_ei; thus, the edge e_i can be also represented as (v^1_ei,
v^2_ei).  In directed graphs, the edge is directed from node v^1_ei to
node v^2_ei.  In undirected graphs, the order of the two nodes does
not make a difference; i.e., e_i = (v^1_ei, v^2_ei) = (v^2_ei,
v^1_ei).  Note that without specific mention, we limit our discussion
to undirected graphs in this chapter.  The nodes v^1_ei and v^2_ei are
incident to the edge e_i.  A node v_i is said to be adjacent to
another node v_j if and only i there exists and edge between them.  In
social graphs, different relations such as friendship can be viewed as
edges between nodes, whereas chemical bonds are considered as edges in
chemical compound graphs (we simply regard all chemical bonds as edges
while ignoring their different types).  A graph G = {V, E} can be
equivalently represented as an adjacency matrix, which describes the
connectivity between the nodes.

Definition 2.2 (Adjacency Matrix)    For a given graph G = {V, E}, the
corresponding adjacency matrix is denoted as A of {0, 1}^NxN.  The i,
jth entry of the adjacency matrix A, indicated as A_i,j, represents
the connectivity between two nodes v_i and v_j.  More specifically,
A_i,j = 1 if v_i is adjacent to v_j; otherwise, A_i,j = 0.

In an undirected graph, a node v_i is adjacent to v_j if and only if
v_j is adjacent to v_i; thus, A_i,j = A_j,i holds for all v_i and v_j
in the graph.  Hence, for an undirected graph, its corresponding
adjacency matrix is symmetric.
----

Okay!  An adjacency matrix is a 2-dimensional boolean array, with
dimension length equal to the number of nodes (vertices) in the graph.
Each cell in the array is 1 or True if the coordinates of the cell
represent a connection from one node to another, and 0 or False if
there is no such connection.  This could be expanded to represent 3-
or n- -ended edged graphs with arbitrary directionality, by adding
further dimensions.

Cool!  If this state of mind continues I might type more in.