4.2.6
BollobsRiordan model
a
Here I briey describe one of the rst preferential attachment model that has been rigorously
analyzed3 . First consider the dynamical picture.
Let G1 be the initial graph that consists of unique vertex and one self-loop. Now to
Chapter 5
Processes on complex networks
Up till now we discussed the structure of the complex networks. The actual reason to study
this structure is to understand how this structure inuences the behavior of random processes
on networks. I will talk about
3.2
Conguration model
3.2.1
Denition. Basic properties
Assume that the vector d = (d1 , . . . , dn ) is graphical, i.e., there exits a graph on n vertices such
that vertex 1 has degree d1 , vertex 2 has degree d2 , and so on. We would like to generate a
r
Chapter 4
Preferential attachment model
4.1
Conditional expectations
For the following we will need the notion of the conditional expectation. In this section I discuss
the necessary theory. Actually, we will need only one small instance of this notion, b
Chapter 3
Generalizations of the ErdsRnyi
o
e
random graphs
ErdsRnyi random graphs have two incarnations: The rst one that we studied in much details
o
e
is G (n, p) when the probability of each edge is specied. And the second one is G (n, m) when a
xed n
2.4
2.4.1
The giant component of the ErdsRnyi random graph
o
e
Non-rigorous discussion
We know that if pn = o(1) then there are no triangles. In a similar manner it can be shown that
there are no cycles of any order in G (n, p). This means that most compo
Mathematics of networks
Artem S. Novozhilov
August 29, 2013
A disclaimer: While preparing these lecture notes, I am using a lot of dierent sources for
inspiration, which I usually do not cite in the text. I plan to give a short literature review at some
p
1.2
1.2.1
Graph theory
Denitions, denitions.
A graph G consists of a non-empty vertex set V , and an edge set E of unordered 2-element sets
from V . Formally, a graph G is a non-empty set V together with an irreexive and symmetric
relation R on V ; E deno
1.3
1.3.1
Probability theory
Basics
Probability theory starts with a non-denable notion of experiment, which has possible outcomes.
The set of all possible outcomes is called the sample space and usually denoted by . In all the
problems dealing with proba
Chapter 2
ErdsRnyi random graphs
o
e
2.1
Denitions
()
Fix n and consider the set V = cfw_1, 2, . . . , n, and put N := n be the number of edges on the
2
full graph Kn , the edges are cfw_e1 , e2 , . . . , eN . Fix also p [0, 1] and choose edges according