NÚMERO CÓDIGO PESO NOMBRE

This section contains proofs of all the theorems. First we will prove the existence of a giant

component in supercritical regime.

4.3.1

Proof of Theorem 4.0.1

Proof.

Fixα

0

=ε/10,c=ε

3

, now for large enoughnthe conclusion of the Lemma 4.2.2 holds.

Now we run the DFS algorithm with a sequence of i.id Bernouli(ρ) random variables. We

consider the situation after

εnmany vertex queries (a vertex will be included inU

or not type

queries) of the algorithm. Assume at some time point

t∈[ε

3

n, εn], the set

U

becomes empty.

Then we must have|S ∪ W|=tand

|S|=P

t

i=1

Y

i

. Then by 2 and 3 in Lemma 4.2.3whp

(1 +

3₄ε

)t

n

6p|S|62ε61/3

for small enough

ε. Now since at that point

U

is empty,

NG(S)

⊂ W. The function

g(x) :=

x−

₂x_n2

, is non-decreasing when

x6n, thus it is non-decreasing atx=np|S|₆n/3< n. Now

since we have (1 +

3₄ε

)t6np|S|, hence by Lemma 4.2.2 we havewhp,

|W|_>(1−α

0

)

np|S| −np

2

_|S|

2

2

= (1−α

0

)

np|S| −(np|S|)

2

2n

>(1−

ε

10)(1 +

3ε

4

)t

1−

1

2n(1 +

3ε

4)t

>(1−

ε

10)(1 +

3ε

4

)t

1−

1

6(1 +

3ε

4

)ε

> t,

(4.3.1)

for small enough

ε, contradicting our theorem assumption. Thus

whp

all the vertices that

are being explored in the time frame [ε

3

n, εn] belong to the same epoch and hence the same

component. Again using parts 2 and 3 of the Lemma 4.2.3 we get the size of this component is

bounded below by

ε_p

.

4.3.2

Proof of uniqueness under hereditary degree assumption

First we will prove uniqueness of the giant component under an additional assumption that

we will call hereditary degree assumption. It is interesting to note that if in addition to

A1

and

A2we suppose that the following hereditary property (HD) holds for the graph sequence

G

n

= (V

n

, E

n

), then the giant component will be unique when

p=o(1),np

2

→ ∞,a

n

=o(np),

b

n

=np

2

. In particular we will not requireA3

andb

n

=o(np

3

).

Assumption HD.

For each

β >

0 and

n

>

N(β), every large subgraph

U

⊂

V

n

, say for

|U|>0.9nsatisfy,

max

v∈U

d(v, U)<(1 +β)p|U|.

Proposition 4.3.1.

In addition to the conditions in Theorem 4.0.1 assume thatGsatisfyHD.

Then there is an unique giant component with size greater than or equal to

ε_p

whpand all other

components are of size less than

O((lnn)

2

).

Proof of Proposition 4.3.1.

Let

C(ρ) be a component with size at least

ε_p

.

Recall that

O

Gn

(C(ρ)) :={v∈V\C(ρ) :v

is

nota neighbor of

C(ρ)}. Lemma 4.2.11 gives

|O

G

(C(ρ))|6

n(1−ε+

ε₂2

+εl

n

). At this end, note that all vertices of a component that is not connected to

C(ρ) must belong toO

_G

(C(ρ)). Also the largest connected component of

Per

Gn,ρ

(O

G

(C(ρ)))

is no more than the largest connected component in a setPer

Gn,ρ

(P) whereP

is any set with

sizen(1−ε+

ε₂2

+εl

n

) containing

O

Gn

(C(ρ)). Choose

εsmall enough and

nlarge enough so

that

n(1−ε+

ε₂2

+εl

n

)

>0.9n. Now by

HD

max

v∈P

d(v,P)

<(1 +ε

5

)p|P|

when

n

is large.

We get thatP

is a graph onn(1−ε+

ε₂2

+εl

n

) vertices with maximum degree (1 +ε

5

)p|P|and

each vertex is retained with probability

1+_npε

. It is easy to check for

ε >0 small enough and

n

large enough,

1−ε

2

/4

np(1 +ε

5

_)(1−ε+

ε2 2

+εl

n

)

>

1 +ε

np

.

Thus the subgraph induced by

G

n

on

P

is a graph on

n(1−ε+

ε

2

2

+εl

n

) vertices with

maximum degree (1 +ε

5

)p|P|and each of the vertices is retained with probabilityρthat is less

than1−

ε2

4

/p(1 +ε

5

)|P|, hence we appeal directly to Theorem 1 in (Krivelevich, 2016) and

get that the largest connected component in

Per

G,ρ

(P) is less than

O

(lnn)

2

.

4.3.3

Proving uniqueness under A1, A2, A3

Notice that in the proof of Proposition 4.3.1, we only needed max

v∈U

d(v, U)<(1 +β)p|U|

for a particular set

U, namely for

U

=

O

_G

(C(ρ)). From Lemma 4.2.6 we have that if

G

n

=

(V

n

, E

n

) satisfiesA1,A2and

A3then for any fixed large setU

⊂V

n

, there are not too many

vertices inG

n

that do not satisfyHD.

Proof of Theorem 4.0.3.

The proof is similar to Proposition 4.3.1, except that now we do not

have

max

v∈P

d(v,P)<(1 +ε

5

_)p|P|,

_(4.3.2)

when

n

is large. But since

|P|> n/2 by Lemma 4.2.6, we have the number of elements in

G

that do not satisfy 4.3.2 is at most

_(αp4₎2

(4p+ 12b

n

), withα

=ε

5

. Thus it is sufficient to show

that the probability that at least one vertex is getting selected out of

_(αp4₎2

(4p+ 12b

n

) is going

takes the value zero with probability going to one. Indeed, the probability is equal to

1−1 +ε

np

4 (αp)2(4p+12bn)

= exp

4

(αp)

2

(4p+ 12b

n

) ln

1−1 +ε

np

>exp

−32(1 +ε)

αnp

2

−

96b

n

np

3

→1.

In the second step we used the fact that ln(1−x)

>−2x

for

x

∈

(0,

1₂

). Now proceeding as

CHAPTER 5

Discussions and future directions

We have already seen the effects of simple local conditions on the overall structure of a

network.

Our current aim is to develop methods to analyze various other properties of a

network while assuming simple general conditions such as pseudo-randomness or a converging

graph sequence. For example one aim is to understand the critical percolation behavior on

pseudo-random graphs. We have already seen that this ensembles of graphs are similar to

Erd˝os-R´enyi random graph in many ways. On the other hand the percolation phenomenon is

well understood in the context of Erd˝os-R´eny graphs. Using the results in (Bollob´as et al., 2010)

it is also easy to derive the threshold when the giant component emerges. A natural question is

to investigate what will be the size and structure of the giant component at criticality? We will

discuss this in more detail in Section 5.2. Another direction that we are particularly interested

in is to understand the estimation procedure of parametric models of networks such as ERGM,

GERGM. These models have been used extensively in many applied fields such as political

science, biology etc. and we are particularly interested to understand under what conditions

the estimation is consistent. More particularly what should be the choice of sufficient statistics

in a model or how the base measure affects the estimation procedure? Lastly, one of our main

goal is to understand the limiting behavior of graphs in sparse regime. We have seen an unified

limit theory makes it lot easier to connect and analyze various properties of the network in the

dense regime. Unfortunately, the techniques are much different in case of sparse graph and the

same limit theory does not work in the sparse regime. Currently we are analyzing the ”critical

phenomenon” or the size of critical component in a multi-type inhomogeneous random graph.

For a single type this was done in (Bollob´as et al., 2007) and the size of the critical component

was derived in (van der Hofstad, 2013; Bhamidi et al., 2010), we are incorporating an additional

“environmental” effect on the connectivity structure. We defer the details to Section 5.1.

In document Autoevaluación con fines de mejoramiento de la carrera de Ingeniería en Sistemas Computacionales de la Facultad de Ciencias Matemáticas y Físicas de la Universidad de Guayaquil (página 110-115)