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American Journal of Engineering Research (AJER) 2014
w w w . a j e r . o r g
Page 281
American Journal of Engineering Research (AJER)
e-ISSN : 2320-0847 p-ISSN : 2320-0936
Volume-03, Issue-04, pp-281-294
www.ajer.org
Research Paper Open Access
Probabilistic Stochastic Graphical Models With Improved
Techniques
1Swati Kumari,
2Anupama Verma
Assistant Professor, JRU, Department of Computer Science Engineering, Ranchi - 834002
Assistant Professor, JRU, Department of Computer Science Engineering, Ranchi β 834002
Abstract: - The web may be viewed as a directed graph. A HTML page can be treated as node and hyperlink as an edge from one node to another this will form a directed graph. But the nature of this graph is evolving with
time, so we require evolving graph model. The Gn,p and Gn,N are static models. In these models, graph G = (V,E)
is not changing with time. Sometimes we need a graph model which is time evolving Gt = (Vt,Et). There are two
characteristic functions of evolving graph model [5] first gives the number of vertex added at time t+1, whereas second gives set of edges added at time t+1. Evolving copying model is different from traditional Gn,p in the
following:
1. Independently chosen edges do not show the results found on the web.
2. This model captures the evolving natures of web graph.
In this paper we review some recent work on generalizations of the random graph aimed at correcting these
shortcomings. We describe generalized random graph models of both directed and undirected networks that
incorporate arbitrary non-Poisson degree distributions, and extensions of these models that incorporate
clustering too. We also describe two recent applications of random
graph models to the problems of network robustness and of epidemics spreading on contact networks. This
paper studies the random graph models. Concentration is made over the paper stochastic models for web graph
by Ravi Kumar et. al. [5]. During the proof of the results given in the paper some more precise results have been found. Some of results are here which can be compared with results of the papers. The improvements are
presented here along with results in the original paper for comparison.
Keywords: - Random Graph, Martingale, Web Graph.
I. INTRODUCTION Definition 1. A HTML page can be treated as node and hyperlink as an edge from one node to another this will
form a directed graph called web graph.
At present this web graph has billion vertices [5], and an average degree of about 7. In this thesis we
discuss different random graph models. These observation suggest that web is not well modelled by traditional
models such as Gn,p. A lot of models for the random graph have been proposed like Gn,p and Gn,N models [6]
etc. These models do not ensure the power-law for degree of vertices and do not explain the abundance of
bipartite cliques observed in the web graph. The copying graph model [5] explains these and also covers the
evolving nature of the web graph.
Evolving copying model is different from traditional Gn,pin the following: (a) Independently chosen edges do not show the results found on the web.
(b) This model captures the evolving natures of web graph.
In this model during every time interval one vertex arrives. At time step t, a single vertex u arrives and the i-th
out-link of u is then chosen as follows. With probability Ξ±, the destination is chosen uniformly at random from
Vt, with remaining probability the outlink is taken to be the ith outlink of prototype vertex p. where Vt is set of
vertex at time t.
American Journal of Engineering Research (AJER) 2014
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A. Problem Definition
The problem is to develop random graph model that explain the different nature of web graph. We have to study
probabilistic techniques, different models for random graph and find more results based on the models
B. Related Work
The "copying" model analysed in this thesis were first introduced by Kleinberget. al. [4]. Motivated by
observations of power-laws for degrees on the graph
of telephone calls, Aiello, Chung, and Lu [1] propose a model for "massive graph"(Henceforth the "ACL
model"). In 2000, Ravi Kumar et. al. proposed a numberof models in his paper "Stochastic models for the web graph" [5]. This thesis hassome more precise results, and also the proof of different results available in thepaper
"Stochastic models for the web graph".
C. Contribution
(1) The web graph is so large that the study of different nature of this graph is difficult without a good modeling.
(2) The evolution of different social networking site can be explained with the help of this modeling.
D. Results and Organization
Results in Paper Mywork done
|E[Nt,0|Nt-k,0]- E[Nt,0|Nt-
(k+1),0]|β€ 2
|E[Nt,0|Nt-k,0]- E[Nt,0|Nt-
(k+1),0]|β€ 1
t+Ξ±
1+Ξ±-Ξ±2-ln π‘ β€ E[Nt,0]β€
t+Ξ±
1+Ξ± E[Nt,0] =
t
1+Ξ±
Pr| Nt,0 - E[Nt,0]β₯ l| β€ exp (-πβπ
4π‘)
Pr| Nt,0 - E[Nt,0]β₯ l| β€ exp (-πβπ2π‘
)
E[Nt,0]= ππ, 0π‘β1π=0 =
t
1+Ξ±
t
1+Ξ±
Pt = limπ‘ββ
E[Nt,0]
π‘ =
1
1+Ξ±
|E[Nt,i|Nt-k,iNt-k,i-1] -
|E[Nt,i|Nt-(k-1),iNt-(k-1),i-1]| β€ 2
|E[Nt,i|Nt-k,iNt-k,i-1] - |E[Nt,i|Nt-
(k-1),iNt-(k-1),i-1]| β€ 1for Nt-k,I =
Nt-(k+1),i
II. PROBABILISTIC TECHNIQUES Preliminaries
We define a discrete probability space (Ξ©, π) where Ξ© is discrete (finite or countable infinite) set and π is
probability such that
π: Ξ© β [0,1]meansβ Ο Ο΅ Ξ© π (Ο) Ο΅ [0, 1] and
π π = 1π
In this thesis we will discuss only discrete probability spaces.
Definition 2. Given Ξ©, π , π΅ β Ξ©, Define conditional probability over (Ξ©, π(π΅))as follows.
π π π΅ = 0 ππ π β π΅,π(π)
π(π΅) ππ πππ΅
Where π π π΅ is the probability of happening of Ο when B has already happened? Now the probability of happening of A when B has already happened is denoted by P(A|B) and is given by
π π΄ π΅ = π(π|π΅)πππ΄
= π π|π΅ + π π|π΅
πππ΄βπ΅πππ΄βπ΅πΆ
= 0 + π π|π΅
πππ΄βπ΅
= π(π)
π(π΅)πππ΄βπ΅
=1
π(π΅) π π πππ΄βπ΅
American Journal of Engineering Research (AJER) 2014
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β π π΄ π΅ =π(π΄βπ΅)
π(π΅)
Expectation of Random Variable
Expectation of a random variable is its basic characteristic. The expectationof random variable is weighted
average of the values it assumes, where each valueis weighted by the probability that the variable assumes that
value.
Definition 3. A random variable X is a real-valued function over the sample space Ξ©; that is X :Ξ© β R. The
expectation of random variable X is defined as
πΈπ = π ππ π(ππ )π
The expectation is finite If π |ππ π|π(ππ )converges; otherwise, the expectationis unbounded.
For example:A random variable X that takes the values 2π with probability 1
2π for j = 1,2,β¦β¦
The expectation of X
πΈ π = 1
2π
β
π =1
2π β β
Here the expectation is unbounded.
Conditional Expectation
Definition 4.The conditional expectation [3] of X relative to B is defined whenPB > 0 as
πΈ π π΅ = π ππ π(ππ |π΅)π
Definition 5.Let (πΊ1; p1) and (πΊ2; p2) be probability distributions and (Ξ©; p) iscalled the joint distribution of
them if the following conditions are satisfied.
π(π1 , π2 ) = 1(π1 ,π2)πΞ©
π(π1 , π2 ) = π2
π1
π2 = π(π2)
π(π1 , π2) = π1
π2
π1 = π(π1 )
where
Ξ© = Ξ©1 Γ Ξ©2 = (π1 , π2 |π1πΞ©1 , π2πΞ©2
Definition 6.Two random variable X and Y are independent if and only if
ππ π = π₯ β© π = π¦ = ππ π = π₯ . ππ(π = π¦)
for a values of x and y. Similarly, random variables X1,X2,β¦,Xk are mutuallyindependent if and only if, for any
subset I β [1, k] and any values xi, iπI,
ππ ππ = π₯π
πππΌ
= ππ(πππΌ
ππ = π₯π)
If the events ππ and ππ are pairwise independent then
ππ ππ ππ = ππ[ππ]
for all iβ j
Proof.
ππ ππ ππ = ππ[ππ β© ππ ]
ππ[ππ ]
=Pr[ππ|Pr|ππ ]
Pr[ππ ]= Pr[ππ]
Linearity of Expectation
πΈ π + π = πΈ π + πΈ[π] Proof.
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πΈ π + π = π₯ + π¦ π(π, π)(π₯ ,π¦)πβΓβ
= π₯ + π¦ Pr((π = π₯) β©π¦π₯
(π = π¦))
= π₯ππ π = π₯ β© π = π¦ +π¦π₯
π¦ππ π = π₯ β© π = π¦
π¦π₯
= π₯ππ π = π₯ + π¦ππ π = π¦
π¦π₯
= πΈ π + πΈ[π] General form
πΈ ππ
π
π=1
= [ππ]
π
π=1
πΈ ππ + π = ππΈ π + π Proof.
πΈ ππ + π = π π = π₯ + π Pr π = π₯
π₯πβ
= π π = π₯ Pr π = π₯ + ππ₯πβ
Pr π = π₯
π₯πβ
= ππΈ π + π. 1
= ππΈ π + π If X and Y are two independent random variables, thenE[X,Y] = E[X].E[Y ]
Bernoulli Random Variable
If we run an experiment such that it succeedswith probability p and fails with probability (1-π). Let Y be a
randomvariable such that
π = 1 ππ π π’ππππ π π€π π,
0 ππ πππππ π€π (1 β π)
Here the random variable Y is called a Bernoulli random variable.
Theorem 1. The expectation of Bernoulli random variable is same as the probability of success of the random
variable.
Proof.
E[Y] = 1. π + 0. (1 - π) = π
β E[Y] =Pr[Y=1]
Binomial Distribution Consider a sequence of n independent experiments,each of which succeeds with
probability π. If we represents X as numberof successes in n experiments then X has a binomial distribution.
Definition 7. A binomial random variable X with parameters n and π, denotedby π΅(π, π)is defined by following
probability distribution on π = 0,1,2,β¦ . , π:
ππ π = π = π
π ππ (1 β π)πβπ
0 ππ‘βπππ€ππ π
ππ0,1,2, β¦ . π
Theorem 2.The expectation of binomial random variable is ππ. Proof. The binomial random variable X can be expressed as the sum of Bernoulli random variables.
π = ππ
π
π=1
πΈ π = πΈ ππ
π
π=1
= πΈ[ππ]
π
π=1
from Linearity of expectation
π
π
π=1
from expectation of Bernoulli random variable
= ππ
American Journal of Engineering Research (AJER) 2014
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Geometric Distribution
Sequence of independent trials until the first success is found where each trialsuccess with probability π, This
gives an example of geometric distribution.
A geometric random variable X with parameter π is given by following probabilitydistribution on π = 0,1,2,β¦.
Pr π = π = (1 β π)(πβ1). π
For geometric random variable X equals π there must be π β 1 failures followedby a success.
Definition 8. 1
π‘= β(π)π
π=1 called the harmonic number = ln π + Ξ 1 .
Proof. Since 1
π is monotonically decreasing function, we can write ln π = β«
1
πβ€
1
π
ππ=1
π
π₯=1
βΉ 1
πβ€
1
π= ln π
π
π₯=1
π
π=2
β ln π β€ β(π) β€ ln π + 1
β β π = π ln π + Ξ π
Theorem 3.The expectation of geometric random variable is 1
π
Proof.
πΈ π = π. (1 β π)πβ1
β
π=1
. π
= π. π. (1 β π)πβ1
β
π=1
= π. π. π‘πβ1
β
π=1
(β΅ 1 β π = π‘ 0 < π‘ < 1)
= π. (1 + 2π‘ + 3π‘2 + 4π‘3 + β― )
= π.1
(1 β π‘)2= π
1
π2=
1
π
β΅ π π₯ = 1 + 2π‘ + 3π‘2 + 4π‘3 + β― =1
1 β π‘ π π’π ππ πΊ. π.
πβ² π₯ = 1 + 2π‘ + 3π‘2 + 4π‘3 + β― =1
(1 β π‘)2
Markov's Inequality
Let X be a random variable assumes only non-negative values the for all π‘ > 0
Pr[π β₯ π‘] β€πΈ[π]
π‘
Proof. πΈ π = π₯ππ(π = π₯)π₯πβ
β₯ π₯ππ(π = π₯)π₯β₯π
β₯ πππ(π = π₯)π₯β₯π
πΈ π β₯ πππ π = π₯
π₯β₯π
= πππ(π β₯ π)
πΈ[π] β₯ πππ(π β₯ π)
ππ(π β₯ π) β€πΈ[π]
π
βΉ ππ(π β₯ π‘) β€πΈ[π]
π‘
Chebyshev's Inequality
Pr(|π β πΈ π | β₯ π‘) β€πππ[π]
π‘2
Proof.
Pr π β πΈ π β₯ π‘ = Pr((|π β πΈ π )2| β₯ π‘2)
β€(π β πΈ π )2
π‘2
=πππ[π]
π‘2
Since (π β πΈ π )2 is a non-negative random variable, we can apply Markov'sinequality.
American Journal of Engineering Research (AJER) 2014
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Jensen's Inequality
If f is a convex function, then πΈ[π π ] β₯ π(πΈ π )| Proof. Suppose π has a Taylor series expansion. Expending π = πΈ π and usingthe Taylor series expansion with
a remainder term, yields for some π.
π π₯ = π π + πβ² π π₯ β π +πβ²β² π (π₯ β π)2
2
β₯ π π + πβ² π π₯ β π (β΅ πβ²β²(π) β₯ 0 ππ¦ ππππ£ππ₯ππ‘π¦. )
π(π₯) β₯ π π + πβ² π π₯ β π Taking expectation on both sides
πΈ π π₯ β₯ π π + πβ² π πΈ π₯ β π = π(π)
πΈ[π π₯ ] β₯ π(πΈ π )
Martingale
π- field
Definition 9.A π- field(πΊ, π½) consists of a sample space and a collection ofsubsets π½ satisfying the following
conditions [7].
πππ½
πππ½ β πβ²ππ½
π1 , π2 , β¦ππ½ β π1 βͺ π2 βͺ β¦ππ½
Partition of π΄
Definition 10.π½is a partition of Ξ© if π½ β 2Ξ© and
(i) πΉ = Ξ©πΉππ½
(ii) πΉ β π β πΉππ½
(iii) πΉ β© πΊ = π β πΉ β πΊ π€βπππ πΉ, πΊππ½
Figure 1: Partition of Sample space
Here π½ is partition of Ξ© then corresponding algebra of partition πΈ(π½1) is
π, πΉ, πΊ, π», β¦ , πΉ βͺ πΊ, πΉ βͺ π», β¦ , πΉ βͺ πΊ βͺ π», β¦
Where algebra is smallest set containing π½ that is closed under finite union, intersectionand complementation.
Definition 11. If π½1 , π½2 are partitions of πΊ then π½1 β π½2 if π΄ππΈ(π½1)implies π΄ππΈ(π½2). Where πΈ(π½1) and
πΈ(π½2) are algebra of π½1and π½2 respectively.
Filtration
Definition 12. Given the π- field (πΊ, π½) with π½ β 2πΊ , a filter (sometimes alsocalled filtration) is a nested
sequence π½1 β π½2 β π½3 β¦ β π½π of subsets of 2πΊsuchthat [7] π½0 = π, Ξ©
π½π = 2Ξ©
πππ 0 β€ π β€ π, Ξ©, π½π ππ π π β πππππ. Definition 13. Let (πΊ, π½) be any π β πππππ, and Y any random variable that takeson distinct values on the
elementary events in π½. Then πΈ π π½ = πΈ π π .
Figure 2: Filtration
Notice that the conditional expectation πΈ[π|π]does not really depend on theprecise value of Y on the specific
elementary event. In fact, Y is merely an indicatorof the elementary events inπ½. Conversely, we can write
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πΈ π π = πΈ[π|π(π)] where π(π) is the π β πππππ generated by the events of the typeπ = π¦, i.e., thesmallest
Ο- field over which Y is measurable.
Definition 14. A random variable X over Ο- field (πΊ, π½) is well defined ifπ π1 = π(π2 )for all π΄ππ½ where
π1 , π2ππ΄ and π½ is partition.
It is also called π½ -measurable. A random variable X is π½ -measurable if its valueis constant over each block in
the partition generation π½. Since the partitions generating the Ο- fields in a filter are successively more refined,
it follows that if X,π½π -measurable, it is also π½π -measurable for all π β₯ π. i.e
If π½1 β π½2 and X is well defined over (πΊ, π½1) then X is well defined over(πΊ, π½2) Converse may not be true.
Suppose X is well defined over π½2. How to define a random variable over π½1so that it is well defined.
Let random variable
π π = π π π(π)πππ΄
π (π΄) π€βπππ πππ΄ππ½1 .
β π π = π π π(π|π΄)πππ΄
Proof. π ππ = ππ π π(ππ |π΄)π πππ΄
= π₯ππππ|π πππ΄
πππ|π πππ΄
π(ππ)is same for a partition. So random variable is constant for a particularpartition. Similarly we can proof for
other partitions. Here we assume that the Sample space has been partitioned, the points π1 , π2 , π3with
probability π1 , π2 , π3and valuesπ₯1 , π₯2 , π₯3 respectively are in finer partition π½2
Theorem 4. Defining Y as conditional expectation of X given π½1 denoted byπΈ[π|π½1] = π then
πΈ π = πΈ[π] Proof. πΈ π = π π π(π)ππ Ξ©
= π π π(π)ππ Aπ΄ππ½1
= π π π(π)ππ Aπ΄ππ½1
π΄π π ππ π€πππ πππππππ
π π π(π΄) π΄ππ½1
= π(π΄) π π π(π|π΄)πππ΄π΄ππ½1
= π π΄ . π π π(π)πππ΄
π(π΄)π΄ππ½1
= π π π(π)πππ΄π΄ππ½1
= π π π(π)ππ Ξ©
= πΈ[π] From this result it is clear that for filter π½1 β π½2 β π½3 β¦ β π½π overΟ- field and corresponding random variables
are π0 , π1 , π2 , β¦ , ππthen
π = πΈ π0 = πΈ π1 = πΈ π2 = β― = πΈ[ππ ] Martingale
Definition 15.Let (πΊ, π½, ππ) be a probability space with a fiterπ½0 , π½1 ,π½2 , β¦.Suppose π0 , π1 , π2 , β¦ are random
variables such that for all π β₯ 0, Xi is π½π-measurable. The sequence π0 , π1 , π2 , β¦ , ππ is a martingale provided,
for all π β₯ 0,
πΈ ππ+1 π½π = ππ
Definition 16.The set of random variables ππ , π = 0,1,2β¦ is said to be amartingale [8] with respect to
sequence ππ , π = 0,1,2β¦ if ππ is a function ofπ0 , π1 , π2 , β¦ , ππE[|Zn|] <β,andπΈ[ππ+1 π0 ,π1 , π2 , β¦ , ππ =ππ
If we say ππ is a martingale [8] (without specifying ππ , π = 0,1,2β¦ whenit is a martingale with respect to
itself. i.e. ππ is a martingale if E[|Zn|] <β,andπΈ ππ+1 π0 ,π1 , π2 ,β¦ππ = ππ .
Note:-ππ is martingale with respect to ππ then it is a martingale.
Theorem 5.ππis martingale with respect to ππ then it is a martingale
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πΈ ππ+1 π0 ,π1 , π2 ,β¦ππ = ππ
Proof.we know the identity πΈ π π = πΈ[πΈ[π|π, π]|π] ππ is martingale with respect to ππ then
πΈ ππ+1 π0 β¦ππ = πΈ[πΈ ππ+1 π0 β¦ππ ,π0 β¦ππ |π0 β¦ππ ]
= πΈ[πΈ ππ+1 π0 β¦ππ |π0 β¦ππ] = πΈ ππ π0 β¦ππ
= ππ
Theorem 6. πΈ ππ = πΈ π0 Proof. π0 β¦ππare functions of π0 β¦ππ
πΈ ππ+1 π0 ,π1 , π2 ,β¦ππ = ππ Taking expectation on both the sides.
πΈ ππ+1 = πΈ ππ πΈ ππ = πΈ π0
πΈ π0 is called mean of the martingale.
Example: - Fair bet game:suppose ππ be the outcome of ππ‘β game. ππ is the fortune of gambler after
ππ‘βgame. For given outcome of first n games the gambler's expected fortune after(n + 1)st game is equal to his
fortune before the game. The gambler's expectedwinning on each game is equal to zero. Here π0 , π1 , π2 , β¦ , ππ
is martingale.
Azuma's Inequality
Let π0 , π1 , π2 , β¦ , ππ be a martingale sequence such that for all π
|ππ β ππβ1| β€ π
Pr[|ππ‘ β π0| β₯ π] β€ 2exp(βπ2
2π‘π2)
Where c may depend on k for all tβ₯0 and for any l>0
Proof. Let π½1 β π½2 β π½3 β¦ β π½π be filter sequence and π0 , π1 , π2 , β¦ , ππ be martingalesequence where
π0 = πΈ[ππ] for ππ0,1,2, β¦ , π. Also
πΈ ππ π½π = ππ πππ π < π
If X is π½π measurable (constant for particular partition) and Y is anotherrandom variable then we observe that
πΈ[ππ|π½π] = ππΈ[π|π½π] Since π0 , π1 , π2 , β¦ , ππ is martingale and let ππ = ππ β ππβ1 for π = 1,2, β¦ , π‘.
πΈ ππ π0 , π1 , π2 , β¦ , ππβ1 = πΈ[ππ β ππβ1|π0 , π1 , β¦ , ππβ1] = πΈ ππ π0 , π1 , β¦ , ππβ1 β ππ = 0
Now consider, ππ = βππ
1βππππ
2+ ππ
1+ππππ
2
Using the convexity of the function ππΌππ
ππΌππ β€1 β
ππ
ππ
2πβπΌππ +
1 +ππ
ππ
2ππΌππ
=ππΌππ + πβπΌππ
2+
ππ
2ππ
(ππΌππ + πβπΌππ)
πΈ ππΌππ π0 , π1 , β¦ , ππβ1 β€ πΈ ππΌππ + πβπΌππ
2+
ππ
2ππ
(ππΌππ + πβπΌππ) π0 , π1 , β¦ , ππβ1
=ππΌππ + πβπΌππ
2
β€ π(πΌππ)
2
2
Using Taylor series
β πΈ ππΌππ π0 , π1 , β¦ , ππβ1 β€ π(πΌππ)
2
2
πΈ ππΌ ππ‘βπ0 = πΈ[ ππΌππ ]
π‘
π=1
= πΈ[ ππΌππ ]
π‘β1
π=1
πΈ ππΌππ π0 , π1 , β¦ , ππβ1
β€ πΈ[ ππΌππ ]
π‘β1
π=1
π(πΌππ)2
2
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β€ ππΌ2 ππ2/2π‘
π=1
β€ ππΌ2π‘π2/2 As ππ = π β π Pr ππ‘ β π0 β₯ π = Pr[ππΌ ππ‘βπ0 β₯ ππΌπ ]
β€πΈ ππΌ ππ‘βπ0
ππΌπ
β€ ππΌ2π‘π2/2βπΌπ‘
= πβπ2/(2π‘π2) π€βπππ πΌ =π
π‘π2β Pr[|ππ‘ β π0| β₯ π] β€ 2exp(β
π2
2π‘π2)
III. A SURVEY OF RANDOM GRAPH MODEL FOR WEB MODELING Random graphs
Random graph is a graph which is generated by some random process. Two basic models for
generating random graphs are Gn,Nand Gn,p models. As we know there are many NP- hard computational
problems defined on graphs: Hamiltonian cycle, independent set, Vertex cover, and so forth. One question worth
asking is whether these problems are hard for most inputs or just for a relatively small fraction of all graphs.
Random graph models provide a probabilistic setting for studying such questions.
Gn,NModel of Random Graph
In Gn,Nmodel of random graph n is the count of vertices and N is number of edges in the graph. We consider all
undirected graphs on n vertices with exactly N edges. Since, there are n vertices so possible number of edges
are(π2
). Out of(π2
) many possible edges graph has N edges, this can be selected in(π
(π2
)) manyways. So total
possible graphs are(π
(π2
)).
Generating graph using Gn,Nmodel
One way to generate a graph from graphs in Gn,N is to start with no edges. Choose one of the (π2
) possible edges
uniformly at random and add it to the edges in the graph. Now choose one of the remaining (π2
) β 1 possible
edges independently and uniformly at random and add to the graph. Similarly, continue choosing one of the
remaining unchosen edges independently at random until there are N edges.
Gn,pModel of Random Graph
In Gn,p model of random graph n is the count of vertices and p is the probability of selection of an edge. We
consider all undirected graphs on n distinct vertices v1, v2,β¦,vn. A graph with a given set of m edges has
probability pm(1 β p)(π2
)βπ . Generating graph using Gn,p model
One way to generate a random graph in Gn,p is to consider each of the (π2
) possible edges in some order and then
independently add each edge to the graph with probability p. i.e. Corresponding to each edge out of (π2
) throw a
coin biased with outcome head with probability p. If outcome is head add the edge to the graph, don't add
otherwise. The expected number of edges in the graph is therefore (π2
)p, and each vertex has expected degree (n-
1) p.
The Gn,N and Gn,p models are related: when p = N/(π2
). The Gn,p and Gn,N are static models. In these models,
graph G = (V,E) is not changing with time. Sometimes we need a graph model which is time evolving Gt = (Vt,
Et).
Evolving Model
Evolving Model of graph covers evolving nature of it. There are two characteristic functions of evolving graph model [5] first gives the number of vertex added at time t+1, whereas second gives set of edges added at time
t+1, these are called characteristic function of evolving graph model.
Characteristic functions of evolving graph model fv (Vt,t) and fe (ft,Gt, t). fv (Vt, t) gives number of vertex added
at time t + 1 and fe (ft,Gt, t) gives the set of edges added at time t + 1.
Vt+1 = Vt + fv(Vt,t)
Et+1 = Etβͺfe(ft,Gt,t)
The evolving graph model is completely characterized by fv and fe.
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Linear growth copying model
The Linear growth copying model is evolving model. In each time interval one vertex is added. This is linear
growth, because at timestep t, a single vertex arrives and may link to any of the first t-1 vertices. This model
describes web graph easily. Web graph is time evolving in nature where at timestep one web page may arrive
and is connected to previous one, or a page may be deleted.
Generating graph using linear growth copying model [5]
In each time interval one vertex is added. Edges are added in following way: for a new vertex u out-degree d is
constant. The ith out-degree is decided as- With probability Ξ± destination is chosen uniformly from Vt and with
remaining probability destination is chosen as the ith outlink of prototype vertex p. Where Ξ± is copy factor Ξ±β(0, 1) and the out-degree is d β₯ 1 and fv (Vt,t) = 1. The prototype vertex is chosen once in advance.If Nt,i represents
number of vertices having in-degree i at time t, for simplicity d = 1 and i = 0.
In-degree Nt,0 distribution
Theorem 6.E[Nt,0Nt-k,0] = Nt-k,0Sk,0 + ππ, 0πβ1π=0 .
Proof
E[Nt,0Nt-k,0] = 1 + (1-πΌ
π‘β1)+(1-
πΌ
π‘β1)(1-
πΌ
π‘β2)+β¦
+(1-πΌ
π‘β1)(1-
πΌ
π‘β2)β¦.(1-
πΌ
π‘βπ)Nt-k,0
S0,0=1
Sk,0=Sk-1,0(1-
πΌ
π‘βπ)
E[Nt,0Nt-k,0] =Nt-k,0Sk,0+ π π, 0πβ1π=0
Sk,0= Sk-1,0(1-πΌ
π‘βπ)
E[Nt,0Nt-k,0]=Nt-k,0Sk,0+ ππ, 0πβ1π=0
t-k=1
E[Nt,0N1,0]=N1,0St-1,0+ ππ, 0π‘β2π=0
E[Nt,0]= ππ, 0 π‘β1π=0 (As N1,0=1)
Theorem 7. E [Nt,0Nt-k,0] - E[Nt,0Nt-k+1),0]β€1
Proof.
E[Nt,0Nt-k,0] - E[Nt,0Nt-(k+1),0]= Nt-k,0Sk,0 + ππ , 0π β1π =0 βNt-(k-1),0Sk+1,0- ππ , 0π
π =0 Case 1:
(Nt-k,0=Nt-(k+1),0+1)
=Nt-(k+1),0Sk,0 + Sk,0 β Nt-(k+1),0Sk,0 (1 -πΌ
π‘ β(π +1))-Sk,0
=Sk,0-Sk,0(1-πΌππ‘ β π +1 ,0
π‘ β(π +1))β€1
(Sk,o(1-πΌππ‘ β(π +1,0)
π‘ β(π +1))β€1)
From equations it is probability which is always β€ 1 and Sk,0β€ 1.
Case 2.
(Nt-k=Nt-(k+1),0)
=Nt-(k+1),0Sk,0-Nt-(k+1),0Sk,0(1-πΌ
π‘ β(π +1))-Sk,0
=Sk,o(1-πΌππ‘ β π +1 ,0
π‘ β(π +1))β€1
(Sk,0(1-πΌππ‘ β π +1 ,0
π‘ β(π +1))β€ 1)
From equations ,it is probability which is always β€ 1 and as Sk,0β€1
E[Nt,0Nt-k,0] - E[Nt,0Nt-(k+1),0]β€ 1
This result is an improvement compared to the result given in the paper [5]
E[Nt,0Nt-k,0] - E[Nt,0Nt-(k+1),0]β€ 2
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Theorem 8.E[Nt;0] = π‘
1+πΌ .
Proof. From equation
Sk,0 = Sk-1,0 (1-πΌ
π‘ βπ)
Sk,0 β Sk-1,0 = -Sk-1,0 πΌ
π‘ βπ
(Sk,0-Sk-1,0)(t-k)= - βSk-1,0
(t -k) (Sk-1,0 β Sk,0) = βSk-1,0
(t - (k - 1)) (Sk-1,0 β Sk,0) = (1 + Ξ±) Sk-1,0
(1 + Ξ±) Sk-1,0 = (t -k) Sk,0 - (t - (k + 1)) Sk+1,0
(1 + Ξ±) S0,0 = tS0,0 - (t - 1) S1,0
(1 + Ξ±) S1,0 = (t - 1) S1,0 - (t -2) S2,0
(1 + Ξ±) S2,0 = (t - 2) S2,0 - (t - 3) S3,0
:
(1 + Ξ±) St-2,0 = 2St-2,0 β St-1,0 (1 + Ξ±) St-1,0 = St-1,0 - (t - t) St,0
(1+Ξ±) ππ , 0π‘ β1π =0 =S0,0
βE[Nt,0]=π‘
1+πΌ
This result is an improvement compared to the result given in the paper [5]
π‘ +πΌ
1+πΌβΞ±^2lnt β€ E[Nt,0] β€
π‘ +πΌ
1+πΌ
Theorem 9. Pr[|ππ‘ ,0
β πΈ [ππ‘ ,0]| β₯ π ] β€ 2exp(βπ 2
2π‘)
Proof. Azuma's Inequality
Let π 0,π 1,π 2, β¦be a martingale sequence such that for all π
|ππ
β π π β1| β€ π
whereπ may depend onπ .
Then for all tβ₯0 and any π >0
Pr[|ππ‘
β π 0| β₯ π ] β€ exp(βπ 2
2π‘ π 2)
From theorem 7
|πΈ ππ‘ ,0 ππ‘ βπ ,0 β πΈ ππ‘ ,0 ππ‘ β π +1 ,0 | β€ 1
β Pr[|ππ‘ ,0
β πΈ [ππ‘ ,0]| β₯ π ] β€ exp(βπ 2
2π‘)
This result is an improvement compared to the result given in the paper[5]
Pr[|ππ‘ ,0
β πΈ [ππ‘ ,0]| β₯ π ] β€ exp(βπ 2
4π‘)
πΈ ππ‘ ,0 = π π ,0
π‘ β1
π =0
=π‘
1 + πΌ
π π‘ = limπ‘ ββ
πΈ [ππ‘ ,0]
π‘=
1
1+πΌ.
IV. DISTRIBUTIONS In-degree π΅π ,π distribution
ππ‘ ,π represents number of vertices having in- degree i at time tfor simplicity d = 1 π»ππππ π΅π ,π ππ
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ππ‘ β1,π β 1 π€π
πΌππ‘ β1,π
π‘ β 1+
(1 β πΌ )ππ‘ β1,π
π‘ β 1
ππ‘ β1,π + 1 π€π πΌππ‘ β1,π β1
π‘ β 1+
(1 β πΌ )(π β 1)ππ‘ β1,π β1
π‘ β 1ππ‘ β1,π ππ‘ βπππ€ππ π
defineπ π‘ βπ = πΈ [ππ‘ ,π |ππ‘ βπ ,π
πβ,π β1]where πβ,π β1 = π0,π β1 ,π1,π β1,π2,π β1, π3,π β1,β¦ , ππ‘ β1,π β1for 0 β€ π‘ β
π β€ π‘
The sequence π 0, π 1, π 2, β¦π π‘ forms Doob's martingale [3]
πΈ ππ‘ ,π |ππ‘ βπ ,π
πβ,π β1 = ππ‘ βπ ,π π π ,π + π π ,π
π β1
π =0
πΉ π +1,π β1
where
π 0,π = 1
π π ,π = π π β1,π (1 βπΌ
π‘ β πβ
1 β πΌ π
π‘ β π)
πΉ π ,π β1 =πΌ + 1 β πΌ (π β 1)
π‘ β π
Proof.
πΈ [ππ‘ ,0|ππ‘ β1,π
= π½ , ππ‘ β1,π β1 = πΎ ]
= π½ β 1 πΌπ½
π‘ β 1+
1 β πΌ ππ½
π‘ β 1+ (π½ + 1)
πΌπΎ + 1 β πΌ (π β 1)πΎ
π‘ β 1
+π½ β πΌπ½
π‘ β 1+
1 β πΌ ππ½
π‘ β 1 β
πΌπΎ + 1 β πΌ (π β 1)πΎ
π‘ β 1
= 1 βπΌ
π‘ β 1β
1 β πΌ π
π‘ β 1 π½ +
πΌ + 1 β πΌ (π β 1)
π‘ β 1πΎ
β πΈ ππ‘ ,π |ππ‘ β1,π
ππ‘ β1,π β1 = 1 βπΌ
π‘ β 1β
1 β πΌ π
π‘ β 1 |π
π‘ β1,π+
πΌ + 1 β πΌ (π β 1)
π‘ β 1ππ‘ β1,π β1
πΈ ππ‘ ,π |ππ‘ β2,π
,ππ‘ β2,π β1, ππ‘ β1,π β1 = πΈ πΈ ππ‘ ,π |ππ‘ β1,π
ππ‘ β1,π β1 |ππ‘ β2,π
, ππ‘ β2,π β1,ππ‘ β1,π β1
= 1 βπΌ
π‘ β 1β
1 β πΌ π
π‘ β 1 πΈ ππ‘ β1,π |π
π‘ β2,π,ππ‘ β2,π β1, ππ‘ β1,π β1
+πΌ + 1 β πΌ (π β 1)
π‘ β 1πΈ ππ‘ β1,π β1|π
π‘ β2,π, ππ‘ β2,π β1 ,ππ‘ β1,π β1
πΈ ππ‘ ,π |ππ‘ β2,π
,ππ‘ β2,π β1 = 1 βπΌ
π‘ β 1β
1 β πΌ π
π‘ β 1
1 βπΌ
π‘ β 2β
1 β πΌ π
π‘ β 2 ππ‘ β2,π
+πΌ + 1 β πΌ (π β 1)
π‘ β 2ππ‘ β2,π β1 +
πΌ + 1 β πΌ (π β 1)
π‘ β 1ππ‘ β1,π β1
β΅ πΈ ππ‘ β1,π β1|ππ‘ β2,π
, ππ‘ β2,π β1,ππ‘ β1,π β1 = ππ‘ β1,π β1 πππ
πΈ ππ‘ β1,π β1|ππ‘ β2,π
,ππ‘ β2,π β1, ππ‘ β1,π β1 = πΈ ππ‘ β1,π |ππ‘ β2,π
,ππ‘ β2,π β1
πΈ ππ‘ ,π |ππ‘ β3,π ,ππ‘ β3,π β1ππ‘ β2,π,ππ‘ β2,π β1, ππ‘ β1,π β1
= πΈ πΈ ππ‘ ,π |π
π‘ β2,π, ππ‘ β2,π β1,ππ‘ β1,π β1 |ππ‘ β3,π , ππ‘ β3,π β1ππ‘ β2,π
,
ππ‘ β2,π β1, ππ‘ β1,π β1
= 1 βπΌ
π‘ β 1β
1 β πΌ π
π‘ β 1
1 βπΌ
π‘ β 2β
1 β πΌ π
π‘ β 2 πΈ ππ‘ β2,π |ππ‘ β3,π , ππ‘ β3,π β1ππ‘ β2,π
, ππ‘ β2,π β1,ππ‘ β1,π β1
+πΌ + 1 β πΌ (π β 1)
π‘ β 2πΈ
ππ‘ β2,π β1|ππ‘ β3,π ,ππ‘ β3,π β1ππ‘ β2,π
, ππ‘ β2,π β1, ππ‘ β1,π β1
+πΌ + 1 β πΌ (π β 1)
π‘ β 1πΈ
ππ‘ β1,π β1|ππ‘ β3,π ,ππ‘ β3,π β1ππ‘ β2,π,
ππ‘ β2,π β1, ππ‘ β1,π β1
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β πΈ ππ‘ ,π |ππ‘ βπ ,π
πβ,π β1 = ππ‘ βπ ,π π π ,π + π π ,π
π β1
π =0
πΉ π +1,π β1
where
πΊ π ,π = π
π π ,π = π π β1,π (1 βπΌ
π‘ β πβ
1 β πΌ π
π‘ β π)
πΉ π ,π β1 =πΌ + 1 β πΌ (π β 1)
π‘ β π πππ π β₯ 1
V. SURVEY INFERENCE The synthesis
|π¬ π΅π ,π |π΅π βπ ,π
π΅π βπ ,π βπ = π¬ π΅π ,π |π΅π β π +π ,π
π΅π β π +π ,π βπ | β€ π
Proof.
πΈ ππ‘ ,π |ππ‘ βπ ,π
ππ‘ βπ ,π β1 = πΈ ππ‘ ,π |ππ‘ β π +1 ,π
ππ‘ β π +1 ,π β1 |
= ππ‘ βπ ,π π π ,π + π π ,π
π β1
π =0
πΉ π +1,π β1 β ππ‘ β(π +1),π π π +1,π + π π ,π
π
π =0
πΉ π +1,π β1
= ππ‘ βπ ,π π π ,π β ππ‘ β(π +1),π π π +1,π β π π ,π πΉ π +1,π β1
Case 1:
(ππ‘ βπ ,π
= ππ‘ β(π +1),π β 1)
= |ππ‘ β π +1 ,π π π ,π β π π ,π β ππ‘ β(π +1),π π π ,π (1 βπΌ
π‘ β (π + 1)β
1 β πΌ π
π‘ β (π + 1))
βπ π ,π πΌ + 1 β πΌ π β 1
π‘ β π + 1 ππ‘ β π +1 ,π β1|
= | β ππ ,π + ππ ,π(πΌππ‘β π+1 ,π
π‘ β π + 1 +
1 β πΌ ππ‘β π+1 ,π
π‘ β π + 1 )
β πΌ + 1 β πΌ π β 1
π‘ β π + 1 ππ‘β π+1 ,πβ1|
β€ 2
(β΅ πΌ + 1 β πΌ π β 1
π‘ β π + 1 ππ‘β π+1 ,πβ1 β€ 1)
(β΅ πΌππ‘β π+1 ,π
π‘ β π + 1 +
1 β πΌ ππ‘β π+1 ,π
π‘ β π + 1 β€ 1)
Bound for Case 1 is same as that of paper [3].
Case 2: (ππ‘βπ ,π = ππ‘β(π+1),π + 1)
= |ππ‘β π+1 ,πππ ,π + ππ ,π
βππ‘β(π+1),πππ ,π(1 βπΌ
π‘ β (π + 1)β
1 β πΌ π
π‘ β (π + 1))
βππ ,π πΌ + 1 β πΌ π β 1
π‘ β π + 1 ππ‘β π+1 ,πβ1|
= |ππ ,π + ππ ,π(πΌππ‘β π+1 ,π
π‘ β π + 1 +
1 β πΌ ππ‘β π+1 ,π
π‘ β π + 1 )
β(1 β πΌ+ 1βπΌ πβ1
π‘β π+1 ππ‘β π+1 ,πβ1) β€ 2
Bound for Case 2 is same as that of paper [3]
Case 3:
(β΅ ππ‘βπ ,π = ππ‘β(π+1),π)
= ππ‘β π+1 ,πππ ,π β ππ‘β π+1 ,πππ ,π(1 βπΌ
π‘ β (π + 1)β
1 β πΌ π
π‘ β (π + 1))
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βππ ,π πΌ + 1 β πΌ π β 1
π‘ β π + 1 ππ‘β π+1 ,πβ1
= ππ ,π(πΌππ‘β π+1 ,π
π‘ β π + 1 +
1 β πΌ ππ‘β π+1 ,π
π‘ β π + 1 )
βππ ,π πΌ + 1 β πΌ π β 1
π‘ β π + 1 ππ‘β π+1 ,πβ1
β€ 1 Bound for Case 3 obtained here is an improvement compared to the result givenin the paper [5]
Azuma's Inequality:
Let π0, π1, π2,β¦be a martingale sequence such that for all π
|ππ β ππβ1| β€ π
whereπmay depend onπ.
Then for all tβ₯0 and any π>0
β΅ |πΈ ππ‘,π|ππ‘βπ ,πππ‘βπ ,πβ1 = πΈ ππ‘,π|ππ‘β π+1 ,πππ‘β π+1 ,πβ1 | β€ 2
β Pr|ππ‘,π β πΈ[ππ‘,π| β₯ π] β€ exp(βπ2
4π‘)
VI. CONCLUSION This report studies random graph models using probabilistic techniques.Concentration is made over the
paper Stochastic Model for Web graph by RaviKumar et.al. [5]. during the proof of the results given in the
paper some moreprecise results have been found. Example- for i = 0 report proves |E[Nt,0|Nt-k,0]- E[Nt,0|Nt-
(k+1),0]|β€ 1whereas in the paper it is |E[Nt,0|Nt-k,0]- E[Nt,0|Nt-(k+1),0]|β€ 2. It is also proved that the E[Nt,0] =
t
1+Ξ±. For
general value of i, report proves that|E[Nt,i|Nt-k,iNt-k,i-1] - |E[Nt,i|Nt-(k-1),iNt-(k-1),i-1]| β€ 1for Nt-k,I = Nt-(k+1),iwhereas
result of the paper is |E[Nt,i|Nt-k,iNt-k,i-1] - |E[Nt,i|Nt-(k-1),iNt-(k-1),i-1]| β€ 2. In future one can find improvement in
|E[Nt,i|Nt-k,iNt-k,i-1] - |E[Nt,i|Nt-(k-1),iNt-(k-1),i-1]| for general value of Nt-k,i.
VII. APPENDIX
REFERENCES [1] W. Aiello, F. Chung, and L. Lu. A random graph model for massive graphs. 2000.
[2] B. Bollobas. Random graphs. 1985.
[3] J.L.Doob. What is a martingale. The American Mathematical Monthly Vol 78, No.5 (May 1971), pp.
451-46, 1971.
[4] J. Kleinberg, S. R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins. The web as a graph:
Measurements, models and methods. 1999.
[5] Ravi Kumar, PrabhakarRaghavan, Sridhar Rajagopalan, D. Sivakumar, Andrew Tomkins, and Eli Upfal. Stochastic models for the web graph. 2000.
[6] Michael Mitzenmacher and Eli Upfal. Probability and computing.
[7] Rajeev Motawani and PrabhakarRaghavan. Randomized algorithm. 2009.
[8] Sheldom M. Ross. Probability models for computer science. 2006.