This chapter deals with a basic preliminary constructs and tools needed to set about the survey of web outwardness procedure. It is obvious that the utility of facsimile machine depends on the figure of machines because users have more channels to reach. There are some goods and services that have this belongings, that is, consumer ‘s public-service corporation will be affected by the figure of consumers who consume the same goods and services. In economic science, this positive influence is called web outwardness. The goods and services characterized by such belongings are called web goods. In this chapter, definition, type and beginning of web outwardness are reviewed in Section 1.1. Some tools in this thesis are briefed in Section 1.2.

## Network Externality Concepts

## 1.1.1 Definition of Network Externality

Network outwardness is composed of two economic sciences footings “ web ” and “ outwardness ” . First, the term “ Network ” refers to a group of interacting consumers of similar goods and services. Second, the term “ Externality ” is a well-known term in economic sciences ( Bishop, 2009 ) which means an extra benefit or cost created by unrelated 3rd parties. For illustration, pollutant emitted by a mill creates negative outwardness to nearby occupants for their wellness whereas seting tree creates positive outwardness. Furthermore, some activities create both positive and negative such as singing a vocal, slow drive. Therefore, the term “ Network Externality ” shortly means an extra benefit or cost created by web.

The interaction inside the consumer web creates outwardness to the members of web. Therefore, the larger size of rank in the web generates the greater outwardness, that is, the magnitude of web outwardness additions when the size of web additions. Alternatively, there is another term used, “ web consequence ” , interchangeably. Liebowitz and Margolis ( 1995 ) discourse their differences. In this work, the “ Network Externality ” term is used. In the undermentioned, we review some related footings to web outwardness:

1.1.1.1 Metcalfe ‘s Law

The value of web is relative to the square of web size because a web size has possible alone connexions. See Metacafe ( 1995 ) .

1.1.1.2 Bandwagon Effect

Peoples do something because others are making it, irrespective of their ain belief. This is referred as the “ bandwagon consequence ” . For illustration, one time a peculiar good or service becomes popular, more people tend to purchase it, excessively. See Colman ( 2009 ) .

1.1.1.3 Snob Effect

The value of point additions as the handiness decreases. This is called “ snob consequence ” . For illustrations, art plants, rare casts and coins. This construct relates to the negative web outwardness. See Leibenstein ( 1950 ) .

1.1.1.4 Veblen Effect

Thorstein Veblen ( 1899 ) argued that affluent persons frequently consume extremely conspicuous goods in order to publicize their wealth, thereby accomplishing greater societal position.

1.1.1.5 Cluster Effect

The fold of consumers and manufacturers of a peculiar good or service induces other consumers and manufacturers to relocate at that place. This is known as the “ bunch consequence ” .

## 1.1.2 Type of Network Externality

1.1.2.1 Direct versus Indirect

The term “ Direct ” refers to the web outwardness through the utility of goods such as facsimile machine, e-mail, telephone, etc. While the term “ Indirect ” refers to the web outwardness through its complementary goods and related goods which can in turn the value of the original goods. For case, when figure of iPhone user addition, developers have motive to make more applications which is increase the value of iPhone. See more Katz and Shapiro ( 1994 ) .

1.1.2.2 Local Network Externality The term “ Local ” refers to the consumer ‘s public-service corporation is affected by the size of neighbour instead than full web. For illustration, a determination of coder to take programming package is affected by what package that his neighbour used the most. The use of the whole market has less consequence than the use of his neighbour ( Banerji and Dutta ( 2009 ) ) .

## 1.1.3. Beginning of Network Externality

1.1.3.1 Exchange

Facebook without friend is about useless. The public-service corporation from Facebook depends on the figure of your friends because we have more channels to interact. For some goods, the interactions inside web create extra benefit to the member of web ( Gallaugher ( 2012 ) ) .

1.1.3.2 Knowledge Base

A treatment forum of a peculiar good gives much utile information about the good. For a big web, the experient users normally help new users through an organized treatment forum. Consequently, a new consumer is induced to purchase a good that has a larger web.

1.1.3.3 Switch overing Cost

Imagine you were expert in the Windows OS, will you purchase a MacBook Pro? The reply is “ No ” , because you do n’t desire to pass your clip to larn how to utilize the Mac OS. However, you might purchase it, if you know that the Windows OS is deceasing. Consequently, at the first clip of acquisition, you will take the operating system ( OS ) that has a larger web because it will non decease shortly, ( Shapiro and Varian ( 1999 ) ) .

1.1.3.4 Complementary and Related Goods

Complementary and related goods add more benefit to the good. A 1000000s of applications, vocals and Podcasts in iTunes enhance the value of taking iPhone/iPad/iPod over a rival like the Microsoft Zune. And once more, the larger web normally offers a larger add-on market ( Gallaugher ( 2012 ) ) .

## Mathematical Preliminaries

In this thesis, we develop a dynamic theoretical account for the web goods. We employ mathematical consequences to develop the theoretical algorithms ( theorems/corollaries ) . We utilize the statistical methods to gauge and to prove the parametric quantities to formalize the theoretical mold. Finally, in absence of informations set, we use programming and simulation to formalize the theoretical account and pull a few illations.

## 1.2.1 Mathematical Tools

1.2.1.1 Being of solution of algebraic equations

Definition 1.2.1 Let be such that. Then are said to be coupled lower and upper quasisolutions of if or.

The undermentioned theorem provides the being of solution of algebraic equations.

Theorem 1.6.1 in Ladde et Al ( 1985 )

Assume that and treat a assorted quasimonotone belongings. Suppose further that are coupled lower and upper quasisolutions of and

whenever and. Then there exist drone sequences such that as and are coupled minimum and maximum solutions of such that for any solutions, .

1.2.1.2 Testing stableness of solution procedure

The undermentioned consequences provides a mathematical tool to set up the stableness of the equilibrium provinces ( solutions of algebraic equations associated with the rate maps in dynamic theoretical accounts ) of systems.

Theorem 1.2.2 in Ladde and Sambandham ( 1985 )

Let be a sequence of m-dimensional random vectors such that

Let be the solution procedure of

## ,

where is a sequence of Borel mensurable maps on fulfilling the undermentioned relation

for

Then

for

provided

Corollary 1.2.3 If, and fulfill an inequality

Then

and moreover

## 1.2.2 Statistical Tools

1.2.2.1 Newton-Raphson Method ( Seber and Wild ( 1989 ) )

The Gauss-Newton method for gauging nonlinear parametric quantities can be considered as a particular instance of the more general Newton-Raphson method. The Newton-Raphson method uses a local quadratic estimate of an nonsubjective map.

1.2.2.2 Homogeneity trial ( Endrenyi and Kwong ( 1981 ) )

First, we i¬?t the theoretical account, and so order the remainders as. Second, we compute the ratio of the last squared remainders to the i¬?rst squared remainders. The suggestion for is. Under these considerations, the homogeneousness trial is as follow:

V

1.2.2.3 Normality trial ( Shapiro and Wilk ( 2008 ) )

The random sample, , is drawn from usually distributed population The random sample, , is non drawn from usually distributed population.

We compute where is order statistic for ; is the sample mean. , where

are the expected values of the order statistics of independent and identically distributed random sample from the criterion normal distribution, and is covariance matrix of the order statistics.

1.2.3 Programming and Simulation ( Axelord ( 1997 ) )

To obtain the market equilibrium province of the developed dynamic theoretical account, we utilize the agent-based mold simulation ( ABMS ) . Agent-based mold simulation is a late introduced simulation attack to analyse enter-agent interactions. There are at least three advantages of the ABMS. First, its premises underlying the theoretical account are simple. Simplicity is besides helpful for readers to understand and for research worker to widen the theoretical account. Second, the complex adaptative procedure belongs to the work of simulation. Therefore, complex tools are n’t needed here. Third, because of ABMS, all web properties are traceable from initial province to steady province. The ABMS is more executable. It is based on certain premises and adaptable properties of agents.

## GENERALIZED NETWORK EXTERNALITY FUNCTION

In this chapter, we focus on the development of mathematical mold of web outwardness processes. The debut of the generalised web outwardness map provides a incorporate beginning of a tool for developing and analysing the planning, policy and public presentation of the web outwardness procedure and web goods in a systematic manner. This leads to carry through all bing web outwardness premises as particular instances. We study its belongingss and applications. This survey provides quantitative descriptions, parametric representations of properties and sensitiveness analysis of web outwardness processes. In peculiar, parametric fluctuations characterize planning, policy and public presentation for web goods.

## Introduction

In the survey of web industries and scientific disciplines, the web outwardness map plays a really important function. This map coupled with supply and demand maps is successfully utilized in the survey of economic sciences of web industries ( Shy, 2001 ) . We remark that the construct of web outwardness was introduced by Bell ‘s employee, N. Lytkins ( 1917 ) . Historically, the web outwardness map is motivated by its utility in economic sciences. Furthermore, the development of the research in this country is centered on the augmentations of qualitative belongingss of the web outwardness map ( Economides ( 1996 ) ; Church and Gandal ( 1992 ) ; Economides and Viard, ( 2003 ) ; Gottinger ( 2003 ) ; Bayer and Chan ( 2007 ) ; Ben-Zion and Tavor ( 2006 ) ; von Seggern, ( 2007 ) . A farther relevant historical developments and consequences are summarized in the followers.

The web outwardness map ( Katz and Shapiro, 1986 ) is the map that describes the relationship between web value and its corresponding size. Let be market portion of a web good, be the web outwardness map and be the web outwardness value. From the definition of web outwardness map ( Economides, 1996 ) , we note that the web outwardness value increases when the size of market portion additions, that is, the first derived function of is positive, and therefore

## .

( 2.1.1 )

The early bing research work in the country of web outwardness map is centered on the one-dimensionality premise on outwardness map:

## .

( 2.1.2 )

For more inside informations see Church and Gandal ( 1992 ) , Economides and Viard ( 2003 ) , Gottinger ( 2003 ) , Bayer and Chan ( 2007 ) .

The thought of decreasing return was incorporated into the web outwardness map by Ben-Zion and Tavor ( 2006 ) . Furthermore, the first derived function of map attacks to zero when the market portion is really big, that is,

( 2.1.3 )

Recently, Lin ( 2008 ) has considered the following flexible functional signifier of web outwardness map:

## .

( 2.1.4 )

The look for in ( 1.4 ) was based on the undermentioned qualitative belongingss:

Negatively fringy map, ;

is concave map when ;

is bulging map when ;

is additive map when.

Furthermore, Hans-Werner Gottinger ( 2003 ) has classified the web outwardness map into classs, viz. , additive, logarithmic and exponential functional signifiers. See Table 2. Historical sum-up of web outwardness map under assorted premises. The additive map postulates that, as the web grows, the fringy value approaches to a changeless. The logarithmic map postulates that, as a web grows, the fringy value diminishes. In this preparation, web outwardness at the bound must be either negative or nothing. The exponential map postulates that, as a web grows, the fringy value additions. This type of web outwardness map is referred to as ‘Metcalfe ‘s Law, Robert Metcalfe ( 1995 ) .

Table 2. Historical sum-up of web outwardness map under assorted premises

Premise

Sign of

Graphic Shape

I

Nothing

Line

Two

Negative

Concave

Three

Positive

Convex

In this chapter, we recognize the rapid growing in communicating, scientific discipline and engineering in the twenty-first century. The international activities are significantly increasing. The different types of consumers ( local/global degree ) are able to interact with each other easy and more often. We farther acknowledge the thoughts of Katz and Shapiro ( 1986 ) , Economides ( 1996 ) , Lin ( 2008 ) , von Seggern ( 2007 ) , Ben-Zion and Tavor ( 2006 ) , Gottinger ( 2003 ) and the historical premises sing the assorted signifiers of web outwardness maps. We observe that the group dynamic interactions ( Ladde et al, 2012 ) are traveling to play a important function in web goods in the twenty-first century. The thought of a group dynamic coupled with the web outwardness construct leads to a impression of web outwardness procedure. This farther strengthens our motive to set about a survey of the development of dynamic theoretical account of web outwardness procedure, its cardinal belongingss and significance. In Section 2.2, we present a rule of web outwardness procedure and develop a mathematical theoretical account. Using the mathematical theoretical account of web outwardness procedure, we study the belongingss of web outwardness map in Section 2.3. In Section 2.4, we briefly sketch the significance of the dynamic theoretical account of web outwardness procedure, in peculiar, parametric fluctuation analysis to qualify the assorted facets of web outwardness procedure, viz. , be aftering, policy and public presentation. The function, range and future waies of the research are outline in Section 2.5.

## Network Externality Process

In this subdivision, we officially introduce a few footings: consumer web, web goods and web outwardness procedure. We consider a consumer/user web as a group of interacting consumers/users of similar goods/services/information/knowledge/entities. The similar good under the treatment of consumer web is referred to as a web good. The consumer group interacting procedure of web goods is called a web outwardness procedure. Network outwardness processes act upon the values of web goods. The value of web goods is influenced by both consumer demand-supply maps every bit good as the web outwardness procedure. The influence of web outwardness procedure of the web goods is measured by the consumer/user web size/share. The value of a web good influenced by a web outwardness procedure is called web outwardness value. It is determined by the current market size/share. This procedure leads to the well-known definition of web outwardness map ( Katz and Shapiro, 1986 ) .

Example 2.2. Let us see a game console as a web good. The consumer web is a group of consumers who purchased a game console ( Xbox/PS3/Wii ) . Here, Xbox/PS3/Wii are web goods. The web outwardness procedure is the consumer group interactions such as sharing their games, accoutrements, cognition, experiences and sentiments about the game console. The web outwardness value is the value of a game console generated by this group ‘s interaction procedure.

Example 2.2. Let us see Microsoft Word. We know that when the figure of Microsoft Word users addition, the value of goods will increase. This is due to the fact that they have more channels to portion their files. In this instance, Microsoft Word is a web good.

Example 2.2. Let us see a rare good for illustration an old-timer. When the figure of proprietors additions, the value of goods must diminish. Therefore, the gustatory sensation of valuable characteristics of the goods diminishes. In this instance, the old-timer is a web good.

On the footing of the work ( Ben-Zion and Tavor, 2006 ) , we noted that as the market share/size increases the web outwardness value approaches to one of the two distinguishable critical degrees in a monotone mode. This leads to constructs of lower and upper bounds of the web outwardness procedure.

Definition 2.2. The Lower Limit of Network Externality Value is the greatest lower edge or the infimum of scope of web outwardness map, .

Definition 2.2. The Upper Limit of Network Externality Value is the least upper edge or the supremum of the scope of the web outwardness map, .

Remark 2.2. The Lower and upper bounds of web outwardness values are referred to as stationary/equilibrium provinces of web outwardness procedure.

Definition 2.2. The Excess Network Externality is the difference ( comparative alteration ) between the current province and the lower bound of the web outwardness procedure.

Definition 2.2. The Deficit Network Externality is the difference ( comparative alteration ) between the upper bound and the current province of the web outwardness procedure.

In the undermentioned, we provide illustrations to exemplify the above presented constructs.

Example 2.2. From Example 2.2. , we interpret the lower bound as the minimal value of goods at the minimal degree of market portion and the upper bound as the maximal value of goods at maximal degree of market portion. The extra web outwardness is a grade of the value when the market portion is non low. The shortage web outwardness is the loss of the value when the market portion is non high. For more illustrations, see Shy ( 2001 ) .

Example 2.2. From Example 2.2. , we interpret the lower and upper bounds as the threshold values of the affinity/taste of valuable characteristics. The extra web outwardness is degree of affinity/taste when market portion is non really big. The shortage web outwardness is loss of affinity/taste when market portion is non really low.

On the footing of the above development and based on historical premises on the web outwardness value, Katz and Shapiro ( 1986 ) , Economides ( 1996 ) , Ben-Zion and Tavor ( 2006 ) , Lin ( 2008 ) , Gottinger ( 2003 ) , we province a rule of web outwardness procedure. Furthermore, we develop a mathematical theoretical account for web outwardness procedure. This development provides the quantitative description about the historical premises sing the web outwardness map, the parametric representation of properties of web outwardness procedure and planning, policy and public presentation.

Principle of Network Externality Procedure: The rate of alteration of web outwardness of web goods is straight relative to the merchandise of the extra web outwardness per unit of current/used market portion and the shortage web outwardness per unit of fresh market portion.

Development dynamic theoretical account of web outwardness procedure: Let, and be the web outwardness, the lower and upper bounds of web outwardness procedure, severally, for web goods. Therefore, is the extra web outwardness, and is the shortage web outwardness of the web goods. Consequently, is the extra web outwardness per used market portion, and is shortage web outwardness per fresh market portion. By the rule of web outwardness procedure, the moral force of web outwardness procedure is described by

## .

( 2.2.1 )

where, and is a invariable of proportionality, and the mark of depends on the types of web goods ; , and is the initial value of the web outwardness map of the web goods at its initial market portion, . By work outing this differential equation, we have the closed signifier look for the web outwardness map:

## .

( 2.2.2 )

Remark 2.2. For a freshly introduced web goods, that is comparable to bing web goods, it is moderately realistic to hold an available market portion. For this intent, we can loosen up the sphere of definition limitation in ( 2.2.2 ) . Let and be Numberss between nothing and one, stand foring lower limit and maximal market portions of a given web good, severally. Under this modified sphere, the modified version of ( 2.2.1 ) is obtained by replacing and in ( 2.2.1 ) by and, severally. Hence,

## ,

( 2.2.3a )

and

, for.

( 2.2.3b )

The map in ( 2.2.3b ) is called generalised web outwardness map ( GNEF ) .

Remark 2.2. We can besides see the sphere of ( 2.2.3b ) in footings of market size. Let be the current market size and be figure of possible consumers in the market. The relationship between market size and market portion for the same individual web good can be expressed by, where is the market portion as defined before. The differential equation ( 2.2.3a ) is reduced to

## .

( 2.2.3c )

Remark 2.2. If the market portion has a natural growing that is described by the Verhulst-Pearl population dynamic theoretical account ( Ladde et al, 2012 ) ( its solution is a Sigmoid curve ) . From ( 2.2.3a ) , the web outwardness value will besides hold a natural growing, described by. In other words, the differential equation ( 2.2.3a ) is a comparative growing rate theoretical account ( Huxley, 1932 ; Robert Rosen, 1967 and Ladde et Al, 2012 ) with regard to the market portion and web outwardness value.

Remark 2.2. If we assume, , and, , so the GNEF fulfills the belongingss of a cumulative distribution map ( CDF ) .

Remark 2.2. We farther note that there are five and seven parametric quantities in ( 2.2.2 ) and ( 2.2.3b ) , severally. These parametric quantities play an of import function in analysing: planning, policy and public presentation facets of web goods.

## Properties of the Generalized Network Externality Function

In this subdivision, we present the qualitative belongingss of the GNEF. These belongingss shed a visible radiation on the historical premises ( Economide ( 1996 ) , Ben-Zion and Tavor ( 2006 ) ) that are made about the web outwardness map. In fact, the map determined by differential equation ( 2.2.3a ) possesses all the specified belongingss of web outwardness map in the literature ( Gottinger, ( 2003 ) , Lin ( 2008 ) ) in systematic and incorporate manner.

2.3.1 Admissible Market Share: For web goods, the sphere of GNEF ( 2.2.3b ) and ( 2.2.3c ) are and, severally.

2.3.2 Switch overing Cost: The scope of both GNEF ( 2.2.3b ) and ( 2.2.3c ) for the web outwardness goods is. The parametric quantity can be considered as the exchanging cost/the lower limit threshold for being of the web goods in the market. The construct of exchanging cost is defined by Thompson and Cats-Baril ( 2002 ) as “ the costs associated with exchanging provider ” . For illustration, in the instance of telephone as a web good, the exchanging cost includes the attempts needed to inform friends and relations about a new telephone figure. The operator exchanging cost is related to larning about how to utilize the interface of a new nomadic phone from different trade names. Furthermore, in the instance of electricity as a web good, the cost includes the lost clip due to the paperwork necessary when exchanging to a new electricity supplier. Types of exchanging costs include: issue fees, hunt costs, larning costs, cognitive attempts, emotional costs, equipment costs, installing and start-up costs, fiscal hazard, psychological hazard and societal hazard. In short, the parametric quantity “ ” can be considered as “ planning parametric quantity ” to come in the web goods.

2.3.3 Monotonicity: The debut of web outwardness rule leads to the two types of following constructs: ( I ) the positive fringy web outwardness and ( two ) the negative fringy web outwardness. From the sphere and scope of GNEF, we observe that the looks and in ( 2.2.3a ) and ( 2.2.3c ) are positive. Hence, if so. In this instance, is referred to as a positive fringy web outwardness map. The web outwardness value increases as the market share/size additions, that is, is an increasing map on. On the other manus, if, so. In this instance, is referred to as a negative fringy web outwardness map. Therefore, the web outwardness map decreases as the market share/size additions, that is, is a diminishing map on. For both types of fringy web outwardness procedure, is the monotone map with the greatest lower edge and the least upper edge. See Figure 2. Sketchs of the form of GNEF, illustrating of ( 2.2.3b ) for ( a ) and ( B ) ..

x3.1a.png

x3.2a.png

( a )

( B )

Figure 2. Sketchs of the form of GNEF, illustrating of ( 2.2.3b ) for ( a ) and ( B ) .

In fact, a in writing illustration for carry throughing all belongingss of map in ( 2.2.3b ) with the sphere between and, lower bound, upper bound, inflexion point ( point of decreasing return ) , and increasing/decreasing ( depending on the mark of ) .

Remark 2.3. Example 2.2. and Example 2.2. have positive and negative fringy web outwardness, severally.

Remark 2.3. From the above treatment, we note that the development of web outwardness map in Section 2.2 with possesses all belongingss that were outlined by Lin ( 2008 ) , in systematic and incorporate manner. Furthermore, the types of web goods can be straight verified by the mark of the first derived function. The first derived function of in ( 2.2.3b ) is

## .

( 2.3.1 )

For and, if so, and if so. In short, the mark of invariable of proportionality represents the types of web goods.

In order to exemplify the staying premises in Table 2.1, we need to present the undermentioned notation and deduce some looks. We define

## .

( 2.3.2 )

For and, ( 2.3.1 ) is reduced to

## .

( 2.3.3 )

Therefore, the invariable of proportionality relates to the velocity of alteration or incline at point when, see detail Figure 2. Sketchs of the form of GNEF, exemplifying the function of when, ..

x3.3a.png

Figure 2. Sketchs of the form of GNEF, exemplifying the function of when, .

2.3.4 Law of Decreasing Return: The jurisprudence of decreasing return is utilized in about all facets of economic sciences. This construct is besides used to analyze the web goods ( Ben-Zion and Travor, 2006 ) . In the literature, the web outwardness map is assumed to fulfill decreasing return premise together with. In order to warrant the cogency of this belongings, we need to present a twosome constructs.

Definition 2.3. The Lower-Left Terminal Point ( LLTP ) is a brace of the greatest lower bounds of the sphere and scope of the web outwardness map. Hence, the LLTP of GNEF is.

Definition 2.3. The Upper-Right Terminal Point ( URTP ) is a brace of the least upper bounds of the sphere and scope of web outwardness map. Hence, the URTP of GNEF is.

Let be the mention line linking the two points, and. The equation of is, where, and it is the incline of the line. Furthermore, this incline is the ratio of the scope of web outwardness with the scope of available market portion for the web goods. This can be interpreted as the maximal extra web outwardness per unit scope of the market portion. We introduce as the web outwardness index ( NEI ) of a web goods.

Proposition 2.3. For and.

If, so

If, so

Proof From ( 2.3.2 ) , ( 2.3.1 ) is reduced to, ,

When, .

When, .

For, if, so and if, so.

For, if, so and if, so.

For, by the l’Hopital ‘s regulation,

## .

## .

If, so and if, so.

If, so and if, so.

Remark 2.3. The by-product of Proposition 2.3. is that the rate of alteration of GNEF converges to zero at and from the right and left, severally, whenever. In fact, and.

We further note that depending on the positive or negative fringy web outwardness and for, we have matching two types of Torahs of decreasing returns, viz. , positively and negatively decreasing returns. Furthermore, from the above treatment, the positively and negatively decreasing returns are characterized by

, for and

( 2.3.4a )

( concave up to concave down )

, for and.

( 2.3.4b )

( concave down to concave up )

severally, where is referred as a point of inflexion. See Figure 2. Sketchs of the form of GNEF, illustrating of ( 2.2.3b ) for ( a ) and ( B ) ..

Illustration 2.3. A jurisprudence of positively decreasing return in economic sciences provinces that as a individual increases his/her ingestion of a product/good – while maintaining the ingestion of other merchandises changeless, there is a diminution in the fringy public-service corporation that individual derives from devouring each extra unit of that merchandise ( Samuelson and Nordhaus, 2009 ) . The same empirical economic sciences rule is applicable here, when market portion increases – while maintaining the other factors changeless, there is a diminution in the fringy value of web outwardness, that is, when market portion passes the critical size the 2nd derived function of web outwardness map will be less than zero, ; . A jurisprudence of negatively decreasing return can be illustrated, analogously. The Figure 2. Sketchs of the form of GNEF, illustrating of ( 2.2.3b ) for ( a ) and ( B ) .a farther illustrates the construct of positively decreasing return, that is, geometrically, the point of inflexion at which the concave shape alterations from the concave up to the concave down. The Figure 2. Sketchs of the form of GNEF, illustrating of ( 2.2.3b ) for ( a ) and ( B ) .b illustrates the characteristic of negatively decreasing curve, that is, the point of inflexion at which the concave shape alterations from the concave down to the concave up.

2.3.5 Concave shape: The form and concave shape of GNEF is determined by its 2nd derivative. From ( 2.2.3a ) , the 2nd derived function of is

## .

( 2.3.5 )

The point of inflexion of are derived by.

The concave shape of the GNEF varies depending on its parametric fluctuations. Without loss of generalization, we consider the positive fringy outwardness map, . The instance can be imitated, analogously. In the undermentioned, we analyze a few peculiar instances.

Case I: In this instance, the invariable of proportionality of ( 2.2.3a ) peers to the reciprocal of the web outwardness index, ( or ) . Under this status, in ( 2.2.3b ) is reduced to

## .

( 2.3.6 )

Proposition 2.3. For and.

If, so for all, and.

If, so for all, and is below the line.

If, so for all, and is above the line.

Proof Let, and the equation ( 2.3.6 ) can be written as

## .

( 2.3.7 )

Let and simplify the equation ( 2.3.7 ) into

## .

( 2.3.8 )

## .

( 2.3.9 )

From ( 2.3.5 ) and ( 2.3.9 ) , the 2nd derived function of GNEF can be rewritten as

## .

( 2.3.10 )

Therefore, from ( 2.3.9 ) and ( 2.3.10 ) ,

If so for all, and.

If so for all, and.

If so for all, and.

In this instance, the mark of concave shape will non alter for full sphere ; hence the incline of GNEF is besides monotone. See Figure 2. Sketchs of the form of GNEF, exemplifying the function of. ( a ) ( hence ) , and ( B ) , .a.

Remark 2.3. is tantamount to which means the initial extra web outwardness per used market portion peers to the initial shortage web outwardness per fresh market portion.

Case II: In this instance, for and, the GNEF has S and N shaped graphs, severally. See Figure 2.3b.

x3.5a.png

x3.6a.png

( a )

( B )

Figure 2. Sketchs of the form of GNEF, exemplifying the function of. ( a ) ( hence ) , and ( B ) , .

Based on Table 2.1, there are three premises about the form of web outwardness map, viz. additive, concave and convex maps. The GNEF ‘s form is adjustable to suit all these premises. Hence, we can state that each premise is a particular instance of GNEF, see Figure 2.3a. Furthermore, the GNEF is able to supply a monotone S-shape and N-shape map, see Figure 2.3b.

Example 2.3. The illustration of an N-shaped web outwardness map is the merchandise that develops itself when the web size reaches a certain degree. The smartphones, for illustration, has few applications in the early in which decreasing construct in fringy web outwardness is applied. Subsequently, when web size additions, there are more developers making many new applications. At this point, the fringy web outwardness is no longer decreasing, but exponentially increasing.

## Applications: Planning, Policy and Performance

In this subdivision, we analyze the effects due to parametric fluctuations on the web outwardness procedure and its value. The presented consequences provide a glance of the function and range of parametric fluctuations as control mechanisms/strategies for the planning, policy and public presentation sing web goods/services/information/labor/entity.

2.4.1 From the belongingss of described in Section 2.3, it is obvious that the dynamic theoretical accounts ( 2.2.3a ) and ( 2.2.3c ) of web outwardness procedure carry through all the bing premises used in the literature, Gottinger ( 2003 ) , Lin ( 2008 ) , Ben-Zion and Tavor ( 2006 ) .

2.4.2 The mathematical description of web outwardness procedure provides the foundation and the footing for the parametric sensitiveness analysis of GNEF. The dynamic nature of the web outwardness procedure provides the parametric dependance. The underlying parametric quantities can be characterized and decomposed into different categories of parametric quantities depending on the planning, policy and public presentation of the web outwardness goods/entity. In short, the sphere of parametric quantities can be decomposed into sub-domains sing the planning, policy and public presentation schemes.

2.4.3 The policy shapers are the individuals who set the programs pursed by the house or authorities aims. The developed programs depend on the calculator ‘s estimations from the theoretical account. The house directors have the common aim to hold high web outwardness value or high market portion. From the treatment 2.3.2 and the exchanging cost in Section 2.3, the parametric quantity “ ” ( lower bound of web outwardness ) is a “ planning parametric quantity ” . For and, when parametric quantity “ ” additions, while maintaining the other parametric quantities changeless, the web outwardness value additions for lower market portion and lessenings for the higher market portion. Similar reading can be given for the other parametric quantities, and. See Table 2.2, 2.3, 2.4 and 2.5.

2.4.4 From Proposition 2.3. , and its status, indicates the form of GNEF. Hence, the relationship between the category of parametric quantities and creates the part of parametric quantities that have the tantamount form. See Figure 2.4a-f.

For,

If, so the form of GNEF is shown as Figure 2.3a ;

If, so the GNEF has S-shape shown as Figure 2.3b ;

If, so the GNEF has N-shape shown in Figure 2.3b.

Hence, the policy shapers can use these parts to warrant the form of GNEF as they want. Effectss of these parametric quantities can be utilized by the policy shapers to pull off the web goods. The policies are based on the ultimate end ( s ) of firm/government.

2.4.5 From Section 2.3, we showed that parametric quantities represent the boundary parametric quantities of GNEF, parametric quantity represents its form, and parametric quantities represent its location. The policy shapers of a house can command certain parametric quantities on the footing of the suited policies to carry through their aims. If the policy shapers can command and, they can command the rate of alteration at initial point. For the point under mention line, the higher the higher rate of alteration. For the point above mention line, the lower the higher rate of alteration. That is, the policy shaper can project house ‘s growing by seting the initial market portion. For the little house, the lower initial market portion has higher growing rate. For the big house, the higher initial market portion has higher growing rate. For more in writing representations, see Table 2. Consequence of the initial parametric quantity to the GNEF ‘s form at assorted when.

2.4.6 From the mark of, we have either web outwardness map increasing or diminishing. However, this phenomenon can be interrupted by integrating the distinct clip intercession procedure ( Ladde ( 2005 ) , Korzeniowski and Ladde ( 2010 ) ) . In the mold of web outwardness procedure, this intercession thought is so motivated by the overall policy of network/users or supplier. In fact, the thought of intercession maintains competitive/cooperative behaviour of the comparable web goods. This so avoids monopoly of a market of web goods/service/information. Presently, this work is at the planning phase.

Table 2. Consequence of a parametric quantity “ ” to the GNEF ‘s form at assorted when

Table 2. Consequence of a parametric quantity “ ” to the GNEF ‘s form at assorted when

Table 2. Consequence of a parametric quantity “ ” to the GNEF ‘s form at assorted when

Table 2. Consequence of a parametric quantity “ ” to the GNEF ‘s form at assorted when

Table 2. Consequence of the initial parametric quantity to the GNEF ‘s form at assorted when

under mention line “ ”

above mention line “ ”

( a ) “ ” parametric quantity

( B ) “ ” parametric quantity

( degree Celsius ) “ ” parametric quantity

( vitamin D ) “ ” parametric quantity

( vitamin E ) “ ” parametric quantity

( degree Fahrenheit ) “ ” parametric quantity

Figure 2. The control part of assorted parametric quantities

## Decisions

2.5.1 After careful reappraisal and rating of the web outwardness literature, we officially developed several thoughts, notably, web outwardness procedure and web goods. By utilizing these thoughts, we formulated the Principle of Network Externality. The introduced rule provides a quantitative description of the construct of web outwardness as the dynamic procedure with regard to a market share/size.

2.5.2 The presented dynamic description of web outwardness procedure provides a systematic manner of analysing its well-known and good recognized belongingss in a incorporate manner. In fact, it provides a sufficient status to formalize bing premises in the literature, Gottinger ( 2003 ) , Lin ( 2008 ) , Ben-Zion and Tavor ( 2006 ) . This extends the bing thoughts in a incorporate and consistent mode.

2.5.3 In general, the web outwardness is considered to hold positive fringy web outwardness ; nevertheless, we have shown that for some types of web goods/needs/deficiencies/labor/education, the web outwardness has negative effects. For illustration the users of rare point market will lost their forte, when the market size additions. In short, the marginal of web outwardness map is non merely positive but besides can be negative, depending on types of web goods.

2.5.4 The most important part of the web outwardness procedure is that it provides the sufficient conditions for the bing premises sing the web outwardness map. Furthermore, conditions depend on the parametric quantities, this parametric quantity dependance of theoretical account provides mechanism to do the policy to run into public presentation end of the web goods.

## EMPIRICAL STUDY OF

## GENERALIZED NETWORK EXTERNALITY FUNCTION

In this chapter, we utilize the generalised web outwardness map ( GNEF ) in Chapter 2 to the existent universe informations. We define the two normalized constructs. We use statistical techniques to gauge the parametric quantities in GNEF of banking plus theoretical account.

## Introduction

In the undermentioned, we provide an empirical survey to exemplify the utility of GNEF. Unfortunately, we are unable to happen an expressed informations set with respect to a market portion of a good and its outwardness value. But in the hunt of a information set, We were able to happen two types of banking informations sets from the cardinal bank of the United States of America ( the Federal Reserve or merely “ the Fed ” ) ( hypertext transfer protocol: //www.federalreserve.gov/default.htm ) . Furthermore, the informations sets are with regard to the hebdomadal banking sedimentation and plus of the commercial Bankss in the USA from the January 2008 to January 2010. The studies of the natural informations sets suggest that both informations sets have Sigmoid form curve representation. Figure 3.1a and Figure 3.1b are the US hebdomadal banking plus and sedimentation ( US dollar billion ) , from Jan 2008 – Jan 2010, severally. The single information is modeled in Section 3.2. The US banking sedimentation treated as the market portion by standardization is discussed in Section 3.3. The GNEF for two banking informations sets are developed in Section 3.4.

( a )

( B )

Figure 3. Plot of Weekly US banking ( a ) plus and ( B ) sedimentation ( $ US billion ) from January 2008 through January 2010.

## Statistical Modeling of US Banking Asset and Deposit

In this subdivision, the statistical surveies of the given informations sets are summarized. Furthermore, by using the statistical consequences, we briefly sketch the dynamic mold and parametric quantity appraisal of both informations sets. From Figure 3.1, we can utilize the Sigmoid map to suit them. The brief treatment is as follow. Let

US Banking Asset US Banking Deposit.

( 3.2.1 )

From the secret plans in Figures 3.1, we conclude that the US banking plus and sedimentation informations sets possess the undermentioned deterministic dynamic theoretical accounts:

## ,

( 3.2.2 )

and

T,

( 3.2.3 )

severally, where for, and are positive parametric quantities.

Furthermore, the solutions matching to ( 3.2.2 ) and ( 3.2.3 ) are

( 3.2.4 )

and

## ,

( 3.2.5 )

where and.

We apply the Newton-Rahpson Method ( Seber and Wild ( 1989 ) ) to gauge the parametric quantities and for in ( 3.2.4 ) and ( 3.2.5 ) . The statistical sum-up is outlined in Table 3.1.

Table 3. Estimated parametric quantities of US banking plus and sedimentation theoretical account

Dynamic Model

US Banking

Asset Model

US Banking

Deposit Model

Lower Limit:

( Standard Error )

11.0036

( 1.48×10-2 )

11.4909

( 1.75×10-2 )

Upper Limit:

( Standard Error )

12.0311

( 1.00×10-2 )

13.2204

( 1.41×10-2 )

Location Parameter:

( Standard Error )

2.607×10-4

( 1.75×10-4 )

0.0045

( 1.11×10-3 )

Constant of Proportionality:

( Standard Error )

0.2067

( 1.87×10-2 )

0.0664

( 3.68×10-3 )

Residual Sum of

Square: Roentgen

0.4882

0.4938

Coefficient of

Determination: :

0.9773

0.9906

Remark 3.2. The consequences of the informations adjustment in Table 3.1 includes four parametric quantities, their standard divergences in parenthesis, residuary amount of square and coefficient of finding are besides included. Both theoretical accounts have really low RSS ( about nothing ) and high ( about one ) . The fluctuation of US banking plus and US banking sedimentation can be explained by Sigmoid map 97.73 % and 99.06 % , severally. We note that the standard mistakes in parenthesis corresponding to parametric quantities are recorded in Table 3.1. The Residual criterion mistake ( RSE ) for the US Banking plus theoretical account is 0.0695 with 101 grades of freedom. It needs 18 loops for the convergence. In the instance of US banking sedimentation theoretical account, the residuary criterion mistake ( RSE ) is 0.0699 with 101 grades of freedom. It needs 14 loops to set up the convergence.

Remark 3.2. By detecting the qualitative nature of the above cited informations sets, we construct the information sets, and introduce assorted types of normalized US banking sedimentation theoretical accounts. The developed theoretical accounts will be utilized to present the US banking plus as web outwardness for the banking industry.

( a )

( B )

Figure 3. Curve adjustment of the US banking ( a ) plus and ( B ) sedimentation

## Normalized US Banking Deposit Models

To warrant the use of the US banking plus and sedimentation as the web outwardness and fiscal market portion, severally, we need to analyse the plus as the web outwardness value and the banking sedimentation as the market portion. Because the sphere of in ( 3.2.3 ) is between nothing and one, , we need to modify the information values of US banking sedimentation theoretical account ( USBD Model ) and develop normalized banking sedimentation theoretical accounts. In peculiar, we need an upper bound for banking sedimentation. In the absence of a anterior cognition of absolute least upper edge of US banking sedimentation, we am forced to normalise the given informations set and besides need to develop the dynamic theoretical account for normalized US banking sedimentation. As a consequence of this, based on different normalized banking sedimentation process, we have developed two theoretical accounts.

## 3.3.1 Normalized US Banking Deposit Model ( USBD Model ) – I

We normalize the graduated table of banking sedimentation to hold the lower and upper bounds. We define the extra US banking sedimentation as the difference between the current US banking sedimentation and its lower bound, that is, . A theoretical overall extra US banking sedimentation is defined as the difference between the upper and lower bound of banking sedimentation, that is. Therefore, the normalized banking sedimentation is defined by:

or.

( 3.3.1 )

After the standardization of banking sedimentation, the dynamic equation ( 3.2.3 ) is reduced to:

## .

( 3.3.2 )

Its solution is,

## .

( 3.3.3 )

where. Now in the dynamic equation in ( 3.3.2 ) , its upper bound is one and lower bound is zero. Theoretically, the normalized banking sedimentation must be between zero and one. Unfortunately, the parametric quantities and are unknown. Therefore, we use the calculators and of US banking sedimentation theoretical account from Table 3.1 in ( 3.3.1 ) .

Remark 3.3. The presented standardization procedure in ( 3.3.1 ) lost some information points. This is due to the fact that, we used the calculators and alternatively of the true parametric quantities. In order to minimise the loss of informations points, we introduce another standardization method to obtain the construct of market portion. This is described below.

## 3.3.2 Normalized US Banking Deposit Model ( USBD Model ) – Two

In position of Remark 3.3. , we consider the banking sedimentation as a portion of portfolio, and so we calculate the transform informations set by comparing the banking sedimentation to the US mean gross investing ( USAGI ) . The US mean gross investing from January 2008 through January 2010 is 18.82 billion US dollars ( hypertext transfer protocol: //www.federalreserve.gov/default.htm ) . We assume that USAGI is upper bound for the US banking sedimentation. Under this premise, we define the undermentioned transmutation to normalise the US banking sedimentation as follow:

## .

( 3.3.4 )

Now, by following the statement used in the US banking sedimentation theoretical account ( USBD Model ) – I, we have the dynamic theoretical account for the normalized US banking sedimentation theoretical account as:

## .

( 3.3.5 )

Its solution is,

## ,

( 3.3.6 )

where. Now in the dynamic equation in ( 3.3.5 ) , its upper bound is and lower bound is.

## US Banking Asset Network Externality Models ( USBANE Models )

By using our cognition in the theory of comparative growing, J. Huxley ( 1932 ) and Robert Rosen ( 1967 ) and following the definition of web outwardness, we define the comparative growing of the US banking plus, , with regard to the normalized US banking sedimentation. Therefore, we conclude that the US banking plus can be considered as a map of US banking sedimentation. This thought of course illustrates that the US banking plus as web outwardness procedure with regard to the banking sedimentation as a fiscal market portion. As the consequence of this, we utilize two normalized USBD theoretical accounts to develop corresponding dynamic theoretical accounts for the US banking plus as the web outwardness procedure. Therefore, we have two dynamic theoretical accounts for the web outwardness for the US banking plus.

From the Deterministic US banking plus theoretical account in Section 3.2, the comparative growing theoretical account of in ( 3.2.2 ) with regard to two normalized US banking sedimentation theoretical accounts described by ( 3.3.1 ) – ( 3.3.2 ) and ( 3.3.4 ) – ( 3.3.5 ) are as follow:

## ,

( 3.4.1a )

## ,

( 3.4.1b )

severally, where and are as defined before. These two dynamic theoretical accounts are dynamic theoretical accounts of web outwardness of the US banking plus with regard to the normalized US banking sedimentation as the fiscal market portion. These differential equations are precisely similar to the derived function in ( 2.2.3b ) . Therefore, the comparing of parametric quantities in ( 2.2.3b ) with parametric quantities in ( 3.4.1a-b ) is as follow:

and,

( 3.4.2a )

and,

( 3.4.2b )

severally. Thus, all parametric quantities above can be estimate by the merchandise of Table 3. Estimated parametric quantities of US banking plus and sedimentation theoretical account and the secret plans are shown in Figure 3.3.

( a )

( B )

Figure 3. Curve adjustment of USBANE: indirect attack ( a ) Model-I ( B ) Model-II

The drumhead appraisal of two US banking plus web outwardness theoretical accounts ( USBANE Model ) is based on the estimated parametric quantities in Table 3. Estimated parametric quantities of US banking plus and sedimentation theoretical account in the competition of the parametric quantities of normalized USBD theoretical accounts as the fiscal market portion.

Table 3. Estimated parametric quantities of USBANE Models with USBD Market Share

USBANE

Model I

Model II

Lower Limit:

11.0036

11.0036

Upper Limit:

12.0311

12.0311

Minimal Share:

0

0.6106

Maximal Share:

1

0.7025

Constant of Proportionality:

0.1857

0.0171

Sample Size

85

105

Residual Sum of

Square: Roentgen

0.6242

0.4537

## Model Nosologies

The implicit in nonlinear arrested development theoretical account ( Ritz and Streibig, 2008 ) are:

Correct mean map

Variance homogeneousness ( homoscedasticity )

Normally distribution measuring mistakes

Mutually independent measuring mistakes

Misdemeanor of any one of the above premises could ensue in prejudice calculators and/or falsify standard mistakes. In this subdivision, we examine exemplary diagnostic of nonlinear least square analysis of US banking plus and sedimentation theoretical accounts, ( 3.2.4 ) and ( 3.2.5 ) , severally. This is indispensable to analyse their behaviours of the residuary mistake independence, homogeneousness and normalcy under the application of Newton-Raphson method. We consider some of import secret plans, homogeneousness trial and normalcy trial.

( a )

( B )

Figure 3. Plot of remainders vs clip of ( a ) plus, and ( B ) sedimentation theoretical accounts

Figure 3. Plot of remainders vs clip of ( a ) plus, and ( B ) sedimentation theoretical accounts show the remainders secret plan of plus and sedimentation theoretical accounts. Therefore, we conclude that their remainders are homoscedasticity with regard to clip.

( a )

( B )

Figure 3. Plot of standardised remainders vs fitted value of ( a ) plus, and ( B ) sedimentation theoretical accounts

Figure 3. Plot of standardised remainders vs fitted value of ( a ) plus, and ( B ) sedimentation theoretical accounts exhibit the secret plan of standardised remainders vs the fitted values of plus and sedimentation theoretical accounts. These standardised residuary mistake secret plans demonstrate that there is no additive correlativity. This suggests that the residuary mistakes exhibit the homogeneousness. This averment will be farther examined by the application of statistical trial for homogeneousness as described below.

Homogeneity Test: From residuary secret plans, one can easy pull a decision about the homogeneousness of residuary mistakes. However, we apply the homogeneousness trial developed by Endrenyi and Kwong ( 1981 ) to our residuary mistakes of plus and sedimentation theoretical accounts. The brief description of the trial is as follow: First, we i¬?t the theoretical account, and so order the remainders as. Second, we compute the ratio of the last squared remainders to the i¬?rst squared remainders. The suggestion for is. Under these considerations, the homogeneousness trial is as follow:

V

The statistics and its of the trial are recorded in Table 3.3.

Table 3. Homogeneity trial: statistics and p-values

Homogeneity trial

Asset Model

Deposit Model

0.7459

1.0100

0.2256

0.5102

From the Table 3. Homogeneity trial: statistics and p-values, the of two theoretical accounts are bigger than. This shows that there is no signii¬?cant grounds to reject the homogeneousness of their remainders at the signii¬?cance degree 0.05.

( a )

( B )

Figure 3. QQ Plot of standardised remainders of ( a ) plus, and ( B ) sedimentation theoretical account

Figure 3. QQ Plot of standardised remainders of ( a ) plus, and ( B ) sedimentation theoretical account show the QQ secret plans of the standardised residuary mistakes of plus and sedimentation theoretical accounts. There is somewhat different from consecutive line. We need farther analyze the normalcy of standardised residuary mistakes by using the histogram. The consequences are as follow.

( a )

( B )

Figure 3. Histogram of standardised remainders of ( a ) plus, and ( B ) sedimentation theoretical account

Figure 3. Histogram of standardised remainders of ( a ) plus, and ( B ) sedimentation theoretical account exhibit that histogram of the standardised residuary mistakes of sedimentation theoretical account is more normal form than plus theoretical account. In add-on to this, we utilize the statistical normalcy trial developed by Shapiro and Wilk ( 2008 ) and this trial is as described below.

Normality Test: For this trial, the nothing and alternate hypotheses are:

The random sample, , is drawn from usually distributed population

The random sample, , is non drawn from usually distributed population

We compute where is order statistic, is the sample mean. , where

are the expected values of the order statistics of independent and identically distributed random variables sample from the criterion normal distribution and is the covariance matrix of those order statistics.

The statistics and its of the trial are recorded in Table 3.4

Table 3. Normality trial: statistics and p-values

Normality Test

Asset Model

Deposit Model

0.9783

0.9814

0.0819

0.1489

From the Table 3. Normality trial: statistics and p-values, their are bigger than. This suggests that there is no signii¬?cant grounds to reject the normalcy of their remainders at the signii¬?cance degree 0.05. In drumhead, from the above statistical survey, we conclude that the remainders exhibit independence, homogeneousness and normalcy of Asset and Deposit theoretical accounts.

## Decisions

We utilize the generalised web outwardness map ( GNEF ) in Chapter 2 to the existent universe informations set. In the banking industry, we consider the banking plus as the web outwardness value and banking sedimentation as the market portion. The information sets are with regard to the hebdomadal banking sedimentation and plus of the commercial Bankss in the USA from the January 2008 to January 2010. There are two methods to handle banking sedimentation as market portion, taging the minimum/maximum portion and portfolio methods. From the studies of banking plus and sedimentation, it suggests that both informations sets have Sigmoid form curve. Then their comparative growing is the same as ( 2.2.3a ) in Chapter 2 which leads to GNEF ( 2.2.3b ) . Consequently, there are besides two US banking plus web outwardness theoretical accounts.