Research Article
a
January
2018
Computer Science and Software Engineering
ISSN: 2277-128X (Volume-8, Issue-1)
A Strategic Game Theory Model Based Social Network
Analysis for Multiuser Bandwidth Rate Allocation using
Predictive Artificial Neural Network
Nirmalya MukhopadhyayAssistant Professor, Department of IT, iNurture Education Solutions Pvt. Ltd. under Kaziranga University, Jorhat, Assam, India
Abstract: In this paper I am going to first explain in detail the role of Game Theory over Social Network Analysis. Then I will look into the Predictive model of Artificial Neural network & will explain in details that how this model will be used to develop a mathematical model which will fairly and efficiently allocate the required rate of bandwidth to all the users in a Multiuser Network System. Afterwards, I will propose some newly designed algorithms which will help me in the implementation of the mathematical model. The testing result of the implementation will compare our proposed architecture with the existing model. Finally, I will end this discussion by self-estimating our proposed model and judging the future scope of the same.
Keywords: Strategic Game Theory, Behavioural Game Theory, Nash Equilibrium, Optimal Strategy, Dominant Strategy, Social Network Analysis, Delay Sensitive Multimedia Network, Predictive Artificial Neural Network.
I. INTRODUCTION
Game Theory is the branch of decision theory concerned with interdependent decisions. It is the study of mathematical models of interaction, conflict and co-operations between two or more than two intelligent and rational decision-makers. As the basis of game theory was the analysis of competitive scenarios, the problems are called Games and the participants are called Players. The games that we are talking about are well-defined mathematical objects. The five major elements of a well-defined game are: The number of Players, the set of all possible Actions (or Strategies), Information about allActions available, A Payoff Consequence or Payoff Matrix and descriptions of all players‘ ordinal
preferences. Originally this theory was named as Zero-sum-game, where the gain of one player results in the loss of the
others, who are participating in the same game. John Von Neumaan invented mixed strategy equilibrium for a
Zero-sum-game using Brouer fixed point theorem on continuous mapping into compact convex sets. Later he came with an
Axiomatic theory of expected utility, which allowed Mathematical Statisticians and Economists to make optimal decisions under uncertainty. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define non-cooperative games.
John Forbes Nash (―A Beautiful Mind‖ fame) developed another equilibrium strategy. Although this concept was first used in 1838 by Antoine AugustinCournot in his theory of Oligopoly (A special market structure where the market is dominated by a few number of firms), the establishment of the concept & massive useof the same was came into picture through mixed strategy equilibrium.Cournot‘s theory was based on pure strategy, where a Cournot‘s equilibrium is established when a firm maximizes its profit by choosing an action irrespective of the actions chosen by the other firms. On the other hand, a Nash Equilibrium is simply a solution concept for a non-co-operative games where two or more players are involved and each player knows the equilibrium strategy or dominant strategy of other players i.e., one player can gain anything by changing their own strategy.
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comparative study of social structure, where a detailed observation of interactions between each participant is taken into account and their relationship is maintained under some constraints agreed upon by all the participants. Graph Theory, Network Theory are directly related to SNA. All the nodes in a Graph can be considered as Players or Actors and ties or edges between them are the Relationship or Interactions between the participants. Some renowned sociologists like Georg Simmel, Emile Durkheim first introduced the concept of SNA in their research work, where they observed the pattern of relationship status between a group of individuals (whom we can say Players in Game Theory terminology). Some scholars like Ronald Burt, Edward Laumann, and Harrison White etc. have widespread the idea of SNA into different research areas.
An Artificial Neural Network (ANN) is a mathematical modeling of physical and biological human brains for studying the thoughts and works of natural human brains. Artificial Intelligence is applied to mathematically designed neurons connected together to function just like human brains. The Neurophysiological knowledge of developers helped them to model the neurons easily through mathematical and statistical formulations. The neurocomputing scientists solve the difficult predictive situations through the help of Artificial Neural Networks. The human perceptions are taught to the artificial neurons by the developers so that they can manipulate the inputs of a system as effectively as receptors, effectors and cortex (Central Nervous System) of a human brain to produce outputs similar to human beings. This is also important in SNA, as the interactions between the participants of a Social network are human beings. Now if we want to implement a new model of Game Theory for Social Network Analysis that will act and react just like a human being, we have to use the predictive model of ANN so that the mathematical model apparently looks like an interactive and intelligent human being. In the later part of this paper, we will propose such type of a mathematical model of Game theory with equilibrium conditionswhich will actually analyses a Social Network using Predictive ANN that will effectively and fairly distribute the bandwidth rate for multiuser delay sensitive network as per human perception.
We have time series, i.e., a variable x changing in time xt (t=1,2,...) and we would like to predict the value of x in time (t+h)[15]. The prediction of time series using neural network consists of teaching the net the history of the variable in
a selected limited time and applying the taught information to the future [7]. Data from past are provided to the inputs of
neural network and we expect data from future from the outputs of the network. For more exact prediction, additional information can be added for teaching and prediction, for example in the form of interventional variables (intervention indicators)[1] [6] [31]. However, more information does not always mean better prediction; sometimes it can make the
process of teaching and predicting worse. It is always necessary to select really relevant information, if it is available [11]
[5] [7]
.
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Fig: Teaching of time serieswith intervention indicator
For the task of predicting the indexes, we'll be using the so called multilayer feed forward network[23] [3] [16]
[40]
which is the best choice for this type of application. In a feed forward neural network, neurons are only connected forward[11] [38] [5]. Each layer of the neural network contains connections to the next layer, but there are no connections back. Typically, the network consists of a set of sensory units (source nodes) that constitute the input layer, one or more hidden layers of computation nodes, and an output layer of computation nodes[12] [37] [26] [34] [22]. In its common use, most neural networks will have one hidden layer, and it's very rare for a neural network to have more than two hidden layers. The input signal propagates through the network in a forward direction, on a layer by layer basis[15] [4] [30]. These neural networks are commonly referred as multilayer perceptrons (MLPs). Shown below is a simple MLP with 4 inputs, 1 output, and 1 hidden layer.
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The input layer is the conduit through which the external environment presents a pattern to the neural network
[22] [25] [30]
. Once a pattern is presented to the input layer, the output layer will produce another pattern. In essence, this is all the neural network does - it matches the input pattern to one which best fits the training's output [36]. It is important to
remember that the inputs to the neural network are floating point numbers, represented as C# double type [17] [19] (most of
the time we will be limited to this type).
The output layer of the neural network is what actually presents a pattern to the external environment (the result of the computation). The number of output neurons should be directly related to the type of work that the neural network is to perform [34] [38] [2].
There are really two decisions that must be made regarding the hidden layers[14] [27]: how many hidden layers to
actually have in the network and how many neurons will be in each of these layers. Problems that require two hidden layers are rarely encountered. There is currently no theoretical reason to use neural networks with any more than two
hidden layers, thus almost all current problems solved by neural networks are fine with just one hidden layer [3] [8]. Even
though the hidden layers do not directly interact with the external environment, they have a tremendous influence on the final output, thus you should carefully choose the number of neurons within it. Using too few neurons in the hidden layers will result in so called "under-fitting", which occurs when the hidden layers are not able to adequately detect the signals in a complicated data set [39] [26]. The "over-fitting" problem can occur, when the neural network has so much information processing capacity that the limited amount of information contained in the training set is not enough to train all of the neurons in the hidden layers. There are many ―rule-of-thumb‖ [9] [4] [35]
methods for determining the correct number of neurons to use in the hidden layers, here are just a few of them:
The number of hidden neurons should be between the size of the input layer and the size of the output layer.
The number of hidden neurons should be 2/3 the size of the input layer plus the size of the output layer.
The number of hidden neurons should be less than twice the size of the input layer.
Fig: Components of an Artificial Neuron Network
One can see that there are 3 basic element of a neuronal model:
1. A set of synapses or connecting links, each characterized by a weight or strength of its own: X1,X2,...,Xnwith
corresponding weights: Wk0, Wk1,...,Wkm. As you will see further, the weights represent the "knowledge" that the
neural network contains about a specific training data. Their values will directly affect the output of the neural network[13] [17] [29].
2. An adder for summing the input signals, weighted by the respective synapses of the neuron: Vk = ∑(WkjXj+bk),
where k=[1,r], (r=number of neurons), j=[1,m] (m=number of input synapses). Simply speaking - the input signal X is multiplied by the weight W and summed in the adder with all the other items. The result of this summation V will go to the input of the activation function [17] [32] [1].
3. An activation function for limiting the output of a neuron: Yk = Φ(x). The activation function has an important
role in the schema of a neuron. It generates the output according to the summed input signals calculated in the adder. Summarized, the output signal of each neuron can be defined as follows: Yk = Φ(∑(WkjXj+bk)). It is
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this method requires computation of the Gradient of the error function at each iteration step, we must guarantee the continuity and differentiability of the error function. A commonly used non-linearity that satisfies this requirement is sigmoid[6] [18] [38][28]non-linearity defined by the logistic function: Φ(v) = 1/(1+exp(-αv)), where a is the slope parameter of the sigmoid function. By varying the parameter a, we obtain sigmoid functions of different slopes, as illustrated in the following figure (3 different a values):
II. EXISTING MODELS& ASSUMPTIONS
Game Theory can be formulated using a five tuple. If we consider ‗G‘ as a game, then G can be mathematically defined as follows:
G = [N, Ai, Ia, Ui, Pi]
Where, N is the number of Players participating in the Game G, Ai is the set of available actions or strategies for
Player i, Ia is the set of information available for all available actions, Ui is the values of payoff matrix or payoff
consequences for Player i and Pi is the set of ordinal preferences of strategies for Player i over the available strategies and
strategies chosen by other Players.
Informally, Nash equilibrium is a dominant strategy selection by a Player in a non-co-operative Game for optimal outcome or best strategy. If we denote Nash equilibrium by a*, then it can be defined as follows:
* * *
( ,
)
( ,
)
i i i i i iU a a
U a a
,
a
iA
iWhere,
a
iis the particular strategy taken by the Player I from the set of available actions Ai,a
i is thestrategies of all Players of the Game except Player i.
Basic assumptions:
We consider a model of directed relations, where each tie i!j has a sender i (namely, ego)and a receiver j (referred to as alter). Now if the network is observed under two or more discrete point of time but the time parameter t is continuous
then the process unfolds by the time series of various lengths. These observations are called ‗Network Panel Waves‘ [34]
[15] [6]
for non-network ties.This assumption was proposed already by Holland and Leinhardt (1977), and elaborated by Wasserman (1979 and other publications) and Leenders (1995 and other publications) – but their models represented only reciprocity, and no other structural dependencies between network ties[14] [11] [35].
The actors have to control their outgoing ties. It implies that the actor for a particular tie can‘t modify it unanimously, but the modification depends on other actors‘ attributes, their positions in the network and their perceptions
about the network. It is formally considered that actors have full information about the network and the other actors[32] [26]
[18]
. At a given moment one probabilistically selected actor – ‗ego‘ – may get the opportunity to change one outgoing tie.Holland and Leinhardt (1977) first proposed that only one tie can change at a particular time.
Summarizing the status of the four basic assumptions: the first (continuous-time model) makes sense intuitively; the second (Markov process) is an as-if approximation and it would be interesting in future to construct models going be-yond this assumption; the third (actor-based) is often a helpful theoretical heuristic; and the fourth (ties change one by one) is an assumption which limits the applicability to a wide class of panel data of directed networks for which this assumption seems relatively harmless[35] [20] [17] [22].
Change determination model:
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selected focal actor then may change one outgoing tie (i.e., either initiate or withdraw a tie), or do nothing (i.e., keep the present status quo). This means that the set of admissible actions contains n elements: n 1 changes and one non-change. The probabilities for a choice depend on the so-called objective function. This is a function of the network, as perceived by the focal actor[36] [18] [28]. Informally, the objective function expresses how likely it is for the actor to change her/his network in a particular way. On average, each actor ‗tries to‘ move into a direction of higher values of her/his objective
function[11] [15], subject to the constraints of the current network structure and the changes made by the other actors in the
network; and subject to random influences. The objective function will depend in practice on the personal network of the actor, as defined by the network between the focal actor plus those to whom there is a direct tie (or, depending on the specification, the focal actor plus those to whom there is a direct or indirect– i.e., distance-two – tie), including the covariates[24] for all actors in this personal network. Thus, the probabilities of changes are assumed to depend on the personal networks that would result from the changes that possibly could be made, and their composition in terms of covariates, via the objective function[14] [6] [7] [37] [2] values of those networks.
The name ‗objective function‘ was chosen because one possible interpretation is that it represents the short-term objectives (net result of preferences and structural as well as cognitive constraints) of the actor. Which action to choose out of the set of admissible actions, given that ego has the opportunity to act (i.e., change a network tie)[26] [33] [4], follows the logic of discrete choice models (McFadden, 1973; Maddala, 1983) which have been developed for modeling situations where the dependent variable is a choice made from a finite set of actions.
Specification of the objective function:
The objective function determines the probabilities of change in the network, given that an actor has the opportunity to make a change. One could say it represents the ‗rules for network behavior‘ [22] [29] [1]of the actor. This function is defined on the set of possible states of the network, as perceived from the point of view of the focal actor, where ‗state of the network‘ refers not only to the ties but also to the covariates. When the actor has the possibility of moving to one out of a set of network states, the probability of any given move is higher accordingly as the objective function for that state is higher.
Like in generalized linear statistical models[24] [32], the objective function is assumed to be a linear combination
of a set of components called effects[10] [11],
( , )
( )
i k ki
k
f
x
s
x
For the model selection, an essential part is the theory-guided choice of effects included in the objective function in order to test the formulated hypotheses. A good approach may be to progressively build up the model according to the method of decreasing abstraction (Lindenberg, 1992). An additional consideration here is, however, that
the complexity of network processes, and the limitations of our current knowledge concerning network dynamics[8] [23] [36]
[31]
, implies that model construction may require data-driven elements to select the most appropriate precise specification of the endogenous network effects. For example, in the investigation of friendship networks one might be interested in effects of lifestyle variables and background characteristics on friendship, while recognizing the necessity to control for tendencies toward reciprocation and transitive closure [12] [21].
Interactions:
Like in other statistical models, interactions can be important to express theoretically interesting hypotheses. The diversity of functions that could be used as effects makes it difficult to give general expressions for interactions. The ego -alter interaction effect[16] [23] for an actor covariate, mentioned above, is one example.
Another example is given by De Federico (2004) as an interaction of a covariate with reciprocity. In her analysis
of a friendship network[39] between exchange students, she found a negative interaction between reciprocity and having
the same nationality. Having the same nationality has a positive main effect, reflecting that it is easier to become friends with those coming from the same country. The negative interaction effect[14] [17] [21] [28] [31] was unexpected, but can be explained by regarding reciprocation as a response to an initially unreciprocated tie, the latter being a unilateral invitation to friendship. Since contacts between those with the same nationality are easier than between individuals from different nationalities, extending a unilateral invitation to friendship is more remarkable (and perhaps more costly) between individuals of different nationalities than between those of the same nationality. Therefore it will be noticed and appreciated, and hence reciprocated, with a higher probability. Thus, the rarity of cross-national friendships leads to a stronger tendency to reciprocation in cross-national than same-nationality friendships.
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( )
n i i n i ix
f x
x
,
x
i0
Where the key terms that should be taken care of are defined as follows:
1. According to Marson&Gerla, the fairness index is defined as:
,
{
}
i i j jP
F
Min
P
, where Pi&Pj are powers (i.e. (Throughput/Round-trip Delay)) achieved by the Players i &j.
2. The variance is defined as:
2 1
1
(
)
1
n i iVariance
x
n
3. The Mean (µ) is calculated by:
1
1
n i ix
n
4. The Coefficient of Variation is determined by:
Variance
COV
Mean
2 1 1 11
1
(
)
1
1
n n i i i i n i ix
x
n
n
COV
x
n
1(
1)
n i iCOV
n
x
On the other hand, a commonly adopted method is for the rate controller to minimize the weighted sum of the
distortions or try to maximize the weighted sum of the PSNRs, i.e., the optimization problem becomes:
1
(
)
i
n
i i i
R i
Min
W D R
, subject to constraint1 n i i
R
R
OR, 1(
)
i ni i i
R i
Max
W PSNR R
, subject to constraint1 n i i
R
R
Where R is the available network bandwidth, wiis the weight, Di is the distortion, and PSNRiis the PSNR of the
ithuser. Notice that the solution to the above optimization-based methods is highly related to the selection of the weights
wi. However, in the literature, the weights wi's are usually heuristically[17] determined, e.g., wiis uniformly set to be 1=N. Moreover, such a formulation can only address the efficiency issue, e.g., how to maximize the weighted sum of the
PSNRs or minimize the weighted-sum[11] [40] of the distortions. As such, the fairness issue, which is an important problem
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III. OUR PROPOSAL &MATHEMATICAL MODELING
I have designed a simple Predictive Artificial Neural Network. The network contains six numbers of nodes (which we can assume as Players for a Multiuser Delay Sensitive Bandwidth Allocation Game) and obviously fifteen ((6*(6-1))/2 = 15) numbers of edges (i.e., the interactions between Players). Each node contains some value. I taught the network, and it assigned values for each node after iterative thinking process. The snapshots of the model are as below:
Fig: The Predictive ANN model before the thinking process started.
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Fig: The Predictive ANN model after a lot of iterative thinking process is over.
Now as I proceed with the time it has been observed that the assigned trust values of each node have decreased. It is such basically because of the change in number & type of interactions between each node. The fairness issue has played its role here. It has maximizes the value of N5 node to establish the Nash equilibrium based on the strategy taken by the node N4. Here I propose a new fairness index for any Player xithat participated in the Game and wants to
participate in the Nash equilibrium. This fairness function is the optimal and dominant strategy for the player xibased on
the strategy of Player xj.
Fairness index for xi is:
,
'( )
( )
{
}
'(
)
i i i j
j
f x
F x
Max
f x
2
1
2
1 ,
2
1
2
1
(
)
( )
{
}
(
)
n i i
n i i
i i j n
j j
n j j
x
x
F x
Max
x
x
--- (1)
Whenever any multicast network is considered,it is important to look after the throughput, transmission time and data loss. For the predictive ANN model that I developed, following is the mathematical modeling of the different factors that has significant effects on the Social Network Analysis.
The throughput (X) of Players is defined by,
2
8*
2
3
*{
12*
* *(1 32
)}
3
8
S
X
P
P
R
P
P
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Where, X is the transmit rate in bits/second, S is the packet size in bytes,
R is the Round-Trip Time (RTT) in seconds and
P is the loss events as a fraction of the no. of packets transmitted. Again, R can be calculated as:
R= TR_now – TR_R’--- (3) Where, TR_now is the time the data packet arrives at the receiver, and
TR_R‘ is the receiver report time-stamp echoed in the data packet. The data loss can be calculated by the equation at a low-pass filter as:
1
(1
)
i i i
Y
WY
W X
--- (4)Where, W is considered as a constant & its value is 65000/65536, which corresponds to a corner frequency of
approximately 0.0013𝑝𝑎𝑐𝑘𝑒𝑡𝑠−1.
The loss interval can be calculated from equation 1 as:
3
8*
_
*
2
1
*
_
S
X
recv
R
L o
--- (5)2
_
*
_
{
}
3
(
) *8*
2
X
recv R
L o
S
--- (6)Receivers can adjust L_o by the following equation:
2
_
_ *(
)
_ max
R
L o
L o
R
--- (7)The expected delay until the first feedback message is sent is:
(1
)
[ ]
(1 log )
log
n
n N
T
x
E D
dx
T
N
x
--- (8)The expected number of feedback message can be calculated as:
1
1
[
]
T{
(1
) (1
) }
nT
n
E M
N
N
N
N
--- (9)Where n is the actual no. of receivers
The following exponentially weighted random timer mechanism sets up the feedback timer to expire after
max( *(1 log( ) log( )), 0)
t
T
x
N
--- (10)Where,X is a random variable uniformly distributed in (0, 1), T is the duration of feedback round (i.e., 6*R_max), and N is the estimated upper bound on the no. of receivers.
So, the allocated bandwidth rate for each user will be calculated & distributed as per the following equation:
1
( , )
s e(
)
A t t
Capacity Traffic
Noise
n
1
( , )
s e(
(
*
)
)
A t t
Capacity
Utilization Capacity
Noise
n
1
( , )
s e((1
) *
A t t
Utilization
Capacity
Noise
n
2 255 101
( , )
((1
) *
(ln 10 log
Di))
s eA t t
Utilization
Capacity
n
--- (11) Where, Di
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IV. CONCLUSION &FUTURE SCOPE
In the conclusion part I would like to highlight the fact that I have designed a Predictive Artificial Neural Network based on Strategic Game Theory Model which is considered as a Social Network and has gone through Social Network Analysis to establish Nash Equilibrium which actually distributes the available bandwidth efficiently and fairly to all the participants. I first taught the network to think and calculate the trust values for each participating Nodes and then reassign the values as per their conflict & co-operation while interacting. On the basis of each Player‘s Strategic decisions, the dominant strategy is finalized. Finally, Nash Equilibrium is established between the participants. This equilibrium helped the model to finally assign a fair and effective bandwidth rate to each & every participant.
But, as per my observation, this model is applicable for participants of the same network. For inter-networked system I have to model the things differently. This disadvantage of this proposed system will actually lead it to a better solution for the Social Network Analysis.
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