In Mobile ad hoc networks, Reputation is considered as an important aspect for establishing collaboration between wireless mobile nodes. Computing the level of reputation in MANETs is highly challenging due to the dynamic movement of nodes and their computational overhead. In MANETs, the presence of a selfish node could affect both the reliability and the quality of data disseminated in the network. Hence there is a need for devising a detection mechanism to identify the selfish behavior of nodes. In this paper, we propose a conditional probability based mathematical solution for detecting the selfish nodes in an ad hoc scenario. The performance of the proposed solution is analyzed through ns-2 simulations based on the parameter like Packet Delivery Ratio, Control Overhead and Total Overhead, which is determined by varying the threshold level for selfish node detection. The result facilitates to predict large number of selfish nodes using optimal threshold range.
Keywords: Conditional Probability, Threshold range, Selfish nodes, Random variable, Erlang distribution
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Enabling Collaboration among the wireless mobile nodes in a MANET is considered to be a significant issue that has not been explored in the past decades . Establishing cooperation between the mobile nodes is crucial because of the lack of dynamic movement of mobile nodes in MANET. The network performance may degrade in the existence of malicious nodes called selfish nodes which deny collaborating in order to save its own resources . Hence, a need arises for framing a mathematical model for efficient and effective detection of selfish nodes.
In this paper, we propose an efficient probabilistic mathematical model based on conditional expectation to identify and mitigate selfish nodes. This is accomplished through the manipulation of Conditional Reliability Expectation Coefficient (CREC) for each and every node participating in an ad hoc scenario. The protocol used for our experimental study is the AODV protocol, which is an reactive and tree based protocol.
The rest of the paper is organized as follows: In Section 2, we elaborate on the literature review. In section 3, we discuss the proposed conditional probability based solution for selfish nodes in MANETs. The simulation parameters and evaluation metrics are presented in section 4. The detailed analysis of the solution based on the evaluation metrics are presented in section 5. Finally, we conclude in section 6 with future scope.
From the recent past, numerous mathematical models based on probability were proposed for detecting the existence of the selfish nodes in ad hoc networks. Some of the probabilistic solutions available in the existing literature for detecting misbehaving nodes are listed below:
Thomas M.chen and Varadharajan venkataraman  proposed that the Dempster-Shafer theory of evidence is ideally suitable for detecting malicious nodes because it can predict uncertainty of node co-operation to the maximum extent. They also contributed a Dempster rule of combination that could help in determining a numerical procedure for combining multiple evidences obtained from neighbor nodes. They used two bounds called belief and plausibility for detecting malicious nodes. Their method of detection is based on posterior probability.
Md.Amir Khusru Akthar and G.Sahoo proposed that a node in MANET is selfish when the parameter called P(S|Pos) computed based on Baye's Theorem is less than 0.5.Their model classified the nodes available in the network as regular or selfish nodes. They also used the method of prior probability and continuous Baye's theorem for monitoring the transmission of a regular node towards selfish node. They gave the relationship between the number of nodes and their conditional probability towards selfishness.
Hernandez - Orallo et al  proposed a mathematical detection model to manipulate the time as well as cost of detecting selfish nodes with the aid of watchdogs. In this model the occurrence of communication between any two communications between any two mobile node follows Poisson distribution .The authors used two states to detect the behavior of nodes. They are NOINFO and POSITIVE states. The network was modeled as a time Markov chain represented with the help of a transition matrix in canonical form.
S.Buchegger and J.L.Boudec  proposed a Bayesian approach for determining the reputation of each and every participating node of ad hoc network. They used a Beta distribution which is the conjugate of Bernoulli distributions for determining reputation rating. Initially They took prior is BETA (1, 1) and the uniform distribution on BETA (0, 1). They computed reputations based on first hand observation and the reputation rating published by other nodes participating in the network. They classified nodes as normal and misbehaving based on threshold expression of tolerance. They also addressed the various issues that could arise due to reputation fading.
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C.Zouridzki et al  proposed a trust establishment framework for determining the reliability of packets forwarded by mobile nodes in an ad hoc network. They assumed that the each node forms an opinion about all other nodes with the help of first and second hand information collected from the network. This opinion metric was used as a parameter to detect misbehaving nodes so that the performance of the network does not degrade. They also used trust and confidence values which captures the statistical inference about the reliable delivery of packets.
Extract of the Literature
The Probabilistic approaches present in the existing literature for identifying selfish nodes lack in the following issues. They are:
A mechanism which can identify node's selfishness by means of conditional probability based Erlang distribution has not been proposed.
An approach which makes use of both the conditional and total probability for manipulating the network performance has not been explored.
3. The Proposed Model
Let us consider 'x' as the lifetime of the network, which contains both co-operative nodes (non-selfish nodes) and non co-operative nodes (selfish nodes).
Let 'y' be the random variable for the determination of selfish or non-selfish nodes given by
Y=0, if the nodes are selfish.
Y=1, if the nodes are non selfish.
To Manipulate the MTTS for node in the network, the conditional expectation of x when y is given by routing that when there is absence of selfish nodes in
Where λ is the rate at which the node becomes selfish. On the other hand, if a 'n' nodes becomes selfish, the mean life of the network is the sum of mean lifetime of the 'n' nodes that becomes selfish
Using the theorem of total expectation the MTTS of a node is
Thus, when a=0, the reliability of the network is not affected. But when a=1, the reliability of the network is affected. Given that the selfish node is present (y=1), the network lifetime, 'x' is the sum of two independent exponentially distributed random variables, each with parameter.
Thus the conditional probability of a x given y=1 is exlay distributed,
On the other hand, when there is selfish node, Initially then,
and mdf of the network nodes X is
Thus, the reliability of the node given by
Extensive simulation has been carried out to analyze the impact of conditional probability based mathematical model for detecting and mitigating selfish nodes in MANET. We mainly focused on finding out the minimum and the maximum threshold probability value that can enable highest optimal prediction of existence of selfish nodes. The analysis was also carried out to evaluate the network performance based on evaluation parameters namely Packet Delivery Ratio, Control Overhead and Total Overhead.
5.1 Simulation Setup
We conducted exhaustive simulations using the simulation tool NS-2.33. In the experimental setup, we set the maximum of 1000 packets for detecting selfish nodes which could be sufficient to continue the session up to the end of the simulation time. Some of the simulation parameters used for experimental analysis are simulation area of 1000X1000 Square meters, 100 mobile nodes, simulation time of 1000, packet size of 512 bytes and the traffic model used is a Poisson Constant Bit Rate model.
5. Experimental analysis of Conditional Probabilistic based Mathematical Model (CPMM)
From the fig.1, it is clear that the maximum number of selfish nodes could be detected when a range set of 0.25 is defined. It depicts the maximum threshold detection ranges 0.10 and the minimal threshold detection range is 0.40.
Fig.1: The Chart representing the variation in number of selfish nodes based on different range set for detection.
From fig.2, it is obvious that when the maximum threshold of 0.10 is set the Packet Delivery Ratio increases. This is because of the isolating and mitigating large number of selfish nodes. But, when a minimum threshold of 0.40 is chosen the Packet Delivery Ratio is considerably low to an extent of 14%.
Fig.2: Performance analysis for CPMM based on number of selfish nodes and Packet Delivery Ratio
From fig.3, it is obvious that when the maximum threshold of 0.10 is set the Control Overhead decreases. This is because of the isolating and mitigating large number of selfish nodes. But, when a minimum threshold of 0.40 is chosen the Control Overhead is considerably high to an extent of 26%.
Fig.3: Performance analysis for CPMM based on number of selfish nodes and Control Overhead
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From fig.4, it is obvious that when the maximum threshold of 0.10 is set the Total Overhead decreases. This is because of the isolating and mitigating large number of selfish nodes. But, when a minimum threshold of 0.40 is chosen the Total Overhead is considerably high to an extent of 19%
Fig.4: Performance analysis for CPMM based on number of selfish nodes and Total Overhead
This paper presented an extensive analysis of the impact of Conditional Expectation Based Probabilistic Mathematical Model for detecting and mitigating the existence of selfish nodes. The results predicts that our mathematical model deployed could outperform other mathematical model in terms of Packet Delivery Ratio, Control Overhead and Total Overhead computed by varying the minimum and maximum range of threshold detection. This solution also aided in framing an optimal threshold range in which maximum number of selfish nodes could be detected and mitigated.