Improving Probability Expectation Definition in Subjective Logic

Improving Probability Expectation Definition in Subjective Logic Mehdi Seyrafi 1, Nasrollah Moghadam Charkari 2 1, 2 Electrical and Computer Engineering Department, Tarbiat Modares University,Tehran, Iran m.seyrafi, moghadam}@modares.ac.ir Abstract: Subjective logic extends probabilistic logic by also expressing uncertainty about the probability themselves. This makes reasoning in presence of uncertain or incomplete evidence conceivable. It has useful applications in trust management and reputation systems. Probability expectation is one of the subjective logic operators and definitions. It has an important effect on performance of systems based on subjective logic. In this paper we propose an enhanced definition for expectation in subjective logic. We applied it on a reputation system in MANET. Simulation results show better performance for the system when using enhanced expectation definition. Keywords: subjective logic, expectation, reputation, MANET. 1. Introduction 1.1. Subjective Logic Backed in Dempster-Shafer theory of evidence [1], Josang introduced subjective logic [2, 3], as an uncertainty favoring probabilistic logic. To enable reasoning with argument models in presence of uncertain or incomplete evidence, subjective logic extends probabilistic logic by augmenting the probability values with uncertainty about themselves. Arguments in subjective logic are called opinions. An opinion denoted by ω y:x = ( b y:x, d y:x, u y:x, a y:x ), expresses the relying party y s belief in the truth of statement x. Here b, d, and u and a represent belief, disbelief, uncertainty, and base rate respectively. They meet the following conditions: (1) The base rate is the a priori probability in the absence of committed belief mass. In other words, it is the default willingness of part y to believe truthfulness of statement x when y has no evidence of x. Therefore it determines how uncertainty is viewed as belief. Uncertainty is considered as disbelief for a base rate of 0.0, while it is regarded as belief for a base rate of 1.0. When coming to comparison of opinions or decision making based on them, the opinion should be transmuted into a single value. It is the probability expectation value, which in subjective logic is calculated by: E y:x = b y:x + u y:x a y:x (2) The more evidence about x is gathered by y, the more pise expectation it can compute. Therefore probability expectation E expresses the a posteriori probability, when there is some evidence of the statement x. For example, if the opinion of y about x is ω y:x = (0.8, 0.0, 0.2, 0.5), then its expectation is E y:x = b y:x + u y:x a y:x = 0.8+0.2 0.5 = 0.9. Also a completely uncertain opinion (0.0, 0.0, 1.0, a y:x ) always has an expectation equal to the base rate, i.e. E y:x = b y:x + u y:x a y:x = 0.0 + 1.0 a y:x = a y:x. Base rate then becomes the default expectation for unknown entities (nodes of MANETs, hereon). 1.2. Subjective Logic in Reputation Systems Multi-hop communication is the basis of connectivity in mobile ad hoc networks (MANETs), wireless mesh networks (WMNs) and some other types of networks. It relies on the cooperation of intermediate nodes in forwarding packets toward destination. However, some nodes may seek benefits in refraining from cooperation. Such selfish behavior of nodes may degrade overall performance of a network or even paralyze it. Thus preventing this type of behavior in such networks is an important issue to deal with. Various reputation systems [4-6] have been suggested and evaluated to incentivize nodes to cooperate. These systems generally involve some mechanisms to detect selfish nodes and carry some punishments for them. Isolating selfish nodes will restore network performance for remaining cooperative nodes. In such systems, initially, all nodes are completely uncertain about selfishness of each other. Every node has a kind of default positive reputation to have the chance to start interactions and gather more positive reputation. The past interactions of a node with its neighbors should be gathered as the evidences to discover selfish behavior of the node. The evidence may be obtained from dit interactions with the desired node, or captured ommendations from other neighboring nodes. To aggregate the collected evidence into a single value, handle contradictions and decide whether a node is selfish or not, a logic or framework is required. Subjective logic is an attractive candidate. It has ently been popular in reputation systems. In reputation system proposed in [7], every node collects evidences of its neighboring nodes. The evidences are merged into a subjective logic opinion and the opinion is saved in a reputation table. Upon facing a new evidence (having an interaction with a specific node), it is combined in the current opinion about that node. The new opinion then replaces the old one. This helps avoiding the need to save evidences one by one, and thus decreases memory usage. When a cooperative node eives a request from another node, it checks its reputation table to find out whether the requester is selfish or not. If there is not enough evidence about the requester, it calls neighbors whether they have any 107

useful opinion about the requester. After collecting all ommendations, they are merged into a single opinion using subjective logic cumulative fusion operator [3], to form the final opinion. Different kinds of weighting average operators have been applied to merge the eived ommendations [7-10]. Expectation of not being selfish for the requester node is computed by applying subjective logic expectation operation in equation (2) to the final opinion. If it falls below a predefined threshold, it will be considered as selfish and therefore its service request is rejected; otherwise its request is accepted and service is granted. We enhanced standard subjective logic expectation formula to facilitate rapid detection of selfish nodes. Simulations show using our definition expectation results in considerable improvement in decreasing success rate of selfish nodes compared to former related works. 2. Deficiencies of Current Expectation Definition As stated in section 1, the base rate a is the a priori probability in the absence of any evidence. In the field of MANETs, this means if there is no evidence of the behaviors of a node, its probability expectation to cooperate in forwarding packets from others, is equal to the base rate. Therefore setting a proper value as the base rate is very important. Setting a proper value for base rate requires good knowledge about nodes in the MANET. This is not normally feasible, as administration of nodes may be independent and with different requirements. Therefore base rate is usually set to 0.5. This leads to a value of 0.5 for expectation E, at the startup of operation of MANET. If a node is completely selfish, it does not forward any requested packets. Then its real expectation to forward packets is 0. To hide the selfishness and have some chance to get service from others, selfish nodes may cooperate with a probability γ ϵ [0, 1). In this case, the real expectation for selfish nodes is γ. The cooperation rate of selfish nodes (γ) is not known by others when the network starts up or when a new node joins the network. While gathering more evidence, uncertainty decreases, and computations result in more pise values for expectation. So computed expectation converges to cort value, which is γ. Consider a node y and its neighbor x in the reputation system used in [9]. After node y interacts with node x, it updates its opinion about node x using the following algorithm: If node x accepts the request from node y, the interaction is positive. Then y adds the value δ to its belief about node x (b y:x ), and decreases the uncertainty about x (u y:x ) by δ. If node x drops request from node y, the interaction is negative. Then y adds the value δ to its disbelief about node x (d y:x ), and decreases the uncertainty about x (u y:x ) by δ. The peding algorithm is followed until u y:x reaches zero. Then for upcoming interaction: for positive interactions: (3) for negative interactions: (4) Figure 1 shows how expectation E of selfish node x changes in the view of node y, while interactions are performed. Here we have γ=0.3 and δ=0.01. Figure 1. Expectation (E1) of a selfish node to cooperate using standard subjective logic definition, where cooperation rate γ = 0.2, step size δ = 0.001 and base rate ɑ = 0.5. As can be seen in Figure 1, it takes a period of time to come to true expectation. E changes from the value a at startup, to γ after 1/δ steps, in which δ is the change value applied to opinion parameters after each interaction. In that period, the incort expectation considered for node x, is greater than the real value. It causes node y to accept service requests from selfish node x. This means that it takes some time for node y, to ognize node x as a selfish node. Convergence time is the average value of the time for all cooperative nodes to detect selfish nodes. We seek smaller convergence time to have a better performance of the reputation system. 3. Proposed Improvement In order to decrease the convergence time, we should decrease the effect of a on calculation of E, while more evidence is being collected. To yield this, we replace base rate a in equation (2) with a weighted average of base rate and the proportion of positive interactions amongst total number of interactions: (5) In equation (5), ω 1 and ω 2 are the weighting factors. At startup, when there is no evidence, b and d are zero. So ω 1 108

should be zero and ω 2 (weight of base rate) should be 1 (the maximum value). When enough evidence is collected, b+d becomes 1. At this time the weight of evidences should have reached to its maximum value, while base rate should have lost its effect on expectation. Therefore at this time ω 1 is 1 and ω 2 is 0. To meet the aforesaid criteria for ω 1 and ω 2, we choose ω 1 = 1-u and ω 2 = u. Substituting ω 1 and ω 2 in equation (5), we have: [ ] Despite linear relation of E and u in equation (2), we come to a nonlinear relation in equation (6). Now we consider equation (6) from a different perspective. If we factor out u, we have: Equation (7), suggests that E 2 can be calculated by replacing a with E 1 in equation (2). We can further replace a in equation (6) with E 1 to come to the equation of E 3 with degree 3: The same way we can have E 4 : We use the same scenario described in section 2 for all E 1 to E 4 to create Figure 2. (6) (7) (8) (9) MANETs to find and isolate selfish nodes. It is based on the system defined in [7] and [9]. 4.1. Preliminaries Any request from a node such as forwarding a packet, declaring a route, etc. can be simulated by sending a generic service request, and then eiving a request acceptance reply, request rejection reply or not eiving any reply from the target. When a packet is sent to a neighbor expecting it to forward, if the neighbor does forward it, the originator node can hear that too. The original packet can be regarded as a service request and the forwarded packet can be regarded as a service acceptance reply. If the neighbor selfishly drops that packet, no packet can be heard by originator and it can be regarded as not having any response to service request. We assume there are two types of nodes in network: Selfish nodes, which do not accept service requests from other nodes. However they may try to hide their selfishness by accepting requests with a probability of γ ϵ [0, 1). If a request is not to be accepted, it will be silently dropped; no reply is sent back. Cooperative nodes, which always accept requests from other nodes known to be cooperative. Interactions between nodes fall into three categories: If a request acceptance reply is eived, it is considered a positive interaction. If a request rejection reply is eived, it is considered an uncertain interaction. If no reply is eived within a defined timeout, it is considered a negative interaction. 4.2. Dit Opinion Each node saves an opinion about every one of its dit neighbors in its reputation table. At startup, each node assigns an uncertain opinion ω uncertain =(0,0,1,0.5) to every neighbor. Any positive or negative interaction alters node s knowledge about the neighbor and reduces uncertainty about it. δ ϵ [0, 1] is defined to determine how much change should be applied to opinion after an interaction. Opinions on reputation table are updated using the following algorithm: If the interaction is positive, If u δ, then Figure 2. Expectation of a selfish node to cooperate, using different definitions of expectations, where E1 is the standard subjective logic definition, cooperation rate γ = 0.2, step size δ = 0.001 and base rate ɑ = 0.5. Else, As Figure 2 shows, all expectations converge to final value (γ). But it converges more rapidly for a greater degree of equation E. So we expect less total convergence time in a MANET to find selfish nodes when using E 2 or E 3 instead of standard expectation (E 1 ). We also expect less average success rate for the selfish nodes in the network. If the interaction is negative, If u δ, then Else, 4. A Reputation System as Test-bed We define a subjective logic based reputation system for 109

In case of an uncertain interaction there would be four cases: If b,d δ/2, then Else if b< δ/2 and d δ/2, then Else if b δ/2 and d < δ/2, then Else if b,d< δ/2, then Obviously as number of interactions increases, the uncertainty value decreases to zero, resulting in more accurate opinion. 4.3. Recommendations When cooperative node y eives a service request from node x, it retrieves opinion ω y x from its reputation table. If u y x, information is enough for decision about node x. So node y assumes ω y x as final reputation ω final y x. If u y x > β, node y decides that there is not enough information about node x. So it broadcasts a reputation query packet to its dit neighbors and waits for ommendations from them. Upon eiving a reputation query, each neighbor checks its reputation table for node x. If it has already interacted with node x, its uncertainty about node x is smaller than 1. So it responds node y with its reputation about node x. Node y waits for a period of T seconds to accumulate all ommendations. Then it merges ommendations to build up a single ommended opinion ω y x b y x y x u y x a y x. The merge process is actually a weighted average of ommendations belief, disbelief and uncertainty as follows: b y x d y x u y x (10) Where R denotes the set of ommenders, and V i is the weight factor computed as: E y i (11) E y In equation (11) E y:i is the expectation of y about i, and C i:x is the familiarity value introduced in [9]. It is proportional to how node i is certain about node x, and is calculated as: C i:x = b i:x +d i:x = 1 - u i:x (12) 4.4. Final Opinion As mentioned in previous section, if uncertainty about service requester node is less than the threshold β ( u y x ), information is enough for decision about node x. So final node y assumes ω y x as the final reputation (ω y x ω y x ), otherwise node y calculates the ommended opinion ω y x using ommendations eived from other nodes. In this case, subjective logic cumulative fusion operator is applied to ommended and opinions to compute final opinion: final ω y x ω y x ω y x b y x d y x u y x 4.5. Decision About Selfishness b y x u y x b y x u y x u y x u y x u y x u y x d y x u y x d y x u y x u y x u y x u y x u y x u y x u y x u y x u y x u y x u y x y x (13) After computing final opinion ω final y x, it is time to decide whether node x is selfish or not. This is done by comparing its expectation with the minimum required expectation α. If final E y x, node x is considered as cooperative by node y, its request is accepted and a request acceptance reply packet is sent back to it. Otherwise it is considered selfish and a request rejection reply is sent back. 5. Evaluation We evaluate our definition of expectation by incorporating it on a MANET reputation system in a simulation environment. The reputation system is detailed in section 4. The scenario is run 2 times, first applying standard subjective logic expectation defined in equation (2), and then applying our enhanced expectation definition (E 2 ) in equation (6). We then compare the performance results of the reputation system to find the influence of expectation definitions. 5.1. Simulation Environment OMNET++ 4.1 is used as the platform for simulation. To simulate MANETs, inetmanet framework is used in OMNET++. We use AODV routing protocol on an IEEE 802.11g MAC protocol. For simplicity, the 802.11 MAC is changed to work ideally. No signal attenuation and conflict is considered. Transmission range for all nodes is set to 100 meters. 110

(a) (b) (c) (d) Figure 3. Average success rate of selfish nodes (10 out of total 50 nodes are selfish) (a) γ=0, speed=0m/s (b) γ=0, speed=2m/s (c) γ=0.2, speed=0m/s (d) γ=0.2, speed=2m/s 50 nodes are randomly deployed in an area of 640 640 Table 1. Simulation Parameters meters. 10 nodes are selected to act selfishly. Each node selects a random neighbor every 5 seconds and sends it a service request. Packets are UDP based with 512 bytes fixed data size. To mitigate the effect of random seed selection in software, all experiments in our simulations are repeated 16 times with different random seeds, and the resulted values are averaged. Parameters details are listed in Table 1. Parameter Simulation area Number of mobile nodes Value 640 640 m 50 5.2. Performance Metric We consider success rate of selfish nodes to evaluate the model. Success rate is defined as proportion of service requests accepted by cooperative nodes per total number of requests sent to them by a specific node, at a given period. The more selfish nodes are detected, the less success rate selfish nodes would have. Detection process of selfish nodes causes this rate to decrease more and more for selfish nodes. We average this value from all selfish nodes. The more rapidly it decreases for all selfish nodes, the more efficient is selfishness detection scheme. 5.3. Simulation Results Figure 3 shows average success rate of selfish nodes in the network. As can be seen in all cases, when using enhanced expectation definition (E 2 ), success rate of selfish nodes drops more rapidly compared to using standard subjective logic expectation (E 1 ). When there is no mobility (a) and (c), selfish nodes are detected more rapidly and their success rate will soon drop to value γ. While selfish node moves around, it faces new nodes, which have no evidence of its past behavior. They provide service to the selfish node until they ognize it as a selfish one. Therefore mobility of selfish nodes makes it more difficult for the whole network to detect them. So, as can be seen in (b) and (d), success rate of selfish nodes needs more time to reach the final value. As shown in Figure 3 (b) and (d), the performance is even more improved in the face of mobility. 111 Simulation time 1600 s Media Access Control (MAC) Protocol IEEE 802.11g Transmission range Routing protocol Traffic type Packet rate Packet size Maximum node velocity Mobility model 100 m AODV CBR 1 every 5 sec. 512 bytes 0, 2 m/s Random WP Cooperation rate of selfish nodes γ 0.0, 0.2 Uncertainty threshold β 0.1 Minimum required expectation α 0.3 Reputation query period T Service request timeout (detect negative interaction) 4.5 s 4 s Base rate a 0.5

6. Conclusion In this paper we have defined an enhanced method to compute probability expectation for subjective logic opinions. Our new definition decreases the effect of base rate more rapidly than the standard one, when there is some evidence about the truth of the statement. Simulations on a MANET reputation system show better performance when using this enhanced definition instead of the standard one. Acknowledgement The work reported in this paper has been supported in part by the Informatics Services Corporation (ISC) 1. References [1] G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, 1976. [2] A. Josang, "A Logic for Uncertain Probabilities," International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9(3), pp. 279-311, 2001. [3] Audun Jøsang. Subjective Logic. Draft book, January 2012. (http://folk.uio.no/josang/papers/subjective_logic.pdf, July 10, 2012) [4] Malamati Louta, Stylianos Kraounakis, Angelos Michalas, "A SURVEY ON REPUTATION-BASED COOPERATION ENFORCEMENT SCHEMES IN WIRELESS AD HOC NETWORKS," International Conference on Wireless Information Networks and Systems, July 26-28 2010, Athens, Greece, pp. 90-93. [5] Chengqi, S. and Z. Qian, "Protocols for stimulating packet forwarding in wireless ad hoc networks," Wireless Communications, IEEE, vol. 17(5), pp. 50-55, 2010. [6] A. M. Raafat, M. Fathy, A. Yehia, M. A. Azer, "Cooperation incentives in wireless Ad hoc networks," 2nd International Conference on Education Technology and Computer - ICETC'10, 2010 [7] P. Kane, P.C. Browne, "Using uncertainty in reputation methods to enforcecooperation in ad-hoc networks," Proceedings of the 5th ACM Workshopon Wireless Security, September 2006, pp. 105 113 [8] Venkat Balakrishnan, Vijay Varadharajan, and Uday Tupakula, "Subjective Logic Based Trust Model for Mobile Ad-hoc Networks,"Proceedings of the 4th ACM International Conference on Security and Privacy in Communication Networks(SecureComm 2008), Istanbul, Turkey, September 2008. [9] Yining Liu, Keqiu Li, Yingwei Jin, Yong Zhang, Wenyu Qu, A novel reputation computation model based on subjective logic for mobile ad hoc networks, Future Generation Computer Systems, vol. 27,pp. 547-554, 2011. [10] Hui Lin, Li Xu, Jianliang Gao and Kai Yang, "A subjective logic based dynamic trust mechanism for voice over internet protocol (VOIP) over wireless mesh networks (WMNs)," Scientific Research and Essays, vol. 6(18), pp. 3873-3884, 1September, 2011. 1 www.isc.iranet.net 112