Collusion is the malicious activity mostly frequent in agent-based recommender systems in which two or more agents agree with each other to mutually exchange high positive feedback in order to gain undue advantages by altering the correct computation of reputation measures in their agent communities. Therefore, identification of colluding agents is an important issue and several strategies have been developed to this purpose. Among them, the EigenTrust algorithm is well known, although it is limited by the necessity of knowing a priori which agents are considered as trustworthy and the impossibility of recognizing several groups of colluding agents acting simultaneously and autonomously. The focus of this paper is dealing with the above issues and, to this end, we will present a strategy to support EigenTrust both providing the necessary inputs about pre-trusted agents and recognizing groups of malicious agents. In particular, we combined EigenTrust with a clustering process in order to suitably grouping the agents according to their reputation scores. We carried out a preliminary tests which have shown promising results about the efectiveness of the proposed strategy.
The social skills of software agents make them able to support complex social relationships
inside their communities [
The possibility to measure the trustworthiness of an actor has been widely described in the
literature [
Conveniently, trust can be defined as: ”the quantified belief by the trustor with respect to the
competence, honesty, security and dependability of the trustee within a specified context” [
In particular, the trust is a subjective measure directly computed by the trustor agent about the trustee agent. If a trustor agent does not have a suitable knowledge about the trustee then the trustor must require the opinion of other agents. The measure of the trustworthiness that the whole community place on a given agent is identified with the term reputation (which is an indirect measure) and, in presence of malicious agents implementing selfish behaviors, including deceptions and frauds, the availability of reputation measures is essential to make a good choice of agents with which to collaborate. Unfortunately, the computation of reputation measures in the presence of several diferent groups of colluding agents, each one operating independently, can lead to a misrepresentation of agent reputation within the community, making the task of identifying malicious agents very dificult.
In the literature several strategies to detect malicious actors have been described (see Section 2),
among which an efective proposal is represented by the well-known EigenTrust algorithm [
Preliminary experiments carried out on a test case have provided promising results showing that our proposal is potentially able to identify several groups of colluded agents. Moreover, the obtained experimental results highlighted as the efectiveness of our method depends on the dimension of the agent community and the percentage of colluded agents present therein.
The rest of the paper is organized as follows. In Section 2 we present some related work, while in Section 3 we introduce the proposed strategy to identify colluded agents belonging to several independent groups, In Section 4, we provide an example of application of our method and describe the simulations we carried out to validate it. Finally, in Section 5 some conclusions are drawn and our ongoing researches will be discussed.
Given the multifaceted nature of reputation, it is an hard task that of providing its definition. In the following, we will refer to the definition of reputation derived by [ 20, 21], where reputation of an actor can be assumed to be the expectation about his/her future behaviors estimated on the basis of feedback about past his/her behaviors with regard to a specific context at a specific time made by other actors. In general, reputation influences future choices when they are based on expectations about future behaviors [22] and, in particular, it is central when a direct knowledge of the potential counterpart there not exist or it is not adequate, like large real or virtual communities. Moreover, the reputation knowledge reduces both information asymmetry between parties and the risks of deceptions [23].
To be efective, a reputation system (RS) needs to involve long-living entities, driving future decisions and living on feedback released by counterparts [24]. In turn, feedback tightly depends on their honesty in reflecting the real opinion of the trustor about the trustee and on its accuracy measured as diference with respect to the real trustee’s trustworthiness [20].
Other aspects of a reputation system, particularly relevant in presence of very large communities, relate to information management in a centralized or distributed architecture [25, 26] and/or in a local or global approach [27, 28]. The choice among these aspects is driven by community size, cost, scalability, computational and storage overhead, complexity, and other properties. For a more complete overview on those and other properties of the reputation systems, the interested reader may refer to [29, 30, 31, 32, 33].
Typically, opinions about a trustee (usually in numerical form) are collected and aggregated, often through a weighted average, to update the trustee’s reputation score within the community. In this light, in the popular eBay RS [34], sellers’ reputation scores are updated using feedback issued by buyers, who must also interpret the reputation scores based on their personal attitudes toward risk. Despite its various updates over time, the resilience of this RS is critical being exposed to various threats [35, 36, 37].
Designed over an overlay network, two performing RSs are i) PeerTrust [38], a distributed RS capable to identify reliable and malicious peers by aggregating direct feedback, indirect feedback (taking into account the source reliability), number of peers’ transactions and context and ii) Hypertrust [39], designed to work in large and competitive federations of utility computing infrastructure to discovery resources on the basis of reliability and reputation data.
The EigenTrust [40] RS assumes the transitivity of trust opinions and calculates the overall reputation of peers by aggregating and normalizing the trust representation of peers appropriately weighted by their trustworthiness. Trust values, arranged in a matrix, and converge asymptotically to the matrix eigenvalues. However, several strategies have been proposed to maliciously manipulate EigenTrust [41] scores and, likewise, several strategies have been proposed to improve the resilience of this [42, 43]. Against collusive activities this RS requires the presence of mentors whose opinions are assumed to be trustworthy, but to identify collusive actors there is no unambiguous criterion, e.g., an univocal threshold on reputation scores to divide honest peers from malicious ones.
We highlight how, to the best of our knowledge, [38], [39] and [40] are among the bestperforming RSs, although profoundly diferent both in terms of the application scenario, which influences their design, and the metrics adopted to calculate the actors’ reputation scores, which are in any case based on both direct and indirect contributions.
Finally, we would like to point out the existence of some RSs, such as FIRE [44], specifically designed for benign scenarios in the absence of malicious actors, which generally do not fit well in the real world.
More generally, RSs can be subject to a variety of attacks, of which collusion ones are common [45] and occur when two or more malicious actors secretly agree to engage in illicit activities aimed at changing the perception of their trustworthiness in the rest of the community.
In more detail, the more colluders there are, the less efective are the strategies aimed at identifying them. Such strategies are mainly based on [46]: (i) seeking and evaluating opinions that disagree with most other users; (ii) attending to sudden changes in reputation measures over time; (iii) assessing the accuracy and honesty of users giving feedback. In addition, to avoid specific colluders’ countermeasures, RSs often combine multiple strategies and mechanisms together to increase their resilience toward collusive activities [47]. However, other strategies to detect collusion attacks have been proposed in the literature. For example, modeling the detection of colluders using game theory such as in [48] or with the use of blockchain technology such as in [49] where the blockchain also supports a reputation-based voting scheme in which candidate history and recommended opinion are considered or in [50] where messages and path redundancy are adopted based on the reliability and performance of nodes.
Finally, not rarely more groups of colluders can act in a community at the same time, but only few RSs consider such an issue as in [51] where the search is based on the search of tight peers’ relationships or in [52] where the hypergraph theory is exploited to find whose vertices closely connected through hyperedges to find cluster of nodes tightly connected.
In a nutshell, Eigentrust operates in a social network scenario and its main idea is to calculate the reputation of each member i as the sum of all the trust values that each other user assigns to , weighted with ’s reputation. In practice, can be assumed to be the barycenter of all the . In short, Eigentrust exploits some information about known agents (mentors) considered as trustworthy by assigning them a very high reputation even though, at the same time, the reputation of all other agents is penalized by the algorithm. Although in this way EigenTrust can efectively detect malicious agents (among those agents not pre-trusted) the task of detecting mentors is not trivial.
To deal with this limitation of Eigentrust, our strategy is first to identify the best candidates to be considered as colluding agents and, at the same time, indicate all the other agents (who are assumed not to be colluding) as mentors. Then, a new algorithm for organizing agents into clusters is introduced; agents belonging to the clusters with the worst average reputation scores will be considered as colluded if they assign high trust values to each other belonging to the same cluster, while from the other agents in the community they receive low levels of trust.
We verified that our two-step strategy is very efective in presence of a single cluster of colluding agents, but in presence of multiple clusters of colluding agents, each operating independently from the other clusters, our strategy is able to identify only the one with the worst average reputation as a cluster of colluders. To overcome this drawback, a suitable threshold was introduced into the algorithm in order to consider as colluded all the clusters with an average reputation lower than this threshold. Next, the algorithm works in an iterative way by removing from the community the clusters (i.e., agents) thus identified, then the reputations of the remaining agents will be recalculated in order to identify additional clusters of colluded agents. The iterative process will end when all remaining clusters have an average reputation greater than the threshold used by our algorithm.
It is important to point out two current limitations of our approach. The first limitation is that the threshold under which a group should be considered colluded is decided arbitrarily by the social network administrator; while in some aspects this is reasonable, we are operating for automatically extrapolating information from the social community to make this process less arbitrary. The second limitation is related to the clustering algorithm employed, which in advance requires to know the number of clusters to be formed. Again, we are operating to overcome this limitation both by verifying the performance of other clustering algorithms that do not require such information in advance, and by trying to extrapolate from the social network the information necessary to reliably estimate the number of clusters to be used. Both of the above issues are the subject of our ongoing research.
Let us consider a social network of members, each one identified in an univocal manner by an integer , 1 ≤ ≤ . In such a context, the trust perceived by the member about the member is represented by the real number , 0 ≤ ≤ 1, ∀, = 1, . . . , , while the reputation that has in the social network is computed as the sum of all the trust values , = 1, . . . , , referred to a trustor , suitably weighted by the reputation of . More formally: (1) (2) = ∑︀=1 ∑︀ =1 , = 1, . . . , .
Let us denote with = [ ] the trust matrix. It can be assumed to be the transpose of the weighted adjacency matrix = [ ] of a directed graph . In each node corresponds to a user and a non negative value is assigned to the edge (, ) to represent the trust value placed by the member on the member . Therefore, the Equation (1) can be reformulated as: = , ‖‖1 = 1, where = (1, . . . , ) is the reputation vector and the 1-norm is defined as the sum of the absolute values of the vector components. However, to guarantee the reputation uniqueness, the vector in (2) must be normalized. Moreover, we impose equal to 1 the sum of the trust values assigned by each member to the other social network members because, in this way, the matrix will result column-stochastic.
A solution to the eigensystem problem given by Equation (2) can be equivalently reformulated in the computation of the stationary distribution for a Markov chain [53], which is represented by the matrix , named transition matrix. In this case, if we assume that has only positive elements, by applying the Perron Frobenius Theorem, then = 1 = ( ) is a simple eigenvalue of (the other ones are less than 1 in modulus), and there exists a unique vector ∈ R, ‖‖1 = 1, such that = ( ) = that is a unique positive reputation vector.
A modified version of (2) is the well-known PageRank model: ︀( + (1 − ) )︀ = (3) where 0 ≤ ≤ 1, is the vector of all ones, while is a non-negative vector with unitary 1-norm, that is = 1 and the vector is known as teleportation vector. In the original PageRank algorithm = 1 so that when ̸= 0, 1, the existence and uniqueness of the solution to (3) is confirmed.
In the next subsection, we propose a strategy that allows to detect malicious users based ob the matrix .
In this Section we present the proposed strategy to provide the EigenTrust algorithm with the inputs on pre-trusted agents and the information to identify groups of malicious agents, directly from the trust matrix = [ ] as defined in Equations (1) and (2).
More in detail, let , a pair of agents classified as malicious because we assume they collude when (i) and are high and they are similar from each other and (ii) diferently, the sum of the remaining trust values belonging to the rows and is low. In other words, in the matrix the values of and are high while these users receive low values from the majority of the other members.
To group the social network users, a Spectral clustering [54] has been adopted. This algorithm exploits the spectral transformation of a data similarity matrix to separate data in an Euclidean space and arrange them in a more complex way. A main advantage of this algorithm is to not make strong assumptions on the form of clusters so that it is suitable to solve a wide range of problems. Moreover, it is computationally eficient also on large data sets because when the similarity is chosen (a not trivial step) only the solving of a linear problem is required, without risking local minima or algorithm restarts.
From a practical viewpoint, for a set of data points (i.e., agents), assigned the similarity ≥ 0 between all the agents pairs , , we cluster similar data points into more groups. We represent the data as a graph = (, ), where each vertex represents an agent and each edge represents the similarity value between vertices and . In particular, two agents are connected if the similarity is positive and, in this case, the weight of the edge will be .
In other words, two agents will be connected if they will have a high similarity degree. Now, the clustering problem can be restated based on the similarity graph in order to find the partition of such that edges occurring between diferent groups will be denoted from very low weights (this means that they are dissimilar). Furthermore, the edges within a group will be denoted from high weights, to mean that the agents belonging to the same cluster are similar among them.
Based on the construction of the similarity graphs we model the local neighborhood relationships between the data points. In such a manner the link between users belonging to + the same category (malicious or honest) will be evident, with = 2 , where 0.1 + | − | + = 2 . Note that by construction, is a symmetric matrix and its elements 0.1 + | − | are characterized by high values if the users and are similar, low values if the users and are not similar.
More deeply, in such a way, we exploit the key principle of the Spectral Clustering techniques of considering the data as vertices of a graph and weighting the connections by the similarity between two vertices. On this basis, we assume the data of the training set as an approximation of a topological space which can be studied through the spectral properties of a matrix called the Laplacian, in order to perform an appropriate partitioning.
Accordingly to [54], the applied Spectral Clustering algorithm is the following. Input: Similarity matrix ∈ R× , number of clusters to construct. 1. Compute not normalized Laplacian = − where is the degree matrix and is the weighted adjacency matrix equal to .
(∑︀ 2) 21 2. Compute the normalized Laplacian = − 21 − 21 . 3. Compute the first eigenvectors 1, . . . , of . 4. Let ∈ R× be the matrix containing the vectors 1, . . . , as columns. 5. Form the matrix ∈ R× from by normalizing the rows to norm 1, that is set = . 6. For = 1, . . . , let ∈ R be the vector corresponding to the -th row of . 7. Cluster the points ()=1,..., with -means algorithm into 1, . . . , .
Output: Cluster 1, . . . , with = {| ∈ }.
The choice of the number of clusters provided in input to the algorithm will depend on both the number of users and the nature of social network. Moreover, the average reputation associated with each cluster is computed, according to Equation (1), as: ˜ = ∑︀∈ , ||
∀ = 1, . . . , .
Observe as the malicious users’ reputation is generally very low, although it can vary significantly based on context and the actions they carried out. In this respect, to discriminate honest from colluded users, based on some preliminary tests, the threshold has been set to = 0.46/ by taking into account the number of network users.
Exploiting we have identified, the clusters 1, . . . , with ˜ ≤ , with ≤ as clusters of colluding agents. Then the agents belonging to the clusters from 1 to identified as malicious colluding agents are removed from the network and the reputations of the remaining agents is recalculated as well as the average reputations of the remaining clusters. The process is reiterated until all the remaining clusters will have a reputation greater than .
To test the correctness of our method we carried out some preliminary tests simulating a little social network of 500 users. We did not limit ourselves to considering only honest or malicious users, because in reality users often have multiple characteristics, often not well defined, representing this a real challenge to test our method. For example, we considered malicious colluding users who receive medium-high trust values or those who receive high trust values from only a few. At the same time, we considered honest users who receive low trust values from the majority of users. In addition, we tried generating the test social network using diferent statistical distributions but found no significant diferences, probably due to the small size of the social network. This aspect will be clarified in future research when medium to large networks will be tested.
More specifically, with Matlab we generated a dataset and the results obtained varied according to the percentage of malicious colluding agents in the network. As shown in the Table 1 and as we can see better in the Figures 1 and 2, the detected malicious clusters, therefore the malicious agents, compared to what is identified by the strategy, increase as the percentage of efective ones increases. While, the percentage of false positives, i.e. of honest agents evaluated as malicious by the proposed strategy, decreases as the number of efective malicious agents in the community increases.
Finally, with the information obtained regarding the malicious users present in the community, we were able to calculate the reputation of the users by applying EigenTrust. The previously identified malicious ones can be considered as non-pretrusted.
Let us consider the case in which the actual malicious ones are 25%. The Figure (3) represents the initial reputation obtained by solving Equation (2) and the reputation obtained by applying Eigentrust. We desire to highlight how Eigentrust significantly lowers the reputation of malicious users, while the reputation of honest users, as we know, recognized as false positive, is slightly distorted, not marking a clear diference between very good and less good honest users.
However, we consider this as the result of preliminary tests that were only meant to verify the potential of the proposed strategy. In this regard, we can consider the results obtained by applying the proposed strategy as promising.
Number of users
In this paper, we have focused on the problem of identifying in a social network, more groups of colluded agents.
To this aim, among the systems proposed in the literature, we have considered the wellknown algorithm Eigentrust, that is recognized as one of the most efective solution to measure the reputation in a set of social agents. However, the EigenTrust algorithm is limited by the necessity of knowing a priori which agents are considered as trustworthy and the impossibility of recognizing several groups of colluding agents acting simultaneously and autonomously.
To address the problems above, in this paper we have proposed a diferent strategy, able to suggest, in an automatic fashion, information about the trustworthiness of agents, as required by EigenTrust, and combining a clustering stage with EigenTrust to group agents exploiting their reputation scores to detect groups of colluded agents. Then a recursive strategy has been implemented to identify each group of colluded agents and, iteration by iteration, cleaning the social network from the presence of such malicious actors.
Preliminary tests, preformed by simulating a little social network, have highlighted that our method is efective in identifying several groups of colluded agents. Results have shown as such efectiveness is connected to the dimension of the social community and the percentage of colluded agents present therein.
Forthcoming research will be focused on testing our method on very large community also to test its scalability, accuracy and complexity, refine the choice of threshold and study a strategy for automatically assigning the number of clusters to be examined, also by testing other clustering algorithms.
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