Charif Haydar
email: charif.alchiekhhaydar@loria.fr

Azim Roussanaly
email: azim.roussanaly@loria.fr

Anne Boyer
email: anne.boyer@loria.fr

Individual Opinions Versus Collective Opinions in Trust Modelling

Keywords: Trust modelling, Subjective logic, Recommender system, Collective trust

Social web permits users to acquire information from anonymous people around the world. This leads to a serious question about the trustworthiness of the information and the sources. During the last decade, numerous models were proposed to adapt social trust to social web. These models aim to assist the user in becoming able to state his opinion about the acquired information and their sources based on their trustworthiness. Usually, opinions can be based on two mechanisms to acquire knowledge: evaluating previous interactions with the source (individual knowledge), and word of mouth mechanism where the user relies on the knowledge of his friends and their friends (collective knowledge). In this paper, we are interested in the impact of using each of these mechanisms on the performance of trust models. Subjective logic (SL) is an extension of probabilistic logic that deals with the cases of lack of evidence. It supplies framework for modelling trust on the web. We use SL in this paper to build and compare two trust models. The first one gives priority to individual opinions, and uses collective opinions only in the case of absence of individual opinions. The second considers only collective opinions permanently, so it always provides the most complete knowledge that leads to improving the performance of the model.

I. INTRODUCTION

Web 2.0 provides a highly connected social environment. It allows data exchange among anonymous people from all around the world. Acquiring information from such sources raises the question about its reliability and trustworthiness. Modelling social trust into computational trust appeared to overcome the trustworthiness problem (for both information and resources). Today, computational trust is integrated in many domains and contexts such as social networks, recommender systems [START_REF] Massa | Using trust in recommender systems: an experimental analysis[END_REF], [START_REF] Golbeck | Computing and applying trust in web-based social networks[END_REF], file sharing [START_REF] Kamvar | The eigentrust algorithm for reputation management in P2P networks[END_REF], etc.

We consider social trust as the belief of an individual, called truster, that another individual, called trustee, has the competence and the willingness to either execute a task to the favour of the truster, or to assist him to execute it. The assistance can simply be recommending another individual to execute the task. The truster tries to acquire information and constructs his own belief about the trustee before deciding to cooperate with him. Building truster's opinion about the trustee is mainly derived by two means; the first is by exploiting previous interactions between both of them, so the truster relies on his own knowledge about the trustee (individual opinion). The second uses the word of mouth mechanism, where the truster exploits the collective knowledge of his trustee friends and their friends (collective opinion). Local trust models are those which exploit individual opinions as they are available, and collective opinions otherwise [START_REF] Massa | Using trust in recommender systems: an experimental analysis[END_REF], [START_REF] Josang | Trust network analysis with subjective logic[END_REF], [START_REF] Golbeck | Computing and applying trust in web-based social networks[END_REF].

Our objective in this paper is to show that collective opinions can be fruitful and efficient to improve the trust based recommender system's performance even in the presence of individual opinions. We show this by comparing two trust based recommender systems: the first relies on a classical local trust model, and the second uses permanently collective opinions. Both of them are based on the subjective logic (SL) [START_REF] Josang | Subjective logic[END_REF], which is an extension of probabilistic logic, based on the belief theory [START_REF] Dempster | The DempsterShafer calculus for statisticians[END_REF], [START_REF]Advances in the Dempster-Shafer theory of evidence[END_REF]. SL provides a flexible framework form modelling trust.

The object of our comparison is the dataset stackoverflow [START_REF]Stack overflow website[END_REF]. It is a social website based on a question answering platform to assist users to find answers to their questions in diverse domains (programming, mathematics, English language, cooking, etc.). We assume that proposing an answer is a proof of willingness to assist the person asking. Therefore, our objective is to find the user capable to provide the most relevant answer.

The paper is organized as follows: in Section II, we present the general framework, starting by presenting social trust and computational trust. In II-B, we introduce subjective logic and some of its operators. In Section III, we describe both individual and collective trust models that we propose. In Section IV, we describe the used dataset, and present our evaluation method. In Section V, we discuss our results. Finally in Section VI, we present our conclusions and future work.

II. GENERAL FRAMEWORK

The objective of trust is to find the appropriate person to cooperate with in order to achieve a given task or context. Truster's decision about to cooperate or not is influenced by many factors such as: the context, the completeness his opinion about the trustee, the emergency of the task for him and many more. In the following section, we present a real life example about trust in order to explain this phenomena, and some factors that can influence the cooperation decision.

Suppose that Alice wants to paint her house. She publishes this information and receives three offers from three professional candidates (Eric, Fred and George) willing to do the job for her. She already knows Eric because he painted her clinic sometime ago. Alice does not know neither Fred nor George. If Alice is satisfied by the job of Eric in her clinic, she might hire him for the house directly, and ignore the offers of Fred and George. Nevertheless, if Alice is perfectionist, she will investigate about them. Alice can ask her friends (Bob and Caroline) about Fred and George.

Suppose that Bob says that Fred is a good professional. Caroline says that she recently hired George to paint her house and she is not satisfied about his work, whereas her sister Diana has hired Fred and was satisfied. Note that even though Alice trusts Bob and Caroline, she will not ask any of them to paint her house, because she thinks that they lack competence. Even so, they are still capable to play an important role as advisers or recommenders.

After the suggestions of Bob and Caroline, Alice will eliminate George and choose between Eric and Fred.

In this scenario, Alice asked her friends only about the candidates that she herself does not know. The scenario could have been changed if she asked them also about Eric. Bob could say for example that Eric is good for concrete walls used in Alice's clinic, but he is not very competent for wooden walls like those of Alice's house. This information can be sufficient to convince Alice to hire Fred instead of Eric.

This example shows the limit of direct interactions manner, and that the word of mouth may be useful to enrich the knowledge of the truster about the trustee. It can lead to sharpen his decision even when he thinks that his own acquired knowledge is sufficient to take a decision. Furthermore, this scenario allow us to distinguish four types of trust relationships; these types are also discussed in [START_REF] Josang | Trust network analysis with subjective logic[END_REF]:

1) Direct trust: trust is the result of interactions between exclusively the truster and trustee, such as the relations "Alice Bob" and "Alice Eric". 2) Indirect trust: the two persons do not know each other. Trust is established due to trustee intermediate persons, such as the relation "Alice Fred". 3) Functional trust: the expectation of the truster is that the trustee accomplishes the task himself, such as the relation "Alice Eric", "Alice Fred" and "Alice George". 4) Referential trust: the expectation of the truster is that the trustee will recommend someone to accomplish the task, such as the relation "Alice Bob" and "Alice Caroline". Note that the recommendation of Caroline is also based on her referential trust in her sister Diana. In other words, no obligation for the trustee in referential trust to base his recommendation on a functional trust relation. Normally a series of referential trust relations must end with one functional trust relation [START_REF] Josang | Semantic constraints for trust transitivity[END_REF].

Fig. 1 illustrates the trust network used by Alice to make her decision.

In the next section, we discuss the formalization of social trust for the social web, and compare the different models that exist. • Quantitative, also called global-trust or reputation: the system computes a score for each user, this score represents his global trustworthiness. This score is considered when any other user needs to interact with this user [START_REF] Ziegler | Spreading activation models for trust propagation[END_REF].

• Qualitative, also called local-trust or relationship: it takes into account the personal bias. It is represented as user to user relationship. It is the trustworthiness of a user Y from the point of view of one single user X [START_REF] Ziegler | Spreading activation models for trust propagation[END_REF].

• Process driven (system): it represents the trust of the users in the system [START_REF] Kwan | Trust and online reputation systems[END_REF].

This work focuses on the qualitative axes. Most local trust models [START_REF] Massa | Using trust in recommender systems: an experimental analysis[END_REF], [START_REF] Abdul-Rahman | Supporting trust in virtual communities[END_REF], [START_REF] Krukow | Towards a Theory of Trust for the Global Ubiquitous Computer: A Dissertation Presented to the Faculty of Science of the University of Aarhus in Partial Fulfilment of the Requirements for the PhD Degree[END_REF], [START_REF] Mui | Computational Models of Trust and Reputation: Agents, Evolutionary Games, and Social Networks[END_REF] tend to formulate local trust problem in the form of a trust network. A trust network is a directed weighted graph where vertices represent users and edges represent trust relationships. Models differ by their notation of edges, and their strategies in traversing the network to compute trust between two unconnected users. This operation is called Trust propagation. It is fundamental in local trust models, as it allows to estimate how much a user A (called source node) should trust a user B (called destination node).

Computational trust is applied to many fields in artificial intelligence, recommender systems, file sharing, Spam detection, networks security, etc. Most computational models are fitted to their application fields and context. Basically, we identify two categories. Models dealing only with trust relationships, and models dealing with trust and distrust relationships.

The first category contains numerous models such as [START_REF] Tang | Managing trust in peerto-peer networks[END_REF], [START_REF] Richardson | Trust management for the semantic web[END_REF], [START_REF] Zaihrayeu | IWTrust: improving user trust in answers from the web[END_REF], [START_REF] Falcone | A fuzzy approach to a belief-based trust computation[END_REF], [START_REF] Almenrez | PTM: a pervasive trust management model for dynamic open environments[END_REF], [START_REF] O'donovan | Trust in recommender systems[END_REF]. The main disadvantage of this category is that models do not distinguish between distrusted and unknown persons. Social systems have to give chances to new and unknown users to prove their trustworthiness, whereas it must be more severe in blocking distrusted and malicious users [START_REF] Burke | Identifying attack models for secure recommendation[END_REF]. Unknown users are often new users, a system unable to distinguish them from distrusted users risk to be very severe with them, so discourage the evolution of the trust network, or to be so tolerant even with distrusted users, so less efficient.

Models in the second category distinguish between unknown and distrusted people. Models in [START_REF] Cock | A many valued representation and propagation of trust and distrust[END_REF], [START_REF] Guha | Propagation of trust and distrust[END_REF], [START_REF] Kuter | Using probabilistic confidence models for trust inference in web-based social networks[END_REF], [START_REF] Josang | Trust network analysis with subjective logic[END_REF], [START_REF] Victor | Gradual trust and distrust in recommender systems. fuzzy sets and systems[END_REF], identify three possible cases: trust, distrust and ignorance. Authors in [START_REF] Victor | Gradual trust and distrust in recommender systems. fuzzy sets and systems[END_REF] classify these models into two groups; gradual models [START_REF] Cock | A many valued representation and propagation of trust and distrust[END_REF], [START_REF] Guha | Propagation of trust and distrust[END_REF], [START_REF] Victor | Gradual trust and distrust in recommender systems. fuzzy sets and systems[END_REF] and probabilistic models [START_REF] Kuter | Using probabilistic confidence models for trust inference in web-based social networks[END_REF], [START_REF] Josang | Trust network analysis with subjective logic[END_REF]. Gradual representation of trust is more similar to the human way in expressing trust, whereas probabilistic representation is more meaningful mathematically.

We use subjective logic [START_REF] Josang | Trust network analysis with subjective logic[END_REF], [START_REF] Josang | Subjective logic[END_REF] (SL) in our models. Our choice is motivated by many factors. SL considers trust ignorance and distrust relationships, which is compatible with our need to distinguish between unknown and distrusted people. Most other trust models consider the creation and the evolution of trust links as an external issue, they describe and deal with existing links. SL is more transparent about this issue, trust relationships in SL are based on the accumulation of interactions between a couple of users. It proposes many operators that allow to integrate many aspects and factors of trust, which make it one of the most generic and flexible trust models.

It is based on the belief theory [START_REF] Dempster | The DempsterShafer calculus for statisticians[END_REF], [START_REF]Advances in the Dempster-Shafer theory of evidence[END_REF] which offers the capacity to aggregate many beliefs coming from many sources (even contradictory ones), which corresponds to the case when a user has to aggregate the opinions of many friends of him about a given problem.

The following section II-B is dedicated to explain the structure and some operators of subjective logic.

B. Subjective logic

Subjective logic (SL) [START_REF] Josang | Subjective logic[END_REF] is an extension of probabilistic logic, which associates each probability with a degree of uncertainty. Subjective logic allows to build models that treat with situations of incomplete evidences.

Belief theory [START_REF] Dempster | The DempsterShafer calculus for statisticians[END_REF], [START_REF]Advances in the Dempster-Shafer theory of evidence[END_REF] is a special case of probability theory dedicated to treat incomplete knowledge. The sum of probabilities of possible cases can be less than 1. Subjective logic [START_REF] Josang | A logic for uncertain probabilities[END_REF] offers a belief calculus using a belief metrics called opinion. The opinion of an individual U about a statement x is denoted by: ω U x = (b, d, u, a) where: b, d, u ∈ [0, 1] are respectively the belief, disbelief and uncertainty of U about x. The sum of the three values equals to one (i.e b + d + u = 1). Base rate a ∈ [0, 1] is the prior probability. Basically, base rate is a statistical measure applied in cases of evidences' absence. For example, when we know that the percentage of a disease x in a given population is 1%, then the base rate of x's infection is 1%. When we meet a new individual who did not make a test for the disease, a priori we assume that the probability that he is infected is 1%. In social trust cases, while no a priori statistics are present, we consider that unknown person has a half chance to be trustworthy. So we use a base rate a = 0.5. In subjective logic, the base rate steers the contribution of the uncertainty in the computation of the probability expectation value according to 1:

E(ω U x ) = b + a × u (1)
The opinion in subjective logic is based on the accumulation of successful and failed experiences. After each experience, U updates his opinion about x consistently with experience's outcome. According to this description, opinion can be represented as a binary random variable. Beta distribution is normally used to model the behaviour of this kind of variables. By consequence, the opinion corresponds to the probability density function (PDF) of beta distribution. PDF is denoted by two evidence parameters α and β that can be written as functions of the number of successful and failed experiences respectively.

α = r + W × a β = s + W × (1 -a) (2)
where r the number of successful experiences (evidences). s is the number of failed experiences. W is the noninformative prior weight which ensures that the prior (i.e., when = s = 0) Beta PDF with default base rate a = 0.5 is a uniform PDF (normally W = 2).

The expectation value of beta PDF is:

E(Beta(p|α, β)) = α α + β = r + W a r + s + W (3)
In subjective logic, the mapping between the opinion parameters and the beta PDF parameters is given as follows:

b r (r + s + W ) (4)
d = s (r + s + W ) (5)
u = W (r + s + W ) (6)
Table I shows an example of the evolution of an opinion with successive interactions. In the first line of Table I, we see the case of absence of evidences (experiences). The opinion is completely uncertain (u = 1). In this case, according to 1, the expectation value equals to the base rate value. The arrival of new experiences, will make the uncertainty decreases, regardless if these experiences are successful or failed. Successful experiences will augment the belief, whereas failed experiences will augment the disbelief.

Subjective logic opinions can be illustrated in the interior of an equilateral triangle. The three vertices of the triangle are called belief, disbelief, and uncertainty. The uncertainty axis links the uncertainty vertex with the opposite edge (the beliefdisbelief edge), the uncertainty value of the opinion is plotted on this axis considering that its contact with the edge beliefdisbelief represents the value 0, whereas the contact with the uncertainty vertex represents the value 1. In the same way we describe the belief and the disbelief axis.

The opinion is represented by the intersection point of the three projections on the three axis (belief, disbelief and certainty) as shown in the example in Fig. 2. The bottom of the triangle is the probability axis, the probability expectation value is the projection of the opinion point on the probability axis with respect to the line linking the uncertainty vertex with the base rate point on the probability axis. Fig. 2 illustrates an example of opinion mapping in subjective logic. The opinion is represented by a point inside the triangle. The point is the intersection of the projection of the three values b, d, and u on the axis of belief disbelief and uncertainty respectively. the probability expectation value E(x) is the projection of ω x on the probability axis directed by the axis linking a x with the uncertainty edge. Subjective logic proposes the uncertainty favouring transitivity. This operator enable the user A to receive the opinion of a friend C his trustee friend B, or to ignore the opinion of B in case of A distrust Formally the operator is given by 7

ω A B = b A B , d A B , u A B , a A B ω B C = b B C , d B C , u B C , a B C ω A B ⊗ ω B C =        b A:B C = b A B .b B C d A:B C = b A B .d B C u A:B C = d A B + u A B + b A B .u B C a A:B C = a B C (7)
2) Opinion fusion: Suppose in the previous example that A has another trustee friend D who also trusts C. A has two separate sources of information about C.

Subjective logic proposes two main types to fuse B's and D's opinions about C:

ω C B ⊕ω C D =          b C B⋄D = b C B .u C D +b C D .u C B u C B +u C D -u C B .u C D d C B⋄D = b C B .u C D +b C D .u C B u C B +u C D -u C B .u C D u C B⋄D = u C B +u C D u C B +u C D -u C B .u C D (8)
This operator allows the user to aggregate the opinions of his trustee friends, regardless if their opinions were contradictory or not.

III. PROPOSED MODELS

The aim of our models is to predict the most relevant answer to a given question within a list of answers. Basically, trust models consider that the question owner will trust more the answers written by trustworthy people, so they try to retrieve these users. We have developed two trust aware models. Both of them use subjective logic. We refer to them as individual trust model (ITSL), and collective trust model (CTSL). ITSL is a classical local trust model, so it exploits only individual opinions when they are available, otherwise it exploits collective opinions. CTSL exploits collective opinions all the time.

A. Individual trust subjective logic (LTSL)

This model is basically based on the model proposed in [START_REF] Josang | Trust network analysis with subjective logic[END_REF]. It consists of building a local trust network between users. The edges of this network are SL opinions of users about each other. Formally, we represent the trust network as a graph G = (V, E) where V represents the set of vertices (users), and E represents the set of edges (direct trust relationships). Suppose that a user a asks a question q, a set of users R will propose many answers to him. The aim of the trust model is to compute a score for each user r ∈ R using the trust network. The trust model estimates that a will accept the answer proposed by the highest score member of R. Local trust computes the score according to 9:

score(r) = e(a, r) if e(a, r) ∈ E j ⊕[e(a, f j ) ⊗ e(f j , r)] elsewhere (9)
where: e(a, r) is the direct opinion (edge) of a in r. f j is a member of F , the set of the direct friends of a, formally: f j ∈ F : ⇐⇒ e(a, f j ) ∈ E. Σ 0≤j≤N ⊕ is the aggregation of multiple (exactly N ) opinions. Note that e(f j , r) itself can be composed of the opinions of the friends of f j .

In stackoverflow, when a user A asks a question, he receives a list of answers from many users. A can accept only one answer. Unaccepted answers are not necessarily bad ones. They might be simply not good enough compared to the accepted one. They even might be better but arrived too late and A has already accepted another satisfactory answer. Basically, while we do not have an explicit reaction from A towards the unaccepted we suppose four hypotheses to treat them: 1) rigorous hypothesis: unaccepted answers are considered as failed interactions. 2) ignoring hypothesis: unaccepted answers are not considered at all. 3) independent subjective hypothesis: in both previous methods, the interaction value is either +1 (successful), or -1 (failed). In this method, we introduce relatively successful/failed interactions. We use the rates of community towards the answer to estimate a subjective successful/failure of the interaction. In fact, the thumb-up represents a successful interaction with an unknown user, same thing for the thumb-down with a failed interaction. The global reaction of the community towards the answer is subjective opinion resulting from members' interactions with the answer. We consider the expectation value of the community's opinion as the value of the partially successful/failure of the interaction between the person asking and the replier. 4) dependent subjective hypothesis: regarding to the fact that a user can give a thumb-up for an answer because it is better/worse than others, the attribution of thumbup and thumb-down can be relative too. The reason why we propose another subjective method where our certainty is influenced by the global number of thumb-up and thumb-down attributed to all answers of the same question. In this case, the opinion about an answer is dependent on the the other opinions about the other answers.

Certainty j = j th 2 + ann i=an0 i th
where th is an absolute value of thumb (up or down). j is the current answer. n is the number of answers of the current question. The default non-informative prior weight W is normally defined as W = 2 because it produces a uniform Beta PDF in case of default base rate a = 1/2. The three components of the opinion are: belief j = uncertainty j × j th up j th where j th up is the number of thumbs up attributed to the answer. disbelief j = uncertainty j × j th down j th where j th down is the number of thumbs down attributed to the answer.

uncertainty j = 1 -certainty j
Finally we compute the expectation value of the resulting opinion and consider it as the value of the relative success/failure interaction.

To predict the accepted answer of a given question q asked by the user A, we identify R the set of users who contributed answers to the current question. Then, we traverse the graph (trust network) to compute the local trust between the owner of the question and each of them. We assume that A will accept the answer of the most trustee user within R. According to this model, A consults his friends only about members of R with whom he has no direct interactions, otherwise considers only his own opinion. Consulted friends repeat the same strategy in consulting their friends. The drawback of this model is when A has only one interaction with a member r of R, this might be not enough to evaluate him. A may have a friend B who has had many interactions with r so more apt to evaluate r. According to this model A will not ask B about his opinion in r.

The aim of A is to rank R by the trustworthiness of its members. Whenever he has no information about a member r of R, A will ask his friends about their opinions in this very member. So the task of friends is to evaluate r without any farther information. The pseudo code 3 shows how this model works in demanding friends' opinions.

B. Collective trust

This model is based on collective opinions instead of personal opinions. In the previous model, collective opinions were used only in the case of absence of personal opinions. In this model, collective opinions are used in all cases. This semantically means that A will ask his friends about all the members of R, so even those who he already knows. Formally:

score(r) =              (a, r)⊕ j ⊕[e(a, f j ) ⊗ e(f j , r)] if e(a, r) ∈ E j ⊕[e(a, f j ) ⊗ e(f j , r)]
elsewhere [START_REF] Kwan | Trust and online reputation systems[END_REF] This model assumes that direct interactions are frequently unable to assure sufficient information about users. In the previous model, user could supply a personal opinion about another user once he has at least one interaction with him. We think that this affects the quality of the opinion, because of the lack of experience. In the current model, user aggregate his opinion with the his friends' opinions, each friend's opinion is conditioned by the trust given to him by the active user.

Example:

Back to the same example in Section II. Fig. 5 illustrates trust network extracted from the described relations in the example. So when A asks a question to which she get replies from E, F and G, then R = E, F, G. A needs to rank the members of R to identify the most trustworthy member.

IV. EXPERIMENTAL WORK

We use the dataset of the website stackoverflow. The website offers a question answering forum for multiple domains, mainly but not limited to computer science. The available data contains 30 domains. Users subscribe to the website by domain, so one user can have multiple accounts, according to the number of domains in which he participates. The total number of accounts is 374,008 for about 153,000 users.

The user asks a question in a given domain, and associates a set of keywords to his question, then he receives many answers. He chooses the most relevant answer to him and attributes an "accepted answer" label to it. Nevertheless, users can keep proposing new answers. Subsequent users who have the same problem as the person asking can take advantage of the answers and rate them on their usefulness by attributing thumb-up or thumb-down. In the available dataset, we have access to only the total number of thumbs-up and the total number of thumbs-down an answer has, but no information about suppliers' identities. The website, offers the possibility to order answers by relevance, where the accepted answer is put in the top of the list, followed by the other answers ordered by the difference between thumbs-up count and thumbs-down count. Our work aims to use trust based models to predict the accepted answer over the set of available answers. Total number of questions in current dataset equals to 371,594, for a total number of answers 816,487. We divide the questions of each domain in five equivalent sets. Then, we apply a crossing test in five iterations, in each iteration we use four sets for learning and building the trust network and the fifth for testing the prediction quality.

Evaluation Metrics

We consider the problem of finding the accepted answer as a list ranking problem with one relevant item. Mean reciprocal rank (MRR) is a quality metrics used to evaluate systems that have to give out a ranked list with only one relevant item. Reciprocal rank (RR) of question is 1/r where r is the rank given by the evaluated algorithm to the accepted answer. Mean reciprocal rank is the mean value of RR's to all questions. The value of this metrics varies between 0 and 1, where 1 is the best precision score.

MRR is a good indicator to the performance of prediction algorithms for ranked lists. Nevertheless, we think that it is not perfectly adapted to our case. MRR is usually used for systems that have to predict a list of items within which a relevant item exists. We are trying to find the accepted answer by reranking an existing list of answers. Remark the case when the algorithm ranks the relevant item in the last position of the list, the algorithm is recompensed for at least having chosen the item within the list. In our case, the list is predefined, so the algorithm should not be recompensed for ranking the relevant item at the end of the list. The range of RR values is [1/r, 1], we propose a modified version where the value varies between 1 if the relevant item is in the top of the list, and 0 if it is at the end of the list. We call this metrics mean predefined lists rank (MPLR), where predefined lists rank PLR is given by the formula:

P LR = N -r N - 1
where: N is the size of the list. MPLR is the average of PLRs for all questions. We employ a modified competition ranking strategy, so the ranking gap is left before the ex aequo items. For example if two items on the top of the list have the same score, they are considered both second, and no item is put at the top of the list.

V. RESULTS AND DISCUSSIONS

Only questions with accepted answer and more than one proposed answer are appropriate for our test. The corpus contains 118,778 appropriate questions out of the 371,594 questions of the corpus. In real life, regret can assist to re-establish trust. The structure of local trust systems does not possess any mechanism to reconsider relationship after a bad integration with a destination user (which can be occasional), collective opinions allow the reconsideration of the relation with this user if he was trustee by intermediate friends of source user.

Regarding the four hypotheses about treating unaccepted answers in individual trust, we find that probabilistic methods are slightly better than both rigorous and ignoring hypotheses.

In the collective trust model, the three hypotheses that try to infer from unaccepted answers surpass the performance of the forth that neglects these information (ignoring hypothesis). We conclude that unaccepted answers can be profitable, and then should not be neglected. Extracting information from these answers is possible thanks to the flexibility of subjective logic. This framework proves again its capability to deal with incomplete evidence cases.

On the other hand, in global trust, all users contribute in evaluating the trustworthiness of each user. A unique score is attributed to each user. The score is based on all his interactions (as trustee), so a source user can access more information about the destination user (compared to collective trust). The disadvantage of global trust is its weakness against malicious group attack. If a group of malicious users cooperate to highly rate each other they can improve their reputation score and disturb the system [START_REF] Mobasher | Toward trustworthy recommender systems an analysis of attack models and algorithm robustness[END_REF].

Collective trust model can provide source users with more information about destination users compared to local trust, and it controls who can participate in computing the score of destination user. Theoretically, collective trust model is more resistant to group attack compared to global trust model. In addition, collective trust is user biased, this means that a destination user has not the same score every time, and that the score is dependent to the source user too.

VI. CONCLUSION AND FUTURE WORKS

We showed in this paper the superiority of collective opinions over individual ones in a trust model. They are usually more complete, reliable and informative. So, depending on them, it leads the user to take more robust decisions.

The model proposed lies between local and global trust. Collective trust supplies its users with more complete information than classical local trust. Besides, it relies, theoretically, on more trustworthy information than global trust.

The first argument was proven experimentally on real dataset in this work. Our future work will focus on comparing the performance of a collective trust model and global trust model, when injecting malicious group attacks in the dataset.

Fig. 1 :

 1 Fig. 1: Trust network

Fig. 2 :

 2 Fig. 2: Subjective logic Opinion

1 :for all f ∈ riends do 7 :Fig. 3 :

 173 Fig. 3: Individual trust function

1 : 3 : 6 :Fig. 4 :

 1364 Fig. 4: Collective trust function

For

  the individual trust model, scores are computed as follows: score(E)= e(A,E) score(F)= [e(A,B) ⊗e(B, F )]⊕[e(A, C) ⊗ e(C, D) ⊗ e(D, F )] score(G)= e(A,C) ⊗e(C, G) As for the collective trust model, the scores of F and G do not change, but the score of E becomes as follows: score(E)= e(A,E) ⊕ [e(A,B) ⊗e(B, E)]

Fig. 5 :

 5 Fig. 5: Trust graph

TABLE I :

 I Opinion evolution with successive interactions

	No	state	r	s	belief disbelief	uncertainty
	0	no interaction	0 0	0	0	1
	1	successful interaction	1 0	1/3	0	2/3
	2	failed interaction	1 1	1/4	1/4	2/4
	3	successful interaction	2 1	2/5	1/5	2/5

Table

  II illustrates the MRR scores of both models, and table III illustrates MPLR scores. MPLR scores are, of course, lower than those of MRR. Nevertheless both tables lead to the same conclusions.It is obvious that the collective trust has a considerably better performance than individual trust on this dataset. This is because of collective opinions that rely on more complete

TABLE II

 II

		: MRR results
	method	Individual trust SL Collective trust SL
	Rigorous	0.57	0.88
	Ignoring	0.58	0.75
	Dependent probabilistic	0.62	0.87
	Independent probabilistic	0.617	0.86

TABLE III

 III Trustee friends enrich collective opinions by more knowledge that leads them to be more reliable and accurate than individual ones. These results show the limit of individual opinions and local relationships, because direct interactions can be poorly informative, and relying only on them can lead to inaccurate decisions. An individual in a social environment needs always to integrate and interact within communities to be more informed, and more capable to adjust his decisions.

		: MPLR results
	method	Individual trust SL Collective trust SL
	Rigorous	0.37	0.85
	Ignoring	0.36	0.69
	Dependent probabilistic	0.442	0.84
	Independent probabilistic	0.438	0.83
	evidences than individual ones.


Francesco Di Maio
email: francesco.dimaio@polimi.it

Samuele Baronchelli

Enrico Zio

MINIMAL CUT SETS IDENTIFICATION OF NUCLEAR SYSTEMS BY EVOLUTIONARY ALGORITHMS

Fault Trees (FTs) for the Probabilistic Safety Analysis (PSA) of real systems suffer from the combinatorial explosion of failure sets. Then, minimal cut sets (mcs) identification is not a trivial technical issue. In this work, we transform the search of the event sets leading to system failure and the identification of the mcs into an optimization problem. We do so by hierarchically looking for the minimum combination of cut sets that can guarantee the best coverage of all the minterms that make the system fail. A multiple-population, parallel search policy based on a Differential Evolution (DE) algorithm is developed and shown to be efficient for mcs identification, on a case study considering the Airlock System (AS) of CANDU reactor. coherent if it can be expressed without any *  of complemented literals X , non-coherent otherwise.

INTRODUCTION

Fault Tree (FT) is a tool widely used in Probabilistic Safety Assessment (PSA) of Nuclear Power Plants (NPPs) [START_REF] Nureg | PRA Procedures Guide, Vols 1&2[END_REF]NASA, 2002;Zio, 2007]. Traditionally, FTs are used for quantifying various probabilistic measures (including probabilities and/or frequencies of sequences, safety margins, importance factors and sensitivity indices) [START_REF] Kumamoto | Probabilistic risk assessment and management for engineers and scientists[END_REF][START_REF] Epstein | Can we trust PRA?[END_REF]. The size of the system may challenge the FT analysis, in practical situations: even for the Airlock System (AS) of a CANDU nuclear reactor with only 9 components [START_REF] Lee | Petri Net Modeling for Probabilistic Safety Assessment and its Application in the Air Lock System of a CANDU Nuclear Power Plant[END_REF], the minimal cut sets (mcs) identification problem gives rise to a FT structure function composed by 2 9 =512 minterms (i.e., a product of the literals α representing each component state, 1 failed, 0 safe) , 16867 cut sets  (i.e., a combination of components failures leading the system into failure state) and cut set chart (i.e., a table with all the minterms as columns and the cut sets as rows) of 8635904 elements. To overcome the problem, research efforts have developed in two directions: one looking for approximations of the probabilistic measures of interest obtained by considering only some selected mcs; another one developing computational methods to easily more efficiently assess the probabilistic measures from the exact mcs. One example of approximation consists in considering only small order mcs (i.e., mcs formed by a small number of elements) [Rauzy, 2001], which in principle capture the main part of the top-event probability. Another truncation process selects only the mcs with probability of occurrence larger than a given threshold. However, mcs truncation can have direct consequences on the safety level of the NPP, because it is not known how many are the mcs neglected (because of big or of small probability) in the estimation of the risk/safety indicators of interest. For this reason, it has been pointed out that mcs exact identification (rather than truncation) is one of the technical issues to be tackled in the development of PSA for risk-informed decision making, e.g. for maintenance, service inspections and safety margins quantification in new NPPs design [Fleming, 2003;[START_REF] Duflot | [END_REF][START_REF] Zio | Safety Margins Confidence Estimation for a Passive Residual Heat Removal System[END_REF].

A first attempt in developing computational methods for limiting the mcs combinatorial explosion of FTs without approximation has been to encode the Boolean formulae derived by the FTs into binary decision diagrams (BDDs) [Akers, 1978]. One of the major advantages of a BDD is that it provides exact values for probabilistic measures and it does not need any kind of truncation or approximations. However, BDD is highly memory consuming and very large models, such as FTs of NPP systems are beyond capability [START_REF] Rauzy | Exact and truncated computations of prime implicants of coherent and non-coherent fault tree[END_REF]. Another attempt for identifying mcs is the Dynamic Flowgraph Methodology (DFM), which is a directed graph based approach to model and analyze the behavior of dynamic systems [START_REF] Garrett | The dynamic flowgraph methodology for assessing the dependability of embedded software systems[END_REF]. The main drawback is scalability, that is, the fact that realistic modeling causes a combinatorial explosion as the number of states in the system increases [Bjorkman, 2013]. In order to tackle this challenge, a DFM has been solved by a BDD (based on meta-products or on zero-suppressed BDD) [Bjorkman, 2013]. Also Petri nets suffer from the combinatorial explosion of the number of states, when applied to complex systems [START_REF] Labeau | Dynamic reliability: towards an integrated platform for probabilistic risk assessment[END_REF].

We present a novel approach to tackle this issue of exact mcs identification of coherent and noncoherent FTs based on a Hierarchical Differential Evolution (HDE) algorithm. The ordinary DE algorithm has been demonstrated to be an efficient, effective and robust method for the identification of prime implicants (PIs) in simple non-coherent structure functions [Di Maio et al., 2013]. Here it is applied within a hierarchical scheme to deal with its computational limitations and avoid any approximation in the identification of mcs of complex coherent structure functions. With the proposed scheme, we look for the minimum combination of cut sets that can guarantee the best coverage of all the minterms that make the system fail: during the first step of the iteration process, a multiple-population, parallel search policy is implemented to expedite the convergence of the second step of the exploration algorithm.

The paper is organized as follows. Section 2 is devoted to recalling some base terminology (FT, Boolean Formulae, coherent and non-coherent structure functions, minterms, etc.). In Section 3, the HDE technique for mcs identification is presented. In Section 4, it is applied to the FT of a CANDU Airlock System (AS) and its results are compared with those obtained with a DE algorithm.

Conclusions and remarks are given in Section 5.

TERMINOLOGY

In this Section, we introduce the terminology used throughout the article with reference to FT analysis. The causal relations that lead to the FT top event can be described by a set of Boolean formulae built over a set of variables (literals) α 1 , α 2 , α 3 ,... α n , and connectives (and, or, not, k-outof-n), whose semantics is defined by means of truth tables. By manipulation of the truth tables, the top event can be expressed in terms of primary events (e.g. components failures in our case of interest). The simplest way to express the structure function Φ, which relates the top event to the primary events, is in terms of minimal cut sets (mcs) *   . A mcs is a combination of primary events (cut set  ), which if all, and only all, verified cause the top event to occur. Then, a mcs *  is one of the 2 n products of literals (minterms), whose occurrence ensures the failure of the system * ( ) 1    , while no proper subset of *  is a cut set [Epstein, 2005]. A structure function Φ is

CANDU AIR LOCK SYSTEM ANALYSIS

The Airlock System (AS) of a Canada Deuterium Uranium (CANDU) reactor is a safety system required to keep the pressure of the inner side of the reactor vault lower than the outer side in order to avoid the dispersion of contaminants out of the reactor bay, in case of accident. The system consists of a vessel in the containment wall of the reactor vault, with two doors in order to allow the inspection of the vault: one door opens towards the inside of the reactor vault, the other towards the outside; so, at least one airlock door, whose seals are inflated via the air system, must be closed by a latch with sufficient pressure in the seals to fulfill its safety function.

A FT has been developed for analyzing a scenario that involves a Design Basis Accident (DBA) occurred in 2011 in the AS of a CANDU NPP [START_REF] Lee | Petri Net Modeling for Probabilistic Safety Assessment and its Application in the Air Lock System of a CANDU Nuclear Power Plant[END_REF]. During the accident, the inflation of the seals switched to the back-up air supply tank and the FT top event is the incapability of the AS to maintain the pressure boundary [START_REF] Lee | Petri Net Modeling for Probabilistic Safety Assessment and its Application in the Air Lock System of a CANDU Nuclear Power Plant[END_REF]. The possible causes for this top event can be: the pressure equalizer valve fails (V1), doors fail to close because latches are not locked (D1)

and seals are cracked or cannot be inflated (S1). The pressure equalizer valves are designed to equalize the pressure between the reactor bay and the service side and, therefore, to allow controlled flow between the reactor bay and service side. The pressure equalization can fail due to gear box failure (G1) that may limit the vents from opening and closing, to the presence of leakages in the piping system (P1/P2) or to the failure of the exhaust pipe (E1). The airlock doors must be closed by a latch, otherwise the pressure equalizer valves and seals cannot be called in operation on demand. In addition, the possibility is considered that the back-up tank is already empty (T1) or fails to engage (T2) when the inflation of the seals is switched to the back-up air supply system. The basic failure events that can give rise to the AS failure are listed in Tab. 1.

Basic Failure Events ID Code

1.

Pressure equalizer valve is failed V1

Doors fail to close and lock D1

3.

Seals are cracked S1

4.

Gearbox fails G1

5.

The piping system presents minor leakages P1

6.

The piping system presents major leakages P2

7.

Exhaust pipe fails open E1 8.

Back up tank is empty T1

Back up tank fails to engage T2

Table 1. Basic failure events and ID code for the considered DBA in a CANDU AS

The FT for the DBA here considered is shown in [START_REF] Lee | Petri Net Modeling for Probabilistic Safety Assessment and its Application in the Air Lock System of a CANDU Nuclear Power Plant[END_REF]. The structure function, whose expression is

          1 AND 1 OR 1 AND 1 OR 1 OR 1 OR 1 OR 2 OR 2 OR 1 G E T S V P V T P D     
, entails 497 minterms leading to the system failure and 16867 cut sets can be found. The true solution opt x comprises 7 mcs *  ({D1}, {P2}, {T2}, {V1}, {E1,G1}, {P1,T1}, {S1,T1}) [Lee et al., 2012]. The results of the identification of the mcs of the CANDU AS obtained by the proposed HDE are presented in the following paragraphs.

HIERARCHICAL DIFFERENTIAL EVOLUTION FOR MCS IDENTIFICATION

We treat the problem of mcs identification as a set covering problem (SCP) [START_REF] Beasley | A genetic algorithm for the set covering problem[END_REF].

In the context of mcs identification, the SCP is the problem of covering each one of the minterms by a group of cut sets of minimal cost. We define the cost of a cut set Π as the number of literals α associated with system components included in the cut set (literal cost). Each solution of the SCP, ˆ, opt x is represented by a specific combination of independent variables, or, mathematically speaking, by a R-dimensional vector 12 ( , ,..., )

R x x x x 

(hereafter called chromosome within the jargon of the differential evolution (DE) optimization method (Appendix A)) where a value of 1 in the i-th vector position x i implies that Π i is chosen to be in the cover; a value of 0, otherwise [Sen, 1993].

The novelty of the Hierarchical Differential Evolution (HDE), here proposed for mcs identification, builds on in the application of a two-step Differential Evolution (DE) optimization [START_REF] Wang | A Modified Binary Differential Evolution Algorithm[END_REF]. In the proposed hierarchical framework, the first optimization is fed with subsets Γ i , i=1, 2, ...,S, of the whole set Ω, where the i-th subset Γ i is generated by randomly assigning to it N cut sets Π of Ω in a way that each cut set belongs to only one subset, i.e.,

for i ij j      
, and the union of all the subsets is equal to all the cut sets, i.e.

1 n i i     .
For each of the subsets i  , i=1, 2, ...,S, 1a) we build a cut set chart, using all the minterms as columns and the cut sets Π belonging to i  as rows 2a) we build the cost vector, where to each Π is assigned its literal cost 3a) we perform the DE optimization 4a) we find the best individuals {Π} i .

The second DE optimization is performed on the new subset   select (for each potential solution g x ) three randomly chosen individuals for reproduction (Eq. 1a) create for each target vector g x a noisy vector g v using mutation process create a trial vector 1 g u  mixing target and noisy vectors, g

x and g v , respectively (Eq. 3a)

compare target vector g x with each related trial 1 g u  and eventually replace (Eq. 4a)

end memorize the Π contained in the i best solutions   i  end create an initial population of NP potential solutions x composed by   i  , 1,2,..., iS  for 1: gG 
select (for each potential solution g x ) three randomly chosen individuals for reproduction (Eq. 1a) create for each target vector g x a noisy vector g v using mutation process create a trial vector 1 g u  mixing target and noisy vectors, g x and g v , respectively (Eq. 3a)

compare target vector g x with each related trial 1 g u  and eventually replace (Eq. 4a) end

First level DE

Second level DE memorize the best solution found ˆopt x

Three performance indicators are used to judge the goodness of the results. In the evaluation, the optimizations are repeated a number of times (20 in our case), to account for the inherent stochasticity of the search algorithm. The three performance indicators are:

-Cpu: cpu time (expressed in seconds) necessary to converge to the solution ˆopt x .

-Success rate (Sr): percentage of trials for which the true optimum opt x is found.

-Accuracy (λ): the larger λ, the larger the accuracy of the solution [Tvrdìk, 2006] as:

                           11 10 0 if 1 0 = 11 if 10 log otherwise opt t opt opt x x x            

DE Results

We apply the DE approach proposed in [Di Maio et al., 2013]. A "One complement" fitness function [START_REF] Shackleford | A High-Performance, Pipelined, FPGA-Based Genetic Algorithm Machine[END_REF] is embedded into the evolutionary algorithm: the cost of the trial solution ˆopt

x is mapped into a binary fitness function made up by two parts where the most important digits are determined as the complement to one of the uncovered faulty minterms, whereas the least important digits are determined as the complement to one of the sum of the costs of the cut sets included in the trial solution. In this way, we obtain that a complete subset of cut sets that covers all faulty minterms has surely a larger fitness than any other incomplete subset. For the ease of clarity, with respect to the AS of the CANDU, since the columns of its cut set chart are 497 (that is, equal to the number of minterms), 9 bits code the maximum number of uncovered columns, whereas the

sum of the cost of all the 16867 cut sets is equal to 103298 so that 17 bits code the cost part of the trial solution. In Fig. 1 the calculation procedure of the "One Complement" fitness function is

shown for the best solution: the uncovered columns are equal to zero, while the total cost of the best solution is equal to 10 (4 cut sets contain only one basic event and 3 contain 2 basic events); the complement to one of 0 on 9 bits is equal to 511, and the complement to one of 10 on 17 bits is equal to 131061; joining together this two parts of the fitness function gives a fitness value for opt x equal to 67108853. It is seen that, in this real case with a large number of minterms and cut sets, the NP value is critical for finding the true mcs list: for population sizes NP=30 and 100, the DE cannot succeed in identifying the correct mcs (Sr=0), whereas with NP=300 the Sr increases only to 5%. Even the success rate for NP=500 (Sr=75%) cannot be acceptable for industrial applications or regulatory purposes.

Tab. 8 reports the results obtained with NP=700 and limiting G to MAXGEN =2000 which are sufficient to achieve convergence and allow saving computational time, i.e. better cpu performance, whilst expediting the convergence towards the optimal mcs ˆopt x because of a larger population. In fact, when NP=700, the Sr and λ indicators outperform those achieved with smaller population sizes.

NP 700 Cpu [s] 9353.48 Sr 100 % λ 11

Tab. 8. Performance indicators for the DE performed with NP=700

It is important to notice that, with this setting:

1) Sr turns out to be equal to 1, which means that the algorithm always finds the true optimum solution ˆ; opt x

2) cpu indicator shows that reducing the value of MAXGEN allows for an effective practical application.

In Fig. 2, the evolution of the difference Δ between the fitness values of opt x and ˆopt x during 20 different trials of the iterative search for the optimal solution ˆopt x is shown on a semilogarithmic plot for NP=30, 100, 300, 500 and 700: the mean values of the differences Δ at each generation are plotted in continuous line, with error bars of the minimum and maximum fitness values at each generation.

Fig. 2. Fitness function convergence using DE with different NP values

The convergence of the algorithm improves in value and number of generations for larger population sizes because of a better exploration of the search space, as confirmed by the Sr values of the indicators for the different NP values in Tables 4 to 8 and in Figs. 3 and4.In particular, in Fig. 3, the boxplots of the ˆopt x fitness values obtained by the ordinary DE algorithm (with population of 30, 100, 300, 500 and 700 chromosomes) after MAXGEN generations are plotted.

For the sake of clarity, in Fig. 4 the plotted boxplots are the zoom of Fig. 2 with respect to population sizes of NP=100, 300, 500 and 700. It can be noticed that increasing the population size NP moves the mean fitness value of the population towards the fitness value of the true optimum solution equal for this case study to 67108853. Moreover, the increase of the number of individuals in the population gives rise to distributions that are shrinked on the opt x fitness value, which makes the result more reliable.

In conclusion, NP reveals to be a critical parameter for the efficiency of the search of the mcs by the DE algorithm: the broader the search space (i.e., the number of cut sets) the larger NP is needed to explore it. On the other hand, Tabs. 4 to 8 show that the cpu indicator increases less than linearly with the population size, whereas it is linearly dependent on the number of generations. Therefore, a compromise must be sought to improve the cpu by using a larger population for a shorter number of generations.

HDE Results

In order to apply the HDE technique to the CANDU FT, we have to partitioned Ω into S subsets Γ i , where S is set to 10 and N=1687 for eachsubset. The cut set chart and cost vector are built for each subset as shown in Section 3.1. Then, the DE algorithm embedded into the first optimization stage is run with a population size NP equal to 500 and the maximum generation number MAXGEN equal to 700. In Tab. 9 the mean cpu time required for performing the optimization on a single i-th subset is shown. The total time required by the HDE optimization is equal to 1233s, which is significantly reduced compared with the previous DE (see Tabs. 4 to 8).

The faster convergence obtained by the HDE compared with previous DE with NP=700 is shown in Fig. 5, where the evolution of the difference Δ between the fitness values of opt x and ˆopt x is shown on a semilogarithmic plot. For DE, parameters NP and MAXGEN were set equal to 700 and 2000 respectively, while for HDE they are equal to the values of Tabs. 9 and 10.

CONCLUSIONS

The exact identification of the mcs of FTs is an important task in PSA. method has been applied for the analysis an airlock system of a CANDU reactor. The superior HDE results becomes evident as the number of basic events in the FT grows.

APPENDIX A Differential Evolution (DE)

For solving the so defined SCP we resort to Differential Evolution (DE) [START_REF] Wang | A Modified Binary Differential Evolution Algorithm[END_REF], which belongs to the class of Evolutionary Algorithms (EAs) [START_REF] Storn | Differential evolution -a simple and efficient heuristic for global optimization over continuous spaces[END_REF][START_REF] Holland | Adaptation in Natural and Artificial Systems[END_REF].

DE entails three phases called mutation, crossover and selection. In the first phase, at the g+1-th generation, for each gene x r in the chromosome vector information. This increases the probability of maintaining some good property from the target vector and avoids drastic changes during the generation of new solution.

During the selection process, the population is modified by substitution. Referring to minimization, if the fitness, i.e., the cost, of 1 g u  is less than the fitness of g x , the first will be a member of the next generation g+1, replacing the target vector, and the trial vector is discarded

1 1 1 ( ) ( ) g g g g g u if fitness u fitness x x x otherwise          
The selection criterion in DE is greedy and for sure the following generation is better or at least equal to the previous generation.

  cut sets included in the best individuals   i  found at the end of the first optimization. In detail, 1b) we build a new cut set chart, using all the minterms as columns and the cut sets   perform the DE optimization 4b) we find the mcs *  of the systemThe pseudo-code is shown below. for 1: iS  sample without replacement N cut sets  from  populate the i-th subset i  end for 1: iS  create an initial population of NP potential solutions x containing a selection of the cut sets  in the i

Fig. 1 .

 1 Fig. 1. Procedure for the calculation of the fitness function for the best solution of the CANDU AS

Fig. 3 .Fig. 4 .

 34 Fig. 3. Boxplot of the fitness values obtained with the DE-based algorithm with NP=30, 100, 300, 500 and 700

  Cpu time for the first step of the HDE optimizationAt this first optimization stage, it is not possible to apply the Sr and λindicators because the true solution optx does not belong to any of the subsets. With respect to the computational demand of the first stage of the HDE optimization, it is worth considering that on ordinary computers (e.g., Intel® Core™ i5.2500 CPU @3.30GHz) it is possible to treat at least 5 subsets at the same time. Thus, the total time approximately required for the first step is 1214.40 s.The number of cut sets found by the first step of the optimization is 100, among which the ordinary DE embedded in the second step of the optimization procedure has to search for the optimal mcs set opt x . As in the first step, the cut set chart and the cost vector associated to the new cut sets are defined, and a DE search is launched with NP and MAXGEN equal to 500 and 200, respectively (Tab. 10). The mcs found by the HDE are the same as those reported in Tab. 3, proving that the HDE is capable of identifying the mcs. The cpu time required for the second step optimization Cpu time required by the second step of the HDE optimization

Fig. 5 .

 5 Fig. 5. Fitness function convergence using DE and HDE

  of NP different chromosomes at the g-th generation, a probability estimation vector

  r-th genes of the three randomly chosen individuals, with probability estimation vector, the corresponding genes of the noisy vector 1 least one bit of the trial individual is inherited from the mutant individual so that DE is able to avoid duplication individuals and effectively search within the neighborhood; this contributes to maintain the diversity inside the perturbed population, shuffling old and new

Table 2 .

 2 Values of the parameters F, CR and b used in the DEThe analysis is performed for population sizes NP=30, 100, 300 and 500. The only stopping criterion is the generation number G set equal to MAXGEN =5000. Performance indicators for the DE optimizations are shown in Tab. 4, 5, 6 and 7, for NP=30, 100, 300 and 500, respectively.

	NP	30
	Cpu [s]	2046.63
	Sr	0 %
	λ	4.50
	Tab. 4. Performance indicators for the DE performed with NP=30

Tab. 7. Performance indicators for the DE performed with NP=500

  It becomes non-trivial for systems that are composed by large numbers of components. In this paper, we have addressed this

		10 5
										DE
	t ness( xopt )	10 3 4 10								HDE, 1 st step HDE, 2 nd step
	t ness( 7 xopt ) --	10 1 10 2
	= -
	"	10 0
		10 -1
		0	200	400	600	800	1000	1200	1400	1600	1800	2000
						Generation Number

issue by proposing a novel HDE algorithm. This amounts to transferring the mcs identification into a hierarchical optimization problem: during the first step, a multiple population parallel DE search policy is used to expedite the convergence of a second step of DE exploration. The proposed