Adding trust

In this section we extend the base model with the requirements for a trust-based recommendation system.

Adding agent experiences

The model assumes that agents have (private) option experiences. The experience of agent a with option p at time t is described as a tuple ha, p, t,Prefe,Prefii where

(i) Prefe : Inte → [0, 1] describes the explicit preference of agent a for option p as a

mapping of the setInteof so-called explicit attributes into values from [0, 1],

(ii) Prefi:Inti→ [0, 1] describes agents a’s implicit preferences for option p by assigning

[0, 1]-values to the internal attributesInti.

Note that our model slightly differs from Battiston et al. (2006) by adding the time- attribute and omitting the (tuple-dependent) decay-attribute. Furthermore, in Battiston et al. (2006) it is assumed that agents have a limited memory only, and can remember only the m most recent experiences for some agent-independent constant m. It is also assumed that the value of experiences decays over time.

Let E be the set of all option experiences of agents. We assume that agents have recorded at most one experience with each option.

Adding the agent network

As described in Montaner et al. (2002a), agents may rely on other agents to support their decisions. The agents on which some agent a is relying are called a’s neighbors. The agent-neighbor relation is represented as an edge from the agent to the neighbor. Each edge has associated a so-called trust value, expressing to what extent the agent relies on the opinion of that neighbor. As a consequence, this network for agent connections is a directed, weighted graph.

The tuple ha, b, ti represents the edge from agent a to agent b with associated trust value t ∈ [0, 1]. Let C be the set of all such agent relations.

The partial function Trustis introduced for convenience to record all trust values

between agents as follows: Trust(a, b) = t ⇔ ha, b, ti ∈ C. Furthermore, we will use a → b when ha, b, ti ∈ C for some trust value t.

Note that the neighbor relation → is irreflexive and in general not symmetric of

transitive. Also the functionTrustmay not be consistent in the sense that:

a → b ∧ b → c ⇒Trust(a, c) =Trust(a, b) ·Trust(b, c) In a later section we will define the closure of this relation.

5.5. AGENTS RELIABILITY 69

What agents will reveal

When an agent is confronted with a new option, it may want to consult its neighbors to form its opinion of this new product. An agent only has an opinion of some product if there is a corresponding experience in E. The product opinions (reflecting product interest or product preference) of agent a are recorded in the agent interest function

v(a) : P → [−1, 1].

Suppose agent a has experience ha, p, t,Prefe,Prefii ∈ E with option p. Rather than

revealing the actual functions Prefe and Prefi, agent a will aggregate these personal

functions into single scoresAggre(Prefe)andAggri(Prefi)by applying aggregate functions

Aggre and Aggri respectively. The public interest value of agent a in option p then is

calculated as follows:

v(a)(p) =Aggr(Aggre(Prefe),Aggri(Prefi))

whereAggris an aggregation function combining the external and internal scores. In Mon-

taner et al. (2002a) the Ordered Weighted Average (OWA) operator Yager (1988) is sug- gested to aggregate to detail values into a total score, by introducing a partial order on the details. Note that an alternative method may be to use AHP, for example Dolan (2008).

The aggregation functionAggrto aggregate the internal and external score may be chosen,

as in Montaner et al. (2002a), as the weighted average of the individual scores.

When asked about its opinion on option p, agent a will not only reveal the value

v(a)(p) (if agent a has an opinion about option p at all), but also give an indication of

the relevance of this opinion. The value of experiences decays over time. We assume that Decay(t) is the decayed value of tuple ha, p, t,Prefe,Prefii ∈ E. A most recent experience

has decay value 1; the older the experience, the closer the decay value to 0. A typical

definition of the decay function is Decay(t) = 2−twhere t is measured is some convenient

unit, for example in years.

In Battiston et al. (2006) the agent preferences obtained from thevfunction are stored in a special table. In our approach, the table also contains the decay values:

ha, p, t,Prefe,Prefii ∈ E ⇒ ha, p,v(a)(p), Decay(t)i ∈ Prefs

Agent similarity

Agents may want to compare themselves to other agents, for example to investigate initial trust values. This is only possible by using publicly available information, such as the agent opinion function introduced in previous section. Typically, a new agent a may select a set ¯P of preferred options, and then check other agents opinions for these options. Then typically, this agent is obtaining the restrictionv(b)[ ¯P] of functionv(b)to domain ¯P from each checked agent b. Using agent similarities, agent a will choose the most similar checked agents b as its preferred peers, and add them as neighbors to the set C via the tuple ha, b, si where s is the similarity score obtained between a and b.

The similarity Sim(a, b)of agent a with agent b is expressed as a similarity between

their option interestsv(a)andv(b). In Battiston et al. (2006) this interest is expressed using the Manhattan distance dManh(a, b) = |v(a) −v(b)| (for details on the notation of operators on function, see Appendix A):

Sim_{Manh(a, b) = 1 −}|v(a) −v(b)|

|P|

The normalization factor |P| (number of elements in set P) is used to guarantee that the similarity values are within the requested interval [0, 1]. In Montaner et al. (2002a) Clark’s

distance is proposed to relate agents to each other: Sim_{Clark(a, b) =} v(a) −v(b) v(a) +v(b) ₂

Note that in Information Retrieval applications the cosine measure for similarity is most popular:

Simcos(a, b) = v(a) •v(b) kv(a)k₂· kv(b)k₂

In the sequel, we assume that Sim(a, b) refers to one of these alternatives. Agent a

may also involve a (personal) partial option weighting function δ : P 99K [0, 1] for which kδ[a]k₁= 1as follows:

Simw(a, b) =Sim(δ ·a, b)

using the normalization of the similarity result with the δ-values for the domain of function a. Another possibility is that the similarity function takes the decay values of the opinions into account.