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In this article the authors will focus on the state of the art on information filtering and recommender systems based on trust. Then the authors will represent a variety of filtering and recommendation techniques studied in different literature, like basic content filtering, collaborative filtering and hybrid filtering. The authors will also examine different trust-based recommendation algorithms. It will ends with a summary of the different existing approaches and it develops the link between trust, sustainability and recommender systems.

Methodology of this study will begin with a general introduction to the different approaches of recommendation systems; then define trust and its relationship with recommender systems. At the end the authors will present their approach to “trust-based recommendation systems”.

The purpose of this study is to understand how groups of users could improve trust in a recommendation system. The authors will examine how to evaluate the performance of recommender systems to ensure their ability to meet the needs that led to its creation and to make the system sustainable with respect to the information. The authors know very well that selecting a measure must depend on the type of data to be processed and user interests. Since the recommendation domain is derived from information search paradigms, it is obvious to use the evaluation measures of information systems.

The authors presented a list of recommendations systems. They examined and compared several recommendation approaches. The authors then analyzed the dominance of collaborative filtering in the field and the emergence of Recommender Systems in social web. Then the authors presented and analyzed different trust algorithms. Finally, their proposal was to measure the impact of trust in recommendation systems.

Let G=(V,E) be a graph. The complement of G is the graph G¯:=(V,[V]2 \ E) where [V]2 is the set of pairs {x,y} of distinct elements of V. If K is a subset of V, the restriction of G to K is the graph G↾K:=(K,[K]2∩E). We prove that if G=(V,E) is a graph and k is an integer, 2 ≤ k ≤ v−2, then there is a k -element subset K of V such that e(G¯↾K)≠e(G↾K), moreover the condition k ≤ v−2 is optimal. We also study the case e(G¯↾K) ≢ e(G↾K)(mod p) where p is a prime number. Following a question from M.Pouzet, we show this: Let G=(V,E) be a graph with v vertices. If e(G) ≠ e(G¯) (resp. e(G)=e(G¯)) then there is an increasing family (Hn)2≤n≤v (resp. (Hn)2≤n≤v−2) of n -element subsets Hn of V such that e(G↾Hn)≠ e(G¯↾Hn) for all n. Similarly if e(G) ≢ e(G¯) (mod p) where p is a prime number, p>v−2, then there is an increasing family (Hn)2≤n≤v of n -element subsets Hn of V such that e(G↾Hn) ≢ e(G¯↾Hn)(mod p) for all integer n∈{2,3,…,v}.