1. Trust Management for the Semantic Web Matthew Richardson1†, Rakesh Agrawal2, Pedro Domingos By Tyrone Cadenhead

2. Overview The semantic web is a large, uncensored system to which anyone may contribute. Raises question of how much credence to give to each source. Tackle by employing concept of “web of trust”, in which each user maintains trusts in a graph of a small number of other users. Compose these trusts into trust values for all users. Each user receives a personalized set of trusts.

3. Overview Anyone can be an information producer or consumer of anyone else’s information. Major issue of how to decide the trustworthiness of each information source. Sheer magnitude and diversity of sources make it impossible to have all information be consistent and of high quality.

4. Model One solution is to establish the degree of belief in a statement that is explicitly asserted by one or more sources on the semantic web. User’s belief in a statement should be a function of her trust in the sources providing it. b i = f( j t ij ). Authors propose solution based on recursive propagation of trust. If A has trust u in B and B has trust v in C, then A should have some trust t in C that is a function of u and v. S={users}, R={trust}, aRb by transitive closure.

5. Model Beliefs: any user, i, may assert her personal belief b i in the statement in the range [0,1]. The collection of personal beliefs in a statement is a column vector b. Trusts: user, i, may specify a personal trust, t ij, for any user j. Trust is in range [0,1]. t ij may be different from t ji. The collection of personal trusts is a NxN matrix T. Merging: we want to compute for any user, their belief in a statement given by Merged beliefs. The trust between any two users is given by the merged trust matrix.

6. Matrix T, Vector b t1,1t1,2t1,3t1,4t1,5 t2,1t2,2t2,3t2,4t2,5 t3,1 t4,1 T5,1t5,2T5,3T5,4T5,5 1 2 3 4 5 Tb

7. Algorithm (1) Path Algebra Interpretation Assumption that a merged belief depends only on the paths of trust between the user and any other user with a personal belief in the statement. Algorithm: – Enumerate all paths between the user and every user with a personal belief in the statement. – Calculate the belief associated with each path by applying a concatenation function to the trusts along the path and also the personal belief held by the final node. – Combine those beliefs with an aggregation function.

8. Path Algebra Interpretation Let о represent the concatenation function, and ◊ represent the aggregation function. E.g. t ik оt kj is the amount that user i trusts user j via k. If ◊ is addition and о is multiplication, then ◊( k: t ik оt kj ) = t ik t kj. Matrix operation C = AB such that C ij = ◊( k: A ik оB kj )

9. Local Belief Merging Let well-formed decomposable path problems be defined for which ◊ is commutative and associative, and о is associative and distributive over ◊. Algorithm: In step 2, the user needs only the merged beliefs of her immediate neighbors, which allows her to merge beliefs locally.

10. Definitions Let ◊ be addition, and о be multiplication. Commutative: a + b = b + a Associative: (a + b) + c = a + (b + c) Associative: (a * b) * c = a * (b * c) Distributive: a * (b + c) = (a * b) + (a * c)

11. Cycles It is improbable a web of trust will be acyclic. A combination function is cycle-indifferent if it is not affected by introducing a cycle in the path between two users. With cycle indifference, the aggregation over infinite paths will converge, since only the(finite number of) paths without cycles affect its calculation.

12. Algorithm 2: Probabilistic Interpretation Imagine a random knowledge-surfer hopping from user to user in search of beliefs. At each step, the surfer probabilistically selects a neighbor to jump to according to the current user’s distribution of trusts. With probability equal the current user’s belief, the random surfer says “yes, I belief in the statement”. Otherwise it says “No”. When choosing which user to jump to, the random surfer will, with probability λ i [0,1], ignore the trusts and instead jump directly to the original user, i.

13. Probabilistic Interpretation ij is the probability that, at any given step, user i ’s random surfer is at user j. i is the probability that, at any given step, user i ’s random surfer says “yes, I belief in the statement”. This is based on random walks on a Markov chain. The convergence properties of such random walks are well studied: and will converge as long as the network is irreducible and aperiodic.

14. Computation User i’s trust in user j is the probability that her random surfer is on a user k, times the probability that the surfer would transition to user j, summed over all k. And is the probability that user i’s random surfer says “yes”. This is the probability that the random surfer is on a given user times that user’s belief in the statement.

15. Cont’d User i selects a neighbor probabilistically according to her distribution T i, and then, with probability (1 - λ), accepts the neighbor’s (merged) belief, and with probability λ accepts her own belief. In Matrix form: is This says that a user may compute her merged trusts knowing only the merged trusts of her neighbors.

16. Definitions Aperiodic: there exists an integer k > 1 that divides all cycles. Irreducible: graph remains unchanged after a reduction algorithm is applied. Idempotent: multiple applications of the operation does not change the result. ◊(x,x) = ◊(x). Transitive Closure: given set S, binary relation R, aRb. S={set of humans}, R={parent of}, transitive closure of R is aRb means a is the ancestor of b.