Trust Management for the Semantic Web Matthew Richardson1†, Rakesh Agrawal2, Pedro Domingos

1. Trust Management for the Semantic WebMatthew 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. bi = f( jtij). • 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 bi 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, tij, for any user j. Trust is in range [0,1]. tij may be different from tji. 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 T b

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. tikоtkj is the amount that user i trusts user j via k. • If ◊ is addition and о is multiplication, then ◊( k: tikоtkj) = tiktkj. Matrix operation C = A•B such that Cij = ◊( k: AikоBkj)

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 λi [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. • iis 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 Ti, 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.