Transcript Lecture 18

Trust Management for the Semantic
Web
Matthew Richardson1†, Rakesh Agrawal2, Pedro Domingos
By
Tyrone Cadenhead
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.
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.
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(j tij).
• 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.
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 .
Matrix T, Vector b
T
b
t1,1
t1,2
t1,3
t1,4
t1,5
1
t2,1
t2,2
t2,3
t2,4
t2,5
2
t3,1
3
t4,1
4
T5,1
t5,2
T5,3
T5,4
T5,5
5
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.
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)
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.
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)
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.
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.
Probabilistic Interpretation
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.
•
ij
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.
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.
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.