Hierarchical Model of Trust in Contexts
Jan Samek and Frantisek Zboril
Brno University of Technology, Faculty of Information Technology,
Bozetechova 2, 612 66 Brno, Czech Republic
{samejan,zborilf}@fit.vutbr.cz
http://www.fit.vutbr.cz
Abstract. Trust may be an important part of agent’s reasoning when
resources, services or information are sought inside a multiagent system.
Decision which agent to contact when more acceptable possibilities is
naturally done with respect to expected outcome of forthcoming cooperation, so the partner with the best quality should be probably contacted
first. Typically for one agent or device a trust may be evaluated in many
contexts. Particular kinds of trust need not to be independent and trust
in one aspect could influence trust in another aspect. We present a model
of trusts as a multilevel computational graph where trusts are computed
in conjunction on some trusts at preceding levels.
1
Introduction
When a trust pays important role in a decision process it should be properly
evaluated on the bases of previous experience of agent with particular subjects
of matter. Studies and applications on trust and reputation principles usually
use uniform context environment [5,7], which is a simplified model of classical
sociologist trust concept known from human societies. In a uniform context environment, entities trust in system is evaluated into one dimensional value, when
trust is a number from an interval < 0, 1 > (for example), when the lower value
from interval (0) means that entity is fully untrustworthy and the upper value
(1) means that entity is fully trustworthy. This uniform context environment
concept is sufficient for many models [6,5,10] and applications [9,14,16,2] and it
is well studied.
In our research, we concentrate to multiple context trust environment [5],
which is more complex in trust evaluating and it is closer to real word trust concept. In a multiple context environment, entities trust is always associated with
some context [3,4,12]. In the multiple context environment we can’t simply say
that entity is good or bad (0 or 1) in providing such services, we have to say that
some entity in providing some service is good/bad (trustworthy/untrustworthy),
in another service is good/bad and so on.
Our hierarchical model of trust in contexts (HMTC) approach addresses some
open questions in context sensitive trust scenario like context transference [13,5],
incomplete information [8] or side effects [15]. With using multilevel context
graph, we are able to model relation between different contexts and make decision
about agent qualities in different aspects of his/her future behaviour.
F. Zavoral et al. (Eds.): NDT 2010, Part II, CCIS 88, pp. 356–365, 2010.
c Springer-Verlag Berlin Heidelberg 2010
Hierarchical Model of Trust in Contexts
357
This HMTC concept is based on our previous research [11], where trust-based
models for multiagent system were studied from different aspects. In [13] we
introduced agent reasoning framework based on trust and reputation principles
and in [12] we dealt with trust and reputation evolution in multiple context
scenario.
In this paper we present a model of trust hierarchy as a multilevel graph,
which we specify in Section 2. Computations on this model that are performed
when some particular trusts are modified are issue of Section 3. In section 4 we
present some experimental results and in section 5 we provide conclusions and
future work directions.
2
Hierarchical Model of Trust in Contexts
Structure, behaviour and properties of the model will be introduced during the
following sections. First of all we state some premises. This model is intended
to be used for agent’s reasoning which is based on some beliefs about some
qualities of other elements in the system. Our concept supposes that each element in the system has different qualities which are not completely independent
and these qualities are correlated together with using specification and generalisation – some qualities or attributes in lower abstraction level can create
one or more common qualities in higher abstraction level. Entities in the system can be rated in these different qualities and agent’s reasoning is done by
combination of different beliefs in different qualities with using connections in
such qualities. In this paper, we do not describe how a hierarchical model of
qualities for different entities is made and how qualities are connected. We suppose that model is created empirically from knowledge of the real system of
interest.
2.1
Usage of MHTC for Trust Modeling
The HMTC is primary designed for usage when agents are reasoning with respect
to their beliefs about trust to some elements in the system. It is supposed that
the highest layer stands for trust to the agent or the system element itself.
Then follows layers which bring more and more specific trust domains from
upside down. The last layer contains contexts that are most specific and usually
measurable or observable form the trust point of view.
HMTC is especially used in interpretation and evaluation phases of trust
reasoning in multiagent systems [13]. After some interaction or monitoring other
entities (agents) in system, we need to evaluate result of interaction. In multiple
context environments requires this evaluation requires at least two decisions:
1. outcome of interaction (positive/negative),
2. context of interaction.
The HMTC graph should be very useful in decision about context of interaction
because each interaction or observation may belong to many different contexts.
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This can be described on example, where two different agents (agent A and agent
B) in a system communicate together: A asks B for information about temperature in his/her environment, B response that his/her ambient temperature is
20 degrees Celsius. At first, there is context communication, second context is
provide temperature data, next context should be quality of providing temperature data which can be deduced from precision of received temperature data and
latency of response. There are many contexts, which can be recognized in this
interaction event; however some of these are irrelevant for trust model purposes.
After decision in which context of interaction was made and which context is
relevant, we can update node trust in MTC graph in most relevant contexts and
improve trust interval in their neighbours.
2.2
Model Basics
Hierarchical model of trust in contexts (HMTC) is a graph where N is a set
of nodes and E is a set of edges. HTMC is a multilevel graph and there exist division of the node set into disjunction non-empty sets N1 , N2 . . . Nn so
that N = N1 ∪ N2 · · · ∪ Nn , which define n levels of graph. As well as for the
node set there exist division of the edges set to the sets E1 , E2 . . . En−1 , then
E = E1 ∪ E2 · · · ∪ En−1 Edge connects pair of nodes from the node set and we
denote edge between nodes u, v ∈ N as (u, v) = eu,v . Edges are allowed only
between two nodes in neighbour levels:
∀(u, v) : (u, v) ∈ Ei →(((u ∈ Ni ) ∧ (v ∈ Ni+1 ))
∨ ((u ∈ Ni+1 ) ∧ (v ∈ Ni )))
(2.1)
The graph is bidirectional so if there is a edge eix then there is a edge in opposite
i+1
−i
i+1
i
i
i
direction e−i
x . In our case edges ea,b = (na , nb ) and eb,a = (nb , na ) both
belong to the set Ei and because Ei ⊂ E then they also belong to the edge
set E.
2.3
Structure and Behaviour of the Model
The structure of the model consists of the following components: set of nodes N ,
set of edges E, set of time moments T , trust function ρ and weight function w.
HM T C = (N, E, T, ρ, w)
(2.2)
Our model is a discrete system which behaviour is defined in some time moments
which constitute a time set T . We denote particular time moments as T =
{t1 , t2 . . . ti }. Trust function maps every node to the interval < 0, 1 > from with respecting time:
ρ : N × T →< 0, 1 > .
(2.3)
Hierarchical Model of Trust in Contexts
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Weight function maps every edge to the interval < 0, 1 > from :
w : E →< 0, 1 > .
(2.4)
Definition of these functions includes restriction that weight of edges in forward
direction is equal to weight of vertex in opposite direction:
∀eix : w(eix ) = w(e−i
x ).
(2.5)
Also there is a restriction, than for all non-leaf nodes is valid that node’s incoming
edges weights sum is equal to 1:
∀nix ∈ N − Nn :
w((nl , nl+1 )) = 1.
(2.6)
∀nl+1
Trust function is defined using the weight function in such a way that for every
non-leaf node the trust is computed as a sum of multiplications of children nodes
trusts and corresponding weights. Information of the trust is brought when a leaf
node is evaluated with a value. So we suppose that the ith leaf nodes are set to
a constant value at time t : ti < t < ti+1 that we denote as τ (i, t). So the trust
assignment function is defined as:
ρ(nla , t)
=
∀ela,b ,nl+1
b
τ (nla , t)
w(ela,b )ρ(nl+1
b , t) 1 ≤ l < n
otherwise
(2.7)
Figure 1 describes simple HMTC graph with three levels, six nodes and six
edges: N = N1 ∪ N2 ∪ N3 where N1 = {n1 }, N2 = {n2 , n3 }, N3 = {n4 , n5 , n6 };
E = {en1 ,n2 , en1 ,n3 , en2 ,n4 , en2 ,n4 , en3 ,n5 , en3 ,n6 }. Set N3 denote leaf nodes set,
these nodes are evaluated by function τ , other nodes are computed by trust
function ρ as shown in figure 1.
Fig. 1. Simple HMTC graph example
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Extended Behavior and Trust Evaluation
Basic model structure suppose that every leaf nodes trust is updated directly
from the environment (denoted by function τ ) and non-leaf nodes’ trust can be
only updated indirectly with computation by equation (2.7). This computation
is called up-direction toward to that only parent nodes can be computed on
the bases of child nodes’ values. In some cases it would be useful to update
non-leaf nodes directly, for example on the basis of observed knowledge about
another agent from the environment. It brings extended trust computation which
can be made in down-direction which is based on knowledge about parent and
neighbours nodes trusts.
In this paper, we do not address how and why such thing happens, but we will
study what possible change causes to the trust model. We study situation when a
trust of a ith node nli of the lth layer is changed. We begin with description of an
extended trust interval and after this we illustrate example of down-direction and
up-direction re-computation of trust intervals for a parent and a children node.
3.1
Extended Trust Interval
The above mentioned model would be sufficient when trust in every context is
known exactly. Next model will use an interval which is an inclusion on the interval < 0, 1 >. In this case the trust function is defined as ρ : N × T → 2<0,1> .
Such mapping uses power set of the interval, but in fact the mapping is done
only to an interval with some minimum and maximum. For this reason we
better define two more trust evaluation functions ρmin : N × T →< 0, 1 > and
ρmax : N × T →< 0, 1 > that determines minimum and maximum of the interval
for each node. Also, the formulae ∀n ∈ N : ρmin (n) ≤ ρmax (n) must be fulfilled.
At the start of the model run each node is set to the most general interval:
∀n ∈ N : ρmin (n, t1 ) = 0, ρmax (n, t1 ) = 1. Each trust phenomena is unknown at
the very beginning and the agent has no idea about the real trust value. As the
agent system is executed, information about particular trust can be specified.
3.2
Up-Direction Computation
Up-direction computation is used in a case, when parent node trust is computed
based on knowledge about all child nodes trusts. Example of this computation
direction is described in figure 2. A1 ’s trust is computed as his/her previous trust
is intersected with a new computed trust via all his child nodes B1 . . . Bn by the
equations:
n
ρmin (A1 , tj+1 ) =
wi ρmin (Bi , tj )
i=1
ρ
max
(A1 , t
j+1
)=
n
(3.1)
wi ρ
max
j
(Bi , t )
i=1
ρ(A1 , tj+1 ) = max(0, ρmin (A1 , tj+1 )), min(1, ρmax (A1 , tj+1 ))
(3.2)
Hierarchical Model of Trust in Contexts
361
Fig. 2. Part of context graph for “up-direction” computation
3.3
Down-Direction Computation
Down-direction computation is used in a case when child node trust is computed
based on knowledge about parent and neighbours nodes trusts.
Example in figure 3 illustrates situation, where trust in node B1 is computed via parent node A1 and all A1 children (B2 . . . Bn ) in the next level
without the node itself. For these example, we compute extended interval limits
ρmin (B1 , tj+1 ) and ρmax (B1 , tj+1 ) as follows:
ρmin (B1 , tj+1 ) =
ρ
max
(B1 , t
j+1
)=
ρmin (A1 , tj ) −
ρmax (A1 , tj ) −
n
i=2
wi ρmax (Bi , tj )
i=2
wi ρmin (Bi , tj )
w
n1
(3.3)
w1
ρ(B1 , tj+1 ) = max(0, ρmin (B1 , tj+1 )), min(1, ρmax (B1 , tj+1 ))
(3.4)
In a case when a computed node has more then one parent, computation is
done via all his/her parents by equations 3.3 and 3.4. All the partial results
are aggregated by intersection. In the example in figure 3, when the trust of
node C1 is needed to be re-computed, it is done by the parents B1 and B2 .
The re-computed partial results ρ(C1B1 , tj+1 ) and ρ(C1B2 , tj+1 ) are consequently
intersected:
(3.5)
ρ(C1 , tj+1 ) = ρ(C1B1 , tj+1 ) ∩ ρ(C1B2 , tj+1 )
4
Experimental Results
The goal of our experiments is to determine how many nodes of our model are
impacted (re-computed from their previous values) with a direct node updating
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J. Samek and F. Zboril
Fig. 3. Part of context graph for “down-direction” computation
Fig. 4. Example of HMTC graph for WSN node
event from an environment in different graph levels. For this re-computation,
we use recursive update algorithm, where child and parent nodes (of node Ni )
trust have to be re-computed when node trust (Ni ) was directly updated. For
this experiment, we construct HMTC graph for a scenario, where sensor nodes
in wireless sensor network (WSN) are evaluated in different contexts.
Setup of our HMTC graph shown in figure 4. Our HMTC contains a root
node, 10 non-leaf nodes and 4 leaf nodes. Level of graph is 4 (n = 4). All nodes
trusts are initialized to interval < 0, 1 >. In our experiments, we split all nodes
into three sets by their level (L): root node (L = 1), non-root and non-leaf
nodes (1 < L < n), leaf-nodes (L = n). For this three sets, we simulate node
Hierarchical Model of Trust in Contexts
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Fig. 5. Experimental results
trust direct updating (for each node from set) from 0.0 to 1.0 (respectively from
< 0.0, 0.0 > to < 1.0, 1.0 > with using the interval representation) by step 0.05.
We observers how many nodes are re-computed with using up/down direction
recursive computation. For each nodes set and each simulation step we compute
average value of re-computed nodes.
Graph in figure 5 shows result of our experiments. This result uncovers some
interesting characteristics of HMTC:
1. Direct trust update in leaf nodes always causes the same number of nodes
re-computation irrespective to amount of trust change.
2. Updating of non-leaf node trust into values near extreme values (0 and 1)
causes the greatest number of nodes re-computation. The number of recomputations depends on edge weight configuration. On the other hand,
updating trust of node into middle value of possible trust interval (trust
about 0.5, which can be described as ignorance in fuzzy trust modeling [1])
causes minimal number of nodes re-computation.
3. In corresponding to item 2. above, we state that updating of root node trust
into values which are near to extreme values (0 and 1) causes re-computation
of all the other nodes in graph irrespective to edge weights configuration.
5
Conclusion and Future Work
In this paper we propose Hierarchical Model of Trust in Contexts which is primary used for agent reasoning in multiple context trust scenario. This model is
represented by multilevel graph, where each node of graph represents different
context of trust and each edge models correlation between different contexts.
Structure, behaviour and computations in this model were also presented.
We suppose that HMTC is useful for modelling trust and reputation principles
more precisely with respect to contextualization. HMTC allows us to improve
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J. Samek and F. Zboril
agent decision making and reasoning in large scale multi-agent societies, where
selection of competent agents for interaction and communication may be difficult.
In our future work we will concentrate to extended trust computation when
different sources or events updates the nodes, where collision of trust intervals
can arise (intersection is an empty set). This requires determining types of events
which can update contexts in graphs and define how different events can be
handled. There is also open question how to deal with history of update events
and their impact to node trusts.
Acknowledgment
This work was partially supported by the grants GACR 102/09/H042, BUT
FIT-10-S-1 and the research plan MSM0021630528.
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