A Coalitional Game Model for Heat Diffusion Based Incentive Routing and Forwarding Scheme.

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A Coalitional Game Model for Heat Diffusion
Based Incentive Routing and Forwarding Scheme
(Work in Progress)
Xiaoqi Li, Wujie Zheng, and Michael R. Lyu
Department of Computer Science and Engineering
The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
{xqli,wjzheng,lyu}@cse.cuhk.edu.hk
Abstract. We propose an incentive routing and forwarding scheme that
integrates a reputation system into a monetary payment mechanism to
encourage nodes cooperation in wireless ad hoc networks. For the first
time in the literature, we build our reputation system based on a heat
diffusion model. The heat diffusion model provides us a way of combining
the direct and indirect reputation together and propagating the reputa-
tion from locally to globally. Further, we model and analyze our incentive
scheme using a coalitional game, which is not the usual non-cooperative
game like others. We further prove that under a proper condition this
game has a non-empty stable core. From the evaluation we can see that
the cumulative utility of nodes increases when nodes stay in the core.
Keywords: Coalitional Game, Incentive Routing, Heat Diffusion.
1Introduction
The nature of wireless ad hoc networks is to let nodes cooperative together thus
improve the connectivity of the whole network or execute some specific functions
inside the network. However, nodes in this kind of networks may belong to
different individuals or authorities and have their own interests. They may not
want to help others forward routing and data packets, since that will cost their
own energy and bandwidth. Consequently, it is necessary to provide incentive
mechanisms to encourage cooperations among the nodes.
Incentive routing schemes for enforcing selfish agents to cooperate in wireless
networks have been studied for years. One category of solution is using monetary
incentives, either virtually or practically. Payment schemes need to be designed
and usually are analyzed by game theoretic methods. In these schemes, the in-
termediate nodes declare their costs for forwarding packages. Then the routing
protocol selects the lowest cost path (LCP) based on the declared costs. After-
wards the payments are rewarded to nodes on and sometimes off the LCP with
the amount no less than their declared costs. However, a problem arises when
nodes may purposely declare a higher cost to take advantage of the payment
algorithms. So currently more research is focused on how to avoid cheating and
achieve effective and also economic payments.
L. Fratta et al. (Eds.): NETWORKING 2009, LNCS 5550, pp. 664–675, 2009.
c ? IFIP International Federation for Information Processing 2009

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A Coalitional Game Model665
Another category is employing reputation systems to stimulate the nodes to
cooperative. The common idea of these systems is that each node in the network
will monitor the behaviors of its neighbors. If the neighbors are observed for not
executing some functions properly, their reputations will be decreased and they
will be under the threat of being blocked from the network. The key challenges
are how to combine the direct neighborhood reputations together and propagate
them from locally to globally. In these systems, game theoretic methods can also
be used to analyze the effectiveness of the threatening mechanism. In fact we
consider that the method of using monetary incentives and reputation systems
are not mutually exclusive and they can be combined together to design a more
flexible incentive scheme.
On the other hand, the above schemes are usually modelled as non-cooperative
games. However, in wireless networks nodes cannot perform routing and forward-
ing behaviors individually. They must cooperate together to complete one task,
so it is natural to think about modelling the wireless network behaviors as a
cooperative game.
In this paper, we are going to model the routing and forwarding procedures
in wireless networks as a cooperative coalitional game with transferable payoff,
and propose an incentive routing and forwarding scheme that combines the idea
of payment mechanism and reputation system together. Then we analyze that
the game has a non-empty core, which is a stable status in cooperative game
just like the Nash Equilibrium in a non-cooperative game.
Regarding the combination of reputation and payment schemes, we first need
to obtain a combined and globalized reputation, and then smoothly map this
reputation value to a certain amount of payment. On the basis of it, we also
need to design an incentive payment scheme integrating reputation and cost
together. In the formulation of a coalitional game the key challenges are: 1) how
to write the value function of the coalition which represents the collective payoff
of the coalition; 2) how to find the solution of this game where every node has a
satisfying payoff share, so that it will not deviate from some stable status. Some
games may not have such a stable solution. The objective of our paper is to solve
the above questions.
We list our major contributions as follows: First, we design an incentive rout-
ing and forwarding scheme that integrates reputation information into a payment
mechanism. Second, we introduce a heat diffusion model to combine the direct
and indirect reputations together and propagate them from locally to globally in
the way how heat diffuses. Third, unlike others, we model this incentive scheme
using a coalitional game method. A characteristic value function of the coalition
is designed, and we prove that this game has a core solution.
The rest of this paper is organized as follows. We first give the background of
the heat diffusion model in Section 2. After describing some technical prelimi-
naries in Section 3 we will propose our incentive routing and forwarding scheme
in Section 4. The scheme is analyzed in Section 5, then we show some evaluation
results in Section 6. In the end, some related work and conclusions are given in
Section 7 and Section 8 respectively.

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666X. Li, W. Zheng, and M.R. Lyu
2 Background of Heat Diffusion Model on Weighted
Directed Graph
2.1Motivation
A reputation system usually needs to address two problems: 1) how to combine
subjective direct reputations with indirect reputations from neighbors to make
them become more objective; and 2) how to propagate the reputation from
locally to globally. Previously there are different solutions to these problems
such as [1] and [2]. In this work, we will employ the heat diffusion model to
fulfill the requirements.
In nature, heat always flows from high temperature positions to low temper-
ature positions via conductive media. A heat diffusion model describes this phe-
nomenon that heat can diffuse from one point to another through an underlying
manifold structure in a given time period. The higher the thermal conductiv-
ity of the medium, the easier the heat flows, which implies that the two end
points have some cohesive relations. Diffusion behaviors are also affected by the
underlying geometric structures. Some achievements have been made based on
the heat diffusion model such as classification in machine learning field, page
ranking in information retrieval [3] and marketing candidates selection in social
computing [4], but to our best knowledge, there is no previous work that has
been performed on the incentive routing in wireless networks.
We see that in the process of heat diffusion, each node’s heat comes from all
of its incoming links and diffuses out to its successors as long as it can. If we
diffuse heat on a weighted directed graph, the amount of heat a node can get
depends not only on the heat of its neighbors but also on the weights of the
links connecting them. The higher the weight, the more thoroughly the heat can
be diffused. Therefore, if we let the weight be the direct reputation value of the
link, then the amount of heat will be the overall reflection of the underlying
reputation information. The course of heat diffusion through all possible links
can also be deemed as a propagation of the reputations.
2.2Heat Diffusion on Weighted Directed Reputation Graph
We construct a heat diffusion model on the reputation graph G = (M,E,R),
where M = {1,2,...,m} is the node set. E = {(i,j)| i and j are in communication
range and the transmission direction is from i to j}. The heat only flows from i
to j if (i,j) ∈ E. R is the reputation set {rij| rijis the direct reputation of edge
(i,j)}. We use fi(t) to describe the heat value of node i at time t, beginning
from an initial distribution of heat fi(0) at time zero. f(t) denotes the vector
consisting of fi(t).
The heat diffusion modelling is as follows. Suppose, at time t node i diffuses
HD(i,t,Δt) amount of heat to its subsequent nodes. We assume that: a) the
heat HD is proportional to the time period Δt; b) HD is proportional to the
heat of node i; c) each node has the same ability to diffuse heat; and d) node i
intends to distribute HDuniformly to each of its subsequent nodes, but the actual

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A Coalitional Game Model667
heat it can diffuse is proportional to the corresponding reputation weight of the
edge. On the basis of the above considerations, we state that node i will diffuse
λpikfi(t)Δt/liamount of heat to each of its subsequent node k, where liis the
outdegree of node i and λjis the thermal conductivity, which is the heat diffusion
coefficient representing the heat diffusion ability. In the case that the outdegree
of node i is zero, we assume that this node will not diffuse heat to others. Then
the total amount of heat node i will diffuse is?
j during a period of Δt. We also have the following assumptions: a) HR is
proportional to the time period Δt; b) HRis proportional to the heat of node j; c)
HRis zero if there is no link from node j to i. Based on the above considerations,
we obtain HR(i,j,t,Δt) = λjfj(t)Δt. As a result, the heat that node i receives
between time t and t + Δt will be equal to the sum of the heat flowing from
all its neighbors pointing to it, which is?
have λpjifj(t)Δt/lj= λjfj(t)Δt. So we get λj= λpji/lj. To sum up, the heat
difference at node i between time t and t + Δt will be the amount of heat it
receives deduced by what it diffuses. The formulation is therefore:
k:(i,k)∈Eλpikfi(t)Δt/li.
On the other hand, each node i receives HR(i,j,t,Δt) amount of heat from
j:(j,i)∈Eλjfj(t)Δt. Since the amount
of heat that j diffuses to i should be equal to the amount i receives from j, we
fi(t + Δt) − fi(t) = λ
⎛
⎝
?
j:(j,i)∈E
pji
lj
fj(t) − μi
?
k:(i,k)∈E
pik
li
fi(t)
⎞
⎠Δt,
(1)
where μiis a flag to identify whether node i has any outlinks. If node i does not
have any outlinks, μi= 0; otherwise, μi= 1. To find a closed form solution to
Eq.(1), we then express it in a matrix form:
f(t + Δt) − f(t)
Δt
⎧
⎩
= λHf(t),where(2)
Hij=
⎨
pji/lj,
−(μi/li)?
(j,i) ∈ E,
i = j,
otherwise.
k:(i,k)∈Epik,
0,
(3)
Solving the above equation, we get
f(t) = eλtHf(0) (4)
The matrix eλtHis called the diffusion kernel, showing that the heat diffusion
process continues infinite times from the initial heat diffusion step.
3 Technical Descriptions
Before presenting our incentive scheme and coalitional game we first give some
technical notations. Our game is based on the bi-directional weighted graph
G = (M,E,P) described in Section 2. Suppose that s is the source node and

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668X. Li, W. Zheng, and M.R. Lyu
Fig.1. Illustration of Notations of the Coalitional Game
d is the destination node, then the player set of this coalitional game is N =
M \{s,d}. Coalition is denoted by any non-empty subset T ⊆ N, and the overall
payoff of the coalition is denoted by v(T) ∈ R. Then the game is expressed by
Γ =< N,v >. Players will form into coalitions to help establishing the highest
effective path between s and d with the lowest cost under the constraint that each
intermediate node’s heat is higher than a threshold θ. If there’s a tie in the total
cost, s will break the tie by choosing the path with the highest heat. The source
can freely choose the value of θ to meet its requirement on reputation. The larger
the value θ is, which means the source has a higher demand on reputations, the
higher payments it will expend. All the paths established inside the coalition T
connecting s and d compose the path set Psd(T).
Initially, s will load a certain amount of initial heat f(0) and diffuse it on
the reputation graph, then at time t, each node will be diffused fi(t) amount
of heat. Correspondingly each source s has an initial balance of h(0), and the
payment to each node hi(t) is paid by it according to fi(t). Every node evolving
in the routing or forwarding procedure will cost its energy. Since the cost for
sending/receiving routing and data packets are different [5], and the cost for
data transmission is usually larger than that of routing packets transmission, we
denote the routing and forwarding cost respectively by ci(r) and ci(f) ∈ R+,
and ci(r) < ci(f) for all i ∈ N. Please see Fig. 1 for the illustration of notations.
4 Incentive Routing and Forwarding Scheme
The basic idea of achieving incentives is that nodes will be paid when they
help others forwarding data or routing packets. Unlike other payment schemes
that reward the nodes according to their claimed cost, our incentive routing and
forwarding scheme pays the nodes by their reputations. The higher a node’s
reputation is, the higher payment it can get. The payment is given by the source
node. The payment may be in the form of virtual currency like [6] or any other
practical form. In our paper we assume that there is such a payment form and
a payment operation daemon in the network.
In the scheme, the source node s will originate the heat diffusion process start-
ing from itself. Then after collecting the forwarding cost of all the