© Hindawi Publishing Corp. FIRST HITTING TIMES OF SIMPLE RANDOM WALKS ON GRAPHS WITH CONGESTION POINTS

We derive the explicit formulas of the probability generating 
functions of the first hitting times of simple random walks on 
graphs with congestion points using group representations.


Introduction.
Random walk on a graph is a Markov chain whose state space is the vertex set of the graph and whose transition from a given vertex to an adjacent vertex along an edge is defined according to some probability distribution.The probability distribution might depend on vertices, and the case of the uniform distribution over incident edges is called a simple random walk.Many researches have been done on various aspects of random walks such as transience or recurrence, asymptotic behavior of transition probabilities, convergence rates to its stationary distributions, and convergence to a boundary and harmonic functions [2,6].
Random walks can describe the structure of graphs, groups, and related objects and the structure of computer networks or electric networks [3,8].It is quite useful to devise probabilistic algorithms of random walks on graphs which reflect combinatorial problems when deterministic methods to analyze them are known to be difficult [10].It is well known that random walks play a crucial role in the design of randomized algorithms (off-or online) [4,11,15].
The first hitting time (also called the first passage time) is the time taken to reach a vertex for the first time starting from another vertex.This is one of the classical problems on Markov chains, and one can investigate the probabilistic properties of the first hitting time, the stopping time, or transition probabilities [1,12].The expected hitting times for random walks on graphs have been computed using the relation between electrical networks and random walks [9,13,14] and for random walks on finite groups using group representations [5].The group representation approach has been quite powerful in measuring the convergence rate of random walks to its stationary distributions [5,7].
In this paper, we apply group representation technique for a simple random walk on finite graphs with cutvertices (or congestion points), which can be decomposed into several finite groups.Finite graphs with congestion points may capture the delays in computer, telecommunication, and the transportation systems, and the probability generating functions would be useful in performance evaluation of stochastic models, which arise from those various systems.

Preliminaries.
Let G be a finite group with the identity e.Let V be a vector space of dimension d over C of complex numbers, and let GL(V ) be the group of invertible matrices of rank d which represent invertible linear transformations from In other words, there is no subspace W of V except for {0} and V such that ρ(x)W ⊂ W for all x ∈ G. Denote by G the set of all irreducible representations of G.For each x ∈ G, put χ ρ (x) = Tr(ρ(x)), where Tr(ρ(x)) = i (ρ(x)) ii , that is, the trace of ρ(x).The complex-valued function χ ρ is called the character of ρ.For a finite abelian group G, every irreducible representation of G is of dimension 1 and hence χ ρ (x) = ρ(x).For a function µ defined on G, the Fourier transform µ : G → C is defined by and the Fourier inversion formula is given by where |G| denotes the number of elements in G.
Let µ be a probability distribution on G. Suppose that E = {x ∈ G | µ(x) > 0} generates G; in other words, the set of all finite products of some elements in E is equal to G. Then the neighbor of x is given by N x = {gx, g ∈ E}.Consider the transition matrix P = (P x,y ) x,y∈G given by which defines a random walk which moves from x to y with probability µ(yx −1 ).Define the convolution µ * µ by and hence (µ * µ)(x) is the probability that a random path arrives at x in two steps when it starts from e. Similarly, µ * k (x) is the probability that the path arrives at x in k steps starting from e.In particular, a simple random walk is defined by (2.6) Diaconis [5] considered another random walk which starts from x and is absorbed the first time it hits S(⊂ G) whose elements are called sinks.In other words, if a random path arrives at S once, then it permanently stays there.Let a k (t) be the probability of arriving at t at time k (i.e., in k steps) in the restricted random walk with S ≠ φ.Then a k (t) can be interpreted as follows: (1) if t ∉ S, then it is the probability of the path being at t at time k without having hit any sites in S; (2) if t ∈ S, then it is the probability of first being absorbed at t at time k.Let b k (t) be the probability of arriving at t at time k in the unrestricted random walk (S = φ), namely, b k (t) = µ * k (tx −1 ).Let A(t, z) and B(t, z) be probability generating functions given by and for t ∉ S, The dual group Z n consists of characters of the form and the Fourier transform of µ is defined by (2.12) From Fact 2.1, the generating function of the first hitting time from x to t is and the generating function of the first hitting time from x to t without having hit s ≠ t on Z n is (2.14) For notational convenience, Proof.Consider a random path on L from x to 0 without having hit n.
If x is equal to n−1, then each path from x to 0 without having hit n should move the first step to n − 2 with probability 1/2.In other words, (2.17) For 1 ≤ x ≤ n−2, each path from x to 0 without having hit n is either a path from x to 0 without passing n−1 on the way or a path which hits n−1 before arriving at 0 and then restarts a random walk from n − 1 to reach 0 without passing n; (2.18) In other words, For 1 ≤ y ≤ n−1, every random path on L from x to y without having passed any of the endpoints of L, 0, or n can be regarded as a path from x to y generated by a random walk on Z n = {0,...,n − 1} with a sink 0 and a probability distribution µ defined by µ(1) = µ(−1) = 1/2.It is necessary to observe that a path on L from x to 0 without having hit n − 1 is a path on Z n from x to 0 without having hit n − 1.And a path on L which hits n − 1 before arriving at 0 is a path on Z n which hits n − 1 before arriving at 0. For 0 ≤ y, t ≤ n − 1, let T x,y be the first hitting time on Z n from x to y and let T t x,y be the first hitting time on Z n from x to y without having hit t.Equation (2.14) implies that for (2.20) Equations (2.19) and (2.20) imply that for 1 (2.21) Equations (2.17) and (2.21) imply that

.24)
Remark 2.4.Note that in the above proof we also have derived the generating function C n x,0 (z) of the first hitting time from x to 0 without having hit n on L and the symmetry implies C 0 x,n (z) = C n n−x,0 (z).

On graphs with congestion points.
In this section, we consider a simple random walk on a lollipop and a dumbbell, which obviously have cutvertices.A cutvertex is a vertex such that the graph without this vertex is not connected.
Those cutvertices represent typical congestion points which cause delays in various traffic systems such as computer networks, telecommunication, and the transportation systems.We are interested in the first hitting time which can be interpreted as the performance time of such models.Since they have relatively simple structures, the probability generating functions can be derived using basic probabilistic argument.But it is interesting as well to apply group representation technique for these models since they can be decomposed into several cyclic groups Z n .First we define the functions where f (z) and g(z) were defined by (2.15).Theorem 3.1.Consider a simple random walk on the lollipop P , which consists of a circle with 2n vertices and a line segment with n + 1 vertices, as in Figure 3.1.The generating function of the first hitting time from 0 W to 0 on the lollipop P is given by Proof.Let T x,y be the first hitting time from x to y and let T t x,y be the first hitting time from x to y without having hit t on P .Let F x,y (z) = ∞ k=0 Pr(T x,y = k)z k and F t x,y (z) = ∞ k=0 Pr(T t x,y = k)z k .Note that each path from 0 W to 0 should pass through n W = n before arriving at 0 and restart from n to 0; hence Since each path from 0 W to n W can be regarded as the path from 0 to n on Z 2n with a probability distribution µ defined by µ(1) = µ(−1) = 1/2, (2.13) implies that Hence, we obtain Note that each path from n to 0 moves the first step to one of three directions: (n + 1) W , (n − 1) W , and n − 1, with probability 1/3.We have Similar to (3.3) and (3.4), we obtain (3.7) Hence, we have On the other hand, each path from n − 1 to 0 either arrives at 0 without having hit n or hits n before arriving at 0, and then restarts random walk to reach 0 from n = n W , and that the paths on P from n − 1 to 0 without having hit n and from n − 1 to n without having hit 0 can be regarded as the paths on L in Lemma 2.3.Hence, we have (3.9) Equations (3.6), (3.8), and (3.9) and Lemma 2.3(i) imply which, together with (3.5), implies the theorem.Theorem 3.2.Consider a simple random walk on the dumbbell Q, which consist of a line segment with n+1 vertices and two circles each with 2n vertices, as in Figure 3.2.The generating function of the first hitting time from 0 W to 0 E on the dumbbell Q is given by

.11)
Proof.Let T x,y be the first hitting time from x to y and let T t x,y be the first hitting time from x to y without having hit x,y = k)z k .Note that each path from 0 W to 0 E should pass through n E before arriving at 0 E and restart a random walk from n E to 0 E ; we have (3.12) Since each path from 0 W to n E on the dumbbell Q can be regarded as a path from 0 W to 0 on the lollipop P , Theorem 3.1 implies Note that a path starting from n E and ending at 0 E moves the first step randomly to one of three directions: (n − 1) E , (n + 1) E , and 1, with probability 1/3.In other words, ( By symmetry, Since each random path from (n−1) E to 0 E either arrives at 0 E without having hit n E or hits n E before arriving at 0 E and then restarts random walk to reach 0 E from n E , we have Note that each path from (n − 1) E to 0 E without having hit n E on Q can be regarded as a path from n − 1 to 0 on L in Lemma 2.3; Similarly, we have Hence, we have On the other hand, each path from 1 to 0 E on Q should pass through 0 and restart from 0 = n E to 0 E ; we have (3.20) Since each path from 1 to 0 either arrives at 0 without having hit n or hits n and then restarts from n to 0, But those paths on Q from 1 to 0 without having hit n and from 1 to n without having hit 0 can be regarded as those paths on L in Lemma 2.3, that is, Note that each path from n to 0 on Q can be regarded as the path from n to 0 on P and formula (3.10) implies (3.23) Hence, we have (3.24) Formulas (3.20) and (3.24) imply (3.25) and hence (3.14), (3.19), and (3.25) imply the theorem.

Conclusions.
It is not hard to modify our argument to derive the probability generating functions on necklace graphs which consist of several circles and some lines, and also on more general finite graphs which consist of several Cayley graphs.The probabilistic algorithms of random walks on such graphs may give rich information on probabilistic properties of several combinatorial models, such as networks and traffic congestion problems, in terms of hitting times.
Since we have derived the explicit formulas of the probability generating functions in the previous sections, we can efficiently compute the distribution of the first hitting time; namely, the probability, the average, variance, and higher-order moments using Taylor series expansion (see, e.g., [16]).Let p k denote the probability of the first hitting time from x to y being k.The probability generating function, defined by gives the value p k as where A (k) (0) denotes the kth derivative of A(z) at z = 0.Moreover, we can compute the average, variance, and higher-order moments using higher derivatives, for example,

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Some computational softwares, such as Maple or Mathematica, allow us to compute them very quickly with high accuracy.For example, using Maple and Theorem 3.1, the expectations of the first hitting time on the lollipop have been computed up to n = 12 as shown in Figure4.1.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation