P. Tetali. Random walks and the effective resistance of networks. J. Theor. Prob. 4 (1991), 101--109.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... is a constant with =2 ff then x Gamma x;2R Gamma1=2 Delta Gamma( 2 Gammaff) log R which implies (1. 1) The proof that fi 1 ff uses the connection between random walks and electrical networks see [DS] If ( Gamma ; a) is any finite weighted graph then (see [Tet]) T y E T x = G )R e (x; y) 1:4) Here R e (x; y) is the effective resistance between x and y in the network where the edge fx ; y g has conductivity a x 0 . If we collapse the vertices in B(x; R) to a single vertex y, and discard the second term in (1.4) then we obtain an ....

P. Tetali. Random walks and the effective resistance of networks. J. Theor. Prob. 4 (1991), 101--109.

....Group Computing by Graph Transformations II . Laboratoire associ e au CNRS 1 Introduction Random walks have been studied extensively, and have many applications such as generation of random spanning trees[1, 2, 12] token management schemes[8, 11] effective resistance of electrical networks[3, 10, 5], and on line algorithms[4] In this paper, we consider a connected simple undirected graph G = V; E) together with a positive real valued map w over E, w(e) being called the weight of e. A discrete time random walk (or Markov chain) on G is defined as follows. At each step, a particle, located ....

P. Tetali. Random walks and the effective resistance of networks. Journal of Theoretical Probability, 4:101--109, 1991.

....R ab is the effective resistance, as computed by means of Ohm s law, between vertices a and b. The beginner s handbook when studying random walks on graphs from the viewpoint of electric networks is the book of Doyle and Snell (1984) Another mandatory reference in this context is the article of Tetali (1991) where a clever use of the superposition principle and the reciprocity theorem for electric networks yields the following formula for the expected hitting times: E a T b = 1 2 X z C(z) R ab R bz Gamma R az ] 1) where C(z) C zz X x:zx C zx and where the C zz s do not intervene in ....

....proof of one half of proposition 9 58 of Kemeny et al. 1976) in the context of general random walks on graphs. b) N = 3; oe 1 = oe 2 = oe 3 = leads to: E i T j E j T k Gamma E i T k = 1 2 X z C(z) R ij R jk Gamma R ik ] 4) Now, the triangle inequality for resistances given in Tetali (1991) states that R ij R jk Gamma R ik = 2 N ij k C(k) so that replacing this equation into the right hand side of (4) we get E i T j E j T k Gamma E i T k = N ij k (k) which is the second half of the proposition 9 58 mentioned above. A similar equation in Chung (1967) theorem 3, ....

Tetali, P. (1991) Random walks and the effective resistance of networks, Journal of Theoretical Probability, 4, 101-109.

....1, hence the bound ECC n(n Gamma 1) 2 . We shall prove our stronger upper bound by showing that there is a tradeoff between the two quantities, m and R span , and they cannot both attain their maximum value simultaneously. We use the following lemma, which is due to Foster [6] see also [8]) Lemma 2 The sum of effective resistances along the edges of a connected graph is n Gamma 1. X (u;v)2E R[u; v] n Gamma 1 Now we are ready to prove Theorem 1. Proof: Consider T , the minimum resistance spanning tree of G. Edges in T are called twigs, and other edges are called links. If ....

P. Tetali. Random walks and the effective resistance of networks. Journal of Theoretical Probability, 4:101-109, 1991. 4

....of 1 ffl. We do not know whether the extra structure of our graph G 0 can be exploited to improve over the approximation ratio obtainable for metric TSP. 3 Random walks and electrical resistance There is a well known correspondence between random walks and resistance of electrical networks [9, 6, 16]. View each edge of G as a resistor of 1 ohm. The effective resistance between vertices u and v, denoted by R[u; v] is the voltage that develops at u if a current of 1 amp is injected into u, and v is grounded. In [6] it is shown that the effective resistance captures the commute time. Namely, ....

.... resistance lemma) plays a fundamental role in our paper: 5 Lemma 4 For any graph G = G(V; E) with c connected components, the sum of excess resistances along its edges satisfies: X (u;v)2E ffi [ u; v) X v2V 1 d v 1 Gamma c Proof: We present without proof an identity due to Foster [12, 16]: Lemma 5 Let G = G(V; E) be a graph with n vertices and c connected components. The sum of effective resistances along the edges of G satisfies: X (u;v)2E R[u; v] n Gamma c We return to the proof of Lemma 4. By definition of the excess resistance: X (u;v)2E R[u; v] X (u;v)2E ( 1 ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991. 11

.... R[u; v] between vertices, where a graph is regarded as an electrical network in which each edge has unit resistance (see [11] Electrical resistance obeys the triangle inequality R[u; v] R[v; w] R[u; w] Chandra et al. 7] show that Lemma 2 For any two vertices C[u; v] 2mR[u; v] and Tetali [16] shows Lemma 3 For any two vertices H[u; v] 1 2 X w dw (R[u; v] R[v;w] Gamma R[u; w] We use the following lower bound on the resistance between two vertices. Lemma 4 If u is not adjacent to v, then R[u; v] 1=d u 1=d v . Otherwise, R[u; v] 1= d u 1) 1= d v 1) 2 Lemma 4 ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991.

....connections between virtual resistance and electrical resistance in graphs. 2. 3 Properties of virtual resistance There are tight connections between behavior of random walks on graphs, and the interpretation of the graph as an electrical network, where each edge represents a resistance of 1 ohm [DS, CRRST, tetali]. The effective resistance between vertices u and v, denoted by R[u; v] is the voltage that develops in u if a current of 1 amp is forced into u, and v is grounded (by Ohm s law) The effective resistance of a graph, is defined as R = max u;v2V [R[u; v] If the graph is disconnected, then R = 1. ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991.

.... following identity (see [5] for a proof) relates the commute time and the effective resistance: Lemma 2 For any connected graph with m edges, and any two vertices, C[u; v] 2mR[u; v] Another identity characterizes the effective resistance in terms of the number of returns to the origin (see [19] for a proof) Lemma 3 Let d u denote the degree of vertex u . For any connected graph and any two vertices, R[u; v]d u is exactly the expected number of visits at u (including the start) in a random walk starting at u and stopping upon hitting v . As a consequence of the two lemmas, we obtain ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991.

....v in G 0 is bounded from above by the effective resistance between them in G. It is well known that resistances satisfy the triangle inequality R(u; w) R(u; v) R(v;w) 2.2) which follows from the following useful formula from [5] C(u; v) 2 jEj R(u; v) 2. 3) There is also a formula from [15] for H(u; v) in terms of resistances, but it is more complicated: H(u; v) 1 2 X w2V dw Gamma R(u; v) R(v;w) Gamma R(u; w) Delta ; 2.4) where dw is the degree of w. The main lemma in the proof of Theorem 1.1 involves estimating the resistances. A combination of the above identities ....

P. Tetali, Random walks and the effective resistance of networks, J. Theoret. Probab. 4 (1991) 101--109.

....1, hence the bound ECC n(n Gamma 1) 2 . We shall prove our stronger upper bound by showing that there is a tradeoff between the two quantities, m and R span , and they cannot both attain their maximum value simultaneously. We use the following lemma, which is due to Foster [7] see also [8]) Lemma 2 The sum of effective resistances along the edges of a connected graph is n Gamma 1. X (u;v)2E R[u; v] n Gamma 1 Now we are ready to prove Theorem 1. Proof of Theorem 1: Consider T , the minimum resistance spanning tree of G. Edges in T are called twigs, and other edges are ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991. 4

....for low order additive terms) In this respect it is nearly tight, for the path. To bound the cover time, we bound max u;v [D[u; v] over all d regular graphs, and then use Theorem 1.2. Lemma 3.9. For any connected d regular graph, max u;v [D[u; v] n d Gamma 2) 2 =2. Proof. Tetali [12] shows that D[u; v] P z d z (R[v; z] Gamma R[u; z] where d z is the degree of vertex z. For d regular graphs, d z = d. Hence D[u; v] d P z (R[v; z] Gamma R[u; z] By the triangle inequality for resistances, R[v; z] Gamma R[u; z] R[u; v] For convenience of notation, we let R denote ....

P. Tetali, Random walks and the effective resistance of networks. Journal of Theoretical Probability, 4:101-109, 1991.

.... to me) of the mean commute interpretation was given by Chandra et al. [10] One can combine the commute formula with the general identities of Chapter 2 to obtain numerous identities relating mean hitting times and resistances, some of which are given (using bare hands proofs instead) in Tetali [30]. The connection between Foster s theorem and Lemma 6 was noted in [13] Section 4. The spectral theory is of course classical. In devising a symmetric matrix one could use i p ij or p ij Gamma1 j instead of 1=2 i p ij Gamma1=2 j there doesn t seem any systematic advantage to a ....

P. Tetali. Random walks and the effective resistance of networks. J. Theoretical Probab., 4:101--109, 1991.

.... Computing by Graph Transformations II . y Laboratoire associ e au CNRS 1 Introduction Random walks have been studied extensively, and have many applications such as generation of random spanning trees[1, 2, 12] token management schemes[8, 11] effective resistance of electrical networks[3, 10, 5], and on line algorithms[4] In this paper, we consider a connected simple undirected graph G = V; E) together with a positive real valued map w over E, w(e) being called the weight of e. A discrete time random walk (or Markov chain) on G is defined as follows. At each step, a particle, located ....

P. Tetali. Random walks and the effective resistance of networks. Journal of Theoretical Probability, 4:101--109, 1991.

....of our approach is that our results are robust : minor perturbations in the graph (such as the deletion or addition of a few edges) usually do not change the electrical properties of the graph substantially. Following appearance of a preliminary version of this paper (Chandra et al. 1989) Tetali (1991) has extended our ideas to establish a number of new relations between hitting times and effective resistance. The rest of this paper is organized as follows. In section 2 we relate electrical resistance to commute and cover times. Section 3 studies the electrical resistance and the cover time of ....

P. Tetali, Random walks and effective resistance of networks. J. Theoret. Probability 4(1) (1991), 101--109.

....starting from node i. The sum (i; j) H(i; j) H(j; i) is called the commute time: this is the expected number of steps in a random walk starting at i, before node j is visited and then node i is reached again. There is also a way to express access times in terms of commute times, due to Tetali [63]: H(i; j) 1 2 (i; j) X u (u) u; j) Gamma (u; i) 2:1) This formula can be proved using either eigenvalues or the electrical resistance formulas (sections 3 and 4) b) The cover time (starting from a given distribution) is the expected number of steps to reach every node. If ....

....(c) with two nodes s and t nailed down at 1 and 0. Then the force 22 L. Lov asz pulling the nails is 1 R st = 2m (s; t) The energy of the system is 1 2R st = m (s; t) Note that equation (2. 1) can be used to express access times in terms of resistances or spring forces (Tetali [63]) Proof. By construction (b) OE st (v) is the voltage of v if we put a current through G from s to t, where the voltage of s is 0 and the voltage of t is 1. The total current through the network is P u2 Gamma(t) OE st (u) and so the resistence is R st = 0 X u2 Gamma(s) OE st (u) 1 A ....

P. Tetali, Random walks and effective resistance of networks, J. Theoretical Probability 1(1991), 101--109.

....not sensitive to the existance of a constant number of vertices of degree d Sigma 1 (except for low order additive terms) In this respect it is nearly tight, for the path. To bound the cover time, we would like to bound max u;v [D[u; v] over all d regular graphs, and then use Theorem 1. From [9] it follows that for regular graphs D[u; v] ave w (C[v; w] Gamma C[u; w] Now principles similar to those employed in lemmas 11 and 12 can be used to bound the average commute time. We obtain (we omit the details of the proof) Corollary 6, that for any d regular graph, max v [E v [G] 2 ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991.

....is applied to u and v. It can be shown that the effective resistances between vertices relate to the corresponding commute times through the following result, proved by Chandra et al. 4] Lemma 2.2 For any two vertices u and v C(u; v) 2jEjR(u; v) Lemma 2. 2 was later generalized by Tetali [11] to the following result for hitting times. Lemma 2.3 For any two vertice u and v H(u; v) 1 2 X w2V dw (R(u; v) R(u;w) Gamma R(v; w) where dw is the degree of w. We will also need the fact that effective resistances are subadditive. This a long known fact, but if one so wishes one can ....

P. TETALI, Random Walks and the Effective Resistance of Networks, J. Theor. Probab. 4 (1991), 101-109.

....of Lemma 4.2 and Theorem 4.4 below are proved in [19] using Pitman s occupation measure identity. We begin with some simple facts about exit frequencies. The first result is from Aldous [1] several related formulas can be derived using relations to electrical networks, as in [9] 11] or [20]. Lemma 4.1 The exit frequencies x for the naive stopping rule Omega j in reaching state j from state i are given by xk = k Gamma H(i;j) H(j;k) Gamma H(i;k) Delta : More generally, the exit frequencies for the naive stopping rule Omega from initial distribution oe are given by ....

P. Tetali, Random walks and effective resistance of networks, J. Theoretical Prob. No. 1 (1991), 101-109.

.... A useful tool in the analysis of random walks is the notion of electrical resistance R[u; v] between vertices (see [11] Electrical resistance obeys the triangle inequality R[u; v] R[v; w] R[u; w] Chandra et al. 7] show that Lemma 2 For any two vertices C[u; v] 2mR[u; v] and Tetali [15] shows Lemma 3 For any two vertices H [u; v] 1 2 X w dw (R[u; v] R[v; w] Gamma R[u; w] We use the following lower bound on the resistance between two vertices (see [8] for proof) Lemma 4 If u is not adjacent to v, then R[u; v] 1=d u 1=d v . Otherwise, R[u; v] 1= d u 1) 1= d v ....

P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991.

....electric network. From the connections between random walk and electric networks we know that at the exit times, i.e. at the times, when the random walker leaves a generation of the tree forever, the transition probabilities are described by the splitting of an electric current (see [3] 4] [12]) For periodic trees the effective conductances, the current, and thus the stationary random periodic tree at the exit times can be calculated, i.e. the tree as it is watched by the random walker at exit times PERIODIC TREES 3 after a long time of walking. In the present paper we use the ....

....lower than twice the mean reciprocal of the escape probability from the root u of (u) where the mean is taken over all types u of exit vertices. Remark 4.2. The interpretation of Remark 4. 1 c) is further illuminated by the following heuristic argument based on the first hitting time formula of [12]: Let be a weighted, labeled, finite tree with root a. We short its leaves and denote the resulting vertex by b. Each edge (x; y) is weighted with a conductance c(x; y) and c(x) P y c(x; y) For each vertex x of let U x; denote the potential between x and b, if the current from a to b ....

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Tetali, P. (1991). Random Walks and the Effective Resistance of Networks, J. Theor. Prob. 4, 101-109 (Christiane Takacs) Institut f ur Mathematik, Universit at Linz, Altenbergerstraße 69, 4040 Linz E-mail address, Christiane Takacs: Christiane.Takacs@jk.uni-linz.ac.at

....that the expected lengths of the two types of tours from x to x are the same, proving the lemma. 2 Remark. For those readers who are accustomed to thinking of random walks in terms of electrical circuits, we note that the lemma follows also from the following formula which appears in Tetali [17]: HG (x; y) mRG (x; y) 1 2 X z d(z) R G (y; z) Gamma RG (x; z) in which m is the number of edges of G, and RG (u; v) is the effective resistance between u and v when G is regarded as an electrical network with a unit resistor on each edge. A strategy S for the demon will be called a ....

P. Tetali, Random walks and effective resistance of networks, J. Theoretical Prob. No. 1 (1991), 101-109.

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P. Tetali. "Random walks and the effective resistance of networks". Journal of Theoretical Probability, 4:101-109, 1991.

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P. Tetali. Random walks and the effective resistance of networks. Journal of Theoretical Probability, pages 101--109, 1991.

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P. Tetali, Random walks and effective resistance of networks, J. Theor. Prob. 1 (1991), 101-109.

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