Random Walks and Foster s Resistance Theorem

Transcription

1 Random Wals and Foster s Resistance Theorem Chen Qin March 14, 2013 Abstract Given any simple connected graph G =< V,E > such that each edge e E has unit resistance, Foster s Resistance Theorem states that the summation of effective resistances over all edges equals n 1. That is, R e = n 1 e E Further, given a general graph with resistances r e corresponding to edges e E, the theorem states: e E R e r e = n 1 Tetali[1] proved Foster s Resistance Theorem by considering traversals and then appling the Reciprocity theorem. In this paper, we go through that proof. 1 Introduction Doyle and Snell [1] discovered a connection between electrical networs and random wals in graphs. At the time, the discovery was novel and led to breathroughs in both avenues of research. Tetali [2] related the effective resistance R e for e E to traversals in a random wal. This idea was inspired by the commute time [3]; and, along with the Reciprocity Theorem and Kirchoff s Current Law, the idea was used to prove Foster s Resistance Theorem. 2 Preliminaries Consider a simple networ G =< V,E > where every edge e E has the same resistance r e. For a pair of vertices v i,v j V, we define P ij = Prob[v i can reach v j in one step] = 1/d(v i ) if (v i,v j ) E and P ij = 0 otherwise. 1

2 3 Theorems Theorem 3.1. In a simple networ where each edge has unit resistance, the effective resistance R i,j between any pair of vertices v i,v j V is equal to the expected number of traversals along any specific edge (v i,v ) before arriving at v j in a random wal from v i to v j. Proof. Let U i,j be the expected number of times we hit before reaching j on a random wal from i to j. To hit, we must first hit a neighbor of and then cross over an edge to. Therefore, = w since for w N() we have P w = 1 d(w). Next, we divide both sides by d() to get: wp w = w d() = 1 Uw ij d() d(w) w w d(w) Let V j denote the voltage difference between and j. Then, we have: Applying Kirchoff s Current Law, we get: ( ) V j = Uij d() V j = w 1 d() V wj where v w v i v w v j Applying the definition of effective resistance over edge R ij = V ij r ij where V w = V j V wj, we find: This completes the proof. i w = (V j V wj )/1 = Uij d() Uij w d(w) Theorem 3.2. The effective resistance R ij is equal to the expected number of edge traversals out of v along any specific edge (v,v w ) E where v v i v v j. Proof. Let U denote the expected number of visits to in a random wal that starts at v i, goes to v j, and returns bac to v i. It follows that the U = +Uji Applying, we get 2

3 Recall that U d() U = +Uji = V jd()+v i d() = V j d() V i d() = V ij d() R ij = U d() is the expected number of visits to along any specific edge (,z) where z N() in our random wal. The theorem follows. Given a graph G =< V,E >, suppose we consider two electrical setups in which we give special attention to four vertices i, j,, and w with (i,j) E and (,w) E. In the first setup, setup 1, we apply a battery across vertices i and j with a voltage difference V 1 i,j that admits a unit current from to w. In the second setup, we apply a battery across vertices and w with voltage difference V 2,w that admits a unit current from i to j. Theorem 3.3. In any single source connected graph, by the Reciprocity Theorem [4] we have V 1 i,j = V 2,w. We now consider two setups for the given graph. The first has source v i, sin v j, and a voltage is applied so that the current flowing from v to v j is a unit current. In the second setup we have source v, sin v j, and a voltage is applied so that the current flowing from v i to v j is a unit current. Let V ij and V j be these respective voltages. By and the previous theorem, we have: V ij = V j Uij d() = Uj i d(i) The right equation refers to random wals starting from v and v i respectively and ending at v j. The expected number of times we reach v j from one of its neighbours when v v j is exactly 1. Therefore, v N(v j ) d() = 1 v i v j We apply the result above to the Reciprocity Theorem to get: 3

4 v N(v j ) d() = v N(v j ) Since j = 0, we can sum over all vertices to get: v i V v N(v j ) Rewriting the previous equation, we get: (v i,v j ) E [ U j i d(i) = d() d() = 1 d() = n 1 /d()+uj i /d(i)] = n 1 Applying Theorem 3.2, we complete Tetali s proof of Forester s Resistance Theorem[4]: 4 General networ (i,j) E R i,j = n 1 In order to extend Foster s Resistance Theorem to general networs, we begin by defining conductance c e = 1/r e for edges e E and 0 along all other edges. For a vertex x, let c(x) be the sum of the conductances of edges incident on x. Rewriting, we see: We have: Theorem 4.1. V j = Uij c() <v i,v j > E R ij r ij = n 1 Proof. The proof starts by applying the Reciprocity Theorem[4] Multiplying both sides by c ij we get v N(v i ) /c() = Uj i /c(i) c ij/c() = v i N(v j ) 4 U j i c ij /c(i) = 1 if j

5 v j G {v i,v j } G By theorem 3.2, we obtain the proof. References v i N(v i ) c ij/c() = n 1 [ c ij/c()+u ji c ji/c()] = n 1 [1] Doyle P.G. and Snell J.L., Random Wals and Electrical Networs, The Mathematical Association of America(1984). [2] Tetali, P. Random Wals and the Effective Resistance of Networs. J. Theor. Prob. 4, , [3] Chandra A.K., Raghavan P., Ruzzo W.L., Smolensy R., and Tiwari P., The Electrical Resistance of a Graph Captures its Commute and Cover Times, Proceedings of the 21st Annual ACM symposium on Theory of Computing, May(1989). [4] Hayt W.H. and Kemmerly J.E., Engineering Circuit Analysis,McGraw-Hill,3rd ed.(1978). 5