Comparison of methods to determine point-to-point resistance in nearly rectangular networks with application to a hammock network

Considerable progress has recently been made in the development of techniques to exactly determine two-point resistances in networks of various topologies. In particular, two types of method have emerged. One is based on potentials and the evaluation of eigenvalues and eigenvectors of the Laplacian matrix associated with the network or its minors. The second method is based on a recurrence relation associated with the distribution of currents in the network. Here, these methods are compared and used to determine the resistance distances between any two nodes of a network with topology of a hammock.


Introduction
The computation of two-point resistances in networks is a classical problem in electric circuit theory and graph theory, with applications in the study of transport in disordered media [1][2][3], random walks [4], first-passage processes [5] and lattice Green's functions [6]. In recent decades, and especially in recent years, the problem has received widespread interest in the mathematical, physical, engineering and chemical sciences because of its relevance to such a broad range of problems. A nice interpretation of the two-point resistance R ij between nodes i and j in a graph was given by Klein   These equations are a special case of the formula for a resistor network consisting of T nodes and resistance r i,j = r j,i between nodes i and j, thus The resistance between nodes α and β (R α,β ) can be written as [10] R α,β = (3.5) where the λ i are the non-zero eigenvalues of L T and Ψ i = (ψ i,1 , ψ i,2 , . . . , ψ i,T−1 ) are the corresponding orthonormal eigenvectors. The two-dimensional Laplacian of an M × N rectangular network with free boundaries can be written in terms of two one-dimensional Laplacians in the form (3.6) where L free M is the Laplacian for an M-node chain with free boundaries and U M is a unit matrix of dimension M. The Laplacian for other combinations of boundary conditions can be similarly written. The eigenvalues of the two-dimensional lattice matrix are therefore sums of the eigenvalues of the one-dimensional chain matrices and the eigenvectors are products of the corresponding eigenvectors.
The 'hammock' Laplacian cannot be decomposed in this way, so that the eigenvalues and eigenvectors are far more difficult to determine. A similar problem arises if only one additional vertex is added to the rectangle, giving rise to a 'fan' network [21]. However, for the 'fan' only MN of the MN + 1 equations (3.3) are independent so setting the potential of the extra vertex to zero eliminates the corresponding row and column and the Laplacian may be replaced by the resulting minor which may be decomposed in the form (3.6). The resulting equations are independent and all of the eigenvalues are non-zero and are to be included in the sum (3.5). This method is due to Izmailian, Kenna and Wu who used it to determine the 'cobweb' [12] and 'fan' [21] resistance.
If the free boundary conditions of the 'hammock' are replaced by periodic ones, the resulting topology is known as a 'globe'. The same problem arises in that the Laplacian cannot be decomposed. Izmailian & Kenna [13] showed that a solution was to replace the Laplacian by the minor L obtained by deleting the rows and columns corresponding to both added nodes. However, a rather complicated correction needs to be made. The resulting formula for the resistance between any two nodes α and β other than the node 0 is This expression is evaluated for the 'hammock' in §5.1.

Method B
Method B was introduced by Tan [17]. See also Tan et al. [18]. Let I k (i) be the upward current in column k between nodes at heights i − 1 and i. We consider the 'hammock' to be a rectangle with N columns and M + 2 rows with zero resistance in the top and bottom rows. The rows are labelled i = 0 to M + 1 (figure 2).  Figure 2. Rectangular network with the top and bottom rows of resistors having zero resistance. s = 6, p = 4, q = 4, t = 10, x 1 = 3, x 2 = 11, y 1 = 3, y 2 = 6, M = 9, N = 17. In method B, the input node α ≡ N p and the output node β ≡ N q .
Suppose current J is injected in column k = z at height i = y. In §5.2, we derive the following relation between the currents in three adjacent columns: The eigenvalues and eigenvectors of L free m are known [10]. Next, we define a matrix Ψ , the rows of which are the eigenvectors of L free M+1 , and use it to obtain transformed current vectors X k ≡ Ψ I k , k = 1, 2, . . . , N. Applying Ψ to the matrix form (5.37) of equation (3.8) shows that for each row i, X k (i) satisfies a separate second-order recurrence on the column index k (see (5.46)). This may be solved in the standard way with two parameters in each of the three regions of k delineated by the boundaries and the input and output nodes. Having determined the parameters by imposing the boundary conditions, the currents are obtained from the resulting X k using the inverse transformation I k = Ψ −1 X k . The potential difference, and hence the resistance, between the input and output nodes is obtained by summing the potential differences (determined by the currents) along a path between the nodes via the common node i = M + 1 (see equation (5.33)).

Results for the resistance of the M × N 'hammock' network between two arbitrary nodes
We next present some new results for the 'hammock' network coming from each approach. Then we present details of the derivations using each method. In the final section, we compare the advantages and disadvantages of each approach.

Main result
Let u i = 2 + 2h[1 − cos((i − 1)π/(M + 1))] and let λ i be the greater solution of where h = r/s the horizontal-to-vertical resistance ratio. With the resistance of the 'hammock' between N p and N q is found to be R x 2 ,y 2

Resistance between two nodes on the same radial line
Without loss of generality, we take the line to be k = 0 and set p = q = 0 so that Note that 2s + 1 = 2x 1 − 1 and 2t + 1 = 2N − 2x 2 + 1.

Resistance between two nodes on the same transverse line
Setting y 1 = y 2 = y, so that S i (y 1 ) = S i (y 2 ) = S i (y), the numerator of the summand in (4.3) becomes (α − 2β + γ )S i (y) 2 . Further, if we set p = q, the distance between the input and output nodes is In this case, If, furthermore, s = t, then the input and output nodes are symmetrically placed relative to the radial line boundaries and R x 2 ,y Equations (4.3), (4.6) and (4.8) comprise the main new results of this paper. We next present their derivation using the two methods. This will facilitate a comparison between the two approaches in §6.

5.
Derivation of the general form (4.3) by two different methods 5.1. Method A: using the Laplacian approach [13] We begin with the expression (3.7) for the point-to-point resistance in terms of the Laplacian L ij of the rectangular part of the 'hammock'. The nodes on the rectangular part are labelled by and c 0 is given by can be transformed to where Λ k and ψ k,i are eigenvalues and eigenvectors of the second minor L of the Laplacian. The second minor of the Laplacian for the 'hammock' network may be factored in a similar way to the rectangular network. For the example of figure 4  The eigenvalues and eigenvectors of L free N and L DD M are well known [13], Noting that the sum i = 1 to N is equivalent to the sum x = 1 to N with y = 1, we start by evaluating The last equality is valid for all integer values of y' in the range y ≤ 2M + 1, which is clearly our case. The two required sums now follow: and where in (5.3) the input node α = (x 1 , y 1 ) and the output node β = (x 2 , y 2 ) and we have used (5.14). Substituting Σ 1 and Σ 2 into equation (5.2), the required resistance now takes the form

Method B: using the recursion-transform technique of Tan [17]
We use the k, s, t, p, q notation defined in §4. This choice enables the use of symmetry and produces more symmetric coefficients (4.5). Suppose that current J is input at N p and flows out at N q . Let I k (i) be the resulting radial current in the ith resistor from the lower edge of column k in the direction of increasing i ( figure 5). Using Ohm's law, the potential difference between N p and N q may be measured along a path from N q to the common node i = M + 1 and then to N p with the result Here, z = q or −p and y = y 1 or y 2 . When i = 1 in (5.34), I k (i − 1) = 0. The sum of the voltage differences round the loop is zero so using Ohm's law where r i = r for 1 ≤ i ≤ M, r 0 = r M+1 = 0. Combining these equations where h i = r i /s. With h = r/s, equation (5.37) may be written in matrix form where U m is an m-dimensional unit matrix, y is a column matrix with ith element i,y = δ i,y+1 − δ i,y and For k = t, we only use the loop ABEDA in figure 5 to obtain the boundary equations or in matrix form with a similar equation for k = −s.

The recurrence relation
Let χ i ≡ ϕ i−2 = (i − 1)π/(2M + 2). W M+1 has eigenvalues w i = 2 cos(2χ i ) and eigenvectors ψ i , i = 1, 2, . . . , M + 1. The jth component of ψ i is given by [10] Let Ψ be the matrix with ith row ψ i and define X k = Ψ I k . Ψ is invertible with general element of the inverse (5.43) Using (5.33) R x 2 ,y 2 where for i > 1 Multiplying (5.41) on the left by Ψ , noting that Ψ W M+1 is diagonal with diagonal elements w i , and taking the ith component (5.46) where u i = 2h + 2 − hw i and Applying Ψ to (5.41) and taking the ith component

Solving the recurrence relation
For k = z, the general solution of (5.46) is a linear combination of λ k i andλ k i , where λ i andλ i are solutions of (4.1) in terms of which λ i +λ i = u i and λ iλi = 1. The coefficients depend on the region: Matching the solutions at k = q and k = −p, Substituting in the boundary equations (5.48) The final two relations arise from the k = q and k = −p radial lines where the current J is input and output. Using (5.46) with k = z = q and secondly with k = z = −p, in the second case replacing J by −J Solving equations (5.52)-(5.54) for A i andĀ i and substituting in (5.49) gives for 1 < i ≤ M + 1 and Now χ 1 = 0 and λ 1 = 1 so the above expressions are indeterminate when i = 1, but this may be resolved by taking limits.

Summary and discussion
We have derived the resistance between two arbitrary nodes of the 'hammock' network using the two different methods, A and B. Instead of focusing on the potentials as in the Laplacian approach of method A, the recursive strategy in method B is to obtain a relation between the vertical currents in three adjacent columns. Besides different starting strategies, we use different coordinate notations for the different approaches. The coordinates used in method B enable the use of symmetry and lead to symmetric coefficients (4.5).
Each approach has its advantages and disadvantages in general. For method A, the formula (3.5) for the two-point resistance is valid for an arbitrary network. The two-point resistance can be computed for cubic lattices in any spatial dimension (as the Laplacian for d-dimensional regular square lattices can be represented as the sum of d one-dimensional Laplacians, with known eigenvalues and eigenvectors) under various boundary conditions [10], for example free or periodic. Thus the resistance problem is one of the few non-trivial problems which can be solved exactly in high dimensions. Once the eigenvalues and eigenvectors of the Laplacian are known, the resistance between two arbitrary points is given by a very simple summation formula (3.5). While the determination of the eigensystem is straightforward to obtain for hypercubic lattices in any dimensions, the approach cannot readily deal with other complex graphs. However, for the square lattice with one or two added nodes, the Laplacian may be replaced by its first or second minors, respectively, for example, as in method A.
In terms of applications to the 'hammock' network, conversion to a rectangular network is an essential part of both methods. In method A, this is so that the decomposition (3.6) may be used. In method B, the 'hammock' is extended to a full rectangle with N columns and M + 2 rows with zero resistance in the top and bottom rows. If r i is the value of the resistors in row i of the rectangle, then r 0 = r M+1 = 0 and otherwise r i = r so that the same recursion (5.37) can be used for all rows. This extension is not possible in method A as the coefficients are conductances and would be infinite in the top and bottom rows. Instead, the contribution of the two additional nodes is first separated to yield (3.7).
Both methods use the eigenvectors and eigenvalues of the Laplacian L free m of the linear chain of length m with free boundaries. Method A further requires the eigensystem for the Laplacian of a chain with Dirichlet-Dirichelet boundary conditions. This leads to a double sum (5.24) and in order to arrive at the final formula one of the sums has to be removed using a non-trivial identity (5.25). Reference is made to Wu [10] for the proof of the identity. The summation which occurs in the final formula is the starting point of method B (5.33) and the summand involves the transformed current vector and the inverse of the eigenvector matrix (see (5.44)). The former requires the solution of a recurrence relation with constant coefficients and the latter involves a trivial summation (5.45). Finally, method A requires reference to previous calculations (3.7) and (5.25), whereas method B is virtually self-contained using only Ohm's law and Kirchhoff's laws.
The Laplacian approach of method A has so far delivered analytical formulae for the two-point resistances for classes of graphs such as regular two-dimensional square lattices under different boundary conditions [10]; higher dimensional regular square lattices [10]; regular square lattices with a single additional node; so-called cobweb [12] and fan [21] networks; and regular square lattices with two additional nodes-the so-called globe network [13].
Method B has previously been applied to the fan [22], cobweb [23] and globe networks [19]. The method has also been used on the regular square lattice but the potential difference along the top edge is non-zero and has to be calculated by interchanging the x-and y-axes; alternatively, the required potential difference may be determined along a vertical path followed by a horizontal path (J.W. Essam 2014, unpublished data).
Method B could also be applied to problems where the horizontal resistance depends in different ways on the row index i. The simplest of these would be r i = ir which would apply to a fan network embedded in the plane where the length of the resistor wires would be proportional to the distance from the apex. The method as presented here would then require finding the eigensystem of a tridiagonal matrix with elements depending on r i . This can be avoided by working with the vector I(i) ≡ {I 1 (i), I 2 (i), . . . , I N (i)} which when transformed would lead to a second-order recurrence relation with coefficients depending on i.