SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Rajat Vadiraj Dwaraknath
Stanford University
rajatvd@stanford.edu
Ishani Karmarkar  
Stanford University  
ishanik@stanford.edu
Aaron Sidford  
Stanford University  
sidford@stanford.edu
We provide new algorithms and conditional hardness for the problem of estimating effective resistances in  $n$ -node  $m$ -edge undirected, expander graphs. We provide an  $\widetilde{O}(m\epsilon^{-1})$ -time algorithm that produces with high probability, an  $\widetilde{O}(n\epsilon^{-1})$ -bit sketch from which the effective resistance between any pair of nodes can be estimated, to  $(1 \pm \epsilon)$ -multiplicative accuracy, in  $\widetilde{O}(1)$ -time. Consequently, we obtain an  $\widetilde{O}(m\epsilon^{-1})$ -time algorithm for estimating the effective resistance of all edges in such graphs, improving (for sparse graphs) on the previous fastest runtimes of  $\widetilde{O}(m\epsilon^{-3/2})$  [Chu et. al. 2018] and  $\widetilde{O}(n^2\epsilon^{-1})$  [Jambulapati, Sidford 2018] for general graphs and  $\widetilde{O}(m + n\epsilon^{-2})$  for expanders [Li, Sachdeva 2022]. We complement this result by showing a conditional lower bound that a broad set of algorithms for computing such estimates of the effective resistances between all pairs of nodes require  $\widetilde{\Omega}(n^2\epsilon^{-1/2})$ -time, improving upon the previous best such lower bound of  $\widetilde{\Omega}(n^2\epsilon^{-1/13})$  [Musco et. al. 2017]. Further, we leverage the tools underlying these results to obtain improved algorithms and conditional hardness for more general problems of sketching the pseudoinverse of positive semidefinite matrices and estimating functions of their eigenvalues.
In a weighted, undirected graph  $G$  the effective resistance between a pair of vertices  $a$  and  $b$ , denoted  $r_G(a, b)$ , is defined as the energy of a unit of electric current sent from  $a$  to  $b$  in the natural resistor network induced by  $G$ . Effective resistances arise for a broad set of graph processing tasks and have multiple equivalent definitions. For example,  $r_G(a, b)$  is proportional to the expected roundtrip commute time between  $a$  and  $b$  in the natural random walk induced on the graph and when  $\{a, b\}$  is an edge in the graph, it is proportional to the probability that the edge is in a random spanning tree.
Effective resistances also form a particular class of metrics on the vertices [1,2] and are a key measure of proximity between vertex pairs. Correspondingly, effective resistances can arise in a variety of data analysis tasks. For example, effective resistances have been used in social network analysis for measuring edge centrality in social networks [3] as well as for measuring chemical distances [4].
Effective resistances have a broad range of algorithmic implications. Sampling edges of a graph using effective resistance is known to efficiently produce cut and spectral sparsifiers (sparse graphs which approximately preserve cuts, random walk properties, and more) [5-7]. Effective resistance-based graph sparsifiers have also been applied to develop fast graph attention neural networks [8], to design graph convolutional neural networks for action recognition [9], to sample from Gaussian graphical models [10], and beyond [11, 12]. Effective resistances have also been used in algorithms for maximum flow problems, [13-16, 16-18], sampling random spanning trees [19-21], and graph partitioning [22, 23]. More recently, effective resistances have also been used to analyze the problem of oversquashing in GNNs and in designing algorithms to alleviate oversquashing [24-26] and have been applied to increase expressivity when incorporated as edge features into certain GNNs [27].
Algorithms. Given the broad utility of effective resistances, there have been many methods for estimating and approximately compressing them [5, 28-31]. In this paper, our main focus is the following effective resistance estimation problem. (We use  $x \approx_{\epsilon} y$  as shorthand for  $(1 - \epsilon)y \leqslant x \leqslant (1 + \epsilon)y$  and assume all edge weights in graphs are positive. See Section 2 for notation more broadly).
Definition 1 (Effective Resistance Estimation Problem). In the effective resistance estimation problem we are given an undirected, weighted graph  $G = (V,E,w)$ , a set of vertex pairs  $S\subseteq V\times V$ , and  $\epsilon \in (0,1)$  and must output  $\tilde{r}\in \mathbb{R}^S$  such that with high probability (whp.),  $\tilde{r}_{(a,b)}\approx_{\epsilon}r_{G}(a,b)$  for all  $(a,b)\in S$ .
The state-of-the-art runtimes for solving the effective resistance estimation problem on  $n$ -node,  $m$ -edge graphs are given in Table 2. To contextualize these results, consider the special case of estimating the effective resistance of a graph's edges, i.e., when  $S = E$ . This special case appears in many of the aforementioned applications, e.g., [13-15, 19, 20]. The state-of-the-art runtimes for this problem are  $\widetilde{O}(n^2\epsilon^{-1})$  due to [28] and  $\widetilde{O}(m\epsilon^{-1.5})$  due to [30]. A major open problem is whether improved runtimes, e.g.,  $\widetilde{O}(m\epsilon^{-1})$  (which would subsume prior work), are attainable.
One of the main results of this paper is resolving this open problem in the case of well-connected graphs, i.e., expanders. Expanders are a non-trivial, previously studied, special case that is often the first step or a key component for developing more general algorithms [32]. In particular we provide an  $\widetilde{O}(m\epsilon^{-1}\bar{\kappa}(G))$  time algorithm where  $\bar{\kappa}$  is a measure of the graph's connectivity. Previously, the only non-trivial improvement in this setting was an independently obtained runtime of  $\widetilde{O}(m + n\epsilon^{-2}(\bar{\kappa}(G))^{3/2})$  for graphs with  $\bar{\kappa}(G) = \widetilde{O}(1)$  [29]. We improve upon the result of [29] for sparse enough graphs and, as we explain in Section 1.1, we can almost match the result of [29] (up to an  $m^{o(1)}$  factor) in dense graphs.
Interestingly, we obtain our main result by providing new effective resistance sketch algorithms.
Definition 2 (Effective Resistance Sketch). We call a randomized algorithm an  $(\mathcal{T}_{\mathrm{s}},\mathcal{T}_{\mathrm{q}},s)$ -effective resistance sketch algorithm if given an input  $n$ -node,  $m$ -edge undirected, weighted graph  $G = (V,E,w)$  and  $\epsilon \in (0,1)$  in time  $O(\mathcal{T}_{\mathrm{s}}(G,\epsilon))$  it creates a binary string of length  $O(s(G,\epsilon))$  from which when queried with any  $a,b\in V$ , it outputs  $\tilde{r}_{a,b}\approx_{\epsilon}r_G(a,b)$  whp. in time  $O(\mathcal{T}_{\mathrm{q}}(G,\epsilon))$ .
Effective resistance sketching algorithms immediately imply algorithms for the effective resistance estimation problem. We obtain our result by obtaining an  $(\tilde{O}(m\epsilon^{-1}), \tilde{O}(1), \tilde{O}(n\epsilon^{-1}))$ -effective resistance sketch algorithm for expanders (see Section 1.1 for a comparison to prior work).
Lower Bounds. Given the central role of effective resistance estimation and the challenging open-problem of determining its complexity, previous work has sought complexity theoretic lower bounds for the problem. [33] showed a conditional lower bound of  $\Omega (n^{2}\epsilon^{-1 / 13})$  for the problem by showing that an algorithm that computes effective resistances in  $(n^{2}\epsilon^{-1 / 13 + \delta})$  for some  $\delta >0$  time could be used to obtain a subcubic algorithm for the triangle detection problem, that we define below.
Definition 3 (Triangle Detection Problem). Given an  $n$ -node undirected unweighted graph  $G = (V, E)$ , determine whether there are distinct  $a, b, c \in V$  with  $\{a, b\} \in E$ ,  $\{b, c\} \in E$  and  $\{c, a\} \in E$ .
Currently, the only known subcubic algorithms for the triangle detection problem leverage fast matrix multiplication (FMM) and therefore their practical utility (in the worst case) is questionable.
Theorem 1 (Informal, [34]). Given an algorithm which solves the triangle detection problem in subcubic time, we can produce a subcubic algorithm, which only performs combinatorial operations and uses the triangle detection algorithm, for Boolean matrix multiplication (BMM) and additional problems which currently are not known to be solvable subcubicly without FMM.
This theorem implies that any subcubic algorithm for triangle detection that doesn't use FMM implies a subcubic BMM algorithm that doesn't use FMM. Consequently, subcubic triangle detection is a common hardness assumption used to illustrate barriers towards improving non-FMM based methods, e.g., the effective resistance estimation algorithms of this paper.
<sup>1</sup>While our algorithms for the effective resistance estimation problem (Definition 1) were obtained independently, our writing and discussion of effective resistance sketch algorithms (Definition 1) was informed by their paper. We provide a more complete comparison of our work with prior results in Table 1.
In this paper we take a key step towards closing the gap between the best known running times for effective resistance estimation and lower bounds by improving the conditional lower bound of  $\Omega (n^{2}\epsilon^{-1 / 13})$  to  $\widetilde{\Omega} (n^{2}\epsilon^{-1 / 2})$  for randomized algorithms. We show this lower bound holds even for expander graphs, and hence our effective resistance estimation algorithm (as well as [28]) are optimal up to an  $\epsilon^{-1 / 2}$ -factor among non-FMM based algorithms, barring a major breakthrough in BMM.
Broader Linear Algebraic Tools. The effective resistance between vertex  $a$  and vertex  $b$  in a graph  $G$  has a natural linear algebraic formulation. For all  $a, b \in V$  it is known that  $r_G(a, b) = \vec{\delta}_{a,b} \mathbf{L}_G^\dagger \vec{\delta}_{a,b}$ , where  $\mathbf{L}_G \in \mathbb{R}^{V \times V}$  is a natural matrix associated with  $G$  known as the Laplacian matrix and  $\vec{\delta}_{a,b} = e_a - e_b$  (see Section 2 for notation). Thus, sketching effective resistances can be viewed as problems of preserving information about subsets of entries of the pseudoinverse of a Laplacian.
Both our algorithms and lower bounds develop more general tools for handling related problems for more general (not-necessarily Laplacian) matrices. On the algorithmic side, we show our techniques can also lead to algorithms and data structures for computing certain quadratic forms involving well-conditioned SDD and PSD matrices. On the hardness side, we show our techniques also improve triangle detection hardness bounds for estimating various properties of the singular values of a matrix.
Paper Organization. In the remainder of the introduction we provide a more precise statement and comparison of our results in Section 1.1. In the remainder of the paper we cover preliminaries in Section 2, present upper bounds in Section 3, and present lower bounds in Section 4. Omitted proofs and additional discussion of related proofs are deferred to the supplementary material.
Algorithms. Here we outline our main algorithmic results pertaining to effective resistance sketching and estimation, and in Section 3 we describe a extensions of our work to broader linear algebraic problems involving SDD and PSD matrices. Our main algorithmic result is a new efficient effective resistance sketch for expanders, a term which is used to refer to graphs with  $\widetilde{\Omega}(1)$ -expansion.
Definition 4 (Expander). For  $\alpha > 0$ , we say that a graph  $G = (V, E, w)$  has  $\alpha$ -expansion if  $\alpha \leqslant \phi(G)$ , where  $\phi(G)$  denotes the conductance of  $G$  and is defined as
$$
\phi (G) := \min  _ {S \subseteq V, S \notin \{0, V \}} \frac {\sum_ {\{u , v \} : u \in S , v \in V \setminus S} w _ {u , v}}{\min  \left(\sum_ {u \in S} d _ {u} , \sum_ {v \in V \setminus S} d _ {u}\right)}, \text {w h e r e} d _ {u} := \sum_ {\{u, v \} \in E} w _ {u}.
$$
Theorem 2. There is an  $(\widetilde{O}(m\epsilon^{-1}), \widetilde{O}(1), \widetilde{O}(n\epsilon^{-1}))$ -effective resistance sketch algorithm for graphs with  $\widetilde{\Omega}(1)$ -expansion.
Table 1 summarizes and compares our Theorem 2 to previous work on effective resistance sketches, including naive algorithms to explicitly compute the pseudoinverse of the Laplacian of  $G$ , which can be computed in  $O(n^{\omega})$  time using FMM or  $\tilde{O}(mn)$  time using a Laplacian system solver (labeled Solver). A  $(\mathcal{T}_{\mathrm{s}}, \mathcal{T}_{\mathrm{q}}, s)$  effective-resistance sketch algorithm implies an  $O(\mathcal{T}_{\mathrm{s}} + |S| \mathcal{T}_{\mathrm{q}})$  algorithm for the effective resistance estimation problem. Hence, Theorem 2 implies the following.
Theorem 3 (Effective Resistance Estimation on Expanders). There is an  $\widetilde{O}(m\epsilon^{-1} + |S|)$  time algorithm which solves the effective resistance estimation problem for graphs with  $\widetilde{\Omega}(1)$ -expansion.
Effective resistance sketches are a common approach to solving the effective resistance estimation problem; but there are also approaches to the problem that do not explicitly construct effective resistance sketches. Table 2 summarizes prior work on effective resistance estimation more broadly.
There has been a long line of research on the problem of computing sketches and sparsifiers of graph Laplacians [5,28,30,36] (i.e., computing a sparse graph  $G'$  such that quadratic forms in the Laplacian of  $G'$  approximately preserve quadratic forms in the Laplacian of  $G$ ). Building on this work, [30] showed there is an algorithm which processes a graph  $G$  on  $n$  nodes and  $m$  edges in  $O(m^{1 + o(1)})$  time and produces a sparse sketch graph  $H$  with only  $\widetilde{O}(n\epsilon^{-1})$  edges such that  $r_G(a,b) \approx r_H(a,b)$ .
$2\omega \leqslant n^{2.37188}$  [35] denotes the fast matrix multiplication constant.
for all  $a, b \in V$ . Consequently, any algorithm which runs in  $\widetilde{O}(m\epsilon^{-c})$  on expanders can be improved to run in  $\widetilde{O}(m^{1 + o(1)} + n\epsilon^{-(c + 1)})$  on expanders simply by running the algorithm on  $H$  instead of  $G$ .
Table 1: Effective Resistance Sketch	Table 2: Effective Resistance Estimation
Citation	Tp	Tq	s	Citation	Runtime	S Restrictions
FMM	nω	1	n2	FMM	nω + |S|	None
Solver	nm	1	n2	Solver	nm + |S|	None
[5]	mε-2	ε-2	nε-2	[5]	n2ε-2	S = V × V
[28]	n2ε-1	1	nε-1	[28]	n2ε-1	S = V × V
[29]	m + nε-2	1	nε-1	[19]	m + (n + |S|)ε-2	None
This Paper	mε-1	1	nε-1	[37]	m + |S| ε-2	None
[30]	mε-1.5	S = E
[29]	m + nε-2 + |S|	None
This Paper	mε-1 + |S|	None
Summary of prior work on Effective Resistance Sketch (Table 1) and Effective Resistance Estimation (Table 2) algorithms on  $n$ -node,  $m$ -edge expanders. All time and space complexities are reported up to to  $\widetilde{O}(\cdot)$ . The methods of This Paper and [29] apply only to expanders; however, the remaining works apply to general graphs. As discussed, when  $m^{1 + o(1)} + n\epsilon^{-(c + 1)} = o(m\epsilon^{-c})$ , any runtime dependence on  $m\epsilon^{-c}$  in the table can be improved to a dependence on  $m^{o(1) + 1} + n\epsilon^{-(c + 1)}$ .
Lower Bounds For the effective resistance estimation problem, [33] showed that any combinatorial algorithm (i.e., an algorithm that uses only combinatorial operations in the sense of Theorem 1) which solves the effective resistance estimation problem for  $S = V \times V$  in  $\tilde{O}(n^2\epsilon^{-1/13 + \delta})$  for some  $\delta > 0$ , would imply a combinatorial subcubic deterministic algorithm which detects a triangle in an  $n$ -node undirected unweighted graph. We improve on their result, as follows.
Theorem 4. Given a combinatorial algorithm which solves the effective resistance estimation problem for  $S = V \times V$  on graphs with  $\widetilde{\Omega}(1)$ -expansion in  $\widetilde{O}(n^2\epsilon^{-1/2 + \delta})$  time for  $\delta > 0$ , we can produce a randomized combinatorial algorithm which solves the triangle detection problem on an  $n$ -node graph in  $\widetilde{O}(n^{3 - 2\delta})$  time whp.
Theorem 4 implies an  $\widetilde{\Omega}(n^2\epsilon^{-1/2})$  randomized conditional lower bound for the problem of estimating effective resistances of all pairs of nodes in an undirected unweighted expander graph, while [33] shows only an  $\Omega(n^2\epsilon^{-1/13})$  lower bound. As we show in Section 4, by a simple reduction we can extend any  $\widetilde{\Omega}(n^2\epsilon^{-c})$  lower bound for the all-pairs effective resistance problem to a  $\widetilde{\Omega}(m\epsilon^{-c})$  lower bound for the all-edges effective resistance problem. Consequently, our result also yields a  $\widetilde{\Omega}(m\epsilon^{-1/2})$  randomized lower bound for the problem of estimating effective resistances of all edges in an undirected expander graph.
In addition to conditional lower bounds for effective resistance estimation, we also improve on existing conditional lower bounds for the problem of estimating spectral sums that we define below.
Definition 5 (Spectral Sum). For  $f: \mathbb{R}^+ \to \mathbb{R}^+$  and  $\mathbf{A} \in \mathbb{R}^{n \times n}$  with singular values  $\sigma_1(\mathbf{A}) \leqslant \sigma_2(\mathbf{A}) \leqslant \dots \leqslant \sigma_n(\mathbf{A})$ , we define the spectral sum  $S_f: \mathbb{R}^{n \times n} \to \mathbb{R}^+$  as  $S_f(\mathbf{A}) := \sum_{i=1}^{n} f(\sigma_i(\mathbf{A}))$ .
[33] showed that for several spectral sums  $S_f$ , any combinatorial algorithm that outputs  $Y \approx_{\epsilon} S_f(\mathbf{A})$  in  $(n^{\gamma} \epsilon^{-c})$  time on an  $n \times n$  PSD matrix would imply an  $O(n^{\gamma + \alpha c})$  time combinatorial algorithm which solves the triangle detection problem, where the scaling  $\alpha$  varies depending on the specific  $S_f$  (see Table 3). We build on their results to show improved randomized conditional lower bounds for several spectral sum estimation problems, as presented in Theorem 5 below.
Theorem 5. Given a combinatorial algorithm which on input  $\mathbf{B} \in \mathbb{R}^{n \times n}$  outputs a spectral sum estimate  $Y \approx_{\epsilon} S_f(\mathbf{B})$  in  $O(n^{\gamma} \epsilon^{-c})$  time with  $\gamma \geqslant 2$  for the spectral sums in Table 3, we can produce a randomized combinatorial algorithm that can detect a triangle in an  $n$ -node graph whp. in  $\widetilde{O}(n^{\gamma + \alpha c})$  time, where  $\alpha$  is a scaling that depends on properties of the function  $f$  (see Table 3 for values of  $\alpha$  for several spectral sums.)
Spectral Sum	[33]	This Paper
TD Runtime	Lower Bound	TD Runtime	Lower Bound
Schatten 3-norm	nγ+4c	n2ε-1/4	nγ+5c/2	n2ε-2/5
Schatten p-norm, p ≠ 1, 2,	nγ+13c	n2ε-1/13	nγ+10c	n2ε-1/10
SVD Entropy	nγ+6c	n2ε-1/6	nγ+5c	n2ε-1/5
Log Determinant	nγ+6c	n2ε-1/6	nγ+5c	n2ε-1/5
Trace of Exponential	nγ+13c	n2ε-1/13	nγ+10c	n2ε-1/10
Table 3: Runtimes for the triangle detection (TD) problem in an  $n$ -node graph using algorithms that produce  $(1 \pm \epsilon)$  multiplicative approximations to various spectral sums in  $O(n^{\gamma} \epsilon^{-c})$  time. The second columns contain the best achievable runtimes for  $\gamma = 2$  that do not use FMM, barring a breakthrough in subcubic triangle detection. All runtimes are reported up to  $\widetilde{O}(\cdot)$ .
General notation. We use  $\mathbf{A}_{i,j}$  to denote the  $(i,j)$ -th entry of  $\mathbf{A}$ . For  $\mathbf{A} \in \mathbb{R}^{n \times n}$  we use  $\lambda(\mathbf{A})$  for its spectrum,  $\lambda_i(\mathbf{A})$  and  $\sigma_i(\mathbf{A})$  for its  $i$ -th smallest eigenvalue and singular value respectively, and  $\rho(\mathbf{A}) := |\lambda_n(\mathbf{A})|$  for its spectral radius.  $\| \cdot \|_p$  denotes the  $\ell_p$ -norm. When  $\mathbf{A}$  is PSD,  $\lambda_{\min}(\mathbf{A})$  denotes its smallest nonzero eigenvalue and  $\kappa(\mathbf{A}) := \lambda_n(\mathbf{A}) / \lambda_{\min}(\mathbf{A})$  denotes its pseudo-condition number. We use  $\langle \cdot, \cdot \rangle$  for the Euclidean inner product,  $\mathbb{1}$  for the all ones vector, and  $e_i$  for the  $i$ -th standard basis vector. We define  $\vec{\delta}_{i,j} := e_i - e_j$  and  $[k] := \{1, \dots, k\}$ . We use  $x \approx_{\epsilon} y$  as shorthand for  $(1 - \epsilon)y \leqslant x \leqslant (1 + \epsilon)y$ . For  $v \in \mathbb{R}^n$ , we use  $v[i:j]$  for the sub-vector from index  $i$  to  $j$ . We use  $v \perp w$  to indicate that  $v, w \in \mathbb{R}^n$  are orthogonal (i.e.,  $v^\top w = 0$ ).
Graphs. We use  $G = (V, E, w)$  to denote a weighted undirected graph on  $V$  with edges  $E$  and edge weights  $w \in \mathbb{R}_{>0}^{E}$  (or  $G = (V, E)$  if unweighted). We use  $\mathbf{A}_G$  to denote its (weighted) adjacency matrix  $(\mathbf{A}_G)_{u,v} = w_{u,v}$  for  $u, v \in V \times V$  and  $\mathbf{D}_G$  to denote its diagonal (weighted) degree matrix  $(\mathbf{D}_G)_u = \sum_{\{u,v\} \in E} w_{u,v}$  for  $u \in V$  (treated as  $w_{u,v} = 1$  if  $G$  is unweighted). We define  $\mathbf{L}_G := \mathbf{D}_G - \mathbf{A}_G$  as its graph Laplacian.  $d_{\max}(G)$  and  $d_{\min}(G)$  refer to the max and min diagonal entry in  $\mathbf{D}_G$ . We may drop the argument or subscript  $G$  if it is clear from context. The effective resistance between nodes  $i, j \in V$  is denoted  $r_G(i,j) = \vec{\delta}_{i,j}^\top \mathbf{L}_G^\dagger \vec{\delta}_{i,j}$ . We assume all input graphs are connected, as effective resistances can be computed separately on connected components. We use  $\mathbf{B}_G$  to denote the  $E \times V$  edge-incidence matrix of  $G$ , where  $(\mathbf{B}_G)_{e,u} = 1$  and  $(\mathbf{B}_G)_{e,v} = -1$  for all  $e \in E$ , and  $e = \{u,v\}$ , with  $\mathbf{B}_{Ge,l} = 0$  for all  $l \neq u$  and  $l \neq v$ .
Symmetric Diagonally Dominant (SDD) Matrices A matrix  $\mathbf{M} \in \mathbb{R}^{n \times n}$  is SDD if it can be decomposed as  $\mathbf{M} = \mathbf{D}_{\mathbf{M}} - \mathbf{A}_{\mathbf{M}}$ , where the  $\mathbf{D}_{\mathbf{M}}$  is a diagonal matrix with non-negative entries and  $\mathbf{A}_{\mathbf{M}}$  is a matrix with zeros on the diagonal such that  $d_{i,i} > \sum_{j=1}^{n} |a_{i,j}|$ . We define the normalization of  $\mathbf{M}$  as  $\mathbf{N}_{M} := \mathbf{D}_{M}^{-1/2} \mathbf{M} \mathbf{D}_{M}^{-1/2}$ . Throughout this paper, we assume, without loss of generality that  $\mathbf{D}_{M}$  has strictly positive entries on the diagonal (otherwise, we can simply remove an entire row and column of zeros). We use  $d_{\max}(\mathbf{M})$  and  $d_{\min}(\mathbf{M})$  to denote the max and min entry in the diagonal of  $\mathbf{D}_{M}$  respectively. We may drop the argument or subscript  $\mathbf{M}$  if it is clear from context.
Spectral Graph Theory. Our results leverage well-known spectral graph theory results. In particular, our algorithm complexities are parameterized by the normalized pseudo condition number of a graph (or SDD matrix).
Definition 6 (Normalized (pseudo-)condition number). We define the normalized (pseudo-)condition number of an SDD matrix  $\mathbf{M} \in \mathbb{R}^n$  as  $\bar{\kappa}(\mathbf{M}) \coloneqq \lambda_n(\mathbf{N}_M) / \lambda_{\min}(\mathbf{N}_M)$ .
To connect the normalized condition number to expander graphs, we can apply Cheeger's inequality, which guarantees that if  $G$  has  $\alpha$ -expansion for some  $\alpha = \tilde{\Omega}(1)$ , then  $\bar{\kappa}(\mathbf{L}_G) = \lambda_n(\mathbf{N}_{L_G}) / \lambda_2(\mathbf{N}_{L_G}) \leqslant 4 / \alpha^2 = \tilde{O}(1)$ .
Theorem 6 (Cheeger's Inequality [38]). Let  $G = (V, E, w)$  be an undirected graph. Then,  $\frac{1}{2}\lambda_2(\mathbf{N}_L) \leqslant \phi(G) \leqslant \sqrt{2\lambda_2(\mathbf{N}_L)}$ .
In addition, we leverage the fact that effective resistances can be expressed in several equivalent expressions. In particular, it is known that:
$$
r _ {G} (i, j) = \vec {\delta} _ {i, j} ^ {\top} \mathbf {L} _ {G} ^ {\dagger} \vec {\delta} _ {i, j} = \vec {\delta} _ {i, j} D ^ {- 1 / 2} \mathbf {N} _ {G} ^ {\dagger} D ^ {- 1 / 2} \vec {\delta} _ {i, j} = (\mathbf {W} _ {G} ^ {1 / 2} \mathbf {B} _ {G} \mathbf {L} _ {G} ^ {\dagger} \vec {\delta} _ {i, j}) ^ {\top} (\mathbf {W} _ {G} ^ {1 / 2} \mathbf {B} _ {G} \mathbf {L} _ {G} ^ {\dagger} \vec {\delta} _ {i, j}),
$$