Discrete curvature on graphs from the effective resistance^*

The classical setting of curvature is differential geometry [7, 43], where it characterizes how much and in which ways a smooth space differs from being flat around a point. Scalar curvature for instance associates a single number to each point on a manifold and allows to distinguish the qualitatively different cases of positive curvature (like the surface of a sphere), zero curvature (like the flat plane) and negative curvature (like a saddle point), as illustrated in figure 1. The scalar curvature is a measure of how much the volume of (small) -balls around a point differ from the volume of -balls in Euclidean space of the same dimension. Ricci curvature, on the other hand, associates a tensor to each point on a manifold, and reflects the difference of volume growth between geodesics emanating from the point in two tangential directions compared to Euclidean growth. While the Ricci curvature includes 'directional' information not present in the scalar curvature, the latter can be retrieved as the trace of the former. We refer the readers to [7, 43] for references on curvature and differential geometry.

While discrete spaces such as graphs lack the differential structure required for classical definitions of curvature, it is still widely acknowledged that these spaces exhibit relevant geometric features and consequently, there have been many attempts at formulating a theory of discrete curvature that works in these generalized settings as well [5, 9, 17, 30, 64, 69]. The idea of discrete curvature is perhaps most easily understood in the case where the discrete spaces under consideration are closely related to some smooth space, for instance a graph which can be isometrically embedded into a manifold (see section 4.1) or random geometric graphs which are natural 'coarse' representations of some smooth space (see section 4.3). However, even when there is no clear underlying or latent smooth space involved, discrete notions of curvature can be formulated. Importantly, most contributions on the subject of discrete curvature not only attempt to generalize the definition of curvature, but also try to translate the multitude of curvature-related results over to this new setting. For more information on different approaches to discrete curvature and their theory, we refer the reader to [9, 46, 60, 67]. In section 4.2 we give a short introduction to combinatorial curvature, OR curvature and FR curvature, as these particular notions are related to our proposed curvature.

Here, we introduce the preliminaries on graphs and effective resistances required to define the resistance curvature; further details can be found in [20]. While we mainly work in the setting of weighted graphs, it is important to note that a weighted graph is nothing more than a set together with some notion of similarity defined between (some) pairs of elements of this set. This type of data is of course very general and widely available, which broadens the applicability of the resistance curvature to more than just graph theory and network science. For instance, any dataset with a similarity kernel—or a parametrized family of kernels to cover different scales, as is often the setting in topological data analysis [14, 32]—will naturally have an associated weighted graph whose resistance curvature then says something about the geometry of the underlying dataset. This is illustrated by the analysis in section 4.3, where we find that the resistance curvature (calculated via an associated graph) retrieves the flat curvature of a point-set in the Euclidean plane.

A weighted graph $G=(\mathcal{N},\mathcal{L},c)$ consists of a set of n nodes $\mathcal{N}$ and a set of m links $\mathcal{L}$ that connect pairs of nodes ⁵ , denoted by an unordered tuple $(i,j)\in \mathcal{L}$ or simply i ∼ j. Furthermore, each of the links is assigned a positive weight through the function $c:\mathcal{L}\to {\mathbb{R}}_{ > 0}$ and we write c_ij for the weight of a link (i, j). We furthermore restrict our attention to simple, undirected graphs without self loops.

One convenient way to describe and study weighted graphs is by their associated Laplacian matrix. For a given n-node graph, the Laplacian matrix is an n × n matrix Q whose rows and columns are identified with the node set $\mathcal{N}$, and with entries

$$\begin{equation*}{(Q)}_{ij}{:=}\begin{cases}-{c}_{ij}\quad \hfill & \quad \text{if}\enspace i\sim j\hfill \\ {k}_{i}\quad \hfill & \quad \text{if}\enspace i=j\hfill \\ 0\quad \hfill & \quad \text{otherwise},\hfill \end{cases}\end{equation*}$$

where k_i := ∑_j∼i c_ij is the weighted degree of node i, i.e. the total weight of all links incident on node i. We will also be interested in the combinatorial degree of a node i, which is defined as d_i := ∑_j∼i 1 and is independent of the link weights. Apart from being a convenient matrix representation of the data associated to a weighted graph, the Laplacian matrix is also a central object in (spectral) graph theory [16, 55, 56] with broad applications [18, 39, 72, 81, 82].

The Laplacian matrix is positive semidefinite and its rank depends on the number of connected components β(G) in the graph as [16]

$$\begin{equation*}null(Q)=\beta (G)\quad \text{or}\quad rank(Q)=n-\beta (G),\end{equation*}$$

where the kernel is spanned by vectors which are piecewise constant on the components of G; we write G_i to denote the unique connected component that contains a given node i. In context of this article, the Laplacian is mainly relevant for its relation to the effective resistance. This resistance is defined based on the Moore–Penrose pseudoinverse of the Laplacian Q^†, which is determined by the equations Q^† Q = QQ^† = proj(ker(Q)^⊥) and can be calculated, for instance, by inverting the nonzero eigenvalues of the Laplacian matrix.

The effective resistance between pairs of nodes is defined based on this pseudoinverse Laplacian as

$$\begin{equation}{\omega }_{ij}{:=}{({\mathbf{e}}_{i}-{\mathbf{e}}_{j})}^{\mathrm{T}}{Q}^{{\dagger}}({\mathbf{e}}_{i}-{\mathbf{e}}_{j}),\end{equation} \tag{ 1 }$$

for any i, j in the same connected component ⁶ , where e_i is the ith unit vector. The resistance matrix Ω is the n × n matrix containing all pairwise effective resistances, as (Ω)_ij = ω_ij . The concept of effective resistances originates from the theory of electrical circuits, where it captures the resistance exerted by the whole network on a current flowing between any two nodes; more precisely, if a unit current flows through the graph from node i to j, where it passes through the links with resistances 1/c and obeys Kirchhoff's laws, then the effective resistance is measured ⁷ as the voltage difference between i and j (resistance = voltage/current) [23, 77]. The effective resistance has many mathematical properties and one of its most celebrated properties is that it is a metric between the nodes of a graph, see also [20]:

Theorem 1 (resistance is distance [38, 49]). The effective resistance is a metric between the nodes of a graph.

Rayleigh's monotonicity principle [24, 77]—which says that adding new paths or shortening existing paths between two nodes can only decrease the effective resistance between these nodes, and vice versa for removing paths—leads to the following intuitive interpretation of this distance measure: if the resistance distance ω_ij is large, this reflects that nodes i and j are not well connected in the network—there are few and mainly long (low weight) paths connecting i, j—, whereas a small ω_ij means that they are well connected—there are many short paths between i, j. The effective resistance thus reflects a more integrated notion of distance compared to the shortest path distance, since it takes into account all the different paths between two nodes and how they are interconnected.

In this article, we will mainly be interested in the product of the link weight and the effective resistance between the end nodes of the link, c_ij ω_ij which we call the relative resistance ⁸ of (i, j). This relative resistance reflects the importance of a link for connectivity of the graph. A link which is very redundant—i.e. by connecting two nodes in group of tightly interconnected nodes—will have a low relative resistance, whereas removing a link with high relative resistance would significantly reduce the connectivity between its endpoints. In [73], it was shown that sampling links proportional to their relative resistance yields statistically representative sparse samples of a graph, which is related to the concept of statistical load/leverage in numerical linear algebra [25].

To further quantify the notion of 'link importance' we introduce a well-known relation between relative resistances and random spanning trees—since the relative resistance of a link only depends on the connected component containing the link we will briefly assume the underlying graph to be connected. A spanning tree T of G is a connected subgraph on all the nodes which is a tree (i.e. has n − 1 links) and the set of all spanning trees is denoted by $\mathcal{T}$. A random spanning tree T is a random element of $\mathcal{T}$ with probability to equal any specific spanning tree T given by

$$\begin{equation*}\mathrm{Pr}[\mathbf{T}=T]=\frac{c(T)}{{\sum }_{{T}^{\prime }\in \mathcal{T}}c({T}^{\prime })}\quad \text{with}\enspace c(T){:=}\prod\limits _{(i,j)\in \mathcal{L}(T)}{c}_{ij}.\end{equation*}$$

In other words, a random spanning tree is sampled proportional to c(T), the product of its link weights. This relates to the relative resistance as, see also [53]:

Theorem 2 ([13, 48]). The relative resistance of a link equals the probability that this link is contained in a random spanning tree, c_ij ω_ij = Pr[(i, j) ∈ T].

For graphs on multiple connected components, the same result holds in terms of random spanning forests which consist of a random spanning tree on each of the connected components independently. Theorem 2 is an important tool to study the relative resistance, and we will make use of the following properties (see appendix A.1).

Property 1 (relative resistance properties). The relative resistance of a link (i, j) is bounded by 0 < c_ij ω_ij ⩽ 1, with equality in the upper bound if and only if it is a cut link. The sum over all relative resistances equals ${\sum }_{(i,j)\in \mathcal{L}}{c}_{ij}{\omega }_{ij}=n-\beta$, which is known as Foster's theorem.

We discuss a few examples to support the introduced definitions; we describe some classes of graphs where positive, zero and negative curvature occurs generically and show some results for Erdős–Rényi (ER) random graphs. Figures 2 and 3 illustrate the examples.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Example 1  (tree graphs). As introduced, a tree graph is a connected graph with m = n − 1 links. Since every link in a tree graph is a cut link, the bounds in properties 1–3 hold with equality, and we find that

$$\begin{equation*}{c}_{ij}{\omega }_{ij}=1\qquad {p}_{i}=1-\frac{{d}_{i}}{2}\qquad {\kappa }_{ij}={c}_{ij}(4-{d}_{i}-{d}_{j})\end{equation*}$$

for all nodes i and links (i, j). In other words, except for the leaf nodes and the links connected to them, all nodes and links have nonpositive node/link resistance curvature.

Example 2  (path graphs). A path graph is a tree with two leaf nodes forming the ends of the path, and all other nodes having degree two. Following the result for tree graphs we find that

$$\begin{equation*}\begin{cases}{p}_{i}=\frac{1}{2}\quad \hfill & \quad \text{at}\;\text{the}\;\text{end}\;\text{nodes}\hfill \\ {p}_{i}=0\quad \hfill & \quad \text{otherwise}\hfill \end{cases}\qquad \begin{cases}{\kappa }_{ij}={c}_{ij}\quad \hfill & \quad \text{if}\enspace i\enspace \text{or}\enspace j\;\text{is}\;\text{an}\;\text{end}\;\text{node}\hfill \\ {\kappa }_{ij}=0\quad \hfill & \quad \text{otherwise}\hfill \end{cases}.\end{equation*}$$

In other words, except for the end nodes and the links connected to them, all nodes and links have zero curvature. The case of a two-node path is slightly different as we get κ_ij = 2c_ij .

  Example 3 (cycle graphs). As noted before, the symmetries of a graph can have implications for the resistance curvature. In appendix C we show that the relevant notion of symmetry is node (link) transitivity which says that any node (link) can be mapped onto any other node (link) by some structure-preserving map, i.e. an automorphism. One example of a node and link transitive graph is the cycle graph, for which the node and link resistance curvatures equal (see appendix C):

$$\begin{equation*}c{\omega }_{ij}=\frac{n-1}{n}\qquad {p}_{i}=\frac{1}{n}\qquad {\kappa }_{ij}=\frac{4c}{n-1}\end{equation*}$$

for all nodes i and links (i, j) and constant link weight c. In other words, all nodes and links are positively curved. Some more examples of transitive graphs will be discussed in section 4.1. Figure 2 shows an example of tree, path and cycle graphs and their curvatures.

Example 4 (ER random graph). An ER graph is a random graph where every pair of nodes is connected with probability ρ. Figure 3 shows how the mean link curvature $\frac{1}{m}{\sum }_{i\sim j}{\kappa }_{ij}$ and resistance curvature distributions evolve in these random graphs as a function of the connection probability (density) ρ. An intuitive explanation for the observed evolution of the mean link curvature is as follows: ER graphs with very small ρ typically consist of a collection of two-paths and disconnected nodes (graph A in figure 3) with a mean link curvature of +2 since κ_ij = 2 for a two-path. For increasing ρ, the ER graph will be a collection of trees or locally tree-like components (graphs B, C) resulting in a decreasing κ, and eventually the graph will become a dense connected graph (graph D) with increasing κ until it reaches mean link curvature +2 for the complete graph at ρ = 1. For the node resistance curvature we find a similar explanation: small and medium-density ER graphs consist mainly of disconnected nodes and 2-paths with p = 1 and p = 1/2 respectively, or a collection of trees with p_i = 1 − d_i /2, which results in the 'discrete distribution' of p on half integers as observed in figure 3. For increasing density ρ, we observe that the locally tree-like graphs have a 'continuous distribution' of possible node curvatures which narrows and converges to p ≈ 1/n when the density increases to 1; this can be explained by the high uniformity of dense ER graphs in combination with Foster's theorem for β = 1. In section 4.3 we treat ERGs as another example of random graphs.

Definitions (2) and (3) for the node and link resistance curvature are fairly simple to state and easy to manipulate, but they lack a geometric character that intuitively relates them to curvature. Here, we discuss an alternative definition of the resistance curvatures in terms of resistance distances in the neighbourhood around a node or link.

A distribution on a graph is a nonnegative function on the nodes $f:\mathcal{N}\to \left[\right.0,\infty \left.\right)$ with unit sum ${\sum }_{i\in \mathcal{N}}f(i)=1$. A distribution determines a random node N ∼ f, which is a random element of the node set with probability of being equal to any particular node given by Pr[N = i] = f(i); we also say that N is distributed according to f. The average distance between two random nodes N and M, distributed according to f and g, can be defined as

$$\begin{equation}\mathbb{E}({\omega }_{NM}){:=}\sum\limits _{i,j\in \mathcal{N}}\,f(i)g(j){\omega }_{ij}\quad \text{with}\enspace N\sim f\;\text{and}\;M\sim g.\end{equation} \tag{ 4 }$$

An important class of distributions follows from the diffusion (heat) equation on graphs ¹¹ : df(t)/dt = −Q f, where Q is the graph Laplacian. If the initial state f(0) is a distribution then its evolution f(t) = exp(−Qt)f(0) remains a distribution; f(t) can also be interpreted as the distribution of a continuous-time random walker with initial distribution f(0). As we will discuss in more detail in section 4.2.2, the diffusion of a distribution on a single node f = e_i is concentrated on the neighbourhood of this node for small times; we will denote this by ρ _t,i := exp(−Qt)e_i which for small times satisfies ρ _t,i ≈ (I − Qt)e_i . In appendix B.5 we show that the node and link resistance curvature satisfy the following alternative definitions:

Proposition 1. The node and link resistance curvature are equal to

$$\begin{equation}{p}_{i}=\underset{t\to 0}{\mathrm{lim}}\left(1-\frac{1}{4t}\mathbb{E}({\omega }_{{N}_{t}{M}_{t}})\right)\quad \text{with}\enspace {N}_{t},{M}_{t}\sim {\boldsymbol{\rho }}_{t,i}\enspace \text{independently}\end{equation} \tag{ 5 }$$

$$\begin{equation}{\kappa }_{ij}=\underset{t\to 0}{\mathrm{lim}}\frac{1}{t}\left(1-\frac{\mathbb{E}\left({\omega }_{{N}_{t}{M}_{t}}\right)}{{\omega }_{ij}}\right)\quad \text{with}\enspace {N}_{t}\sim {\boldsymbol{\rho }}_{t,i},{M}_{t}\sim {\boldsymbol{\rho }}_{t,j}.\end{equation} \tag{ 6 }$$

In other words, the node resistance curvature is related to the average distance between nodes in its neighbourhood and the link resistance curvature is related to the average distance between the neighbourhoods of the end nodes of the link. Equivalently, this corresponds to the average distance between two independent random walkers (as defined by the diffusion equation) starting at a node, and to the average distance between two independent random walkers starting at the end nodes of a link. In section 4.2.2 we will show that expression (6) is closely related to the definition of OR curvature. In [21] we found that $var(\mathbf{f}){:=}\frac{1}{2}{\mathbf{f}}^{\mathrm{T}}{\Omega}\mathbf{f}$ can be interpreted as the variance of a distribution on a graph; this allows to rewrite (5) as

$$\begin{equation}{p}_{i}=\underset{t\to 0}{\mathrm{lim}}\left(1-\frac{1}{2t}var({\boldsymbol{\rho }}_{t,i})\right).\end{equation} \tag{ 7 }$$

The diffusion variance thus grows as 2t around nodes with zero node resistance curvature. This is equal to the variance growth for diffusion or, equivalently, the mean square displacement of Brownian motion on the real line (of zero curvature) [26].

Proposition 1 provides a (geo)metric interpretation of the resistance curvatures and in appendix B we introduce some further alternative characterizations. In the remainder of this article we focus on definitions (2) and (3) and their implications, but a further interpretation of proposition 1 and the definitions in appendix B and their consequences would be an interesting line of future research.

Positive curvature (platonic graphs). Platonic graphs are the graph skeletons of the platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron), which are regular tilings of the two-sphere ${\mathbb{S}}^{2}$. By transitivity of these graphs, we know from appendix C that the resistance curvatures will be constant and positive, equal to p_i = 1/n and κ_ij = 2ρ, where $\rho =m/\left(\genfrac{}{}{0pt}{}{n}{2}\right)$ is the link density. This is in correspondence with the constant positive curvature of ${\mathbb{S}}^{2}$.

Zero curvature (infinite lattice). The rectangular/triangular/hexagonal lattices are infinite graphs that correspond to (the graph skeleton of) the regular tilings of the Euclidean plane. The effective resistance in these lattices has been studied extensively and it was shown that the relative resistance of all links is equal to 2/d, with d the degree of the respective lattice [24, 29, 77]. Consequently, the lattice graphs have constant zero resistance curvatures p_i = 0 and κ_ij = 0, in correspondence with the zero curvature of ${\mathbb{R}}^{2}$.

Negative curvature (infinite regular tree). The infinite regular tree or Bethe lattice [57] corresponds to a regular tiling of the hyperbolic plane ${\mathbb{H}}^{2}$. Like all trees, the relative resistance of every link is equal to one which means that these graphs have negative resistance curvatures p_i = 1 − d/2 and κ_ij = 2c(2 − d) when degree d > 2, in correspondence with the constant negative curvature of ${\mathbb{H}}^{2}$.

Some examples of these constant curvature graphs are shown in figure 1. We remark that the correct curvature in regular tilings is not always reproduced correctly by the established notions of discrete curvature. As noted in [47] for instance, the hexagonal lattice is often assigned a negative curvature because it is locally tree-like.

As a second argument, we show that the node resistance curvature p is related to combinatorial curvature (through random spanning trees) and that the link resistance curvature is related to OR and FR curvature (through tight two-sided bounds).

4.2.1. Node resistance curvature and combinatorial curvature

Combinatorial curvature [37, 46] measures discrete curvature for graphs embedded in the plane (or other surfaces), i.e. with the nodes and links drawn in the plane such that the links only intersect at their endpoints. These drawings partition the plane into faces which are connected regions of the plane bordered by links that form a cycle in the graph. The combinatorial curvature is then defined as [46]

$$\begin{equation*}{p}_{i}^{(\mathrm{c}\mathrm{o})}{:=}1-\frac{{d}_{i}}{2}+\sum\limits _{\text{face}\;f\ni i}\frac{1}{{d}_{f}},\end{equation*}$$

where f ∋ i are the faces that contain node i in their boundary. For tree graphs, only a single unbounded face exists which does not contribute to the combinatorial curvature such that every node in a tree graph has combinatorial curvature 1 − d_i /2 equal to the node resistance curvature. Following the connection between relative resistances and random spanning trees, we moreover find that the resistance curvature of a node i in a general graph equals the expected combinatorial curvature of i in a random spanning tree; in other words (see appendix D.1):

$$\begin{equation}{p}_{i}=\mathbb{E}[{p}_{i}^{(\mathrm{c}\mathrm{o})}(\mathbf{T})]\;\text{with}\;\text{random}\;\text{spanning}\;\text{tree}\;\mathbf{T}.\end{equation} \tag{ 8 }$$

4.2.2. Resistance curvature and Ollivier–Ricci curvature

Ollivier [64] introduced a notion of curvature for metric spaces with an associated Markov chain. This notion of discrete curvature can readily be applied to graphs where the shortest path or geodesic distance is a natural metric $d:\mathcal{N}\times \mathcal{N}\to \mathbb{R}$, and with lazy random walks taking the role of a Markov chain ¹² μ_t . For every node i, the Markov chain determines a function ${\mu }_{t,i}:\mathcal{N}\to [0,1]$ on the nodes, which gives the probability to find the random walker at a node one step after being at i (see later for some examples). This function can be thought of as a 'ball' around i in the graph. The metric d can be lifted to provide a notion of distance between these balls by making use of the 1-Wasserstein distance W₁:

$$\begin{equation*}{W}_{1}({\mu }_{t,i},{\mu }_{t,j}){:=}\underset{P}{\mathrm{min}}\,\mathrm{tr}(PD),\end{equation*}$$

where the minimum is taken over all nonnegative n × n matrices P with 'marginals' ∑_j (P)_xj = μ_t,i (x) and ∑_i (P)_ix = μ_t,j (x) and where matrix (D)_ij = d(i, j) is the distance matrix of the graph. OR curvature measures how much the direct distance d(i, j) between two nodes differs from the distance between balls around these nodes W₁(μ_t,i , μ_t,j ) as a generalization of continuous Ricci curvature to the data $(\mathcal{N},d,{\mu }_{t})$. Lin, Lu and Yau further modified Ollivier's definition as a limit for shrinking balls as [52]

$$\begin{equation*}{\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}{:=}\underset{t\to 0}{\mathrm{lim}}\frac{1}{t}\left(1-\frac{{W}_{1}({\mu }_{t,i},{\mu }_{t,j})}{d(i,j)}\right).\end{equation*}$$

For t → 0, the distances W₁ and d will converge and OR curvature thus measures how much they differ in the first order in t. If the distance between the balls is larger than the direct distance between the points then κ^(OR) will be negative, and the other way around for positive curvature.

One choice for Markov chain μ_t is the lazy random walk

$$\begin{equation*}{\mu }_{t,i}(j)=\begin{cases}1-t{k}_{i}\quad \hfill & \quad \text{if}\enspace i=j\hfill \\ {c}_{ij}t\quad \hfill & \quad \text{if}\;j\sim i\hfill \\ 0\quad \hfill & \quad \text{else}\hfill \end{cases}\quad \text{for}\enspace 0\leqslant t\leqslant {k}_{\text{max}}^{-1}\end{equation*}$$

which can be defined compactly as the vector μ _t,i = (I − Qt)e_i , where I is the identity matrix ¹³ . While the OR curvature has been studied with respect to this Markov chain and the geodesic distance, we have not found it being used in conjunction with the effective resistance distance. We find that OR curvature with respect to the data $(\mathcal{N},\omega ,(I-Qt))$ relates to the link resistance curvature as (see appendix D.2):

Proposition 2. The OR curvature with respect to resistance distance and the lazy random walk is lower-bounded by the resistance curvature as ${\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}\geqslant {\kappa }_{ij}$, with equality if (i, j) is a cut link.

In tree graphs, all links are cut links and the link resistance curvature will thus correspond to the OR curvature throughout the graph. Furthermore, we note that the resistance distance is equal to the geodesic distance in trees.

An alternative Markov chain that is used more often is the normalized lazy random walk, defined by μ _t,i = (I − Q diag(k)⁻¹ t)e_i where diag(k) is the diagonal matrix with the weighted degrees on its diagonal [52, 58]. Similar to the standard lazy random walk, we find the bound

$$\begin{equation}{\kappa }_{ij}^{(\mathrm{L}\mathrm{L}\mathrm{Y})}\geqslant \frac{2({p}_{i}/{k}_{i}+{p}_{j}/{k}_{j})}{{\omega }_{ij}}\end{equation} \tag{ 9 }$$

which suggests a degree-normalized version of the link resistance curvature. The superscript LLY refers to the authors Lin, Lu and Yau of [52] where this variant of OR curvature was first considered.

Remark. The definition of OR curvature, and in particular the Lin–Lu–Yau limit version [52], is closely related to definition (6) as the leading term of the ratio of the distance between shrinking balls around the end nodes of a link and the direct distance between these nodes; especially since we may replace ρ _t,i by μ _t,i in (6). However, while the change from ${\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}$ to κ_ij comes down to 'just' changing shortest-path distance to resistance distance and Wasserstein distance W₁ to average distance, the resulting measure κ_ij does seem to have a unique set of properties that cannot simply be explained by its relation to OR curvature. For instance, the alternative definitions in appendix B and the associated discrete Ricci flow in section 5 are not shared by the OR curvature and the example of hexagonal lattices, misclassified by OR curvature as negatively curved but assigned a zero curvature by the link resistance curvature illustrates that the two definitions disagree in important ways.

4.2.3. Resistance curvature and Forman–Ricci curvature

In [30], Robin Forman introduced a new notion of curvature for CW complexes, a class of discrete/combinatorial spaces that includes graphs. This so-called FR curvature is defined by generalizing the definition of the classical Ricci curvature in terms of the Bochner Laplacian of a manifold to a definition for discrete spaces based on an analogous Bochner Laplacian on these spaces, see also [45]. The FR curvature is expressed in terms of local combinatorial data around a considered point, and Sreejith et al [74] translated Forman's general definition to graphs as

$$\begin{equation}{\kappa }_{ij}^{(\mathrm{F}\mathrm{R})}=2{w}_{i}\left(1-\frac{1}{2}\sum\limits _{k\sim i}\sqrt{\frac{{w}_{ij}}{{w}_{ik}}}\right)+2{w}_{j}\left(1-\frac{1}{2}\sum\limits _{k\sim j}\sqrt{\frac{{w}_{ij}}{{w}_{jk}}}\right),\end{equation} \tag{ 10 }$$

where w_* are some nonzero weights associated to the nodes and links of the graph. Expression (10) clearly resembles the definition of κ in terms of the node resistance curvature and indeed, choosing unit weights yields the following relation (see appendix D.3):

Proposition 3. The FR curvature with respect to unit weights is upper-bounded by the link resistance curvature as ${\kappa }_{ij}^{(\mathrm{F}\mathrm{R})}/{\omega }_{ij}\leqslant {\kappa }_{ij}$, with equality if and only if (i, j) is a cut link.

Remark. If we choose the inverse degree as node weights w_i = 1/k_i and consider graphs with c = 1, then we find the bound ${\kappa }_{ij}^{(\mathrm{F}\mathrm{R})}/{\omega }_{ij}\leqslant 2({p}_{i}/{k}_{i}+{p}_{j}/{k}_{j})/{\omega }_{ij}$ which again features the degree-normalized version of the link resistance curvature as in (9).

Remark. We note one further related notion of discrete curvature: in [75], it is shown that for the shortest-path distance matrix D the equation D w = u determines a function w on the nodes with many properties of a discrete curvature. This relates to resistance curvature by the expression Ωp ∝ u and u^T p = 1 introduced in appendix B. Despite this close relation, the particular properties of the resistance matrix (for instance, its invertibility [20]) result in a solution p with different properties; for instance, most of the alternative definitions in appendix B do not translate to w.

Summary. The relation between the link resistance curvature and Forman and Ollivier–Ricci curvature can be summarized by the two-sided bound

$$\begin{equation*}{\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}\geqslant {\kappa }_{ij}\geqslant {\kappa }_{ij}^{(\mathrm{F}\mathrm{R})}/{\omega }_{ij}\quad \text{with}\;\text{equality}\;\text{for}\;\text{cut}\;\text{links,}\end{equation*}$$

where we stress that these are OR and FR curvatures with respect to particular data: ω as metric for OR and w = 1 as weight for FR. Apart from being an argument for the interpretation of the link resistance curvature as a discrete curvature, this result is relevant in the context of other works that investigate the relation between both curvatures [45, 68, 76].

A natural question for discrete curvatures is whether they converge to continuous curvature on discrete structures that represent finer and finer discretizations of some continuous space. For instance, van der Hoorn et al [80] recently proved convergence of OR curvature to Ricci curvature in the continuum limit of random geometric graphs sampled from Riemannian manifolds. Here, as our third argument, we consider ERGs and find numerical and theoretical evidence that the resistance curvature of these graphs retrieves the underlying zero curvature of the Euclidean plane.

An ERG is a random graph constructed on some domain $\mathbb{D}\subseteq {\mathbb{R}}^{2}$ in the plane from which points are sampled uniformly at random by a Poisson point process with a homogeneous rate λ; the expected number of points is equal to $N=\lambda \,\text{area}(\mathbb{D})$ so we may equivalently fix a desired N. These points are then taken as the nodes of a graph and pairs of nodes are linked if they lie within a certain connection radius (distance) r from each other. All together, a ERG model or ensemble can thus be parametrized by $(\mathbb{D},N,r)$. See [65] for more information on ERGs and their properties and figure 4 for an illustration of their construction and some examples.

Zoom In Zoom Out Reset image size

To simplify the setup for further analysis, we consider ERGs on the disc $\mathbb{D}=\left\{x\in {\mathbb{R}}^{2}:{\Vert}x{\Vert}\leqslant R\right\}$ of radius R, such that nodes at the same radial distance D_i from the boundary are statistically equivalent, and the effective parameters reduce to (R/r, N).

As a first numerical analysis, figure 4 shows the resistance curvatures in an ERG on the disc with R/r = 4 and N = 2 × 10³ nodes. Near the center of the disc, which we call the bulk of the graph, we clearly observe that the curvatures are close to zero, while moving towards the boundary the resistance curvature becomes first negative and then positive at the boundary. Figure 5 shows this result in more detail: the mean and standard deviations of p_i are plotted with respect to D_i /r with data aggregated over 100 samples of an ERG with R/r = 5 and N = 5 × 10³. The main observation in figure 5 is again that p_i is close to zero in the bulk—moreover, it appears to be a zero-mean random variable—and becomes negative and then positive when moving closer to the boundary.

Zoom In Zoom Out Reset image size

We now develop a model for the resistance curvature in ERGs that partly explains these observations. While a full understanding of the relative resistance in ERGs is lacking, in particular concerning dependencies between the relative resistance of different links, we find that the numerical observations in figures 4 and 5 can be explained in great detail using a simple heuristic for the effective resistance. In [83, 84], von Luxburg et al showed that in ERGs with increasing number of nodes, the effective resistance between any pair of nodes i, j converges to the sum of their inverse degrees ${d}_{i}^{-1}+{d}_{j}^{-1}$. While the degree of a node depends on the specific random graph realization, we know by properties of Poisson sampling that the expected degree of a given node i is determined by the area of overlap between the domain and an r-radius disc centered at i, in other words $\mathbb{E}({d}_{i})=\lambda S(i)$ where $S(i){:=}\text{area}(\left\{x\in \mathbb{D}:{\Vert}x-i{\Vert}\leqslant r\right\})$, with expectation over the random graph ensemble. Combining these two results, we propose the following heuristic $\hat{\omega }$ for the effective resistance in ERGs:

$$\begin{equation}{\hat{\omega }}_{ij}{:=}\frac{1}{\lambda S(i)}+\frac{1}{\lambda S(j)}\quad \text{for}\,\text{all}\;\text{links}\;i\sim j.\end{equation} \tag{ 11 }$$

Importantly, this reduces the effective resistance of a link to a purely geometric and local quantity that is only determined by how the r-radius region around the link overlaps with the domain $\mathbb{D}$. In appendix E, we show that this heuristic combined with the assumption that r ≪ R allows to calculate the expected resistance curvature $\hat{p}$ as a function of the distance to the boundary D as

where A(t) is the area of a circular segment at height t and |dA(t)| the change of segment size—as illustrated in the figure on the right—and with expectation taken over the ERG ensemble. While we were not able to find a general closed-form expression for the integral in (12), the expression can be evaluated numerically and is shown in figure 5 by the red line. Importantly, figure 5 shows that formula (12) matches very well with the experimentally observed resistance curvatures (the black line). This agreement suggests that our heuristic captures the main mechanisms behind the convergence of resistance curvature in ERGs.

Due to the appearance of min and max operations, expression (12) is a piecewise function of the boundary distance D. In appendix E, we show that there are three possible regimes for D as a result of the different local geometries around a node, i.e. how much of the connection discs around the node and its neighbours overlap with the domain (see also figure 5). We find the following cases: (A) the boundary regime D ⩽ r where nodes as well as some of their neighbours are influenced by the boundary, (B) the near-boundary regime r ⩽ D ⩽ 2r where nodes are not influenced directly by the boundary (i.e. no overlap between the connection disc and the boundary), but some of their neighbours are, and (C) the bulk regime D ⩾ 2r where nodes are at least two connection radii away from the boundary such that the nodes nor any of their neighbours are influenced by the boundary. Most importantly, in the bulk regime expression (12) simplifies to $\mathbb{E}(\hat{p}(D))=0$ which means that we have zero node resistance curvature $\hat{p}$ in expectation in the bulk of ERGs. In the limit ¹⁴ of r/R → 0, this bulk regime will take up all but a vanishing fraction of the domain and almost all nodes will thus have zero expected resistance curvature.

We remark that the derivation for expression (12) in appendix E is independent of the specific shape of the domain $\mathbb{D}$, i.e. it need not be a disc, and that for ERGs in higher dimensions A(t) would simply be replaced by the volume of a higher-dimensional spherical cap.

Both the numerical results in figure 4 and the heuristic model suggest that the link resistance curvature also converges to zero (in the mean) in the bulk of ERGs. However, the statistical dependencies between the node resistance curvatures at the ends of a link might obstruct convergence for graphs sampled from more general manifolds; a more detailed analysis of p and κ for random geometric graphs is an interesting line of future research.

Ollivier proposed in [64, problem N] to look at the differential equation $\frac{\mathrm{d}}{\mathrm{d}t}d(i,j)=-{\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}d(i,j)$ as a discrete version of the Ricci flow. If we introduce the link resistance curvature κ_ij and resistance distance d(i, j) = ω_ij in this expression, we find

$$\begin{equation}\frac{\mathrm{d}{\omega }_{ij}}{\mathrm{d}t}=-2({p}_{i}+{p}_{j})\quad \text{for}\,\text{all}\enspace i\ne j\;\text{in}\;\text{the}\;\text{same}\;\text{component}\end{equation} \tag{ 13 }$$

which we call the resistance Ricci flow; since ω is metric, we implicitly have dω_ii /dt = 0. Ideally, equation (13) would describe the evolution of a resistance matrix Ω(t) for all t > 0 and starting from any initial resistance matrix Ω₀ := Ω(0). Unfortunately however, Ω(t) is in general not guaranteed to be a resistance matrix as the flow can result in graphs with negative and diverging link weights in finite time as we show below. Nonetheless, we find that the resistance Ricci flow satisfies a number of interesting properties. As the flow (13) affects each connected component independently, we will assume G to be connected in the rest of this section.

If S is a state space (e.g. a vector space) and $P:S\to \mathbb{R}$ a potential defined on this state space, then the evolution ds/dt = −∇P for a flow s(t) ∈ S, assuming that it is well-defined, is called a gradient flow. This name reflects that the resulting flow follows the direction of steepest descent of the potential P, as given by the negative gradient. As a first result, we find that the resistance Ricci flow is a gradient flow (see appendix F):

Proposition 4. The resistance Ricci flow (13) is a gradient flow of the potential $\mathrm{tr}(\frac{1}{2}{\Omega}Q{\Omega})$ defined on symmetric zero-diagonal matrices Ω, as

$$\begin{equation}\frac{\mathrm{d}{\Omega}}{\mathrm{d}t}=-{\nabla }_{{\Omega}}\mathrm{tr}\left(\frac{1}{2}{\Omega}Q{\Omega}\right).\end{equation} \tag{ 14 }$$

Remark. The gradient ∇_Ω is defined on the state space of symmetric zero-diagonal matrices, such that the flow Ω(t) remains symmetric and with zero-diagonal; see appendix F for further details.

The gradient form (14) relates the resistance Ricci flow to the classical diffusion equation for a density x in ${\mathbb{R}}^{n}$, which is a gradient flow for the potential $\mathrm{tr}(\frac{1}{2}{\mathbf{x}}^{\mathrm{T}}Q\mathbf{x})$. Expression (14) could have practical implications for the resistance Ricci flow: since the potential is a decreasing function of time for gradient dynamics as $\mathrm{d}\mathrm{tr}(\frac{1}{2}{\Omega}Q{\Omega})/\mathrm{d}t< 0$, this could be a starting point to study convergence of the resistance Ricci flow.

In appendix F, we furthermore show that the potential can be rewritten as $\mathrm{tr}(\frac{1}{2}{\Omega}Q{\Omega})=2n{\sigma }^{2}$ where ${\sigma }^{2}=\frac{1}{2}{p}^{\mathrm{T}}{\Omega}p$ is a graph invariant that appears in several contexts in the study of effective resistances, see for instance [20] and [21, appendix I]. The gradient expression (14) further adds to the theory of this invariant. As noted in [20], the number 1/(2σ²) also relates to the so-called magnitude (introduced in [51]) of the metric space $(\mathcal{N},\omega )$; this suggests to consider gradient flows of metric spaces with the magnitude as a potential function.

While expression (13) does not provide much insight into how the resistance Ricci flow affects the graph structure, we can also express the flow in terms of Laplacian matrices (see appendix F):

Proposition 5. The resistance Ricci flow (13) is equivalent to the following flow of Laplacian matrices:

$$\begin{equation}\frac{\mathrm{d}Q}{\mathrm{d}t}=2Q\,\mathrm{diag}(\mathbf{p})Q,\quad \mathrm{f}\mathrm{o}\mathrm{r}\,\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\,\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\,\mathrm{L}\mathrm{a}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{n}Q(0)={Q}_{0}.\end{equation} \tag{ 15 }$$

Proposition 5 is remarkable since it shows that the resistance Ricci flow—which was defined based on the proposal of Ollivier—corresponds to a simple flow of Laplacian matrices. By considering the off-diagonal entries of expression (15), we can write the evolution of individual link weights as:

$$\begin{equation}\frac{\mathrm{d}{c}_{ij}}{\mathrm{d}t}=-2\sum\limits _{\begin{subarray}{c}k\sim i,j\\ k\ne i,j\end{subarray}}{c}_{ik}{c}_{kj}{p}_{k}+2{c}_{ij}\left({k}_{i}{p}_{i}+{k}_{j}{p}_{j}\right)\quad \text{for}\,\text{all}\;i\ne j,\end{equation} \tag{ 16 }$$

where the equation for the diagonal (dk_i /dt) follows from the conserved zero row sum. The change of link weights in the resistance Ricci flow is thus caused by two types of processes: there is one term for each shared neighbour k of i and j which can increase or decrease the link weight based on its resistance curvature p_k , and a second term based on the degrees and curvatures of i and j. Other works such as [2] have arrived at similar dynamical rules when modeling dynamic network structures, with a balance between positive link-reinforcing terms and competing link-decaying terms. As a possible variation on the resistance Ricci flow, one might consider the dynamics dQ/dt = 2Q diag(q)Q where q is some function on the nodes; for instance, if q depends on a process taking place on the graph such as in [2], this could model the codependent evolution of structure and processes. We now consider two examples of the resistance Ricci flow.

Example 1 (node transitive graphs). The node resistance curvature in node transitive graphs is constant and equal to p_i = 1/n. Furthermore, from (13) we find that dω_ij /dt = −2/n is equal for all pairs of nodes such that transitivity is conserved by the flow and thus p_i (t) = 1/n as long as this is well-defined. As a consequence, the resistance Ricci flow for node transitive graphs simplifies to dQ/dt = 2Q²/n. Diagonalizing this equation and solving the differential equation dx/dt = 2x² ⇒ x(t) = x₀/(1 − 2tx₀) for each eigenvalue, we find the solution

$$\begin{equation*}Q(t)={\left[I-\frac{2}{n}t{Q}_{0}\right]}^{{\dagger}}{Q}_{0}\quad \text{for}\enspace t< \frac{n}{2{\mu }_{\text{max}}({Q}_{0})}.\end{equation*}$$

This flow diverges in finite time for t → n/(2μ_max). The resistance Ricci flow for node transitive graphs is formally very similar to a matrix flow studied in [54] as a model for structural balance in social networks, which considers the differential equation dX/dt = X² for a matrix X. The entries (X)_ij of this matrix represent pairwise affinities or rivalries, depending on the sign, and finite-time divergence was interpreted in the context of the model as the onset of a structurally balanced configuration of social relations.

Example 2 (path graph). All nodes in a path graph have zero node resistance curvature, except for the end nodes which have p = 1/2. From equation (16) for the evolution of link weights, we then find that the only change occurs for the end links, whose link weight evolves according to

$$\begin{equation*}\frac{\mathrm{d}c}{\mathrm{d}t}={c}^{2}{\Rightarrow}c(t)=\frac{{c}_{0}}{1-{c}_{0}t}\quad \text{for}\enspace t< \frac{1}{{c}_{0}},\end{equation*}$$

and diverges to +∞ for t → 1/c₀. While this result may seem pathological, we can interpret the 'infinite affinity' (which corresponds to ω → 0) as merging the end node and its neighbour into a single node—this is a common operation in the theory of effective resistances. This analysis shows that the resistance Ricci flow of a path graph evolves by merging the end nodes of the path with their neighbours at t = 1/c₀, resulting in a path of decreasing length until just a single node remains.

Finally, we remark that the 'degree-normalized' link resistance curvature 2(p_i /k_i + p_j /k_j )/ω_ij , which naturally appears in correspondence with Ollivier and Forman Ricci curvature (see (9)), also has an associated Ricci flow dω_ij /dt = −2(p_i /k_i + p_j /k_j ) ≡ dQ/dt = 2Q diag(p/k)Q. Initial results suggest that this Ricci flow has additional interesting properties.

We prove the two statements in property 1:

Proof of bounds for the relative resistance. We recall theorem 2 which says that the relative resistance c_ij ω_ij of a link (i, j) is equal to the probability that a random spanning tree contains this link. Since c_ij ω_ij is thus a probability, it satisfies 0 ⩽ c_ij ω_ij ⩽ 1.

We first refine the lower bound. By theorem 1 we know that the effective resistance is a distance and thus that ω_ij > 0 for i ≠ j. We also know that link weights are positive by definition such that we have c_ij ω_ij > 0.

Next, we consider when equality occurs for the upper bound. If (i, j) is a cut link in G, then any subgraph of G without (i, j) is disconnected. Hence, all connected subgraphs (including all spanning trees) of G must contain (i, j) and thus c_ij ω_ij = Pr[(i, j) ∈ T] = 1. Hence, if (i, j) is a cut link then c_ij ω_ij = 1. Conversely, let c_ij ω_ij = 1 for some link and assume that it is not a cut link. But then if we consider the connected graph G' = G\(i, j) from which (i, j) has been removed, we arrive at a contradiction: any spanning tree T of G' is also a spanning tree of G (since ${\mathcal{L}}_{T}\subseteq {\mathcal{L}}_{{G}^{\prime }}\subseteq {\mathcal{L}}_{G}$ and G' is connected) and T does not contain (i, j) since it is a subgraph of G'; however, c_ij ω_ij = 1 implies that (i, j) is in all spanning trees of G which contradicts the existence of such a spanning tree T. Consequently, we know that if ω_ij c_ij = 1 then (i, j) must be a cut link. Together, we have thus shown that ω_ij c_ij = 1 ⇔ (i, j) is a cut link. This completes the proof. □

Proof of Foster's theorem. Assume G is connected. We again use theorem 2 on the relation between c_ij ω_ij and random spanning trees to rewrite the sum over all relative resistances:

$$\begin{align}\hfill \sum\limits _{(i,j)\in \mathcal{L}}{c}_{ij}{\omega }_{ij}& =\sum\limits _{(i,j)\in \mathcal{L}}\mathrm{Pr}[(i,j)\in \mathbf{T}]\hfill \\ \hfill & \stackrel{(\text{a})}{=}\sum\limits _{(i,j)\in \mathcal{L}}\left(\sum\limits _{T\in \mathcal{T}}\mathrm{Pr}[\mathbf{T}=T]{\mathbf{1}}_{\left\{(i,j)\in T\right\}}\right)\hfill \\ \hfill & =\sum\limits _{T\in \mathcal{T}}\mathrm{Pr}[\mathbf{T}=T]\sum\limits _{(i,j)\in \mathcal{L}}{\mathbf{1}}_{\left\{(i,j)\in T\right\}}\hfill \\ \hfill & \stackrel{(\text{b})}{=}\sum\limits _{T\in \mathcal{T}}\mathrm{Pr}[\mathbf{T}=T](n-1)\hfill \\ \hfill & =n-1.\hfill \end{align} \tag{ 17 }$$

In step (a) we make the probability expression explicit and in step (b) we use the fact that any spanning tree of a connected graph has n − 1 links, independent of the tree. In brief, this derivation shows that ∑_i∼j c_ij ω_ij equals the average number of links in a spanning tree, which is constant and equal to n − 1. For a disconnected graph, this result holds for every component independently and as a result we find that the sum over all links in the graph equals n − β(G); this completes the proof. □

Expression (17) can be used for other sums of relative resistances. For instance, since every cycle must at least miss one link in a spanning tree (which is cycle-free) we find that ∑_{(i,j)∈{cycle}} 1_{(i,j)∈T} ⩽ |{cycle}| − 1 for any cycle $\left\{\text{cycle}\right\}\subseteq \mathcal{L}$. Consequently, the sum of relative resistances over a cycle satisfies

$$\begin{equation*}\sum\limits _{(i,j)\in \left\{\text{cycle}\right\}}{c}_{ij}{\omega }_{ij}\leqslant \vert \left\{\text{cycle}\right\}\vert -1\quad \text{for}\,\text{all}\;\text{cycles.}\end{equation*}$$

Similarly, since all spanning trees must have at least one link in a given cut—this is any set of links such that removing the set disconnects the graph ¹⁶ —, we find

$$\begin{equation}\sum\limits _{(i,j)\in \left\{\text{cut}\right\}}{c}_{ij}{\omega }_{ij}\geqslant 1\quad \;\text{for}\;\text{all}\;\text{cuts.}\end{equation} \tag{ 18 }$$

These bounds are an improvement over the straightforward bounds using 0 < c_ij ω_ij ⩽ 1.

We prove the two statements in property 2:

Proof of ∑p_i = β . Invoking Foster's theorem, we can write the sum over all node resistance curvatures as

$$\begin{equation*}\sum\limits _{i\in \mathcal{N}}{p}_{i}=n-\frac{1}{2}\sum\limits _{i\in \mathcal{N}}\sum\limits _{j\sim i}{c}_{ij}{\omega }_{ij}=n-\sum\limits _{(i,j)\in \mathcal{L}}{c}_{ij}{\omega }_{ij}=\beta ,\end{equation*}$$

which completes the proof. □

To prove the node resistance curvature bounds in property 2, we first show a stronger result:

Property 4. The resistance curvature of a node with d_i ⩾ 1 is bounded by

$$\begin{equation*}1-\frac{{d}_{i}}{2}\leqslant {p}_{i}\leqslant 1-\frac{\beta ({G}_{i}{\backslash}\left\{i\right\})}{2}\quad \text{for}\,\text{any}\;\text{node}\;i\in \mathcal{N},\end{equation*}$$

where G_i \{i} is the component G_i with node i removed. Equality is achieved if and only if all links connected to i are cut links, in which case both bounds are equal.

Proof. The lower bound follows immediately from the definition (2) of resistance curvature and the relative resistance bound c_ij ω_ij ⩽ 1 with equality if and only if all incident links are cut links. The upper bound follows by summing the relative resistances over cuts: if removing i (which is possible since d_i ⩾ 1) disconnects G_i into β' := β(G_i \{i}) components, this means that the links incident to node i can be partitioned into β' sets of links ${\left\{{C}_{k}\right\}}_{k=1}^{{\beta }^{\prime }}$ which are cuts of G_i . By the bound (18) for the sum of relative resistances over a cut, we then find

$$\begin{equation*}{p}_{i}=1-\frac{1}{2}\sum\limits _{k=1}^{{\beta }^{\prime }}\sum\limits _{(i,j)\in {C}_{k}}{c}_{ij}{\omega }_{ij}\stackrel{(\text{cut}\;\text{bound})}{\leqslant }1-\frac{\beta ({G}_{i}{\backslash}\left\{i\right\})}{2}.\end{equation*}$$

Next, we consider when equality occurs in the upper bound. First, if all links connected to i are cut links then we know (a) that their relative resistances are equal to 1 and (b) that β(G_i \{i}) = d_i such that p_i ⩽ 1 − d_i /2. Since this is equal to the lower bound, we know that equality must hold. To prove conversely that equality in the upper bound implies that all links connected to i are cut links, we will make use of basic facts about spanning trees and cycles in graphs, see for instance [11]. Since the upper bound for p_i was based on the cut bound (18), we know that if this upper bound holds with equality as p_i = 1 − β(G_i \{i})/2 then the cut bound (18) must also hold with equality. By equation (17) from which the cut bound follows, this implies that ${\sum }_{(i,j)\in {C}_{k}}\mathbf{1}\left\{(i,j)\in T\right\}=1$ for all spanning trees T and all cuts C_k ; in other words, we find that every spanning tree has exactly one link in every cut C_k . If n ⩽ 2 it then easily follows that the link is a cut link, so assume n > 2. Let T be a spanning tree, C_k a cut and ℓ = T ∩ C_k the unique link of T in the cut C_k . If we assume that |C_k | > 1 then we may take another link in the cut ℓ' ∈ C_k and consider the graph T + ℓ', i.e. the tree with this new link added. This graph will have a unique cycle H (see [11]) and, since cycles have length at least 3, we can take a link ℓ'' ∈ H which is different from ℓ, ℓ'. But then the graph T + ℓ' − ℓ'' is again a spanning tree of G (see [11]) which contains both ℓ and ℓ' i.e. two links in the cut C_k . Since this is in contradiction with every spanning tree having exactly one link in the cut C_k , we know that |C_k | = 1 which means C_k is a cut link; this holds for all cuts k = 1, ..., β'. In other words, if the upper bound holds with equality we know that all links connected to i are cut links. This completes the proof. □

Proof of node resistance curvature bounds. The upper bound in property 2 follow as a corollary of property 4 since β(G_i \{i}) ⩾ β(G_i ) = 1. □

Property 4 shows that the resistance curvature is constrained by the combinatorial structure of the graph as neither d_i nor β(G_i \{i}) depend on the weights c in the graph. Furthermore, this gives an example of nodes which have a nonpositive curvature:

Property 5 (cut  nodes). Cut nodes have nonpositive curvature, with zero curvature if and only if their degree is two.

Proof. If i is a cut node, then β(G_i \{i}) ⩾ 2 and thus by property 4 it follows that p_i ⩽ 0. Next, we consider when zero curvature occurs. Any cut node must have d_i > 1 since otherwise i is disconnected or a pendant/leaf node and thus not a cut node. If i is a cut node with d_i = 2, then the two links incident on i must be cut links with unit relative resistance and thus p_i = 0. Conversely, if p_i = 0 for a cut node, then from property 4 we know that β(G_i \{i}) = 2 must hold and that all links connected to i must be cut links. This implies that d_i = 2 as required. □

A related result about cut nodes and the sign of curvature was shown by Fiedler in [28], and follows as a corollary of property 5:

Corollary 1. If G has a cut node then either G is a path graph or G has a node with negative node resistance curvature.

Proof. Let G be a graph with cut node i. Either i has degree d_i > 2 and thus negative curvature by property 5, or degree d_i = 2 and zero curvature. If d_i = 2 we furthermore know that the incident links are cut links; hence, if we consider a neighbour of i, then either (a) this neighbour has degree d = 1, in which case it is a leaf nodes with p = 1/2 > 0, or (b) this neighbour has degree d = 2 and both incident links are cut links, such that p = 0 and we may consider its neighbour recursively, or (c) this neighbour has degree d > 2 and either (c1) all incident links are cut links such that p = 1 − d/2 < 0 or (c2) not all incident links are cut links, but then the upper bound in property 4 is strict and p < 0. Thus, either we encounter a node with negative curvature or we can recursively consider further neighbours of degree d = 2 until we reach a node of degree d = 1 and stop the process, which would correspond to a path graph. This completes the proof. □

We mention one additional identity for the node resistance curvature which will be used in the rest of this appendix: The product of the Laplacian and resistance matrix of a connected graph equals

$$\begin{equation}Q{\Omega}=-2I+2\mathbf{p}{\mathbf{u}}^{\mathrm{T}},\end{equation} \tag{ 19 }$$

where u = (1, ..., 1)^T is the all-one vector and Ω the resistance matrix. This identity follows from Fiedler's identity [20, 28] and the work of Bapat [6] which relate the Laplacian and resistance matrices of a graph. In particular, we note that this implies that diag(QΩ) = 2(p − u).

Proof of property 3: The bounds for the link resistance curvature κ_ij in property 3 follow immediately from the bounds for the node resistance curvature p_i in property 2. □

We can also prove the following stronger result:

Property 6 (link  resistance  curvature  bounds). The link resistance curvature is bounded by

$$\begin{equation*}\begin{cases}{\kappa }_{ij}\geqslant \frac{1}{{\omega }_{ij}}(4-{d}_{i}-{d}_{j})\quad \hfill \\ {\kappa }_{ij}\leqslant \frac{1}{{\omega }_{ij}}\left(6-2\beta ({G}_{i}{\backslash}(i,j))-\beta ({G}_{i}{\backslash}\left\{i,j\right\})\right)\quad \hfill \end{cases}\end{equation*}$$

with both bounds achieved (and equal) if and only if all links incident to i and j are cut links, and where G_i \(i, j) is the graph with link (i, j) removed and G_i \{i, j} the graph with nodes i, j removed.

Proof. The lower bound for the link resistance curvature follows directly from the lower bound p_i ⩾ 1 − d_i /2 on the node resistance curvature in property 2 and the bound c_ij ω_ij ⩽ 1 in property 1, with equality only if all incident links are cut links.

We continue with the upper bound. For simplicity, we further assume that G = G_i is a connected graph—this is without loss of generality since any link will be contained in a unique connected component. We can write the link resistance curvature as

$$\begin{equation}\frac{2({p}_{i}+{p}_{j})}{{\omega }_{ij}}=\frac{1}{{\omega }_{ij}}\left(4-\sum\limits _{l\in {\mathcal{L}}_{ij}}{c}_{l}{\omega }_{l}-2{c}_{ij}{\omega }_{ij}\right),\end{equation} \tag{ 20 }$$

where ${\mathcal{L}}_{ij}{:=}\left\{(x,k)\in \mathcal{L}{\backslash}(i,j)\vert x\in \left\{i,j\right\}\right\}$ are the links incident on (i, j). We note that removing nodes i and j from G is equal to removing the links ${\mathcal{L}}_{ij}$ from G and then removing the nodes i, j from the resulting graph. Hence, every connected component in G\{i, j} is disconnected from G by some cut ${C}_{k}\subseteq {\mathcal{L}}_{ij}$, which gives a partition of ${\mathcal{L}}_{ij}$ into cuts. For each cut, we may then invoke the cut bound (18) which yields

$$\begin{equation*}\sum\limits _{l\in {\mathcal{L}}_{ij}}{c}_{l}{\omega }_{l}=\sum\limits _{k=1}^{\beta (G{\backslash}\left\{i,j\right\})}\sum\limits _{l\in {C}_{k}}{c}_{l}{\omega }_{l}\stackrel{(\text{cut}\;\text{bound})}{\geqslant }\beta (G{\backslash}\left\{i,j\right\}),\end{equation*}$$

with equality if and only if all cuts consist of single links (see proof of property 4) and thus if all links in ${\mathcal{L}}_{ij}$—and thus also (i, j)—are cut links. Second, we know that the relative resistance satisfies c_ij ω_ij ⩾ 0 and that c_ij ω_ij = 1 if and only if (i, j) is a bridge link. Hence in general we have c_ij ω_ij ⩾ [β(G\(i, j)) − 1] where G\(i, j) has the link (i, j) removed; again, equality only holds if this is a cut link. Introducing the bounds for these link terms into (20) yields the proposed upper bound and conditions for equality, and thus completes the proof. □

A first family of definitions for the node resistance curvature follows immediately from identity (19). From the diagonal entries (QΩ)_ii we retrieve definition (2) for the node resistance curvature as

$$\begin{equation*}{p}_{i}=1-\frac{1}{2}\sum\limits _{j\sim i}{c}_{ij}{\omega }_{ij}\quad \text{summarized}\;\text{as}\;\mathbf{p}=\frac{1}{2}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(Q{\Omega})+\mathbf{u}.\end{equation*}$$

Looking at the off-diagonal entry (QΩ)_ix instead, we find

$$\begin{equation}{p}_{i}=\frac{{k}_{i}}{2}\left({\omega }_{ix}-\sum\limits _{j\sim i}\frac{{c}_{ij}}{{k}_{i}}{\omega }_{jx}\right)\quad \text{for}\,\text{any}\;x\ne i.\end{equation} \tag{ 21 }$$

Definition (21) shows that the node resistance curvature of i is related to the difference in (resistance) distance from i to x and the average distance from its neighbours to x; furthermore, this definition is independent of x. Combining definition (2) and expression (21) with x ranging over the neighbours of i with probability c_ij /k_i , we obtain the expression

$$\begin{equation*}{p}_{i}=\frac{1}{2}-\frac{1}{4}\sum\limits _{k,j\sim i}\frac{{c}_{ij}{c}_{ik}}{{k}_{i}}{\omega }_{kj}.\end{equation*}$$

In other words, the resistance curvature of a node is related to the average resistance distance between pairs of neighbours.

In a similar way, if we left-multiply identity (19) by Ω and use the fact that Ωp = 2σ² as follows from (22) in the next section, we find

$$\begin{equation*}{\sigma }^{2}=\frac{1}{4}\sum\limits _{j\sim i}{c}_{ij}{({\omega }_{ix}-{\omega }_{jx})}^{2}\quad \text{for}\,\text{any}\;x\ne i.\end{equation*}$$

This expression shows that σ² is related to the (squared) difference in distance from a node x to the two end points of each link in a graph, independent of x.

Starting again from identity (19) and right-multiplying with the resistance curvature vector p, we find that QΩp = 0 ⇒Ωp ∈ ker(Q). Since ker(Q) = span(u) for a connected graph [16, 21] this means that Ωp is a constant vector and, including the unit-sum property ¹⁷ , that Ωp = 2σ² u. Since the resistance matrix is invertible [20], this can be written as

$$\begin{equation}\mathbf{p}=\frac{{{\Omega}}^{-1}\mathbf{u}}{{\mathbf{u}}^{\mathrm{T}}{{\Omega}}^{-1}\mathbf{u}}\quad \text{and}\quad {\sigma }^{2}=\frac{1}{2{\mathbf{u}}^{\mathrm{T}}{{\Omega}}^{-1}\mathbf{u}}.\end{equation} \tag{ 22 }$$

As noted in section 4.2, Steinerberger [75] defines a discrete node curvature w on graphs based on the equation D w = u for the shortest-path distance matrix D. Taking the resistance distance instead, this would correspond to w = p/(2σ²).

We note that combining (22) and identity (19) for QΩ retrieves Bapat's expression for the inverse resistance matrix of a graph [6]:

$$\begin{equation*}{{\Omega}}^{-1}=-\frac{1}{2}Q+\frac{\mathbf{p}{\mathbf{p}}^{\mathrm{T}}}{2{\sigma }^{2}}.\end{equation*}$$

The effective resistance is a negative type metric which means there exists an isometric embedding $\varphi :\mathcal{N}\to {\mathbb{R}}^{n-1}$ of the square root effective resistance, i.e. with ||φ(i) − φ(j)||² = ω_ij for all i ≠ j. Furthermore, as discovered by Fiedler [28] (see also [20]) this embedding maps the nodes of a graph to the vertices of a hyperacute simplex S in ${\mathbb{R}}^{n-1}$—a simplex with nonobtuse (acute or right) interior dihedral angles. This embedding can be linearly extended to functions on the nodes as

$$\begin{equation*}\varphi (f){:=}\sum\limits _{i\in \mathcal{N}}f(i)\varphi (i)\quad \text{for}\,\text{any}\enspace f:\mathcal{N}\to \mathbb{R},\end{equation*}$$

where a function f is thus the coordinate of a point φ(f) in ${\mathbb{R}}^{n-1}$. As shown by Fiedler [28] (see also [21, appendix I]), we find that the node resistance curvature and σ² satisfy

$$\begin{equation*}{\Vert}\varphi (p)-\varphi (i){{\Vert}}^{2}={\sigma }^{2}\quad \text{for}\,\text{all}\;i\in \mathcal{N}.\end{equation*}$$

Geometrically, this relates to the circumsphere of S—the unique sphere passing through all vertices of S—as follows: the node resistance curvature is the (unit-sum) coordinate of the center of the circumsphere of S and σ is its radius.

In [21] the following variational characterization of p as the solution to an optimization problem is shown:

$$\begin{equation*}\mathbf{p}=\underset{\mathbf{f}:{\mathbf{u}}^{\mathrm{T}}\mathbf{f}=1}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\frac{1}{2}{\mathbf{f}}^{\mathrm{T}}{\Omega}\mathbf{f},\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\,\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}\,\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\,{\sigma }^{2}.\end{equation*}$$

In [21] we furthermore show that the 'variance' of a distribution f (a nonnegative unit-sum vector) on a graph can be measured by the quadratic product $var(\mathbf{f}){:=}\frac{1}{2}{\mathbf{f}}^{\mathrm{T}}{\Omega}\mathbf{f}$. This leads to expression (7) of the node resistance curvature in terms of the variance of distribution f = ρ _t,i .

We prove proposition 1 which expresses the node and link resistance curvature in terms of the average distance between diffusion distributions around the node or link.

Proof of proposition 1. Introducing the definitions of the average distance between random nodes (21) and the neighbourhood diffusion distribution ρ _t,i = exp(−Qt)e_i in the right-hand side of expression (5), we obtain

$$\begin{equation*}\underset{t\to 0}{\mathrm{lim}}\left(1-\frac{1}{4t}\mathbb{E}({\omega }_{{N}_{t}{M}_{t}})\right)=\underset{t\to 0}{\mathrm{lim}}\left(1-\frac{1}{4t}{\mathbf{e}}_{i}^{\mathrm{T}}\,\mathrm{exp}(-Qt){\Omega}\,\mathrm{exp}(-Qt){\mathbf{e}}_{i}\right)\quad \text{for}\enspace {N}_{t},{M}_{t}\sim {\boldsymbol{\rho }}_{t,i}.\end{equation*}$$

By definition of the matrix exponential as the power series $\mathrm{exp}(A)={\sum }_{k=0}^{\infty }{A}^{k}/(k!)$, the leading order term of the neighbourhood distribution in the t → 0 limit is (I − Qt)e_i . Making use of identity (19) for QΩ, we then find

$$\begin{equation*}(I-Qt){\Omega}(I-Qt)={\Omega}+4I-(2\mathbf{p}{\mathbf{u}}^{\mathrm{T}}+2\mathbf{u}{\mathbf{p}}^{\mathrm{T}})t-2Q{t}^{2},\end{equation*}$$

and thus

$$\begin{equation*}\underset{t\to 0}{\mathrm{lim}}\left(1-\frac{1}{4t}\mathbb{E}({\omega }_{{N}_{t}{M}_{t}})\right)=\underset{t\to 0}{\mathrm{lim}}\left({p}_{i}-\frac{1}{2}{k}_{i}t\right)={p}_{i}\quad \text{for}\enspace {N}_{t},{M}_{t}\sim {\boldsymbol{\rho }}_{t,i},\end{equation*}$$

as required. Similarly, for the link resistance curvature we find that the right-hand side in expression (6) is equal to

$$\begin{equation*}\underset{t\to 0}{\mathrm{lim}}\frac{1}{t}\left(1-\frac{\mathbb{E}({\omega }_{{N}_{t}{M}_{t}})}{{\omega }_{ij}}\right)=\underset{t\to 0}{\mathrm{lim}}\left(\frac{2({p}_{i}+{p}_{j})}{{\omega }_{ij}}-\frac{2{c}_{ij}t}{{\omega }_{ij}}\right)={\kappa }_{ij}\quad \text{for}\enspace {N}_{t}\sim {\boldsymbol{\rho }}_{t,i}\;\text{and}\;{M}_{t}\sim {\boldsymbol{\rho }}_{t,j}\end{equation*}$$

as required. This completes the proof. □

We show formula (8) which relates the node resistance curvature and combinatorial curvature. Starting from definition 2 for the resistance curvature and using theorem 2 for the relation between relative resistances and spanning trees, we find

$$\begin{align*}\hfill {p}_{i}& =1-\frac{1}{2}\sum\limits _{j\sim i}{c}_{ij}{\omega }_{ij}\hfill \\ \hfill & =1-\frac{1}{2}\sum\limits _{j\sim i}\sum\limits _{T\in \mathcal{T}}\mathrm{Pr}[\mathbf{T}=T]{\mathbf{1}}_{\left\{(i,j)\in T\right\}}\quad (\text{by}\;\text{theorem}\;\text{2})\hfill \\ \hfill & =\sum\limits _{T\in \mathcal{T}}\mathrm{Pr}[\mathbf{T}=T]\left(1-\frac{1}{2}\sum\limits _{j\sim i}{\mathbf{1}}_{\left\{(i,j)\in T\right\}}\right)\hfill \\ \hfill & =\sum\limits _{T\in \mathcal{T}}\mathrm{Pr}[\mathbf{T}=T]\left(1-\frac{{d}_{i}^{(T)}}{2}\right)=\mathbb{E}[{p}_{i}^{(co)}(\mathbf{T})],\hfill \end{align*}$$

where ${d}_{i}^{(T)}$ is the degree of node i in spanning tree T. This proves formula (8).

We prove proposition 2 which relates the link resistance curvature to OR curvature measured with respect to the effective resistance metric and balls μ_t,i determined by a lazy random walk, i.e. given by the vector μ _t,i = (I − Qt)e_i . We will assume G to be connected, as both κ and κ^(OR) are determined in each component independently.

Proof of proposition 2. We start by bounding the Wasserstein distance between the balls μ_t,i and μ_t,j :

$$\begin{equation}{W}_{1}({\mu }_{t,i},{\mu }_{t,j})=\underset{P}{\mathrm{min}}\,\mathrm{tr}(P{\Omega})\stackrel{(a)}{\leqslant }\mathrm{tr}({\boldsymbol{\mu }}_{t,i}{\boldsymbol{\mu }}_{t,j}^{\mathrm{T}}{\Omega})\stackrel{(b)}{=}{\boldsymbol{\mu }}_{t,i}^{\mathrm{T}}{\Omega}{\boldsymbol{\mu }}_{t,j},\end{equation} \tag{ 23 }$$

where for the inequality (a) we have used that $P={\boldsymbol{\mu }}_{t,i}{\boldsymbol{\mu }}_{t,j}^{\mathrm{T}}$ is a valid matrix for the Wasserstein distance, i.e. with nonnegative entries and the correct marginals, and where equality (b) invokes properties of the trace operator. Introducing the definition of the lazy random walk for μ _t,i and μ _t,j , we obtain

$$\begin{equation*}{W}_{1}({\mu }_{t,i},{\mu }_{t,j})\leqslant {\mathbf{e}}_{i}^{\mathrm{T}}(I-Qt){\Omega}(I-Qt){\mathbf{e}}_{j}\stackrel{(a)}{=}{\omega }_{ij}-2t({p}_{i}+{p}_{j})+2{t}^{2}{c}_{ij},\end{equation*}$$

where in (a) we use the identity QΩ = −2I + 2pu^T as in (19). Introduced into the definition of OR curvature with ω as distance, we then find

$$\begin{equation}{\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}=\underset{t\to {0}^{+}}{\mathrm{lim}}\frac{1}{t}\left(1-\frac{{W}_{1}({\mu }_{t,i},{\mu }_{t,j})}{{\omega }_{ij}}\right)\geqslant \frac{2({p}_{i}+{p}_{j})}{{\omega }_{ij}},\end{equation} \tag{ 24 }$$

establishing the bound in proposition 2.

Next, we show that equality is achieved in the case of cut links. If (i, j) is a cut link, then i is a cut node and thus for any two nodes x, y which are disconnected by removal of i, we have the triangle equality ω_xy = ω_xi + ω_iy , see for instance [49]; similarly, j is a cut node. In particular, the triangle equality will hold for effective resistances between the neighbours of i and the neighbours of j, in other words between the supports $\mathcal{I}{:=}\text{supp}({\mu }_{t,i})$ and $\mathcal{J}{:=}\text{supp}({\mu }_{t,j})$ of the balls around i and j respectively—where for the lazy random walk we also have $i\in \mathcal{I}$ and $j\in \mathcal{J}$ and $j\in \mathcal{I},i\in \mathcal{J}$ because i ∼ j. These effective resistances satisfy ω_xy = ω_xi + ω_ij + ω_jy for all $x\in \mathcal{I}{\backslash}\left\{j\right\},y\in \mathcal{J}{\backslash}\left\{i\right\}$ and consequently, the block matrix ${{\Omega}}_{\mathcal{J}\mathcal{I}}$ has the following low-rank decomposition

$$\begin{equation*}{{\Omega}}_{\mathcal{J}\mathcal{I}}={\boldsymbol{\omega }}_{j}{\mathbf{u}}^{\mathrm{T}}+\mathbf{u}{\boldsymbol{\omega }}_{i}^{\mathrm{T}}+{\omega }_{ij}\left(\mathbf{u}{\mathbf{u}}^{\mathrm{T}}-2{\mathbf{e}}_{i}{\mathbf{u}}^{\mathrm{T}}-2\mathbf{u}{\mathbf{e}}_{j}^{\mathrm{T}}+2{\mathbf{e}}_{i}{\mathbf{e}}_{j}^{\mathrm{T}}\right)\end{equation*}$$

where ω _i is the $\vert \mathcal{I}\vert \times 1$ vector with effective resistances to the neighbours of i, as ${({\boldsymbol{\omega }}_{i})}_{x}={\omega }_{ix}$ for $x\in \mathcal{I}$ and similarly for ω _j , and with u the all-one vectors of appropriate size. Introducing this into the expression for the Wasserstein distance, we retrieve

$$\begin{align*}\hfill {W}_{1}({\mu }_{t,i},{\mu }_{t,j})& =\underset{P}{\mathrm{min}}\,\mathrm{tr}\,(P{\Omega})\hfill \\ \hfill & \stackrel{(\text{a})}{=}\underset{P}{\mathrm{min}}\,\mathrm{tr}\,({P}_{\mathcal{I}\mathcal{J}}{{\Omega}}_{\mathcal{J}\mathcal{I}})\hfill \\ \hfill & =\underset{P}{\mathrm{min}}\left({\mathbf{u}}^{\mathrm{T}}{P}_{\mathcal{I}\mathcal{J}}{\boldsymbol{\omega }}_{j}+{\boldsymbol{\omega }}_{i}^{\mathrm{T}}{P}_{\mathcal{I}\mathcal{J}}\mathbf{u}+{\omega }_{ij}\left[{\mathbf{u}}^{\mathrm{T}}{P}_{\mathcal{I}\mathcal{J}}\mathbf{u}-2{\mathbf{u}}^{\mathrm{T}}{P}_{\mathcal{I}\mathcal{J}}{\mathbf{e}}_{i}-2{\mathbf{e}}_{j}^{\mathrm{T}}{P}_{\mathcal{I}\mathcal{J}}\mathbf{u}+2{\mathbf{e}}_{j}^{\mathrm{T}}{P}_{\mathcal{I}\mathcal{J}}{\mathbf{e}}_{i}\right]\right)\hfill \\ \hfill & \stackrel{(\text{b})}{=}\underset{P}{\mathrm{min}}\left({\boldsymbol{\mu }}_{t,j}^{\mathrm{T}}({\boldsymbol{\omega }}_{j}-2{\omega }_{ij}{\mathbf{e}}_{i})+{({\boldsymbol{\omega }}_{i}^{\mathrm{T}}-2{\omega }_{ij}{\mathbf{e}}_{j})}^{\mathrm{T}}{\boldsymbol{\mu }}_{t,i}+{\omega }_{ij}(1+2{P}_{ji})\right)\hfill \\ \hfill & ={\boldsymbol{\mu }}_{t,i}^{\mathrm{T}}{\Omega}{\boldsymbol{\mu }}_{t,j}-2{\omega }_{ij}{\boldsymbol{\mu }}_{t,j}^{\mathrm{T}}{\mathbf{e}}_{i}{\mathbf{e}}_{j}^{\mathrm{T}}{\boldsymbol{\mu }}_{t,i}+2{\omega }_{ij\,}\underset{P}{\mathrm{min}}\,{P}_{ji}\hfill \\ \hfill & \stackrel{(\text{c})}{\geqslant }{\boldsymbol{\mu }}_{t,j}^{\mathrm{T}}{\Omega}{\boldsymbol{\mu }}_{t,i}-2{t}^{2}{c}_{ij},\hfill \end{align*}$$

where in step (a) we use that a nonnegative matrix P with $\text{supp}(P\mathbf{u})=\mathcal{I}$ and $\text{supp}({P}^{\mathrm{T}}\mathbf{u})=\mathcal{J}$ can only have nonzero entries in $\mathcal{I}\times \mathcal{J}$, in step (b) we introduce the marginals of P and in step (c) we use nonnegativity of P_ji and the fact that ${({\boldsymbol{\mu }}_{t,i})}_{j}={({\boldsymbol{\mu }}_{t,j})}_{i}={c}_{ij}t$ and c_ij ω_ij = 1 since the link is a cut link. Introducing the definitions of μ _t,i and μ _t,j , we obtain

$$\begin{equation*}{W}_{1}({\mu }_{t,i},{\mu }_{t,j})\geqslant {\mathbf{e}}_{j}^{\mathrm{T}}(I-Qt){\Omega}(I-Qt){\mathbf{e}}_{i}-2{t}^{2}{c}_{ij}={\omega }_{ij}-2t({p}_{i}+{p}_{j})\end{equation*}$$

such that

$$\begin{equation*}{\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}=\underset{t\to {0}^{+}}{\mathrm{lim}}\frac{1}{t}\left(1-\frac{{W}_{1}({\mu }_{t,i},{\mu }_{t,j})}{{\omega }_{ij}}\right)\leqslant \frac{2({p}_{i}+{p}_{j})}{{\omega }_{ij}}\quad (\text{if}(i,j)\;\text{is}\;\text{a}\;\text{cut}\;\text{link}).\end{equation*}$$

Combined with the general lower-bound (24) for the OR curvature, this proves that equality must hold between ${\kappa }_{ij}^{(\mathrm{O}\mathrm{R})}$ and κ_ij when (i, j) is a cut link. □

We show proposition 3, which relates the link resistance curvature to FR curvature (10) with a particular choice of weights.

Proof of proposition 3. Starting from the definition of FR curvature (10) with unit weights w = 1, we find that

$$\begin{equation*}{\kappa }_{ij}^{(\mathrm{F}\mathrm{R})}=2\left(1-\frac{1}{2}\sum\limits _{k\sim i}1\right)+2\left(1-\frac{1}{2}\sum\limits _{k\sim j}1\right)=4-{d}_{i}-{d}_{j}\stackrel{(a)}{\leqslant }{\omega }_{ij}{\kappa }_{ij},\end{equation*}$$

where (a) is the lower bound for κ_ij in property 3. This completes the proof. □