Ollivier Ricci-flow on weighted graphs

Ricci flow which was introduced by Richard Hamilton [7] in 1980s is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat. On Riemannian manifolds $M$ with a smooth Riemannian metric $g$, the geometry of $(M,g)$ is altered by changing the metric $g$ via a second-order nonlinear PDE on symmetric $(0,2)$-tensors:

where $R_{ij}$ is the Ricci curvature. A solution to a Ricci flow is a one-parameter family of metrics $g(t)$ on a smooth manifold $M$, defined on a time interval $I$, and satisfying equation (1). Intuitively, Ricci flow smooths the metric, but can lead to singularities that can be removed. This procedure is known as surgery. Ricci flow(and surgery) were used in an astonishing manner in the landmark work of Perleman [17] for solving the Poincaré conjecture, as well as in the proof of the differentiable sphere theorem by Simon Brendle and Richard Schoen [4].

One might image such powerful method can be applied to discrete geometry, where objects are irregular complex networks. In 2019, [13] claim good community detection on networks using Ricci flow defined on weighted graphs. For community detection, as one of the fundamental problems in network analysis, one can refer to [6, 9, 16]. The algorithm in [13] makes use of discrete Ricci flow based on Ollivier Ricci curvature which was introduced in [15, 14] and an analogous surgery procedure to partition networks which are modeled as weighted graphs. The discrete Ricci flow deforms edge weights as time progresses: edges of large positive Ricci curvature (i.e., sparsely traveled edges) will shrink and edges of very negative Ricci curvature (i.e., heavily traveled edges) will be stretched. By iterating the Ricci flow process, the heavily traveled edges are identified and thus communities can be partitioned. This approach has successfully detected communities for various networks including Zachary’s Karate Club graph, Network of American football games, Facebook Ego Network, etc. The paper is beautiful in applications but lack of solid mathematical results/theorems. There are several fundamental questions needed to be addressed:

1. 1.

   What are intrinsic metric/curvature in graphs?

2. 2.

   Is the solution of Ricci-flow equation always exists? At what domain?

3. 3.

   If the limit object of the Ricci-flow exist? Do they have constant curvature?

The goal of this paper is to give a mathematical framework so that these essential questions can be answered rigorously.

We start with a brief explanation of the idea of discrete Ricci flow in [13] which contains two parts: the geometric meaning of Ollivier Ricci curvature and the role of Ricci flow.

In the setting of graphs, the Ollivier Ricci curvature is based on optimal transport of probability measures associated to a lazy random walk. To generalize the idea behind Ricci curvature on manifolds to discrete space, the spheres are replaced by probability measures $\mu_{x},\mu_{y}$ defined on one-step neighborhood of vertices $x,y$. Measures will be transported by a distance equal to $(1-\kappa_{xy})d(x,y)$, where $\kappa_{xy}$ represents the Ollivier curvature along the geodesic segment $xy$. A natural choice for the distance between measures $\mu_{x},\mu_{y}$ is the Wasserstein transportation metric $W_{1}$. Therefore, the Ollivier’s curvature is defined as: $\kappa_{xy}=1-\frac{W_{1}(\mu_{x},\mu_{y})}{d(x,y)}$. By this notion, positive Ollivier Ricci curvature implies that the neighbors of the two centers are close or overlapping, negative Ollivier Ricci curvature implies that the neighbors of two centers are further apart, and zero Ollivier Ricci curvature or near-zero curvature implies that the neighbors are locally embeddable in a flat surface. The Ollivier Ricci curvature can be generalized to weighted graph $(V,E,w)$ where $w$ indicates the edge weights. There has been various generalized versions, while the probability measure may vary the essence using optimal transport theory remains unchanged.

The discrete Ricci flow algorithm in [13] on weighted graphs is an evolving process. In each iteration, the Ricci flow process generate a time dependent family of weighted graph $(V,E,w(t))$ such that the weight $w_{ij}(t)$ on edge $ij$ changes proportional to the Ollivier Ricci curvature $\kappa_{ij}(t)$ at edge $ij$ at time $t$. Ollivier[14] suggested to use the following formula for Ricci flow with continuous time parameter $t$:

where $e\in E$, $\kappa_{e}$ represents the Ollivier Ricci curvature on $e$ and $w_{e}$ indicates the length of edge $e$. Then [13] uses the following formula for Ricci flow with discrete time $t$:

where $d^{t}(i,j)$ is the associated distance at time $t$, i.e the shortest path length between $i,j$ and $\kappa_{ij}(t)$ is the Ollivier Ricci curvature on edge $(i,j)$ at time $t$. Observe such an iteration process, the Ricci flow enlarges the weights on negatively curvatured edges and shrink the weights on positively curvatured edges over time. By iterating the Ricci flow process, edges with high weights are detected so that can be removed by a surgery processes. As a result, the network is naturally partitioned into different communities with relatively large Ricci curvature.

Motivated by work in [13], we propose a theoretic framework for the Ricci flow equations defined on weighted graphs. Since the Ricci flow (2 ) does not preserve the sum of edge length of $G$, which would possibly lead to that the graph becomes infinitesimal in the limit if the initial metric satisfies a certain conditions, see an example in Section 4.2. To avoid this, we define the normalized Ricci flow:

Here we adopt the Ollivier-Lin-Lu-Yau’s Ricci curvature[10], which is the limit version of Ollivier Ricci curvature. Under this normalized flow, which is equivalent to the unnormalized Ricci flow (2) by scaling the metric in space by a function of $t$, the sum of edge length of the solution metric is $1$ in time. To see this, let $\vec{w}(t)=\{w_{e_{0}}(t),\ldots,w_{e_{m}}(t)\}$ be a solution of the unnormalized equation, let $\phi(t)$ be a function of time $t$ and $\phi(t)>0$. Set $\vec{\tilde{w}}(t)=\phi(t)\vec{w(t)}$ and $\sum_{e}\tilde{w_{e}}(t)=1$, then $\phi(t)=\frac{1}{\sum_{h}w_{h}(t)}$. Note the edge curvature $\kappa$ does not change under a scaling of the metric. Thus $\tilde{\kappa_{e}}=\kappa_{e}$ for all edges $e\in E$. Let $\tilde{t}=t$, then

where the last equation is obtained by replacing $w_{e}(t)$ by $\frac{1}{\phi(t)}\tilde{w_{e}}(t)$ for all edges $e$.

On the other side, let $\tilde{w}(t),w(t)$ be solutions of the normalized equation and unnormalized equation respectively, we show that for each edge $e$, $w_{e}=\tilde{w_{e}}\sum_{h}w_{h}(t)$. It suffices to show that $\tilde{w_{e}}\sum_{h}w_{h}(t)$ satisfies equation (2).

Thus, there is a bijection between solutions of the unnormalized and normalized Ricci flow equations.

In Riemannian manifolds, with the establishment of Ricci flow equations, one of important work is to verify whether this equation always has a unique smooth solution, at least for a short time, on any compact manifold of any dimension for any initial value of the metric. In this paper, we study the problem of the existence and uniqueness of solutions and convergence results to the normalized Ricci flow (4) on connected weighted graphs. The difficulty of the problem lies in that there is no explicit expression for $\kappa_{e}(t)$, although $\kappa_{e}$ can be written in terms of the infimum of distance-based 1-lipschitz function, there is no common optimal 1-lipschitz function for all edges $e$, thus it is not easy to estimate the derivative of the right-hand side of Ricci flow. The main Theorem 3 of this paper proves the long-time existence and uniqueness of solutions for the initial value problem involved Ricci flow equations (4) provided that each edge is always the shortest path connecting its endpoints over time. This theorem also implies a same result for the unnormalized Ricci flow (2). We also prove that several convergence results of Ricci flow on path and star graphs. Our results display different possible solutions of Ricci flow for the path of length 2, a graph minor of any finite path resulted by the Ricci flow accompanied with edge operations (see Theorem 4) and prove that Ricci flow on star graph can deforms any initial metric to a constant-curvatured metric (see Theorem 5).

The paper is organized as follows. In section 2, we introduce the notion of Ollivier-Lin-Lu-Yau Ricci curvature defined on weighted graphs and related lemmas; in section 3, we introduce the Ricci flow equation and prove our main theorem; in section 4.2, we mainly apply the Ricci flow on a tree graph and show different convergence results of normalized Ricci flow.

Let $G=(V,E,w)$ be a weighted graph on vertex set $V$ associated by the edge weight function $w:E\to[0,\infty)$. For any two vertices $x,y$, we write $xy$ or $x\sim y$ to represent an edge $e=(x,y)$, $w_{xy}$ is always positive if $x\sim y$. For any vertex $x\in V$, denote the neighbors of $x$ as $N(x)$ and the degree of $x$ as $d_{x}$. The length of a path is the sum of edge lengths on the path, for any two vertices $x,y$, the distance $d(x,y)$ is the length of a minimal weighted path among all paths that connect $x$ and $y$. We call $G$ a combinatorial graph if $w_{xy}=1$ for $x\sim y$, $w_{xy}=0$ for $x\not\sim y$. Next we recall the definition of Ricci curvature defined on weighted graphs.

By the theory of linear programming, the transportation distance is also equal to the optimal solution of its dual problem. Thus, we also have

where $f$ is $1$-Lipschitz function satisfying

It is clear that curvature $\kappa$ is a continuous function about the weight function $w$. On combinatorial graphs, the probability distribution $\mu_{x}^{\alpha}$ is uniform on $x$’s neighbors, the above limit expression for Lin-Lu-Yau’s Ollivier curvature is studied in [10, 3] and it turned out that function $\kappa_{\alpha}:\alpha\to\mathbb{R}$ is a piece-wise linear function with at most three pieces. Therefore one can calculate $\kappa$ easily by choosing a large enough value of $\alpha$. On weighted graphs, the probability distribution $\mu_{x}^{\alpha}$ is determined by weight $w$ and function $\gamma$, the distance $d$ involved is reflected directly by $w$. Some authors used a combinatorial distance $d$ which measures the number of edges in the shorted path instead of the weighted version. For instance, in [12], the authors [12] also simplify the limit expression of $\kappa(x,y)$ to two different limit-free expressions via graph Laplacian and via transport cost. Although the details are different, the curvature definitions are essentially the same. The limit-free expression of $\kappa(x,y)$ in [12] is still true under our definition.

To state this limit-free curvature expression, we need to rephrase the notion of Laplacian in order to adapt to Definition 3. Let $G=(V,w,\mu)$ be a weighted graph, let $f$ represent a function in $\{f:V\to\mathbb{R}\}$. The gradient of $f$ is defined by

According to Definition 3, the graph Laplacian $\Delta$ is defined via:

where $f\in\{f:V\to\mathbb{R}\}$.

The limit-free formulation of the Lin-Lu-Yau Ricci curvature using graph Laplacian and gradient is as follows.

where $\nabla_{xy}f$ is the gradient of $f$, $d$ is the combinatorial graph distance.

Using the weighted distance in Definition 3, the limit expression is simplified to another limit-free version via a so called $\ast$-coupling functions[1]. Let $\mu_{x}:=\mu_{x}^{0}$ be the probability distribution of random walk at $x$ with idleness equal to zero. For any two vertices $u$ and $v$, a $\ast$-coupling between $\mu_{u}$ and $\mu_{v}$ is a mapping $B:V\times V\to\mathbb{R}$ with finite support such that

1. 1.

   $0<B(u,v)$, but all other values $B(x,y)\leq 0$.

2. 2.

   $\sum\limits_{x,y\in V}B(x,y)=0$.

3. 3.

   $\sum\limits_{y\in V}B(x,y)=-\mu_{u}(x)$ for all $x$ except $u$.

4. 4.

   $\sum\limits_{x\in V}B(x,y)=-\mu_{v}(y)$ for all $y$ except $v$.

Because of items (2),(3), and (4), we get

Since $0<B(u,v)$ and $B(x,y)\leq 0$. then

A lower bound of $\kappa(u,v)$ is obtained by using a result of Lemma 3.2 in [1]. We re-organized this result as follows:

In this section, we will describe a continuous Ricci flow process on weighted graphs and prove that this Ricci flow has a unique solution that exists for all time.

Let $\kappa:E(G)\to\mathbb{R}^{|E(G)|}$ be the Ollivier-Lin-Lu-Yau curvature function defined on a weighted graph $G=(V,w,\mu)$ where $w$ is the weight function on the edge set of $G$ and $\mu=\{\mu_{x}:x\in V(G)\}$ be probability distribution function such that for each $x\in V(G)$,

where $\gamma:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a Lipschitz function over $[\delta,1]$ for all $\delta>0$.

Let $X(t)=\langle w_{e_{1}}(t),w_{e_{2}}(t),\ldots,w_{e_{m}}(t)\rangle\in\mathbb{R}_{+}^{m}$ where $t\in[0,\infty)$ and $m=|E(G)|$ denotes the number of edges of $G$. Let $X_{0}\in\mathbb{R}_{++}^{m}$ be an arbitrary vector $\langle w_{e_{1}}(0),w_{e_{2}}(0),\ldots,w_{e_{m}}(0)\rangle$ with $\sum_{i=1}^{m}w_{e_{i}}(0)=1$. Note $X_{0}$ gives the initial metric $\vec{w}(0)$ for graph $G$ with all $w_{e_{i}}(0)$ strictly greater than zero. We then define a system of ordinary differential equations as follows:

Now we introduce the continuous Ricci flow process as follows:

We make three observations:

First, there is no edge getting a zero weight at any time $t$ during the whole Ricci flow process. In the initial weighted graph $G$, $w_{e}(0)\neq 0$ for all edges $e\in E(G)$. Fix an edge $e$, the right-hand side of (9) is bounded below by $w_{e}(-2-2|E(G)|)$ according to Lemma 1, then $w_{e}(t)>w_{e}(0)e^{(-2-2|E(G)|)t}$ which is always positive at finite time.

Second, each edge has a weight assigned to it over time, theoretically, we don’t know if there is an edge meeting the exit condition (I), that is, $w_{uv}(t)>d(u,v)(t)$ for some $uv\in E(G)$. To fix this possible barrier we choose to delete such edges, notice that the resulting graph is still connected. In addition, the only reason for the reduction in the number of vertices is the exit condition (II). Thus, $G$ will not degenerate to a point.

Last, since $\sum_{i=1}^{m}w_{e_{i}}(0)=1$, under the assumption that no edges meeting two exit conditions, we claim that the property $\sum_{i=1}^{m}w_{e_{i}}=1$ is always maintained. To see this, let $T(t)=\sum_{h\in E(G^{t})}w_{h}(t)$, where $G^{t}$ is the resulting graph at time $t$. Sum up both sides of equation (9) over all edges of $G^{t}$, we have

By Theorem 1, the right hand side is a bounded value for all $t$, it follows then that $T(t)-1$ has the following form:

where $c$ is a constant depending on $T(0)$. Since $T(0)=1$, then $c=0$ implies $T(t)=1$ for all $t\geq 0$, done. Therefore, in algorithm 1, for all time $t$ the sum of weight is at most $1$, and for each edge $e$, $0<w_{e}(t)\leq 1$. In order to remain the sum of weight constant $1$, an alternative approach is re-normalize the edge weight after each Ricci flow iteration so that the sum of weight always remains $1$, but sum of weight being $1$ or not does not affect the validity of the following theorem.