Enhanced Forman curvature and its relation to Ollivier curvature
EEnhanced Forman curvature and its relation to Ollivier curvature
Philip Tee ∗ The Beyond Center for Fundamental ScienceArizona State University, Tempe AZ
C. A. Trugenberger
SwissScientific Technologies SA, rue du Rhone 59, CH-1204 Geneva, Switzerland (Dated: February 25, 2021)Recent advances in emergent geometry and discretized approaches to quantum gravity have reliedupon the notion of a discrete measure of graph curvature. We focus on the two main measures thathave been studied, the so-called Ollivier-Ricci and Forman-Ricci curvatures. These two approacheshave a very different origin, and both have advantages and disadvantages. In this work we studythe relationship between the two measures for a class of graphs that are important in quantumgravity applications. We discover that under a specific set of circumstances they are equivalent,potentially opening up the possibility of exploiting the relative strengths of both approaches inmodels of emergent spacetime and quantum gravity.
INTRODUCTION
Curvature is a fundamental concept in general rela-tivity. In the absence of matter, the Einstein equationsdetermine the stationary points of the Einstein-Hilbertaction, the integral of the scalar curvature over the man-ifold. The Einstein-Hilbert action is, however, pertur-batively non-renormalizable. Inspired by the successes oflattice gauge theory, one of the avenues that has been ex-plored to cure this problem is to get rid of the associated“infinities” by regularizing spacetime in terms of simpli-cial complexes and search for an ultraviolet (UV) fixed-point that defines quantum gravity non-perturbatively.While this dynamical triangulation program failed in gen-eral [1], it does much better when a preferred foliation isassumed, so that one is essentially discretizing a Lorentzmanifold. The resulting causal dynamical triangulationsprogram has a rich structure with a much better scal-ing behaviour [1, 2]. An alternative approach is causalset theory [3], in which spacetime is considered as fun-damentally discrete, with the structure of a locally finiteposet describing the causal structure of spacetime (for arecent review see [4]).Simplicial complexes are still piece-wise flat chunks ofspacetime. Recently an approach to discretize geometryby much “wilder” structures, like random graphs (for areview see [5]) has been proposed [6]. The idea of thiscombinatorial quantum gravity approach is for geometricmanifolds to emerge from random graphs in a continuousnetwork transition, for which there is indeed strong ev-idence [7–9]. Contrary to previous discrete approaches,combinatorial quantum gravity is agnostic as to the sig-nature of the metric on the emerged manifolds. Depend-ing on the specific instance these can support a Riemannmetric or both a Riemann and a Lorentz metric.The commonality among these approaches is that adiscrete notion of curvature is needed. For (causal) dy-namical triangulation this is the original Regge curva- ture [10, 11] based on angle deficits. For causal sets,it is the Benincasa-Dowker curvature [12]. For randomgraphs, several notions of combinatorial Ricci curvaturehave been advanced. In the original proposal [6] theOllivier combinatorial Ricci curvature [13–15] was used.This has been recently shown [16, 17] to converge to stan-dard continuum Ricci curvature on random geometricgraphs [18] and is thus a genuine candidate for a dis-cretization of general relativity. A simplified variant ofOllivier curvature has also been recently introduced in[19–21].Ollivier curvature is very intuitive from the geometricpoint of view. However not only one, but several othernotions of graph curvature have been proposed (for ataxonomy with brief description of the rationale for eachwe refer to [16]). Here we focus on another widely stud-ied and used concept of discrete Ricci curvature, Formancurvature [22–24].Forman curvature was defined originally for cell com-plexes, specifically CW complexes. It reduces to a graphcurvature if this is considered as a 1-complex, so thatonly nodes (0-cells) and links (1-cells) are considered.It can be augmented, by considering the graph as a 2-complex, so that closed cycles are considered as bound-ing a 2-cell. However, the focus in prior work was on tri-angles [25]. Consequently even this augmented Formancurvature is not appropriate for applications to quan-tum gravity, since discrete locality requires consideringcycles of length up to 5, as automatically realized in theOllivier curvature. Here we generalize the Forman con-struction for graphs considered as 2-complexes and weshow that the resulting “enhanced” Forman Ricci curva-ture matches the mean field Ollivier curvature on randomgraphs satisfying the independent short-cycle condition[7]. This suggests that the cycle-dependence of discrete,combinatorial curvature measures is unique. a r X i v : . [ g r- q c ] F e b OLLIVIER-RICCI CURVATURE
The Ollivier-Ricci (OR) curvature is a measure ofgraph curvature first introduced by Yann Ollivier [13–15]. It tries to mimick in a discrete setting the geomet-ric property of Ricci curvature as a description of howinfinitesimal balls expand or shrink upon parallel trans-port. Two points in Riemannian space can be used todefine a geodesic through them. Consider now balls ofinfinitesimal radius around these two points. In a posi-tively curved space the average geodesic distance betweenall other points on the two balls is shorter than the dis-tance along the geodesic between the centers. For nega-tively curved space the result is the opposite, and for flatspace the distances are the same.To carry over into discrete graphs the key observationis that a graph is a metric space, and the role of the ballscan be undertaken by unit normalized probability distri-butions that have their support in the neighborhood ofa vertex. Geodesic distance is replaced by the so-called“earth mover” or Wasserstein distance between balls cen-tered around two vertices i and j , denoted b i , b j . In theseballs, “ earth mass”” is assumed to be distributed accord-ing to a unit normalized measure µ i , and a transferenceplan measures the exchange of mass necessary to movethe distributions from b i to b j . The Wasserstein distance W ( µ i , µ j ) is the optimal such transport plan. It is thusdefined as W ( µ , µ ) = inf (cid:88) i ∈ b i ,j ∈ b j ξ ( i, j ) d ( i, j ) , (1)where d ( i, j ) is the graph distance and the infimum has tobe taken over all couplings (or transference plans) ξ ( i, j ).That is to say over all plans on how to transport a unitmass distributed according to µ around i to the samemass distributed according to µ around j , (cid:88) j ξ ( i, j ) = µ ( i ) , (cid:88) i ξ ( i, j ) = µ ( j ) . (2)The OR curvature of an edge ij is then defined as κ OR ( i, j ) = 1 − W ( µ i , µ j ) d ( i, j ) . (3)In the simplest realization the balls b i are chosen as unitballs, and the probability distributions µ i are uniform onthese unit balls. Note, however,that the convergence tocontinuum Ricci curvature on generic random geometricgraphs requires larger, “mesoscopic” balls, in order to“feel” the curvature of the background manifold. Unitballs, instead are sufficient in the flat case, when the cur-vature of the background manifold vanishes at all scales. THE INDEPENDENT SHORT CYCLECONDITION
The Ollivier curvature is very intuitive but also verycumbersome to compute in general since one has to solvea linear programming problem for each edge. Fortu-nately, there exists a class of random graphs for whichthere exists a closed-form expression. Moreover, it turnsout that these are exactly the most important graphs forapplications to combinatorial quantum gravity.First of all we note that the simplest version of OR cur-vature on unit balls, due to its very definition dependsonly on the nearest and next-to-nearest neighbours of avertex on the considered edge. This is a discrete versionof locality; “feeling” the influence of at least up to thenext-to-nearest neighbours of a vertex is the minimum re-quirement of a combinatorial curvature notion that canbe used to define discrete quantum gravity. It turnsout that although the fundamental degrees of freedomin combinatorial quantum gravity are the edges, uponwhich curvature is defined, the physical degrees of free-dom are actually cycles, or loops. These cycles and loopsrepresent a sort of discrete “gauge principle” [6–8]. Lo-cality implies that only triangles, squares and pentagonsmatter.The class of graphs that we shall consider is definedby a network analogue of the statistical mechanical hardcore condition. It is well known that, to avoid an infinitecompressibility of a boson gas when lowering the temper-ature, a hard-core condition must be imposed that keepsthe fundamental degrees of freedom, the particles in thiscase, from overlapping. The independent short-cycle con-dition [6, 7] is the corresponding network analogue. Thephysical degrees of freedom, which in this case are the tri-angles, squares and pentagons, can touch (share an edge)but not overlap (share more than one edge). Formally,this condition can be defined as,
Definition 1
The independent short-cycle condition.Consider the diagram in Fig. 1. The independent short-cycle condition is satisfied by a graph G ( V, E ) , if all ofits closed cycles of length n ≤ do not share more thanone edge. Let C n ( e ij ) represent a closed cycle supportedupon an edge e ij of length n ≤ . For example in Fig.1, C ( e ij ) contains the vertices i, j, l, m and is the set { e ij , e jl , e lm , e mi } . The independent short-cycle condi-tion is satisfied for the graph G if and only if, n ≤ (cid:92) n C n ( e ij ) = e ij , ∀ e ij ∈ E . (4)As is discussed in the caption, the edge graph depictedsatisfies the condition if and only if one of either thedashed square or triangle are present, but not both. Thereader is referred to Kelly et al [7] where it is shownthat it is possible to reduce the condition to a statementexcluding certain subgraphs that occur if it is violated. •• • •••• • ij k lm pnN ( j ) N ( i ) FIG. 1: We depict two vertices i and j and theconnecting edge e ij . There is a triangle (red dashedlines) ( i, k, j ), a square (blue dashed lines) ( i, m, l, j )and a pentagon ( i, n, o, p, j ) sharing the edge e ij . Theblack dashed lines represent links to other vertices inthe neighborhoods of i and j , N ( i ) and N ( j ). Theindependent short-cycle condition is satisfied for theedge e ij , but the presence of the square and the triangledepicted would violate the condition for the edges e jk and e ki as they would support a pentagon ( i, m, l, j, k )and a triangle ( i, k, j ) that share two edges. For theedge e ij this graph satisfies the independent short-cyclecondition, if and only if either the dashed triangle orsquare is present, but not both.Here we shall not pursue the gravitational conse-quences of the independent short-cycle condition. Ratherwe shall focus on the consequences for the mathematicalexpressions of graph curvature. In particular, in [7] it hasbeen shown that, on graphs that satisfy this condition,the OR curvature reduces to a simple closed form, κ OR ( ij ) = (cid:52) ij k i ∧ k j − (cid:104) − (cid:52) ij + (cid:3) ij k i ∨ k j − k i ∧ k j (cid:105) + − (cid:16) (cid:52) ij k i ∧ k j − (cid:52) ij k i ∨ k j (cid:17) ∨ (cid:16) − (cid:52) ij + (cid:3) ij + (cid:68) ij k i ∨ k j − k i ∧ k j (cid:17) , (5)where k i , k j are the degrees of vertices v i , v j , respectively.Further the symbols (cid:52) ij , (cid:3) ij and (cid:68) ij denote the num-ber of triangles, squares and pentagons supported on theedge e ij , and α ∨ β := max( α, β ), α ∧ β := min( α, β ) and[ α ] + := 0 ∨ α for any α, β ∈ R . Remarkably, the inde-pendent short-cycle condition allows one to calculate theOR curvature by simply counting the number of shortloops based on an edge.To expose the dependence of curvature on global com-binatorial quantities, the numbers of cycles based on anedge, we focus on edges which have the same connectiv-ity at their two vertices, by setting k i = k j = (cid:104) k (cid:105) . Theseare the majority of edges when the degree distribution ispeaked. In this case we obtain the even simpler expres- sion κ OR ( ij )= (cid:52) ij (cid:104) k (cid:105) − (cid:20) − (cid:52) ij + (cid:3) ij (cid:104) k (cid:105) (cid:21) + − (cid:20) − (cid:52) ij + (cid:3) ij + (cid:68) ij (cid:104) k (cid:105) (cid:21) + . (6)Of course this expression is exact for all edges of a regulargraph. FORMAN-RICCI CURVATURE
An alternative discrete measure of graph curvature wasintroduced by Robin Forman [22–24] using the topolog-ical constructs of CW (Closure-finite, Weak) cell com-plexes. Forman’s work defines an entire parallel appara-tus of differential forms, Morse theory and Ricci curva-ture to those well understood in the traditional algebraicgeometry of smooth manifolds. This richness of struc-ture is intriguing, but unlike the OR curvature describedin Section , there is no direct relationship of the For-man curvature to that of a smooth manifold in whichthe graph is embedded. The complete treatment is tech-nical and we shall only briefly survey it here. Essen-tially it draws upon an analogy with identities devel-oped by Bochner [26] regarding the decomposition of theRiemannian-Laplace operator on the space of p -forms,Ω p ( M ) defined for a manifold M . This decompositionyields a covariant derivative and a curvature correctionknown as the Bochner-Weitzenb¨ock identity. Its discreteform is used to derive Forman-Ricci (FR) curvature.CW complexes (an excellent standard text is Hatcher[27]) are constructed from p -cells ( p referring to the di-mension of the cell). One constructs a d dimensional CWcomplex by gluing p ≤ d complexes along shared faces.For our purposes we will focus on cell complexes up to p = 2, which are essentially equivalent to graphs, with theaddition that cycles in the graph are assumed to bound a2-cell. This assumption is critical, and often overlookedin definitions of FR curvature. In some literature, theinclusion of these 2-cells is referred to as “augmented”FR curvature, but we view the non-augmented versionas essentially trivial and of no utility in applications toquantum gravity.We define the boundary of a p -cell as the p − (cid:104) p p (cid:105) , the boundary is thecollection of points p and p , and for a general p − cell, α p , it is a proper face of a p + 1 cell β if it is a memberof the boundary set of β , and we write α p < β p +1 , or β p +1 > α p . A p -cell CW complex M over R p , is definedformally a collection of cells α q , q ∈ { , . . . , p } , such thatany two cells are joined along a common proper face, andall faces are contained in the cell complex.An important concept when developing the curvatureof cell complexes is the definition of the neighbors of agiven p -cell [22, 23] introduced by Forman as, Definition 2 α and α are p -cells of a complex M . α , α are neighbors if:1. α and α share a ( p + 1) cell β such that β > α and β > α , or2. α and α share a ( p − cell γ such that γ < α and γ < α . Further, we can partition the set of neighbors of a cellinto parallel and non-parallel. Two p -cells α , α are par-allel neighbors, if one but not both of the conditions inDef. 2 are true, and write α (cid:107) α .With these concepts, FR curvature is defined as a seriesof maps F p : α p → R , for each value of p , and has thefollowing simple form, F p ( α p ) = { β ( p +1) > α p } + { γ ( p − < α p }− { (cid:15) q (cid:107) α p } ,(7)where (cid:15) q is a q -cell that is a parallel neighbor of α p , and q (cid:54) = p . The symbol F p ( α p ) as the number of p − α p , plus the number of p + 1 cells ofwhich α p is part of the boundary minus the number ofparallel neighbors of α p .It is possible to adorn each cell with a weight, and forcompleteness we reproduce here the full version of thisformula for weighted complexes. For each p -cell α p weassociate a weight g α and we denote by ˜ α p its neighborsper Def. 2. Using these definitions we have, F p ( α ) = g α (cid:88) β ( p +1) >α g α g β + (cid:88) γ ( p − <α g γ g α − (cid:88) ˜ α p (cid:107) α p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) β ( p +1) >α p β ( p +1) > ˜ α p √ g α p g ˜ α p g β − (cid:88) γ ( p − <α p γ ( p − < ˜ α p g γ √ g α p g ˜ α p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (8)It is important to note that the last term in this equationis a sum over the absolute difference between p + 1 and p − p = 1, Forman identifies Eq. (7) as the Riccicurvature, defined on the edges of an unweighted graph,and we refer to this as the Forman-Ricci (FR) curvature.We distinguish this particular value of the FR curvaturefor p = 1 by the notation κ Fij = F ( e ij ). For a graphEq. (7) is simple; the vertices and edges constitute the 0and 1 cells, and closed loops in the graph constitute the2-cells. It is this simplicity that underlies the favorablecomputability of FR curvature. The FR curvature is a combinatorial quantity and, asis evident from its definition in Eq. (7), there is no ar-bitrary restriction to the length of cycles that are admis-sible as bounding a 2-cell. The so called “augmented”Forman curvature is obtained by restricting these cyclesto triangles, although this restriction is not present inthe original work of Forman [23]. However, as we ex-plained above, this is not appropriate for applications toquantum gravity, where discrete locality requires takinginto account cycles of length up to 5. We define thus anappropriate “enhanced” Forman curvature κ F R by trun-cating the expansion to cycles of length 5, and note thatDef. 1 similarly covers cycles up to length 5.Assuming that a graph possesses the independentshort-cycle property [7] brings a substantial simplifica-tion also for the enhanced Forman curvature. Indeed, inthis case we can write the following simple expression for κ F Rij that is exact if the graph does not have closed cycleslarger than pentagons, κ F Rij = 4 − k i − k j + 3 (cid:52) ij + 2 (cid:3) ij + (cid:68) ij , (9)where, for a given edge either (cid:52) ij > (cid:3) ij >
0, butnot both. The importance of the independent short-cyclecondition is that it allows one to compute the contribu-tion to the number of parallel edges to e ij from the edges(1-cells) that are not part of any cycle incident upon e ij .This is easily obtained, as the independent short-cyclecondition implies that every cycle incident upon a givenedge consumes precisely one edge from each of the ver-tices v i and v j . As such, the number of parallel edgesthat share a vertex with a given edge e ij is precisely k i + k j − − (cid:52) ij + (cid:3) ij + (cid:68) ij ). To arrive at Eq. (9),we incorporate the number of parallel edges that share a2-cell with e ij being 0 for a triangle, 1 for a square and2 for a pentagon.Eq. (9) further simplifies for edges with the same con-nectivity (cid:104) k (cid:105) at their two vertices, as we have consideredfor the OR curvature in (6), κ F Rij = 4 − (cid:104) k (cid:105) + 3 (cid:52) ij + 2 (cid:3) ij + (cid:68) ij . (10)In this FR case, this expression can be considered asa mean field approximation. When this expression issummed over all edges to obtain the equivalent of theRicci scalar for the graph it becomes exact. As we willsee, for OR curvature this is not so since the proper treat-ment of the [] + terms in Eq. (6) requires correction termsto be applied to the averaged curvature sum. COMPARING OLLIVIER-RICCI ANDFORMAN-RICCI CURVATURE
Considering Eq. (6), the effect of the [] + in the lasttwo terms makes a direct comparison of this expressionwith the FR curvature difficult. To make this comparisoneasier we express the OR curvature in Eq. (6) as a meanfield term, in which all brackets are simply summed upwithout taking into account the “+” subscripts, plus acorrection term. This gives (cid:104) k (cid:105) κ ORij = (cid:104) k (cid:105) κ ORMFij + δ ij ,κ ORMFij = 4 − (cid:104) k (cid:105) + 3 (cid:52) ij + 2 (cid:3) ij + (cid:68) ij , (11) where the correction to the mean field value κ ORMFij isgiven by δ ij = (cid:104) k (cid:105) > (cid:52) ij + (cid:3) ij + (cid:68) ij (cid:104) k (cid:105) − − (cid:52) ij − (cid:3) ij − (cid:68) ij if 2 + (cid:52) ij + (cid:3) ij < (cid:104) k (cid:105) < (cid:52) ij + (cid:3) ij + (cid:68) ij (cid:104) k (cid:105) − − (cid:52) ij − (cid:3) ij − (cid:68) ij if (cid:104) k (cid:105) < (cid:52) ij + (cid:3) ij (12)This shows that the properly enhanced local FormanRicci curvature coincides (up to an overall factor) withthe mean field value of the Ollivier Ricci curvature, (cid:104) k (cid:105) κ ORMFij = κ F Rij . (13)This result is surprising, given that the two discrete cur-vature constructions have completely different origins. Itis a first indication that at least the global combinatorialdependence on the number of cycles is unique for anylocal discrete curvature measure.The correction term vanishes and the two curvaturesbecome essentially identical for graphs with large con-nectivity and sparse cycles. Unfortunately this is not therelevant case for applications to quantum gravity, wherethe emergence of geometry is associated with a conden-sation of short cycles [6, 7, 28]. One exception to thisis the case of a hypercubic mesh, which is by definitiontriangle and pentagon free. In this case both measurescoincide and are numerically zero, matching with the in-tuitive interpretation of this state as a Ricci flat groundstate.To further test this rather surprising result, we cangenerate a variety of random graphs, with varying con-nectivity and edge density, and compute both curvaturevalues to compare the results. For the purposes of thissimulation we have used Erd¨os-R´enyi random graphs [29](in fact, as we are using a fixed graph size they shouldbe correctly termed Gilbert graphs), using a varying linkprobability p in the range 0 .
01 to 0 .
1, after which wemanually enforce the independent short-cycle conditionby removing edges that violate it. For a fixed number N = 100 of vertices this generates graphs with an averagedegree (cid:104) k (cid:105) > p increases, theedge density of the graph will also increase, along withthe density of short cycles. For each edge we compute I ij = (cid:104) k (cid:105) − − (cid:52) ij − (cid:3) ij − (cid:68) ij . From Eq. (12), when I ij > (cid:104) k (cid:105) . In Fig. 2 we plot the average fractional difference between the two curvature measures,( (cid:104) k (cid:105) κ ORij − κ F Rij ) / (cid:104) k (cid:105) κ ORij over all edges in randomly gen-erated graphs against the average value of I ij . For eachlink probability p we generate 10 graphs, to avoid the re-sults being skewed by unusual graph configurations, andwe compute the OR curvature using a combination of theNetworkX python toolkit, as extended by Chien-Chun Ni[30, 31], and the FR curvature using our own library, asthe publicly available libraries do not include longer cy-cles.FIG. 2: For N = 100 vertices we generate randomErd¨os-R´enyi graphs, varying the link probability p . Fora collection of 10 graphs for each value of p , we computethe average absolute fractional difference between (cid:104) k (cid:105) κ ORij and κ F Rij , which we plot against the mean-fieldcondition I ij = (cid:104) k (cid:105) − − (cid:52) ij − (cid:3) ij − (cid:68) ij from Eq. (12).The simulation clearly shows that the two curvaturemeasures differ when I ij < I ij increases andapproaches zero they converge to the same value. This isfully consistent with our analysis and provides supportingnumerical evidence for our main claim. CONCLUSION
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