Coarse Ricci curvature of hypergraphs and its generalization
aa r X i v : . [ m a t h . M G ] F e b COARSE RICCI CURVATURE OF HYPERGRAPHS AND ITSGENERALIZATION
MASAHIRO IKEDA, YU KITABEPPU, AND YUUKI TAKAI
Abstract.
In the present paper, we introduce a concept of Ricci curvatureon hypergraphs for a nonlinear Laplacian. We prove that our definition of theRicci curvature is a generalization of Lin-Lu-Yau’s coarse Ricci curvature forgraphs to hypergraphs. We also show a lower bound of nonzero eigenvaluesof Laplacian, gradient estimate of heat flow, and diameter bound of Bonnet-Myers type for our curvature notion. This research leads to understandinghow nonlinearity of Laplacian causes complexity of curvatures. Introduction
The Ricci curvature of Riemannian manifolds plays an important role to an-alyze the geometric and analytic properties of the manifolds. It was a problemhow to define a concept of Ricci curvature on generic metric spaces. Recently, ongeodesic metric measure spaces, the synthetic notion of lower bound of Ricci curva-ture is defined, called the curvature-dimension condition [22, 36, 37]. On the otherhand, there are some different notions of lower Ricci curvature bound on discretespaces. Even for graphs, we have many notion, Ollivier’s coarse Ricci curvature,Lin-Lu-Yau’s coarse Ricci curvature, the (Bakry-´Emery type) curvature-dimensioncondition, the exponentially curvature-dimension inequality, etc. The former twoare defined by the contraction of L -Wassertein distance. The latter two are definedby holding Bakry-´Emery type functional inequalities. Both coincides in the settingof Riemannian manifolds. Such notions of Ricci curvature on graphs are widelyused to network analysis such as community detection [35].In this paper, we introduce a definition of a coarse Ricci curvature on hyper-graph of Lin-Lu-Yau type. Here, hypergraph is a generalization of graph to beable to represent relations among not only two, but also three or more entities. Onhypergraphs, there is no crucial canonical definition of random walks as on graphs.Hence, we cannot naturally define the curvature notion on hypergraphs in Olliver’smanner. Asoodeh et al [4] introduced a notion of coarse Ricci curvature on hyper-graph by the random walk obtained by reducing hypergraph to a graph with clique This work was partly supported by JST CREST Grant Number JPMJCR1913, Japan, Grant-in-Aid for Young Scientists Research (No.18K13412, No.19K14581), Japan Society for the Promo-tion of Science, and Grant for Basic Science Research Projects from The Sumitomo Foundation(No.200484). expansion. Also there are several other notions of Ricci curvature on hypergraphs.See for instance [13, 14, 20].Our coarse Ricci curvature on hypergraphs is defined by using a nonlinear multi-valued Laplacian called “submodular hypergraph Laplacian”, which introduced by[11, 23, 41]. In [38], they empirically showed that the hypergraph Laplacian definedin [18] outperforms the clique expansion for the quality of solutions of a clusteringproblem. This indicates that the hypergraph Laplacian in [18] is more capable tocapture the geometric structure of hypergraph than the clique expansion. Althoughthe our hypergraph Laplacian is multivalued and nonlinear, in [18], it was proventhat this operator is maximal monotone. This means that our Laplacian is in atractable class of nonlinear operators.Our motivation of this research is to understand whether the nonlinearity ofLaplacian causes some difficulty of curvatures or not. Our nonlinear Laplacian isan appropriate candidate for this attempt. We explain more details about this inthe following.For Riemannian manifolds, though the Ricci tensor needs C smooth structureon them, lower bound condition of Ricci curvature can be described by the metricand measure, that we do not need the smoothness. More precisely, von Renesse andSturm [40, Section 1] proved that for the smooth, complete, connected Riemannianmanifold ( M, g ), a volume measure vol g on it, the Ricci curvature Ric x ( v, v ) for x ∈ M and v ∈ T x M , and any K ∈ R , the following (1)-(5) are equivalent:(1) (lower bound of Ricci curvature) Ric g ≥ Kg , which means that Ric x ( v, v ) ≥ K | v | for any x ∈ M and v ∈ T x M .(2) (convexity of relative entropy) The relative entropy Ent vol g defined in [40,P.924] is the displacement K -convex on the L -Wasserstein space ( P ( M ) , W )defined in [40, P.923–924] (cf.[12]).(3) (transportation inequality) For the measure restricted to the ball of radius r whose center is x ∈ Mm r,x ( A ) := vol g ( B r ( x ) ∩ A ) vol g ( B r ( x )) , the following holds: W ( m r,x , m r,y ) ≤ (cid:18) − K n + 2) r + o ( r ) (cid:19) . (4) (contraction property of the gradient flow of entropy) For the gradient flowΦ : R + × P ( M ) → P ( M ) of Ent m , W (Φ( t, µ ) , Φ( t, ν )) ≤ e − Kt W ( µ, ν )holds for t > µ, ν ∈ P ( M ).(5) (gradient estimate of heat flow) For any f ∈ C ∞ c ( M ), x ∈ M , and t > |∇ h t f | ( x ) ≤ e − Kt h t |∇ f | ( x ) OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 3 holds. Here, h t : L ( M ) → L ( M ) is the heat flow on M .We note that (5) is deeply related to the property of Dirichlet form rather thanthe smoothness of the structure of manifolds. Moreover, the following Bochnerinequality (or Bakry-´Emery curvature dimension condition) is also equivalent tothem(cf. [2, 3]):(6) (Bochner inequality, curvature-dimension condition of Bakry-´Emery type)For any f ∈ C ∞ c ( M ), the following holds:12 ∆ |∇ f | ≥ h∇ ∆ f, ∇ f i + K |∇ f | . Based on these facts, CD space, which is the metric measure space which isnot necessarily manifold and looks like satisfying a lower Ricci curvature condition(with upper bound condition of dimension), is introduced by using the convex-ity of entropy on the L -Wasserstein space (Sturm [36, 37], Lott-Villani [22] forfinite dimension, using not relative entropy but R´enyi entropy). An importantpoint is that the definition is only described in terms of measures and metrics.For the CD space whose dimension is bounded above, many important geometricand functional inequalities such as Bishop-Gromov inequality, Poincar´e inequal-ity and Brunn-Minkowski inequality were proved ([22, 32, 37]). However, Ohta-Sturm showed that the gradient estimate of heat flow does not hold for generic CD spaces([29]).After that, an RCD space [1, 2] is defined, which is a CD space equipped withthe infinitesimal Hilbertianity condition (defined by Gigli [17]) that any Sobolevspace becomes a Hilbert space. On any RCD space, several theorems such as the W -contraction of the gradient flow of the relative entropy, the Bochner inequal-ity (Bakry-´Emery’s curvature dimension condition) and the gradient estimate ofthe heat flow have been proved and these are known as equivalent conditions onmanifolds.Many geometric results known on Riemannian manifolds are also proven suchas Cheeger-Gromoll’s splitting theorem [16], Cheng’s maximum diameter theorem[19], isoperimetric inequalities [9] and so on. Furthermore, some results on RCD spaces include theorems which have not been proved even for Ricci limit spaces. It isknown that both of CD and RCD spaces become geodesic metric spaces.
RCD spacesestablished a firm position as geodesic spaces whose Ricci curvature is bounded frombelow.On the other hand, although several definitions of discrete spaces whose Riccicurvature is bounded below were introduced, there has not been canonical defini-tion. It can been seen that for graphs, coarse Ricci curvatures of Ollivier [30] andLin-Lu-Yau [21] are related to the above (3) or (4), the curvature dimension condi-tion of Bakry-´Emery type [33] is related to (6), exponentially curvature-dimensioncondition is related to the Li-Yau inequality [6, 26, 27], and the definition by Maas
MASAHIRO IKEDA, YU KITABEPPU, AND YUUKI TAKAI [15, 24] and by Bonciocat-Sturm [7] is related to (2). Although all of these defi-nitions stem from the definitions or known facts for geodesic spaces, the relationsamong them are still not clear in discrete spaces. Laplacian is a key ingredientto define the curvature-dimension conditions of Bakry-´Emery type. In spite of theLaplacian on usual graph is defined as a linear self-adjoint operator, this is notdiffusion, thus the property as a differential operator is different from that on
RCD spaces. (However, the exponentially curvature-dimension condition tries to over-come the difficulty. We also remark that though the Laplacian on
RCD space islinear, that of general CD space might be nonlinear. In this sense, graphs can beconsidered to be in intermediate position between them).As mentioned above, in this paper, we introduce a notion of coarse Ricci cur-vature on hypergraph using the resolvent of the hypergraph Laplacian of [18]. Al-though our Laplacian is non-singular and multivalued, under the assumption ofthe lower bound condition of curvature of Lin-Lu-Yau type, we can deduce a di-ameter bound, a lower bound of nonzero eigenvalues, and a gradient estimate ofheat flow (of L ∞ type). These properties do not hold for general CD spaces. Thissuggests that the nonlinearity of Laplacian does not necessarily make the heat flowintractable. The authors believe that this observation will drive us to investigatedeeper properties of curvatures with our submodular Laplacian as a candidate.Our arguments are applicable to more general settings for submodular transfor-mations, which are vector valued set functions consisting of submodular functions.In Appendix A, we give a sufficient condition for a submodular transformation tobe able to straightforwardly generalize the curvature notions and our theorems.This paper is organized as follows: In Section 2, we review fundamental proper-ties of hypergraphs (Section 2.1), submodular hypergraph Laplacian and its resol-vent (Section 2.2), L -Wasserstein distance (Section 2.3). In Section 3.1, we reviewthe definition of the coarse Ricci curvature by Lin-Lu-Yau [21] and rephrase thisdefinition by resolvent of graph Laplacian. Based on this observation, in Section 3.2and Section 3.3, we introduce a difinition of λ -nonlinear Kantorovich difference, and(lower and upper) coarse Ricci curvature on hypergraphs. In Section 4, we showthat for graphs, our coarse Ricci curvature is equal to that of Lin-Lu-Yau. In Sec-tion 5, we show applications of our coarse Ricci curvature to a bound of eigenvalues(Section 5.1), a gradient estimate for heat flow (Section 5.2), and a Bonnet-Myerstype diameter bound (Section 5.3). We give some calculation of our curvatures inSection 6. In Appendix A, we review submodular functions (Section A.1), sub-modular transformations and the submodular Laplacian (Section A.2) and argueabout generalization our curvature notion and theorems to submodular transfor-mations. We also show examples of submodular transformations such as directedgraph, mutual information, directed hypergraphs etc. in Section A.3. OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 5 Preliminaries
Hypergraphs.
A (weighted undirected) hypergraph H = ( V, E, w ) is a tripleof a set V , a set E of nonempty subsets of V , and a function w : E → R > . We call V a set of vertices, E a set of hyperedges, and w a weight function. We remark thatif | e | = 2 for any e ∈ E , H is an undirected weighted graph. We say that H is finiteif the set V is finite. For x, y ∈ V , we write x ∼ y if there exists a hyperedge e ∈ E including x and y . We say that hypergraph H is connected if for any x, y ∈ V ,there is a sequence of vertices { z i } ki =1 such that z = x, z k = y, and z i ∼ z i +1 ( i = 1 , . . . , k − . Throughout this paper, we assume that any hypergraph is finite and connected.Let H = ( V, E, ω ) be a weighted, undirected, connected, finite hypergraph. Wedefine the degree of x by d x := P e ∋ x ω e and the degree matrix by D := diag( d x ).We remark that if the hypergraph H is connected, then d x > x ∈ V ,thus D is non-singular. For a subset S ⊂ V , the volume vol ( S ) of S is defined by vol ( S ) = P x ∈ S d x . For two vertices x, y ∈ V , we consider a distance on V as d ( x, y ) := min { n ; ∃ { z i } ni =0 , z = x, z n = y, z i ∼ z i +1 } . Then, (
V, d ) becomes a metric space. We define the diameter diam ( H ) of thehypergraph H as that of the metric space ( V, d ), i.e., diam ( H ) := max x,y ∈ V d ( x, y ) . We identify the R -valued map on V with the set R V of vectors indexed by V . Weset δ x ∈ R V as the characteristic function at x ∈ V : δ x ( z ) := z = x, . We define the stationary distribution π ∈ R V by π ( z ) = d z / vol ( V ) for any z ∈ V .2.2. Laplacian on hypergraph.
In this subsection, we review a submodular hy-pergraph Laplacian in the sense of Ikeda et al [18] and some properties of the resol-vent of the Laplacian, which were derived via general theory of maximal monotoneoperators. For the details of this theory, see [8], [5], [25], and [34].We define an inner product h , i on R V by h f, g i = f ⊤ D − g = X x ∈ V f ( x ) g ( x ) d − x . On the Hilbert space ( R V , h· , ·i ), we define the (submodular) hypergraph Laplacian L : R V → R V for H = ( V, E, w ) by L ( f ) = Lf := (X e ∈ E ω e b e ( b Te f ) ; b e ∈ argmax b ∈ B e b ⊤ f ) . (2.1) MASAHIRO IKEDA, YU KITABEPPU, AND YUUKI TAKAI
Here, B e is the base polytope for e ∈ E , i.e., the subset of R V defined by B e = Conv( { δ x − δ y ; x, y ∈ e } ) , (2.2)where Conv( X ) for X ⊂ R V is the convex hull of X in R V . We note that thisLaplacian L is a modification of the definition introduced in [11, 23], and a real-ization of the submodular transformation introduced in [41] when the submodulartransformation is a hypergraph. As mentioned in [41, Section 2] or [10, P.15:8], thisLaplacian L is the sub-differential of the convex function Q : R V → R defined by Q ( f ) = 12 X e ∈ E w ( e ) max x,y ∈ e ( f ( x ) − f ( y )) . When the hypergraph H is a graph, by the definition (2.1), our Laplacian L becomessingle-valued L ( f ) = { ( D − A ) f } , where A is the adjacency matrix of the graph H .In this paper, we mainly treat the following normalized Laplacian L : R V → R V ,which is related to random walks or heat diffusion: L ( f ) = L f := L ( D − f ) . By [18, Lemma 14, Lemma 15], L is a maximal monotone operator (or −L is an m -dissipative operator) on the Hilbert space ( R V , h· , ·i ). The resolvent J λ : R V → R V is defined as follows: J λ ( f ) := ( I + λ L ) − ( f ) . Here, A − ( f ) for a multivalued operator A is defined by A − ( f ) := { g ∈ R V ; f ∈ A ( g ) } . By [25, Corollary 2.10], the resolvent J λ is non-expansive, i.e., for any f, g ∈ R V ,and any f ′ ∈ J λ ( f ), g ′ ∈ J λ ( g ), k f ′ − g ′ k ≤ k f − g k holds. This implies that the resolvent operator J λ is single-valued and continuousw.r.t. variable f ∈ R V . We state these properties as a lemma: Lemma 2.1.
The resolvent operator J λ of L is non-expansive, single-valued, andcontinuous w.r.t. a variable f ∈ R V . Since L f is the sub-differential of Q at D − f , it is known that the resolvent J λ is characterized as J λ f = argmin (cid:26) λ k f − g k + Q ( D − g ) ; g ∈ R V (cid:27) . By [25, Lemma 2.11(iii)], the resolvent J λ satisfies the following equation: For any λ, µ > J λ f = J µ (cid:18) µλ f + λ − µλ J λ f (cid:19) . (2.3)The following properties of J λ follows from the specified properties of our Lapla-cian L : OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 7
Lemma 2.2.
For any λ > , any f ∈ R V and any c ∈ R , the following hold: (1) L ( cf ) = c L ( f ) and J λ ( cf ) = cJ λ ( f ) , (2) J λ ( R V ) = R V .Proof. We shall first show (1). For c = 0, (1) is trivial. We assume c >
0. Then,we have argmax b ∈ B e h b , cf i = argmax b ∈ B e h b , f i . Hence, the image of Laplacian of cf becomes L ( cf ) = (X e ∈ E ω e b e ( b Te cD − f ) ; b e ∈ argmax b ∈ B e h b , cf i ) = c (X e ∈ E ω e b e ( b Te D − f ) ; b e ∈ argmax b ∈ B e h b , f i ) = c L ( f ) . We consider − c for c >
0. We haveargmax b ∈ B e h b , ( − c ) f i = − argmax b ∈ B e h b , cf i = − argmax b ∈ B e h b , f i . Hence, L ( − cf ) = ( − c ) L f holds for c > c ∈ R , let g = J λ ( cf ). Then, cf ∈ ( I + λ L )( g ) holds by definitionof J λ . Thus, we have f ∈ ( I + λ L )( c − g ) by c − L ( g ) = L ( c − g ). This implies c − g = J λ ( f ), hence g = cJ λ ( f ).Next, we shall prove (2). By [18, Lemma 15], the range of L is R V . This yieldsthat the range of I + λ L is also R V . Hence, the domain of J λ = ( I + λ L ) − is R V .Then, J λ ( R V ) ⊂ R V . Thus, it suffices to show that for any g ∈ R V , there existsan f ∈ R V such that J λ ( f ) = g , i.e., f ∈ ( I + λ L )( g ). This follows from the factthat the domain of L is R V , hence ( I + λ L )( g ) is not empty. This concludes theproof. (cid:3) By the general theory of maximal monotone operators [25, Theorem 4.2 and The-orem 4.10], the heat semigroup h t := e − t L is defined for the normalized Laplacian L and the following holds: h t f = lim λ ↓ J [ t/λ ] λ f. (2.4)Here, [ a ] for a ∈ R is the maximum integer which is less than or equal to a .We set L f := inf {k f ′ k ; f ′ ∈ L f } . Then, by [25, Lemma 2.11 (ii)], the following holds: k J λ f − f k ≤ λ L f . (2.5)Since L f is a closed convex set by [25, Lemma 2.15], there exists a unique f ′ ∈ L f such that k f ′ k = L f . We set L f as this f ′ . This defines a single-valued MASAHIRO IKEDA, YU KITABEPPU, AND YUUKI TAKAI operator L : R V → R V , called the canonical restriction of L . Then, the identities −L f = lim λ ↓ λ − ( J λ f − f ) = lim t ↓ t − ( h t f − f )(2.6)hold by [25, Lemma 2.22 and Theorem 3.5].We show some useful lemmas. Lemma 2.3.
For any function h ∈ R V and any a ∈ R , the following identity holds: L h = L ( h + aπ ) . Proof.
Let b ∈ B e . Because b is a convex combination of δ x − δ y for x, y ∈ e and D − π is a constant function, we have b ⊤ D − π = 0 . Hence, for any b ∈ B e , we have b ⊤ ( D − ( h + aπ )) = b ⊤ ( D − h ) + a b ⊤ ( D − π ) = b ⊤ ( D − h ) . This implies that L h = L ( h + aπ ) holds. (cid:3) Lemma 2.4.
For any function f ∈ R V and any a ∈ R , the following identity holds: J λ f = J λ ( f − aπ ) + aπ. Proof.
We set g := J λ f and h := J λ ( f − aπ ). Then there exist g ′ ∈ L g and h ′ ∈ L h such that the identities f = g + λg ′ , f − aπ = h + λh ′ hold. Thus, we have( I + λ L )( g ) ∋ g + λg ′ = f = h + λh ′ + aπ ∈ ( h + aπ ) + λ L ( h ) = ( h + aπ ) + λ L ( h + aπ ) = ( I + λ L )( h + aπ ) . Here, the inclusion follows from Lemma 2.3. Therefore, acting J λ to the both sides,we get g = h + aπ because J λ = ( I + λ L ) − is single-valued. (cid:3) L -Wasserstein distance. Let (
X, d, m ) be a metric measure space, that is,(
X, d ) is a complete separable metric space and m is a locally finite Borel measureon X . We set P ( X ) as the set of all Borel probability measures. For µ, ν ∈ P ( X ),a measure ξ ∈ P ( X × X ) is called a coupling between µ and ν if ξ ( A × X ) = µ ( A ) ,ξ ( X × A ) = ν ( A )holds for any Borel set A ⊂ X . We set Cpl ( µ, ν ) as the set of all couplings between µ and ν . Since µ ⊗ ν is a coupling between µ and ν , Cpl ( µ, ν ) is nonempty. We alsodefine P ( X ) by P ( X ) := (cid:26) µ ∈ P ( X ) ; Z X d ( x, o ) µ ( dx ) < ∞ for a point o ∈ X (cid:27) . OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 9
For µ, ν ∈ P ( X ), the L -Wasserstein distance between them, denoted by W ( µ, ν ),is defined as W ( µ, ν ) := inf (cid:26)Z X × X d ( x, y ) ξ ( dx, dy ) ; ξ ∈ Cpl ( µ, ν ) (cid:27) . It is known that W is a metric on P ( X ). We have the following duality formulafor W . Proposition 2.5 (Kantorovich-Rubinstein duality) . For µ, ν ∈ P ( X ) , W ( µ, ν ) = sup (cid:26)Z X f dµ − Z X f dν ; f is a Lipschitz (cid:27) (2.7) holds. We call a Lipschitz function f that realizes the supremum of (2.7) (if it exists)a Kantorovich potential .3.
Definition of coarse Ricci curvatures on hypergraphs
Lin-Lu-Yau’s coarse Ricci curvature on graphs.
We here review Lin-Lu-Yau’s curvature notion for graphs. Let G = ( V, E ) be a simple graph, that is, V is a set and E ⊂ ( V × V \ { ( x, x ) ; x ∈ V } ). Here we do not distinguish { x, y } and { y, x } ∈ E . For x, y ∈ V , x ∼ y means { x, y } ∈ E . Given x, y ∈ V , a sequenceof points { z i } ni =0 is called a path from x to y if z = x , z n = y , z i ∼ z i +1 for i = 0 , · · · , n −
1, and n is called the length of path. The distance d ( x, y ) of x, y ∈ V as the least number of lengths of paths from x to y . A path { z i } ni =0 is said to begeodesic if it realizes the distance between z and z n . A function ω : V × V → R ≥ is a weight function such that ω ( x, y ) > x ∼ y . The degree of x ∈ V is defined by d x = P y ω ( x, y ).For 0 < α < x ∈ V , the function m αx is defined by m αx ( y ) := α if y = x, − αd x ω ( x, y ) if y ∼ x, . We can see m αx as a probability measure on V . It is trivial that m αx ∈ P ( V ) holds.We define the α -lazy coarse Ricci curvature between x and y by κ α ( x, y ) := 1 − W ( m αx , m αy ) d ( x, y ) . Lin-Lu-Yau [21] defined the coarse Ricci curvature κ LLY by κ LLY ( x, y ) := lim α ↑ κ α ( x, y )1 − α . (3.1)They proved the limit (3.1) always exists in [21]. By the Kantorovich-Rubinstein duality (Proposition 2.5), W ( m αx , m αy ) can berephrased by W ( m αx , m αy ) = sup (cid:26)Z X f dm αx − Z X f dm αy ; f is a 1-Lipschitz (cid:27) We remark that m αx can be written as m αx = ( αI + (1 − α ) AD − ) δ x = ( I − (1 − α ) L ) δ x , where A = ( ω ( x, y )) x,y ∈ R V × V is the adjacency matrix of the graph G . By usingthis presentation, we have Z X f dm αx = X z ∈ V f ( z ) m αx ( z ) = f ⊤ m αx = f ⊤ ( I − (1 − α ) L ) δ x = (( I − (1 − α ) L ) Df ) ⊤ D − δ x = h ( I − (1 − α ) L ) Df, δ x i . Thus, W ( m αx , m αy ) can be written as W ( m αx , m αy ) = sup {h ( I − (1 − α ) L ) Df, δ x − δ y i ; f is a 1-Lipschitz } . (3.2)We set λ = 1 − α . Then, we remark that( I − λ L )( g ) = J λ ( g ) + O ( λ )(3.3)holds for any g ∈ R V . Indeed, in the case of the usual graph Laplacian L , forsufficiently small λ , the resolvent J λ = ( I + λ L ) − has a Neumann series expansionas J λ g = g − λ L g + ∞ X k =2 ( − λ L ) k g = ( I − λ L ) g + O ( λ ) . We define the λ -linear Kantorovich difference KD λ ( x, y ) using the resolvent J λ as KD λ ( x, y ) = sup {h J λ Df, δ x − δ y i ; f is a 1-Lipschitz } . (3.4)Then, by the estimate (3.3), we can show W ( m αx , m αy ) = KD λ ( x, y ) + o ( λ ) . (3.5)We prove this estimate regorously in Section 4. The crucial point to extend thedefinition of curvature notion to hypergraphs is that although W ( m αx , m αy ) can-not be extended to hypergraph due to the multivalued property of our Laplacian, KD λ ( x, y ) can be extended naturally due to the single-valued property of the resol-vent J λ for our Laplacian. In the following section, we shall introduce a λ -nonlinearKantorovich difference which is a natural generalization of (3.4). OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 11
Nonlinear Kantorovich difference.
Let K be a positive number. A func-tion f ∈ R V is said to be weighted K -Lipschitz if D − f is a K -Lipschitz functionwith respect to the distance d , i.e., f satisfies (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) d x − f ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kd ( x, y )for any x, y ∈ V . It is equivalent to h f, δ x − δ y i ≤ Kd ( x, y ) for any x, y ∈ V . Werepresent the set of weighted K -Lipschitz functions on V as Lip Kw ( V ). Definition 3.1.
Let λ >
0. For two vertices x, y ∈ V , the λ -nonlinear Kantorovichdifference KD λ ( x, y ) of x and y is defined by KD λ ( x, y ) := sup (cid:8) h J λ f, δ x i − h J λ f, δ y i ; f ∈ Lip w ( V ) (cid:9) . Remark . Formally, we write Z f dµ λx := h J λ f, δ x i . When the hypergraph H is a usual graph, µ λx becomes a measure. Definition 3.1is inspired by the Kantorovich-Rubinstein duality. As shown later, for each λ > λ -nonlinear Kantorovich difference is a distance function on V .We prove that we can restrict the class of functions in the definition of KD λ toa smaller subset. Proposition 3.3.
For a function f , we set k f k ∞ := max z ∈ V (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) d z (cid:12)(cid:12)(cid:12)(cid:12) . Then, the identity KD λ ( x, y ) = sup (cid:8) h J λ f, δ x − δ y i ; f ∈ Lip w ( V ) , k f k ∞ ≤ diam ( H ) (cid:9) holds. Moreover, the subset of Lip w ( V ) g Lip w ( V ) = { f ∈ Lip w ( V ); k f k ∞ ≤ diam ( H ) } is a bounded and closed subset of the finite dimensional Euclid space ( R V , h· , ·i ) ,hence a compact subset.Proof. Let f be a weighted 1-Lipschitz function. We fix y ∈ V satisfying h f, δ y i =min z ∈ V h f, δ z i . We set α := h f, δ y i and F := f − α · vol ( V ) π . Then, as h F, δ x − δ y i = f ( x ) − αd x d x − f ( y ) − αd y d y = h f, δ x − δ y i , this F is also a weighted 1-Lipschitz function. By h F, δ y i = f ( y ) d y − α = 0 , we have h F, δ x i = h F, δ x i − h F, δ y i ≤ d ( x, y ) ≤ diam ( H ) for any x ∈ V . This implies that k F k ∞ ≤ diam ( H ). Moreover, by Lemma 2.4, theidentity J λ F = J λ f − α · vol ( V ) π holds. Thus, by J λ F ( x ) d x − J λ F ( y ) d y = J λ f ( x ) − αd x d x − J λ f ( y ) − αd y d y = J λ f ( x ) d x − J λ f ( y ) d y , we obtain the desired properties.Since g Lip w ( V ) is a subset of the finite dimensional Euclidean space ( R V , h· , ·i ), itis bounded with respect to k · k . Also since f ( v ) /d v = h f, δ v i , L -convergence, L -weak convergence, and point-wise convergence are all equivalent. Hence g Lip w ( V ) isbounded closed, that is, compact. (cid:3) We prove the finiteness of KD λ ( x, y ). Lemma 3.4.
Let λ > . For any x, y ∈ V , ≤ KD λ ( x, y ) < ∞ .Proof. Non-negativeness is obvious since 0 ∈ R V is weighted 1-Lipschitz. Thefiniteness of KD λ ( x, y ) follows from the continuity of J λ and the compactness of g Lip w ( V ) by Proposition 3.3. However, as we show the estimate (3.7) which we uselater, we shall demonstrate a down-to-earth proof here.To prove the finiteness of KD λ ( x, y ), we first show that for any f ∈ Lip w ( V ), L f ≤ vol ( V ) / holds. Since f is weighted 1-Lipschitz, b ⊤ e ( D − f ) = max u,v ∈ e (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) d u − f ( v ) d v (cid:12)(cid:12)(cid:12)(cid:12) ≤ b e ∈ argmax b ∈ B e b ⊤ ( D − f ). We set f ′ = P e ω e b e ( b ⊤ e ( D − f )) ∈ L f .We note that | b e ( x ) | ≤ x ∈ V since b e ∈ B e . Then | f ′ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X e ∋ x ω e b ⊤ e ( D − f ) b e ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X e ∋ x ω e = d x . (3.6)Consequently, we obtain L f ≤ h f ′ , f ′ i = X x ∈ V f ′ ( x ) d − x ≤ X x ∈ V d x = vol ( V ) . Now, we go back to the proof. Define M := max x ∈ V d − / x . For any weighted1-Lipschitz f , Z f dµ λx − Z f dµ λy = h J λ f − f, δ x i + h f, δ x i − h f, δ y i − h J λ f − f, δ y i≤ k J λ f − f k ( k δ x k + k δ y k ) + d ( x, y ) ≤ λ L f ( d − / x + d − / y ) + d ( x, y ) ≤ λ vol ( V ) / M + d ( x, y ) . Because the last quantity is independent of f , we take the suprimum with respectto f to get (cid:3) (3.7) KD λ ( x, y ) ≤ λ vol ( V ) / M + d ( x, y ) < ∞ . The following proposition shows that KD λ is a distance function on V : OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 13
Proposition 3.5.
Let λ > . The following hold: (1) KD λ ( x, y ) = 0 ⇔ x = y . (2) KD λ ( x, y ) = KD λ ( y, x ) . (3) KD λ ( x, z ) ≤ KD λ ( x, y ) + KD λ ( y, z ) .Proof. (1)( ⇐ ) and (2) follow from the definition. First, we prove (1) ( ⇒ ). Weassume that KD λ ( x, y ) = 0. Then, for any f ∈ Lip w ( V ), h J λ f, δ x − δ y i = 0 . On the other hand, the equality of sets { cf ; f ∈ Lip w ( V ) , c ∈ R } = R V (3.8)holds. Thus, by combining (3.8) and the properties J λ ( cf ) = cJ λ ( f ) for any c ∈ R and J λ ( R V ) = R V as in Lemma 2.2, we have h g, δ x − δ y i = 0for any g ∈ R V . By the non-degeneracy of the inner product, we have δ x = δ y ,hense x = y . Next, we prove the triangle inequality (3). For any ǫ >
0, there existsa weighted 1-Lipschitz function f such that KD λ ( x, z ) ≤ Z f dµ λx − Z f dµ λz + ǫ. Thus, we have KD λ ( x, z ) ≤ Z f dµ λx − Z f dµ λz + ǫ = Z f dµ λx − Z f dµ λy + Z f dµ λy − Z f dµ λz + ǫ ≤ KD λ ( x, y ) + KD λ ( y, z ) + ǫ. Since ǫ > (cid:3)
Remark . Although, the λ -nonlinear Kantorovich difference KD λ is a distancefunction on V by Proposition 3.5, it is still unclear how to change this distancewith respect to λ . We only know the following inequality; | KD λ ( x, y ) − KD µ ( x, y ) | ≤ M vol ( V ) / | λ − µ | . In fact, by the equation (2.3), we have h J λ f − J µ f, δ x − δ y i = (cid:28) J µ (cid:18) µλ f + λ − µλ J λ f (cid:19) − J µ f, δ x − δ y (cid:29) ≤ (cid:13)(cid:13)(cid:13)(cid:13) J µ (cid:18) µλ f + λ − µλ J λ f (cid:19) − J µ f (cid:13)(cid:13)(cid:13)(cid:13) · k δ x − δ y k≤ (cid:13)(cid:13)(cid:13)(cid:13) µλ f + λ − µλ J λ f − f (cid:13)(cid:13)(cid:13)(cid:13) · M ≤ M | λ − µ |kL f k≤ vol ( V ) / M | λ − µ | . For any ǫ >
0, there exists a function f ∈ Lip w ( V ) such that h J λ f, δ x − δ y i + ǫ ≥ KD λ ( x, y ). Thus KD λ ( x, y ) − ǫ ≤ h J λ f, δ x − δ y i = h J λ f − J µ f, δ x − δ y i + h J µ f, δ x − δ y i≤ vol ( V ) / M | λ − µ | + KD µ ( x, y ) . Since ǫ > KD λ ( x, y ) − KD µ ( x, y ) ≤ vol ( V ) / M | λ − µ | .Changing the role of µ and λ , we have the conclusion.Next, we prove existence of a weighted 1-Lipschitz function which attains thesuprimum in the definition of KD λ . Proposition 3.7.
For any x, y ∈ V and λ > , there exists a weighted 1-Lipschitzfunction f such that the following identity holds: h J λ f, δ x i − h J λ f, δ y i = KD λ ( x, y ) . (3.9) Proof.
Let { f n } ⊂ Lip w ( V ) be a maximizing sequence of KD λ ( x, y ). As mentionedin Proposition 3.3, without loss of generality, we may assume k f n k ∞ ≤ diam ( H ) forall n . Now g Lip w ( V ) is a compact subset of the finite dimensional Euclidean space( R V , h· , ·i ) by Proposition 3.3 again, thus a sequentially compact subset. Hence, { f n } ⊂ Lip w ( V ) ⊂ R V has a subsequence { f ′ n } which converges to an element f ′ in Lip w ( V ). Because J λ is continuous by Lemma 2.1, we have J λ ( f ′ n ) → J λ ( f ′ ) as n → ∞ . Therefore, KD λ ( x, y ) = lim n →∞ h J λ ( f ′ n ) , δ x − δ y i = h J λ ( f ′ ) , δ x − δ y i holds. This implies the equation (3.9). (cid:3) We call a function satisfying the equality (3.9) a λ -nonlinear Kantorovich poten-tial .3.3. Coarse Ricci curvature on hypergraphs.
In this subsection, we define the λ -coarse Ricci curvature κ λ ( x, y ) on hypergraph and show some properties of it. Definition 3.8.
Let λ >
0. We fix x, y ∈ V . The λ -coarse Ricci curvature κ λ ( x, y )along with x, y on the hypergraph H is defined by κ λ ( x, y ) := 1 − KD λ ( x, y ) d ( x, y ) . The lower coarse Ricci curvature κ ( x, y ) is defined by κ ( x, y ) := lim inf λ ↓ κ λ ( x, y ) λ . In the similar manner, the upper coarse Ricci curvature κ ( x, y ) is defined by κ ( x, y ) := lim sup λ ↓ κ λ ( x, y ) λ . If κ ( x, y ) = κ ( x, y ) holds, we call this the coarse Ricci curvature for x, y , denotedby κ ( x, y ). OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 15
Remark . By the inequality (3.7), the following holds: κ λ ( x, y ) = 1 − KD λ ( x, y ) d ( x, y ) ≥ − λ vol ( V ) / M + d ( x, y ) d ( x, y ) ≥ − λ vol ( V ) / Md ( x, y ) . This estimate implies that κ ( x, y ) satisfies κ ( x, y ) ≥ − vol ( V ) / M/d ( x, y ).It is nontrivial that κ ( x, y ) is finite. We prove this in the following. Lemma 3.10.
For any x, y ∈ V , κ ( x, y ) < ∞ holds.Proof. We fix x, y ∈ V . Let f be a weighted 1-Lipschitz function satisfying f ( x ) /d x − f ( y ) /d y = d ( x, y ). Then KD λ ( x, y ) ≥ h J λ f, δ x i − h J λ f, δ y i = h J λ f − f, δ x i + h f, δ x i − h f, δ y i − h J λ f − f, δ y i = h J λ f − f, δ x i + d ( x, y ) − h J λ f − f, δ y i hold. Thus since1 λ (cid:18) − KD λ ( x, y ) d ( x, y ) (cid:19) ≤ (cid:10) λ − ( J λ f − f ) , δ y (cid:11) − (cid:10) λ − ( J λ f − f ) , δ x (cid:11) d ( x, y ) , we have κ ( x, y ) ≤ (cid:10) L f, δ x (cid:11) − (cid:10) L f, δ y (cid:11) d ( x, y ) < ∞ (3.10)by the equation (2.6). (cid:3) We predict that for any hypergraph H , the identity κ ( x, y ) = κ ( x, y ) holds forany x, y ∈ V . We leave it as a conjecture. Conjecture 3.11.
For any hypergraph H = ( V, E, w ) , the identity κ ( x, y ) = κ ( x, y ) holds for any x, y ∈ V . We prove that Conjecture 3.11 holds for graphs in Section 4.Next, we show a relation between the inferiors of κ ( x, y ) for any pairs of verticesand for adjacent vertices. Lemma 3.12.
Let a be a real number. Then, the following holds: inf x ∼ y κ ( x, y ) ≥ a = ⇒ inf x,y κ ( x, y ) ≥ a. In particular, inf x ∼ y κ ( x, y ) = inf x,y κ ( x, y ) holds.Proof. Let n = d ( x, y ) and { x i } be a shortest path connecting x and y . Then, thefollowing inequality holds for any λ > κ λ ( x, y ) = 1 − KD λ ( x, y ) d ( x, y ) ≥ − P n − i =0 KD λ ( x i , x i +1 ) n = 1 n n − X i =0 (cid:18) − KD λ ( x i , x i +1 ) d ( x i , x i +1 ) (cid:19) . Here, the inequality follows from Proposition 3.5 (3). Hence, we have κ ( x, y ) ≥ n n − X i =0 κ ( x i , x i +1 ) ≥ a. (cid:3) We show another property of the inferior of the lower coarse Ricci curvatures.
Lemma 3.13.
We set κ λ := inf x,y κ λ ( x, y ) and κ := inf x,y κ ( x, y ) . Then, theidentity lim inf λ → κ λ /λ = κ holds.Proof. Because V is a finite set, so is V × V . Thus, κ λ = min x,y κ λ ( x, y ) and κ = min x,y κ ( x, y ) hold. We set { z λ } := { ( x λ , y λ ) } λ and assume that this satisfies κ λ ( x λ , y λ ) = κ λ . By a simple argument, we can show that there is a pair ( x ∞ , y ∞ ) ∈ V × V such that lim inf λ ↓ κ λ ( x ∞ , y ∞ ) λ = lim inf λ ↓ κ λ λ holds due to the finiteness of V × V . Let κ = κ ( x , y ). Then, because κ λ ( x ∞ , y ∞ ) λ ≤ κ λ ( x , y ) λ , by taking the limit λ →
0, we have κ ≤ κ ( x ∞ , y ∞ ) = lim inf λ ↓ κ λ ( x ∞ , y ∞ ) λ = lim inf λ ↓ κ λ λ ≤ lim inf λ ↓ κ λ ( x , y ) λ = κ ( x , y ) = κ . Here, the first inequality is by the definitions of κ , This concludes the proof. (cid:3) Comparison with Lin-Lu-Yau’s coarse Ricci curvature on graphs
In this section, we show that for graphs, κ ( x, y ) = κ ( x, y ) holds for any x, y ∈ V and that this coarse Ricci curvature κ ( x, y ) = κ ( x, y ) = κ ( x, y ) is same as the oneof Lin-Lu-Yau [21] denoted by κ LLY . In this sense, our coarse Ricci curvature canbe regarded as a generalization of Lin-Lu-Yau’s to hypergraphs.
Proposition 4.1.
Let G = ( V, E, w ) be a weighted undirected graph. Then, κ ( x, y ) = κ ( x, y ) holds for any x, y ∈ V . Moreover, κ ( x, y ) = κ LLY ( x, y ) holds for any x, y ∈ V , where κ LLY ( x, y ) is the coarse Ricci curvature introduced by Lin-Lu-Yau [21] .Proof. The definition of Lin-Lu-Yau’s coarse Ricci curvature κ LLY ( x, y ) is κ LLY ( x, y ) := lim α ↑ κ α ( x, y )1 − α , κ α ( x, y ) = 1 − W ( m αx , m αy ) d ( x, y ) . On the other hand, the definition of our curvature κ ( x, y ) (if it exists) is κ ( x, y ) := lim λ ↓ κ λ ( x, y ) λ , κ λ ( x, y ) = 1 − KD λ ( x, y ) d ( x, y ) . OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 17
We evaluate the difference of them with λ = 1 − α . In what follows, we assumethat λ > λ | κ α ( x, y ) − κ λ ( x, y ) | = 1 λ | W ( m αx , m αy ) − KD λ ( x, y ) | d ( x, y )(4.1)holds, it is sufficient to evaluate | W ( m αx , m αy ) − KD λ ( x, y ) | /λ . There exist somepotentials in the both case of W (cf.[39]) and KD λ by Proposition 3.7. We usethese potentials for the following arguments. Let f α be a Kantorovich potential for( m αx , m αy ). Then, we obtain W ( m αx , m αy ) − KD λ ( x, y ) ≤ (cid:18)Z f α dm αx − Z f α dm αy (cid:19) − ( J λ f α ( x ) − J λ f α ( y ))= { ( I − λ L ) f − J λ f } ( x ) − { ( I − λ L ) f − J λ f } ( y ) . Here, because J λ = ( I + λ L ) − holds and λ is sufficiently small, we can take theNeumann series expansion as J λ f = f − λ L f + ∞ X i =2 ( − λ L ) i f. Hence, we have1 λ ( W ( m αx , m αy ) − KD λ ( x, y )) ≤ ∞ X i =2 ( − λ ) i − L f α ( x ) − ∞ X i =2 ( − λ ) i − L f α ( y ) → λ →
0. Hence, we havelim λ ↓ λ ( W ( m αx , m αy ) − KD λ ( x, y )) ≤ . By exchanging the rolls of KD λ and W , we obtain the similar resultlim λ ↓ λ ( KD λ ( x, y ) − W ( m αx , m αy )) ≤ . Consequently, we have lim λ ↓ (cid:12)(cid:12) W ( m αx , m αy ) − KD λ ( x, y ) (cid:12)(cid:12) λ = 0 . (4.2)In particular, because the limit lim α ↑ κ α ( x, y ) / (1 − α ) exists by [21, P.609], thelimit lim λ ↓ κ λ ( x, y ) /λ also exists. This shows that Conjecture 3.11 holds forgraphs. Moreover, by combining the estimate (4.2) and the equation (4.1), wehave κ LLY ( x, y ) = κ ( x, y ) for any x, y ∈ V . (cid:3) Remark . Any arguments so far are applicable for some other situations. Inparticular, by similar arguments, we can show that the Ricci curvature on directedgraphs defined by Sakurai-Ozawa-Yamada [31] is same as a modification of our Riccicurvature to directed graphs. Indeed, because the Laplacian ∆ in their definition is self-conjugate and nonpositive definite operator ([31, Prop. 2.4]) and the measureappears in their definition can be calculated as Z V f dν ǫx = ( I + ǫ ∆) f ( x )([31, Lemma 3.1]), we can accomplish the similar proof as in Proposition 4.1.5. Applications
Eigenvalues of Laplacian.
We call an element µ ∈ R > an eigenvalue of L if there is a function f ∈ R V satisfying L f = µf . Then, the inferior of coarse Riccicurvatures bounds the eigenvalues. Theorem 5.1. If inf x,y κ ( x, y ) = κ , then κ ≤ µ holds.Proof. We assume that L f = µf holds. By multiplying some constant if we need,we may assume that f is weighted 1-Lipschitz. Moreover, without loss of generality,we may assume that f ( x ) /d x − f ( y ) /d y = d ( x, y ) holds for some x, y ∈ V . Then,we have KD λ ( x, y ) ≥ h J λ f, δ x i − h J λ f, δ y i = h J λ f − f, δ x i + f ( x ) d x − f ( y ) d y − h J λ f − f, δ y i = h J λ f − f, δ x i + d ( x, y ) − h J λ f − f, δ y i . Therefore, we have κ ≤ κ ( x, y ) = lim sup λ ↓ κ λ ( x, y ) λ ≤ lim sup λ ↓ λ − (cid:18) − h J λ f − f, δ x i + d ( x, y ) − h J λ f − f, δ y i d ( x, y ) (cid:19) ≤ lim sup λ ↓ d ( x, y ) (cid:0) h λ − ( J λ f − f ) , δ y i − h λ − ( J λ f − f ) , δ x i (cid:1) = 1 d ( x, y ) (cid:0) hL f, δ x i − hL f, δ y i (cid:1) = µd ( x, y ) (cid:18) f ( x ) d x − f ( y ) d y (cid:19) = µ. (cid:3) Gradient estimate.
By using our coarse Ricci curvature, we can show agradient estimate.
Theorem 5.2.
Let κ = inf x,y κ ( x, y ) . Then, for any x, y ∈ V and t > , thefollowing inequality holds: h t f ( x ) d x − h t f ( y ) d y ≤ e − κ t d ( x, y ) . OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 19
Proof.
By multiplying some constant if we need, we may assume that f is weighted1-Lipschitz. We set κ λ := inf x,y κ λ ( x, y ). By the definition of KD λ and κ λ , we have J λ f ( x ) d x − J λ f ( y ) d y = h J λ f, δ x i − h J λ f, δ y i ≤ KD λ ( x, y ) ≤ (1 − κ λ ) d ( x, y ) . Hence, J λ f / (1 − κ λ ) is also weighted 1-Lipschitz. By a similar calculation, wehave J λ f ( x ) d x − J λ f ( y ) d y = h J λ f, δ x i − h J λ f, δ y i = (1 − κ λ ) (cid:18)(cid:28) J λ (cid:18) J λ f − κ λ (cid:19) , δ x (cid:29) − (cid:28) J λ (cid:18) J λ f − κ λ (cid:19) , δ y (cid:29)(cid:19) ≤ (1 − κ λ ) d ( x, y ) . Iterating similar calculations implies that h J nλ f, δ x i−h J nλ f, δ y i ≤ (1 − κ λ ) n d ( x, y )holds for any non-negative integer n . Therefore, by the equation (2.4) we have h t f ( x ) d x − h t f ( y ) d y = lim λ ↓ D J [ t/λ ] λ f, δ x E − D J [ t/λ ] λ f, δ y E ≤ lim inf λ ↓ (1 − κ λ ) [ t/λ ] d ( x, y ) ≤ lim inf λ ↓ e − κλλ [ t/λ ] λ d ( x, y )= e − κ t d ( x, y ) . Here, the second inequality follows from the well-known inequality e xt ≥ (1 + x ) t for given | x | < t > (cid:3) Diameter bound.
We also have a geometric consequence under the upperRicci curvature bound.
Theorem 5.3 (Bonnet-Myers type diameter bound) . We assume inf x = y κ ( x, y ) ≥ κ > holds. Then, the following holds: diam ( H ) ≤ κ . Proof.
Let p, q ∈ V be vertices which satisfy d ( p, q ) = diam ( H ). Then, for f ∈ Lip w ( V ) with h f, δ p − δ q i = d ( p, q ), by the inequality (3.10), we have κ ≤ κ ( p, q ) ≤ (cid:10) L f, δ p − δ q (cid:11) d ( p, q ) ≤ d ( p, q )since such f satisfies |L f ( x ) | ≤ d x for any x ∈ V as the estimate (3.6). (cid:3) Examples
In this section, we calculate some examples. A key formula for calculation is J λ f = argmin (cid:26) k f − g k λ + Q ( D − g ) ; g ∈ R V (cid:27) . Example 6.1.
We consider the hypergraph H = ( V, E, w ), where V := { x, y, z } , E := { xy, yz, zx, xyz } , and ω ( e ) = 1 for any e ∈ E . We here calculate the coarseRicci curvature for this hypergraph H .More precisely, we calculate KD λ ( x, y ) for x, y ∈ V and a sufficiently small λ > f be a weighted 1-Lipschitz function. We set the value f ( x ) = 3 α, f ( y ) = 3 β ,and f ( z ) = 3 γ . By Proposition 3.3, we may assume β = 0. We divide our argumentinto some cases as(1) α > γ > γ > α > γ = 0,(4) γ = α ,(5) α > > γ .Moreover, we divide the cases for the values of J λ f . We remark that 0 ≤ α, γ ≤ f is weighted 1-Lipschitz. We set g = J λ f . Since J λ f → f as λ → g ( x ) = 3 α + 3 a , g ( y ) = 3 b , g ( z ) = 3 γ + 3 c . We define F ( g ) := 12 λ k f − g k + Q ( D − g ) . (1) α > γ >
0. Because λ > J λ f is closed to f . Hence, wemay assume α + a > γ + c > b . Then, the Laplacian L is determined uniquely and b ⊤ xy ( D − g ) = α + a − b, b ⊤ xz ( D − g ) = α + a − γ − c, b ⊤ yz ( D − g ) = γ + c − b, b ⊤ xyz ( D − g ) = α + a − b holds. Hence, we have F ( g ) = 12 λ (cid:0) a + 3 b + 3 c (cid:1) + 12 (cid:0) α + a − b ) + ( γ + c − b ) + ( α + a − γ − c ) (cid:1) . We set r := λ − . Since J λ f is a critical point for F , F a = F b = F c = 0. Hence wehave r ) − − − r ) − − − r abc = − α + γ α + γα − γr . This can be solved for λ >
0, and we have that ( a, b, c ) ⊤ is equal to19 r (1 + r )(3 r + 5) · r )(2 + 3 r ) − r ) + 1 2 + 3(1 + r )2(2 + 3 r ) + 1 3(1 + r )(2 + 3 r ) − r ) + 22 + 3(1 + r ) 3(1 + r ) + 2 9(1 + r ) − − α + γ α + γα − γr . OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 21
Because the inner product h J λ f, δ x − δ y i which we claim can be represented as α + a − b , we have h J λ f, δ x − δ y i = h g, δ x − δ y i = α + a − b = α + 19 r (1 + r )(3 r + 5) { ( − α + γ ) (3(1 + r )(2 + 3 r ) − − (2(2 + 3 r ) + 1))+(2 α + γ ) (2(2 + 3 r ) + 1 − (3(1 + r )(2 + 3 r ) − α − γ ) (2 + 3(1 + r ) − (2 + 3(1 + r ))) } = α − α r (1 + r )(3 r + 5) (cid:0) r + 9 r (cid:1) = 3 αr r + 5 ≤ r r + 5 . Here, the last inequality is obtained by evaluating for α = 1.(2) γ > α >
0. By a similar argument as above, we may assume γ + c > α + a > b .Hence, we have b ⊤ xy ( D − g ) = α + a − b, b ⊤ xz ( D − g ) = γ + c − α − a, b ⊤ yz ( D − g ) = γ + c − b, b ⊤ xyz ( D − g ) = γ + c − b. Thus we have F ( g ) = 12 λ (3 a + 3 b + 3 c ) + 12 (cid:0) ( α + a − b ) + 2( γ + c − b ) + ( γ + c − α − a ) (cid:1) . From F a = F b = F c = 0, we obtain r + 2 − − − r ) − − − r ) abc = − α + γα + 2 γα − γ . This simultaneous equation can also be solved as follows: ( a, b, c ) ⊤ is equal to19 r (1 + r )(3 r + 5) · r ) − r ) + 2 3(1 + r ) + 23(1 + r ) + 2 3(1 + r )(2 + 3 r ) − r ) + 13(1 + r ) + 2 2(2 + 3 r ) + 1 3(1 + r )(2 + 3 r ) − − α + γα + 2 γα − γ . Then, we have h J λ f, δ x − δ y i = α + a − b = α − r )(3 r + 5) { α (3 r + 5) + γr } = r ( α (3 r + 5) − γ )(1 + r )(3 r + 5) . (3) γ = 0. In this case, b = c by the symmetry of H and f . Then we have b ⊤ xy ( D − g ) = α + a − b, b ⊤ xz ( D − g ) = α + a − b, b ⊤ yz ( D − g ) = 0 , b ⊤ xyz ( D − g ) = α + a − b. Thus we have F ( g ) = 12 λ (cid:0) a + 6 b (cid:1) + 32 ( α + a − b ) .F a = F b = 0 leads r − − r ! ab ! = − αα ! . Therefore, we have ab ! = 1 r (2 r + 3) r
11 1 + r ! − αα ! = α r + 3 − ! . Hence, we have h J λ f, δ x − δ y i = α + a − b = 2 αr r + 3 ≤ r r + 3 . The last inequality follows from evaluating the value for α ≤ γ = α . In this case, we also have a = c similarly as the above. Then, we have b ⊤ xy ( D − g ) = α + a − b, b ⊤ xz ( D − g ) = 0 , b ⊤ yz ( D − g ) = α + a − b, b ⊤ xyz ( D − g ) = α + a − b, thus F ( g ) = 12 λ (6 a + 3 b ) + 32 ( α + a − b ) hold. The equations F a = F b = 0 can be written as r − − r ! ab ! = − αα ! , hence we have ab ! = α r + 3 − ! . Consequently, we have h J λ f, δ x − δ y i = α + a − b = 2 αr r + 3 ≤ r r + 3 . The last inequality follows from α ≤ α > > γ . Note that since f ∈ Lip w ( V ), α − γ ≤ α <
1. In this case, wehave b ⊤ xy ( D − g ) = α + a − b, b ⊤ xz ( D − g ) = α + a − γ − c, b ⊤ yz ( D − g ) = b − γ − c, b ⊤ xyz ( D − g ) = α + a − γ − c. OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 23
Thus, we have F ( g ) = 12 λ (cid:0) a + 3 b + 3 c (cid:1) + 12 (cid:0) ( α + a − b ) + 2( α + a − γ − c ) + ( b − γ − c ) (cid:1) . In the same manner as before, we obtain r ) − − − r + 2 − − − r ) abc = − α + 2 γα + γ α − γ . Hence, ( a b c ) ⊤ is equal to19 r (1 + r )(3 r + 5) r )(3 r + 2) − r ) + 2 2(2 + 3 r ) + 13(1 + r ) + 2 9(1 + r ) − r ) + 22(2 + 3 r ) + 1 3(1 + r ) + 2 3(1 + r )(2 + 3 r ) − . Finally we have h g, δ x − δ y i = α + a − b = r (1 + r )(3 r + 5) { α (3 r + 4) + γ } . By comparing the above cases, we can show that the values h J λ f, δ x − δ y i for thecases (1) is less than or equal to 2 r/ (2 r + 3) which is attained for the case (3) and(4). Thus, it is sufficient to compare the cases (2), (3), and (5). We can calculatethe differences as(2) − (3) = 2 r r + 3 − r ( α (3 r + 5) − γ )(1 + r )(3 r + 5) = 2 r r + 3 − αr r + γr (1 + r )(3 r + 5) ≥ r (2(1 − α ) r + 2 − α )(2 r + 3)(1 + r ) , and(2) − (5) = 2 r r + 3 − (cid:18) r (1 + r )(3 r + 5) { α (3 r + 4) + γ } (cid:19) ≥ r (2 r + 3) − αr r ≥ . Then, the most right hand sides are non-negative, because r = λ − is sufficientlylarge and 1 ≥ γ > α in the case (3) and α < γ < KD λ ( x, y ) = 2 λ − λ − + 3 . Consequently, the coarse Ricci curvature κ ( x, y ) exists for x, y ∈ V and becomes κ ( x, y ) = lim λ → λ (cid:18) − KD λ ( x, y ) d ( x, y ) (cid:19) = 32 . Remark . We predict that if we consider the hypergraph H = ( V, E, w ) suchthat V = n , E = 2 V \ {∅ , V } and w ( e ) = 1 for any e ∈ E , the λ -nonlinearKantorovich potential f satisfies that f ( x ) = d x and f ( z ) = 0 ( z = x ). Remark . We predict that κ ( x, y ) = κ ( x, y ) = κ ( x, y )= C ( x, y ) = inf ( (cid:10) L f, δ x − δ y (cid:11) d ( x, y ) ; f : w.1-Lip , h f, δ x − δ y i = d ( x, y ) ) holds. A similar formula holds for graph cases [28]. If this is true, it could besomewhat easier to calculate our curvatures. By assuming this equation once here,we try to show some predicted curvatures in some cases. Example . Let H = ( V, E, w ) be the hypergraph such that V = { x, y, z } , E = { e = { x, y, z }} , and ω e = 1. We consider f : V → R such that f ( x ) = 1, f ( y ) = 0, f ( z ) = 0. Then, we have L f ( x ) = 1, L f ( y ) = − / L f ( z ) = − /
2, and κ ( x, y ) = C ( x, y ) ≤ (cid:10) L f, δ x − δ y (cid:11) = 1 − ( − /
2) = 3 / . Therefore, we predict that the curvature of this hypergraph H is 3 / Example . We consider the hypergraph H = ( V, E, w )as V = { v , v , . . . , v n } , E = 2 V \ {{ v } , . . . , { v n } , ∅} , and ω e = 1. Then, we have E = 2 n − n − d x = d = 2 n − − ( n − − n − for any x ∈ V . Wecount the number of hyperedges e ∈ E including v and v . The number of such e satisfying e = k is (cid:0) n − k − (cid:1) .Let f : V → R be the function satisfying f ( v ) = d , and f ( v i ) = 0 ( i = 2 , . . . , n ).Then, we have L f ( v ) = d . Moreover, for e including v and v such that e = k ,we may choose δ v − ( k − − P i ≥ ,v i ∈ e δ v i v , v ∈ e as b e . Thus, we have L f ( v ) = − n X k =2 k − (cid:18) n − k − (cid:19) = − n X k =2 ( n − k − n − k )!= − n X k =2 n − n − n − k )!( k − − n − n − X ℓ =1 ( n − n − − ℓ )! ℓ != − n − (cid:8) (1 + 1) n − − (cid:9) = − n − − n − . Therefore, we have the following inequality: C ( v , v ) ≤ d (cid:0) L f ( v ) − L f ( v ) (cid:1) = 1 d (cid:18) d + 2 n − − n − (cid:19) = 1 + dd ( n − nn − . By this observation, we predict that the curvature of the complete hypergraph H with n vertices is n/ ( n − Appendix A. More general settings
Our arguments so far are applicable to more general settings for submodulartransformations. Here, submodular transformation is a vector valued set functionconsisting of submodular functions. In this section, we review about submodular
OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 25 functions, submodular transformations, and these Laplacian and show some exam-ples. We also give a sufficient condition for a submodular transformation to be ableto straightforwardly generalize the curvature notions in Section 3 and theorems inSection 5. For more details about submodular transformations, see [41].A.1.
Submodular function.
Let V be a nonempty finite set. A function F : 2 V → R is a submodular function if for any S, T ⊂ V , F satisfies F ( S ) + F ( T ) ≥ F ( S ∪ T ) + F ( S ∩ T ) . An element v ∈ V is relevant in F : 2 V → R if there is a S ⊂ V such that F ( S ) = F ( S ∪ { v } ). We say that v is irrelevant in F if v is not relevant in F . We definethe support supp( F ) of F as the set of elements which are relevant in F . A setfunction F : 2 V → R is symmetric if F ( S ) = F ( V \ S ) holds for any S . We say F isnormalized if F ( V ) = 0. Example A.1.
Let H = ( V, E ) be a hypergraph, and e ∈ E a hyperedge. Then,the cut function F e of e defined as follows is a submodular function: F e ( S ) = e ∩ S = ∅ and e ∩ ( V \ S ) = ∅ , . It is easy to show that a vertex v ∈ V is relevant in F e if and only if v ∈ e .Furthermore, F e is symmetric and normalized.For a submodular function F : 2 V → R , we define P ( F ) = ( g ∈ R V ; X x ∈ S g ( x ) ≤ F ( S ) for any S ⊂ V ) and B ( F ) = ( g ∈ P ( F ); X x ∈ V g ( x ) = F ( V ) ) called the submodular polyhedron and the base polytope respectively. Then, it isknown that B ( F ) is a bounded polytope.The Lov´asz extension f : R V → R of a submodular function F : 2 V → R isdefined by f ( g ) = max b ∈ B ( F ) b ⊤ g. It is known that f ( χ S ) = F ( S ) for any S ⊂ V . Here, χ S is the characteristicfunction of S . In particular, f is indeed an extension of F . It is also known thatthe Lov´asz extension f of a submodular function F is convex.For the Lov´asz extension f of a submodular function F , we set ∂f ( g ) = argmax b ∈ B ( F ) b ⊤ g. Then, it is known that ∂f ( g ) is the sub-differential of f at g . A.2.
Submodular transformation and submodular Laplacian.
Let
V, E benonempty finite sets. A function F : 2 V → R E ; S F ( S ) = ( F e ( S )) e ∈ E is asubmodular transformation if each F e is a submodular function. A submodulartransformation F is symmetric (resp. normalized) if any F e is symmetric (resp.normalized).The Lov´asz extension f : R V → R E of a submodular transformation F is definedby f = ( f e ) such that f e is the Lov´asz extension of F e .For a submodular transformation F : R V → R E , we consider a weight function w : E → R > . Then, we call the quadruple ( V, E, F, w ) a weighted submodulartransformation. We stand for the quadruple as F . We define the degree d x for x ∈ V by d x := P e ∈ E ; x ∈ supp( F e ) w ( e ) and the volume vol ( S ) of S ⊂ V by vol ( S ) := P x ∈ S d x . For x, y ∈ V , x and y is adjacent, denoted by x ∼ y , if there existsan element e ∈ E such that x, y ∈ supp( F e ). By this relation, we can define thedistance function d : V × V → R ≥ and connectivity of F as in Section 2.1.We define the degree matrix D := diag( d x ) x ∈ V ∈ R V × V . We remark that if F is connected, D is invertible.Let F = ( V, E, F, w ) be a submodular transformation. Then, we define thesubmodular Laplacian L : R V → R V by L ( g ) := (X e ∈ E w ( e ) b e b ⊤ e g ; b e ∈ ∂f e ( g ) ) ⊂ R V . We call L := L ◦ D − the normalized Laplacian. We set the inner product h f, g i := f ⊤ D − g and consider ( R V , h· , ·i ) as a Hilbert space. Then, by a similar argumentas in [18, Lemma 14, Lemma 15], the following holds: Proposition A.2.
The normalized Laplacian L is a maximal monotone operatoron the Hilbert space ( R V , h , i ) . More strongly, the normalized Laplacian L is the sub-differential of the convexfunction Q : R V → R defined by Q ( g ) = 12 X e ∈ E w ( e ) f e ( g ) , where g = D − g .By Proposition A.2, we can define the resolvent J λ , the canonical restriction L ,and the heat semigroup h t for the Laplacian L . Then, the straight extension ofLemma 2.2 holds.We define π ∈ R V as π ( x ) = d x / vol ( V ). Then, the following holds: Lemma A.3 ([41, Lemma 3.1]) . We assume that F is normalized, i.e., F e ( V ) = 0 for any e ∈ E . Then, L ( π ) = 0 holds. By Lemma A.3, the similar lemmas as Lemma 2.3 and Lemma 2.4 hold for thenormalized submodular Laplacian L . This implies that by similar arguments, we OARSE RICCI CURVATURE OF HYPERGRAPHS AND ITS GENERALIZATION 27 can obtain the straightforward extensions of definitions and theorems in Section 3and Section 5 for any normalized submodular transformation F with the normalizedsubmodular Laplacian L for F .A.3. Examples.
In [41], Yoshida gave many examples of submodular transforma-tions such as undirected graph (Example 1.1, 1.2, and 1.4), directed graph (Exam-ple 1.5), hypergraph (Example 1.6), submodular hypergraph (Example 1.7), mutualinformation (Example 1.8), and directed information (Example 1.9). We here showanother example:
Example A.4 (directed hypergraph) . A weighted directed hypergraph H is definedas the triple H = ( V, E, w ) of a set of vertices V , a set of hyperarcs E ⊂ V × V ,and a weight function w : E → R > . Here, a hyperarc e ∈ E is an ordered pair( t e , h e ) of a set of tails t e and a set of heads h e . If | t e | = | h e | = 1 holds for any e ∈ E , then H is a usual directed graph. If t e = h e holds for any e ∈ E , then H can be regarded as an undirected hypergraph . Hence, directed hypergraph is ageneralization of directed graph and hypergraph.We define the set function F e : 2 V → R as the cut function for e = ( t e , h e ), i.e., F e ( S ) := S ∩ t e = ∅ and ( V \ S ) ∩ h e = ∅ F e is submodular. Hence, the quadruple F =( V, E, F = ( F e ) e , w ) becomes a submodular transformation. We remark that F is normalized and not symmetric.For this submodular transformation, by a simple calculation from definition of B ( F e ), we have B ( F e ) = Conv( { δ x − δ y ; x ∈ t e , y ∈ h e } ∪ { } ) . (A.1)The base polytope for hypergraph (2.2) is a realization of this for t e = h e . By therepresentation (A.1), the Lov´asz extension f e of F e is written as f e ( g ) = max { max { g ( x ) − g ( y ); x ∈ t e , y ∈ h e } , } . Because any examples introduced in this subsection, F is normalized, i.e., F ( V ) =0. Hence, the similar definitions of coarse Ricci curvatures for F as in Section 3and the similar theorems as in Section 5 hold. Comparing properties of curvaturesfor these examples with them of other curvatures introduced by [31], [14] is aninteresting problem. We leave it for a future work. At first glance, this specialization seems to be strange. However, from the point of view ofsubmodular transformation, this looks natural. Indeed, under the assumption t e = h e , the cutfunction is same as this for undirected hypergraphs References [1] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala,
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RIKEN AIP
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