aa r X i v : . [ m a t h . A T ] J a n HYPERNETWORKS: FROM POSETS TO GEOMETRY
EMIL SAUCAN
Abstract.
We show that hypernetworks can be regarded as posetswhich, in their turn, have a natural interpretation as simplicialcomplexes and, as such, are endowed with an intrinsic notion ofcurvature, namely the Forman Ricci curvature, that strongly cor-relates with the Euler characteristic of the simplicial complex. Thisapproach, inspired by the work of E. Bloch, allows us to canoni-cally associate a simplicial complex structure to a hypernetwork,directed or undirected. In particular, this greatly simplifying thegeometric Persistent Homology method we previously proposed. Introduction
This paper is dedicated to the proposition that hypernetworks canbe naturally construed as posets that, in turn, have a innate interpreta-tion as simplicial complexes and, as such, they are endowed with intrin-sic interconnected topological and geometric properties, more preciselywith a notion of curvature that strongly correlates – not just in thestatistical manner – to the topological structure and, more specifically,to the Euler characteristic of the associated simplicial complex. Thisobservation, that stems from E. Bloch’s work [1], allows us not only toassociate to hypernetworks a structure of a simplicial complex, but todo this is a canonical manner, that permits us to compute its essentialtopological structure, following from its intrinsic hierarchical organiza-tion, and to attach to it a geometric measure that is strongly relatedto the topological one, namely the
Forman Ricci curvature . This ap-proach allows to preserve the essential structure of the hypernetwork,while concentrating at larger scale structures (i.e. hypervertices andhyperedges), rather than at the local. perhaps accidental information
Date : January 15, 2021. attached to each particular vertex or edge. This allows us, in turn, toextract the structural information mentioned above. We should alsomention here that we also proposed a somewhat different approach tothe parametrization of hypernetworks as simplicial and more generalpolyhedral complexes in [13]. While the previous method allows us topreserve more of the details inherent in the original model of the hyper-network, it is also harder to implement in an computer-ready manner,thus emphasizing the advantage of the canonical, easily derivable struc-ture we propose herein. In particular, greatly simplifies the geometricPersistent Homology method proposed in [10] (see also [5], [8]). Weshould further underline that both these approaches have an additionaladvantage over the established way of representing hypernetworks asgraphs/networks, namely the fact that they allow for a simple methodfor the structure-preserving embedding of hypernetworks in Euclidean N -space, with clear advantages for their representation and analysis.2. Theoretical Background
Hypernetworks.
We begin by reminding the reader the defini-tion of of the type of structure we study.
Definition 2.1 ( Hypernetworks) . We define a hypernetwork as a hyper-graph H = ( V , E ) with the hypernode set V consisting of set of nodes, i.e. V = ( V , . . . , V p ), V i = { v i , . . . , v ik i } ; and hyperedges E ij = V i V j ∈ E – hyperedge set of H , the connecting groups of nodes/hypervertices.Note that it is natural to view each hypervertex as a complete graph (or clique ) K n , which in turn is identifiable with the (1-skeleton of) the standard n -simplex . Remark . Note that in [13] we have employed a somewhat moregeneral, but also less common, definition of hypernetwork, where hy-pervertices where not viewed as complete graphs, thus allowing forthe treatment of hypernetworks as general polyhedral complexes, notmerely simplicial ones, as herein (see below).
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Posets.
We briefly summarize here the minimal definitions andproperties of partially ordered sets, or posets that we need in the sequel.In doing this, we presume the reader is familiar with the basic definitionof posets.
Definition 2.3 ( Coverings) . Let ( P , < ) be a poset, where < denotesthe partial order relation on P , and let p, q be elements of P . We saythat p covers q if p > q and there does not exists r ∈ P , such that q < r < p . We denote the fact that p covers q by p ≻ q .While a variety of examples of posets pervade mathematics, the basic(and perhaps motivating) example is that of the set of subsets (a.k.a. asthe power set) P ( X ) of a given set X , with the role of the order relationbeing played by the inclusion. Given the common interpretation ofnetworks, the identification with hypernetworks with a subset of P ( X )is immediate. Definition 2.4 ( Ranked posets) . Given a poset ( P , < ), a rank function for P is a function ρ : P → N such that(1) If q is a minimal element of P , then ρ ( q ) = 0;(2) If q ≺ p , then ρ ( p ) = ρ ( q ) + 1. A poset P is called ranked if there admits a rank function for P . The maximal value of ρ ( p ) , p ∈ P is called the rank of P , and it is denoted by r ( P ).Note that in the definition of ranked posets we essentially (but notstrictly) follow [1] and, while other terminologies exist [6],[15], we preferthe one above for the sake of clarity and concordance with Bloch’spaper.Let us note that if a poset is ranked, than the rank function is unique.Furthermore, if P is a ranked poset of rank r , and if j ∈ { , . . . , r } , wedenote P j = { p ∈ P | ρ ( p ) = j } , and by F i the cardinality of P j , i.e. F i = |P j | .Again, as for the case of posets in general, ( P ( X ) , ⊂ ) represents thearchetypal example of ranked posets, thus hypernetworks represent, inessence, ranked posets, which is essential for the sequel. EMIL SAUCAN
Remark . Many of the hypernetworks arising as models in real-lifeproblems are actually oriented ones (see, for instance [11], [13]). Forthese the poset structure is even more evident, as the order relation isemphasized by the directionality. If, moreover, there are no loops, theresulting poset is also ranked.2.3.
Simplicial Complexes and the Euler Characteristic.
As forthe case of posets, we do not bring the full technical definition of asimplicial complex, bur we rather refer the reader to such classics as[3] or [7].Given a poset P , there exists a canonical way of producing anassociated ordered simplicial complex ∆( P ), by considering a vertexfor each element p ∈ P and an m -simplex for each chain p ≺ p ≺ . . . ≺ p m of elements of P .Since in the present paper we considered only finite hypernetworks/sets,we can define the Euler characteristic of the poset P as being equal tothat of the associated simplicial complex ∆( P ), i.e.(2.1) χ ( P ) = χ (∆( P )) . Note that this definition allows us to define the Euler characteristic ofany poset, even if it is not ranked, due to the fact that the associatedsimplicial complex is naturally ranked by the dimension of the simplices(faces).However, if P is itself ranked – as indeed it is in our setting – thanthere exists a direct, purely combinatorial way of defining the Eulercharacteristic of P that emulates the classical one, in the followingmanner:(2.2) χ g ( P ) = r X j =0 ( − j F j . While in general χ ( P ) and χ g ( P ) do not coincide, they are iden-tical in the case of CW complexes, thus in particular for polyhedralcomplexes, hence a fortiori for simplicial complexes. In particular, weshall obtain the same Euler characteristic irrespective to the model ofhypernetwork that we chose to operate with: The poset model P , its OSETS 5 associated complex ∆( P ), the geometric view of posets a simplicialcomplexes that attaches to each subset of cardinality k , i.e. to eachhypervertex a k -simplex, or the more general polyhedral model thatwe considered in [13]. It follows, therefore, that The Euler characteristic of a hypernetwork is a well defined invari-ant, independent of the chosen hypernetwork model, and as such cap-tures the essential topological structure of the network.
In the sequel we shall concentrate, for reason that we’ll explain indue time, on the subcomplex of ∆( P ) consisting of faces of dimension ≤ ∆ ( P ) of ∆( P ). In particular,we shall show that χ (∆( P )) is not just a topological invariant, it isalso closely related, in this dimension, to the geometry of ∆( P ).2.4. Forman Curvature.
R. Forman introduced in [2] a discretiza-tion of the notion of Ricci curvature, by adapting to the quite generalsetting of CW complexes the by now classical Bochner-Weizenb¨ockformula (see, e.g. [4]). We expatiated on the geometric content of thenotion of Ricci curvature and of Forman’s discretization in particularelsewhere [9], therefore, in oder not to repeat ourselves too much, werefer the reader to the above mentioned paper. However, let us notethat in [9] we referred to Forman’s original notion as the augmented
Forman curvature, to the reduced, 1-dimensional notion that we intro-duced and employed in the study of networks in [14].While Forman’s Ricci curvature applies for both vertex and edgeweighted complexes (a fact that plays an important role in its extendedand flexible applicability range), we concentrate here on the combina-torial case, namely that of all weights (vertex as well as edge weights)equal to 1. In this case, Forman’s curvature, whose expression in thegeneral case [2] is quite complicated, even when restricting ourselvesto 2-dimensional simplicial complexes [9], has the following simple andappealing form:(2.3) Ric F ( e ) = { t > e } − { ˆ e : ˆ e k e } + 2 . EMIL SAUCAN
Here t denotes triangles and e edges, while “ || ′′ denotes parallelism ,where two faces of the same dimension (e.g. edges) are said to beparallel if they share a common “parent” (higher dimensional face con-taining them, e.g. a triangle,), or a common “child” (lower dimensionalface, e.g a vertex). Remark . For “shallow” hypernetworks, like the chemical reactionsones considered in [11], both Forman Ricci curvature and especiallythe Euler characteristic are readily computable but also, due to thereduced depth of such hypernetworks, rather trivial.2.5.
The Gauss-Bonnet formula.
In the smooth setting, there ex-ists a strong and connection between curvature and the Euler char-acteristic, hat is captured by the classical Gauss-Bonnet formula (see,e.g. [4]). While the Forman Ricci curvature, as initially defined in [2]does not, unfortunately, satisfy a Gauss-Bonnet type theorem, sinceno counterparts in dimensions 0 and 2, essential in the formulationof the Gauss-Bonnet Theorem, are defined therein. However, Blochdefined these necessary curvature terms and was thus able to formu-late in [1] an analogue of the Gauss-Bonnet Theorem, in the setting ofranked posets. While in general the 1 dimensional curvature term hasno close classical counterpart, in the particular case of cell complexes,and thus of simplicial complexes in particular, Euler characteristic andForman curvature are intertwined in the following discrete version ofthe Gauss-Bonnet Theorem:(2.4) X v ∈ F R ( v ) − X e ∈ F Ric F ( e ) + X f ∈ F R ( t ) = χ ( X ) . Here R and R denote the 0-, respective 2-dimensional curvature termsrequired in a Gauss-Bonnet type formula. These curvature functions are defined via a number of auxiliary functions , as follows:(2.5) R ( v ) = 1 + 32 A ( v ) − A ( v ) , R ( t ) = 1 + 6 B ( t ) − B ( t ) ; OSETS 7 where A , B are the aforementioned auxiliary functions, which aredefined in the following simple and combinatorially intuitive manner:(2.6) A ( x ) = { y ∈ F , x < y } , B ( x ) = { z ∈ F , z < x } . Since we only consider only triangular 2-faces, the formulas for thecurvature functions reduce to the very simple and intuitive ones below:(2.7) R ( v ) = 1 + 32 deg( v ) − deg ( v ) , R ( t ) = 1 + 6 · = 24 ;where deg( v ) denotes, conform to the canonical notation, the degree ofthe (hyper-)vertex v , i.e. the number of its adjacent vertices.From these formulas and from the general expression of the Gauss-Bonnet formula (2.4) we obtain the following combinatorial formula-tion of the noted formula in the setting of the 2-dimensional simplicialcomplexes:(2.8) χ ( X ) = X v ∈ F (cid:18) v ) − deg ( v ) (cid:19) − X e ∈ F Ric F ( e ) + 24 ;or, after taking into account also Formula (2.8), and some additionalmanipulations:(2.9) χ ( X ) = X v ∈ F (cid:18) v ) − deg ( v ) (cid:19) − X e ∈ F · { t > e } + X v Directions of Future Study The first direction of research that naturally imposes itself as nec-essary and that we want to explore in the immediate future is that ofunderstanding the higher dimensional structure of hypernetworks, thatis by taking into account chains in the corresponding of length greater EMIL SAUCAN than two by studying the structure of the fitting resulting simplicialcomplexes.As we have seen, the (generalized) Euler characteristic is defined foran n -dimensional (simplicial) complex. Therefore it is possible to em-ploy its simple defining Formula (2.2) to obtain essential topologicalinformation about hypernetworks of any depth and, indeed, by suc-cessively restricting to lower dimensional simplices (i.e. chains in thecorresponding poset), explore the topological structure of a hypernet-work in any dimension.Moreover, it is possible to explore not just the topological proper-ties of a hypernetwork, but its geometric ones as well. The simplestmanner to obtain geometric information regarding the hypernetworkis by computing again the Forman Ricci curvature Ric F of its edges.Indeed, Forman Ricci curvature is an edge measure, thus determinedsolely by the intersections of the two faces of the simplicial complex,thus being, in effect, “blind” to the higher dimensional faces of thecomplex. However, it is possible to compute curvature measures forsimplices in all dimensions, as Forman defined such curvatures in alldimensions [2]. While the expressions of higher dimensional curvaturefunctions are more complicated (and we refer the reader to Forman’soriginal work), and their geometric content is less clear than Ricci cur-vature, they still allow us to geometrically filter of hypernetworks inall dimensions, in addition to the topological understanding suppliedby the Euler characteristic. Here again, the simpler, more geometricapproach introduced in [10] combined with the ideas introduced in thepresent paper should prove useful in adding geometric content to theunderstanding of networks in all dimensions.Moreover, while the Euler characteristic, in both its forms, representsa topological/combinatorial invariant and, as such it operates only withcombinatorial weights (that is to say, equal to 1), Forman’s curvatureis applicable to any weighted CW complex, hence to weighted hyper-networks as well. This clearly allows for a further refinement of the OSETS 9 geometric filtering process mentioned above, that we wish to explorein a further study.It follows, therefore, that the combination of the couple of tools aboveendows us with simple, yet efficient means to explore, understand andeventually classify hypernetworks.Another direction of study that deserves future study is that of di-rected hypernetworks, as such structures not only admit, as we haveseen, a straightforward interpretation as posets, but also arise in manyconcrete modeling instances. While in the present paper we have re-stricted ourselves to undirected networks, we propose to further inves-tigate the directed ones as well.Meanwhile, we should bring to the readers attention the fact that wehave previously extended in [12] Forman’s Ricci curvature to directedsimplicial complexes/hypernetworks, and we indicated how this can beextrapolated to higher dimensions in [13].Furthermore, we should emphasize that, it is easy to extend thecombinatorial Euler characteristic by taking into account only the con-sidered directed simplices. (See [12], [13] for detailed discussions on thepertinent choices for the directed faces.) In addition, for 2-dimensionalsimplicial complexes, as those arising from hypernetworks on which weconcentrated in this study, a directed Euler characteristic can be de-veloped directly from Formulas (2.9) and (2.3). One possible form ofthe result formula is(3.1) χ I/O ( X ) = X v ∈ F (cid:18) I / O ( v ) − deg I / O2 ( v ) (cid:19) − X e ∈ F · { t > e } − X v Combinatorial Ricci Curvature for Polyhedral Surfaces and Posets ,preprint, arXiv:1406.4598v1 [math.CO], 2014. [2] F. Forman, Bochner’s Method for Cell Complexes and Combinatorial RicciCurvature , Discrete Comput. Geom. (3), 323–374, 2014.[3] J. F. Hudson, Piecewise Linear Topology , Benjamin, New York, 1969.[4] J. Jost, Riemannian Geometry and Geometric Analysis , Springer, 2011.[5] H. Kannan, E. Saucan, I. Roy and A. Samal, Persistent homology of unweightednetworks via discrete Morse theory , Scientific Reports (2019) :13817.[6] K. H. Rosen, J. G. Michaels, J. L. Gross, J. W. Grossman and Douglas R.Shier (eds.), Handbook of discrete and combinatorial mathematics , CRC Press,Boca Raton, FL, 2000.[7] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology ,Springer-Verlag, Berlin, 1972.[8] I. Roy, S. Vijayaraghavan, S. J. Ramaia and A. Samal, Forman-Ricci curvatureand Persistent homology of unweighted complex networks , Chaos, Solitons &Fractals, , 110260.[9] A. Samal, R. P. Sreejith, J. Gu, S. Liu, E. Saucan and J. Jost, Compara-tive analysis of two discretizations of Ricci curvature for complex networks ,Scientific Report (1): 8650.[10] E. Saucan, Discrete Morse Theory, Persistent Homology and Forman-RicciCurvature , preprint, arxiv:submit/3078713.[11] E. Saucan, A. Samal, M. Weber and J. Jost, Discrete curvatures and networkanalysis , Commun. Math. Comput. Chem., (3), 605-622.[12] E. Saucan, R. P. Sreejith, R. P. Vivek-Ananth, J. Jost and A. Samal, DiscreteRicci curvatures for directed networks , Chaos, Solitons & Fractals, , 347-360, 2019.[13] E. Saucan, M. Weber, Forman’s Ricci curvature - From networks to hypernet-works , Proceedings of COMPLEX NETWORKS 2018, Studies in Computa-tional Intelligence (SCI), Springer, 706-717, 2019.[14] R. P. Sreejith, K. Mohanraj, J¨urgen Jost, E. Saucan and A. Samal, Forman curvature for complex networks , Journal of Statistical Me-chanics: Theory and Experiment (J. Stat. Mech.), (2016) 063206,(http://iopscience.iop.org/1742-5468/2016/6/063206).[15] R. P. Stanley, Enumerative combinatorics, Vol. 1 , Cambridge Studies in Ad-vanced Mathematics, vol. 49, Cambridge University Press Cambridge, 1997.(Corrected reprint of the 1986 original.)[16] M. Weber, E. Saucan and J. Jost, Coarse geometry of evolving networks , JComplex Netw, (5), 706-732, 2018. OSETS 11 Department of Applied Mathematics, ORT Braude College of En-gineering, Karmiel, Israel Email address ::