Curvature Expectation

Riemannian geometry is related to general relativity, probability theory is related to quantum mechanics. Length enters in GR in the form of geodesics, paths of mass points and curvature, the deformation of space through mass. Probability enters in quantum mechanics by seeing solutions of wave or Schrödinger equations in terms of sums of paths (which by the way in the discrete is nothing else than the multi-dimensional Taylor theorem as Feynman-Kac writes exp(i t D) just as a sum of paths. While we don not know how to combine the physical theories, there is a strong connection mathematically and it is extremely powerful in the form of integral geometry. For me this became important in 2012 when struggling with proving that for odd dimensional discrete manifolds the curvature is constant zero. It was still open when writing this paper about discrete Gauss-Bonnet-Chern. It becomes obviou,s once one sees curvature K(v) as the expectation E[i(v)] of Poincare-Hopf indices , where is the Euler characteristic of A. The key is first to prove the Poincare-Hopf result in the discrete, then see curvature as an index expectation. It looks especially elegant with the index formula: in the case of an odd-dimensional q-manifold, the symmetric index is minus the Euler characteristic of the level set in . By the discrete Sard theorem, the level set is a level surface in the (q-1)-sphere S(v) and so a (q-2) manifold which has zero Euler characteristic, so that the symmetric index is constant 0! This of course implies then that the curvature, the expectation is also zero. In 2021, I used index expectation that in general, for any two graphs G,H, the curvature of the Shannon product of G and H is the product of the curvatures of G and H. See this paper. The index formula is especially nice for even dimensional manifolds. The curvature is then 1-E[X]/2, where X is the Euler characteristic of a random co-dimension 1 manifold in S(v). For example, for 4 manifolds, the Gauss-Bonnet-Chern curvature is directly related with the expectation of random 2-dimensional surfaces obtained by taking random level surfaces in the unit sphere. This result motivates in general to look at the functionals on manifolds that gives the Euler characteristic of a codimension k manifolds. This is a statistical problem for discrete manifolds (math 136 notes). The most general result about discrete submanifolds in manifolds is in the “manifolds from partitions” paper. A simple case is that every map g from the vertices of a q-manifold to a finite set produces q-k manifold (if it is not empty).

An other link which was mentioned in the movie was that one can encode Riemannian geometry, in particular lengths in a Riemannian manifold as expectation of some random variable. It follows from Nash’s embedding theorem that there is a natural probability space of functions such that the number of roots of f on a curve is a random variable which has as expectation the length of the curve. There are many nice features about the probabilistic approaches to geometry:

• Probability spaces are deformable. This is in the spirit of modern large language models where “weights’ are nothing else than encoding a probability space.
• Since the discrete and the continuum are covered with the same frame work, one can link the two directly. This works especially well if one looks at integer valued random variables like the number of roots of a function on a curve in the case of length or at the Poincare-Hopf divisors in the case of curvature.
• The frame work is robust. We don’t care for example whether the manifold is a polytop or a smooth manifold. The frame work goes even over to more general sets. In the case of discrete networks, we do not have to care for example whether the networks is related to some manifolds. It just works for all networks.

This presentation originally wanted to feature some experiments I did when looking at as a random variable on the network. Is there a geometric significance of the standard deviation? How big is the correlation between and if v,w are two points. It can not be zero in general because of Poincare-Hopf . For example, if is the 2- point graph vertex set with one connection, then $i_f(v)=0$ if and only if $i_f(w)=1$. The correlation is -1. Here is some code I wrote. It takes as a graph two Pentakis Dodechaedron glued along a vertex. The vertices are colored according to the standard deviation of the index function, the edges are colored according to the correlation between the indices at the two points. The picture of the following code is shown to the right.

UnitSphere[s_,a_]:=Module[{b=NeighborhoodGraph[s,a]},If[Length[VertexList[b]]<2,Graph[{}],VertexDelete[b,a]]];
Generate[A_]:=If[A=={},{},Sort[Delete[Union[Sort[Flatten[Map[Subsets,A],1]]],1]]];
Whitney[s_]:=Generate[FindClique[s,Infinity,All]];   w[x_]:=-(-1)^Length[x]; 
Euler[G_]:=Module[{H},If[GraphQ[G],H=Whitney[G],H=G];Total[Map[w,H]]];
index[s_,f_,v_]:= 1-Euler[Subgraph[s,Select[VertexList[UnitSphere[s,v]],(# /.f)<(v /. f) &]]];
index[s_,f_]:=Module[{v=VertexList[s]},Table[index[s,f,v[[k]]],{k,Length[v]}]];
s=PolyhedronData["PentakisDodecahedron", "Skeleton"];v=VertexList[s];e=EdgeList[s];n=Max[v]-1;
s=UndirectedGraph[Graph[Union[e,Table[e[[k,1]]+n->e[[k,2]]+n,{k,Length[e]}]]]];
v=VertexList[s];e=EdgeList[s];f:=Table[v[[k]]->Random[],{k,Length[v]}];M=1000;i=Table[index[s,f],{M}];
cor=Table[X=Table[i[[m,e[[j,1]]]],{m,M}];Y=Table[i[[m,e[[j,2]]]],{m,M}];mx=Total[X]/M;my=Total[Y]/M;
  cov=(X-mx).(Y-my)/M; varx=(X-mx).(X-mx)/M;vary=(Y-my).(Y-my)/M; sx=Sqrt[varx]; sy=Sqrt[vary];
  e[[j]]->N[cov/(sx*sy)],{j,Length[e]}];
sig=Table[X=Table[i[[m,v[[j]]]],{m,M}]; mx=Total[X]/M; v[[j]]->N[Sqrt[(X-mx).(X-mx)/M]],{j,Length[v]}];
S=GraphPlot3D[s,EdgeRenderingFunction  ->({Hue[#2 /.cor],  Tube[#1,0.05]} &),
                             VertexRenderingFunction->({Hue[#2 /.sig],Sphere[#1,0.15]} &)]