Continuous k-to-1 functions between complete graphs of even order

A function between graphs is k-to-1 if each point in the co-domain has precisely k pre-images in the domain. Given two graphs, G and H, and an integer k ≥ 1, and considering G and H as subsets of R 3, there may or may not be a k-to-1 continuous function (i.e. a k-to-1 map in the usual topological sense) from G onto H. In this paper we review and complete the determination of whether there are finitely discontinuous, or just infinitely discontinuous k-to-1 functions between two intervals, each of which is one of the following: ]0, 1[, [0, 1[and [0, 1]. We also show that for k even and 1 ≤ r < 2s, (r, s) 6= (1, 1) and (r, s) 6= (3, 2), there is a k-to-1 map from K2r onto K2s if and only if k ≥ 2s.peer-reviewe

Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference

2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions

We compute transition amplitudes between two spin networks with dipole graphs, using the Lorentzian EPRL model with up to two (non-simplicial) vertices. We find power-law decreasing amplitudes in the large spin limit, decreasing faster as the complexity of the foam increases. There are no oscillations nor asymptotic Regge actions at the order considered, nonetheless the amplitudes still induce non-trivial correlations. Spin correlations between the two dipoles appear only when one internal face is present in the foam. We compute them within a mini-superspace description, finding positive correlations, decreasing in value with the Immirzi parameter. The paper also provides an explicit guide to computing Lorentzian amplitudes using the factorisation property of SL(2,C) Clebsch-Gordan coefficients in terms of SU(2) ones. We discuss some of the difficulties of non-simplicial foams, and provide a specific criterion to partially limit the proliferation of diagrams. We systematically compare the results with the simplified EPRLs model, much faster to evaluate, to learn evidence on when it provides reliable approximations of the full amplitudes. Finally, we comment on implications of our results for the physics of non-simplicial spin foams and their resummation.Comment: 27 pages + appendix, many figures. v2: one more numerical result, plus minor amendment