Two-dimensional Simplicial Complexes Research Articles

Identifying partial topology of simplicial complexes

This paper investigates partial topology identification of simplicial complexes based on adaptive synchronization. For the nodes of interest, the interactions that they participate in can be accurately reconstructed by designing adaptive controllers and parameter estimators. Particularly, not only pairwise interactions but a higher-order structure can be effectively recovered by our method. Moreover, a new linear independence condition with a rigorous definition is established for parameter estimators to converge asymptotically to the true values. Numerical simulations on a general two-dimensional simplicial complex as well as a real-world structure are provided to show the validity of the result and discuss the influence of different parameters on the identification process.

A Poset-Based Approach to Curvature of Hypergraphs

In this contribution, we represent hypergraphs as partially ordered sets or posets, and provide a geometric framework based on posets to compute the Forman–Ricci curvature of vertices as well as hyperedges in hypergraphs. Specifically, we first provide a canonical method to construct a two-dimensional simplicial complex associated with a hypergraph, such that the vertices of the simplicial complex represent the vertices and hyperedges of the original hypergraph. We then define the Forman–Ricci curvature of the vertices and the hyperedges as the scalar curvature of the associated vertices in the simplicial complex. Remarkably, Forman–Ricci curvature has a simple combinatorial expression and it can effectively capture the variation in symmetry or asymmetry over a hypergraph. Finally, we perform an empirical study involving computation and analysis of the Forman–Ricci curvature of hyperedges in several real-world hypergraphs. We find that Forman–Ricci curvature shows a moderate to high absolute correlation with standard hypergraph measures such as eigenvector centrality and cardinality. Our results suggest that the notion of Forman–Ricci curvature extended to hypergraphs in this work can be used to gain novel insights on the organization of higher-order interactions in real-world hypernetworks.

Spanning surfaces in 3-graphs

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface \mathscr{S} , we show that any two-dimensional simplicial complex on n vertices in which each pair of vertices belongs to at least \frac n3 + o(n) facets contains a homeomorph of \mathscr{S} spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on n vertices with minimum codegree exceeding \frac n3+o(n) contains a spanning triangulation of the sphere.

A universal exponent for homeomorphs

We prove a uniform bound on the topological Turán number of an arbitrary two-dimensional simplicial complex S: any two-dimensional complex on n vertices with at least CSn3−1/5 facets contains a homeomorph of S, where CS > 0 is a constant depending on S alone. This result, a two-dimensional analogue of a classical one-dimensonal result of Mader, sheds some light on an old problem of Linial from 2006.

Comparative analysis of two discretizations of Ricci curvature for complex networks

We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.

Golodness and polyhedral products for two-dimensional simplicial complexes

Abstract Golodness of two-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topological characterizations of Golodness of surface triangulations are given. An answer to the question of Berglund is also given so that there is a two-dimensional simplicial complex which is rationally Golod but not Golod over ℤ / p {\mathbb{Z}/p} .

Optimal Reconstruction Might be Hard

Given as input a point set $$\mathcal S $$ that samples a shape $$\mathcal A $$, the condition required for inferring Betti numbers of $$\mathcal A $$ from $$\mathcal S $$ in polynomial time is much weaker than the conditions required by any known polynomial time algorithm for producing a topologically correct approximation of $$\mathcal A $$ from $$\mathcal S $$. Under the former condition which we call the weak precondition, we investigate the question whether a polynomial time algorithm for reconstruction exists. As a first step, we provide an algorithm which outputs an approximation of the shape with the correct Betti numbers under a slightly stronger condition than the weak precondition. Unfortunately, even though our algorithm terminates, its time complexity is unbounded. We then identify at the heart of our algorithm a test which requires answering the following question: given 2 two-dimensional simplicial complexes $$L \subset K$$, does there exist a simplicial complex containing $$L$$ and contained in $$K$$ which realizes the persistent homology of $$L$$ into $$K$$? We call this problem the homological simplification of the pair $$(K,L)$$ and prove that this problem is NP-complete, using a reduction from 3SAT.

Vertex decompositions of two-dimensional complexes and graphs

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.

Topology of Random 2-Complexes

We study the Linial–Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for p≪n −1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π 1(Y) is free and H 2(Y)=0, asymptotically almost surely. Our other main result gives a precise threshold for collapsibility of a random 2-complex to a graph in a prescribed number of steps. We also prove that, if the probability parameter p satisfies p≫n −1/2+ϵ , where ϵ>0, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as n→∞. We also establish several related results; for example, we show that for p<c/n with c<3 the fundamental group of a random 2-complex contains a non-abelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embeddings and immersions of 2-complexes into a random 2-complex.

On the boundary of a special class of hyperbolic two-dimensional simplicial complexes

A class of hyperbolic two-dimensional complexes X is defined. It is shown that the limit set of the action of π1(X) on the universal covering \({\widetilde{X}}\) of X is equal to the visual boundary \({\partial \widetilde{X}}\) of and that \({\partial \widetilde{X}}\) is path connected and locally path connected. A kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra.

Conserved topological defects in non-embedded graphs in quantum gravity

We follow up on previous work which found that commonly used graph evolution moves lead to conserved quantities that can be expressed in terms of the braiding of the graph in its embedding space. We study non-embedded graphs under three distinct sets of dynamical rules and find non-trivial conserved quantities that can be expressed in terms of topological defects in the dual geometry. For graphs dual to two-dimensional simplicial complexes we identify all the conserved quantities of the evolution. We also indicate expected results for graphs dual to three-dimensional simplicial complexes.

Decompositions of two-dimensional simplicial complexes

We show that the class of Cohen–Macaulay complexes, that of complexes with constructible subdivisions, and that of complexes with shellable subdivisions differ from each other in every dimension d ⩾ 2 . Further, we give a characterization of two-dimensional simplicial complexes with shellable subdivisions, and show also that they are constructible.

Incremental construction properties in dimension two—shellability, extendable shellability and vertex decomposability

We give new examples of shellable, but not extendably shellable two-dimensional simplicial complexes. They include minimal examples that are smaller than those previously known. We also give new examples of shellable, but not vertex decomposable two-dimensional simplicial complexes, including extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability implies the other. We found these examples by enumerating shellable two-dimensional simplicial complexes that are not pseudomanifolds.

Example of a fermion-nondoubling amorphous lattice

An example of an amorphous lattice featuring no fermion doubling is constructed. The lattice represents a bounded two-dimensional simplicial complex with a disk topology, every internal 0-simplex of which belongs to boundaries of an odd number of 1-simplices. The total number of simplices in the complex tends to infinity.

Obstructions to Shellability

We consider a simplicial complex generalization of a result of Billera and Myers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable two-dimensional simplicial complex contains a nonshellable induced subcomplex with less than eight vertices. We also establish CL-shellability of interval orders and as a consequence obtain a formula for the Betti numbers of any interval order.

A Family of Ideals of Minimal Regularity and the Hilbert Series ofCr(Δ)

For a simplicial subdivision Δ of a region inR2, we analyze the dimension of the vector spaceCkr(Δ) ofCrpiecewise polynomial functions (splines) on Δ of degree at mostk. We find an exact sequence which allows us to prove that the dimension series for splines given by5does indeed agree with the bounds on the dimension of the spline space given by Alfeld and Schumaker [1;2]. We give sufficient conditions for the Alfeld–Schumaker bounds to be attained in all degrees, where Δ is a two-dimensional simplicial complex. The conditions are satisfied by the class of complexes considered by6, but also by a much broader class of complexes. Furthermore, for conditions which involve only local geometric data, this result is the strongest possible.