arXiv research

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclas…

Complex equivalence classes found in graph homotopy.

Geometric duality connects graph isomorphism and knot equivalence.

New equivalence relation on ribbon graphs connects to virtual links.

Virtual knots with same writhe polynomial have equivalent intersection graphs.

The paper shows that relaxing assumptions about causal graphs can lead to exponentially large equivalence classes.

Spatial graphs study tangle replacement with equivalence classes.

System uses neural networks to prove program equivalence via rewrite rules.

Study graph products of groups, classifying them up to measure equivalence and rigidity.

Groups of homotopy equivalences of graphs help realize compact subgroups.

New graph types help identify complex relationships.

The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to …

Characterizes Bayesian networks up to unconditional equivalence.

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…

The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.

We show that the Gromov boundary of the free factor graph for the free group Fn with n>2 generators is the space of equivalence classes of minimal very small indecomposable projective Fn-trees without point stabilizer containing a free factor equipped with a quotient topology. Here two such trees are equivalent if the …

Embeddings of mapping tori for end-periodic graph maps are proven.

Proves Khovanov homology has no torsion for bipartite circle graphs.

Every infinitely edge-connected graph has a minor of Farey graph or $T_{\aleph_0}\ast t$.

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not onl…

Defines concordance for spatial graphs and proves sliceness equivalence.

We define an equivalence relation on graphs with signed edges, such that the associated adjacency matrices of two equivalent graphs are congruent over $\mathbb{Z}$. We show that signed graphs whose eigenvalues are larger than $-2$ are equivalent to one of the simply laced Dynkin diagrams: $A_{n}$, $D_{n}$, $E_{6}$, $E_…

The paper defines and proves the existence of train track maps on graphs of groups.

Right-angled Artin groups are classified based on measure equivalence.

Study restricts causal graphs with expert knowledge.

Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.

A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to th…

We study real nonsingular projective cubic fourfolds up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in the middle homology. Moreover, we provide a graph whose vertices represent the equivale…

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeis…

We prove that the criterion for Markov equivalence provided by Zhao et al. (2005) may involve a set of features of a graph that is exponential in the number of vertices.

A new algorithm for robust causal discovery in small sample sizes.

We prove that for some knot-like objects one can easily recognize non-equivalence w.r.t. all Reidemeister moves by studying some equivalence classes modulo only 2nd Reidemeister moves. There are applications to virtual knots, graph-links and looped graphs.

We define $A_k$-moves for embeddings of a finite graph into the 3-sphere for each natural number $k$. Let $A_k$-equivalence denote an equivalence relation generated by $A_k$-moves and ambient isotopy. $A_k$-equivalence implies $A_{k-1}$-equivalence. Let ${\cal F}$ be an $A_{k-1}$-equivalence class of the embeddings of …

New method shows pseudo-Anosov flows on graph manifolds can be simplified.

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We also give an explicit description of the family of entire maximal graphs with a f…

A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call …

This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs".

Polynomial algorithm found for alternating link equivalence.

Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly i…

This paper explains spectral clustering and its equivalence to PCA, breaking it into fully connected and multi-connected cases.

The paper defines conditions for learning causal graphs from data with unobserved variables.

Graphs indistinguishable by GNNs are fully characterized.

New assumptions help identify causal relationships in data.

Constructs graph manifolds with many Anosov flows.

Study proves hyperfiniteness of mapping class group actions on surface graphs.

As previously known, all 3-manifolds of genus two can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an ``elementary transformation'' on these 6-tuples which changes the associated graph but does not change the represented manifo…

New method estimates causal effects without knowing graph structure.

The paper extends foam theory to more complex trivalent graphs.