arXiv research

Simplified Khovanov polynomials for bipartite links.

A new method calculates HOMFLY-PT polynomials for bipartite links.

Improved bipartite link prediction using 2-hop paths.

We define integral odd Khovanov homology of principally unimodular bipartite graph-links.

Positive Thompson links are arborescent tangles.

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links…

This paper improves upper bounds on ribbonlength for certain alternating links.

The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a par…

New method extends knot theory to non-bipartite knots, revealing PDs.

Planar decomposition simplifies HOMFLY polynomial calculation for certain knots and links.

Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.

The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a famous link invariant with many known properties. For example, the HOMFLY polynom…

This article proposes a method to quantify the structure of a bipartite graph using a network entropy per link. The network entropy of a bipartite graph with random links is calculated both numerically and theoretically. As an application of the proposed method to analyze collective behavior, the affairs in which parti…

We study the Seifert surfaces of a link by relating the embeddings of graphs by using induced graphs. As applications, we prove that every link $L$ is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2…

Simplified Khovanov-Rozansky calculus for bipartite knots.

A new model detects common patterns in pollination networks.

Using a model of wealth distribution where traders are characterized by quenched random saving propensities and trade among themselves by bipartite transactions, we mimic the enhanced rates of trading of the rich by introducing the preferential selection rule using a pair of continuously tunable parameters. The biparti…

New method for calculating HOMFLY polynomials in symmetric representations.

We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complement…

We prove that every embedding of $K_{2n+1,2n+1}$ into $\R^3$ contains a non-split link of $n$-components. Further, given an embedding of $K_{2n+1,2n+1}$ in $\R^3$, every edge of $K_{2n+1,2n+1}$ is contained in a non-split $n$-component link in $K_{2n+1,2n+1}$.

We find the minimal number of links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete $k$-partite graphs on 8 vertices. We also look at larger complete bip…

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polyno…

The graph braid group of a complete bipartite graph is the fundamental group of a configuration space of points on the graph, which is a CAT(0) cube complex. We combine an analysis of the topology of links of vertices in this complex, the description of a hidden symmetry among the parameters, and known results from the…

Bipartite networks are currently regarded as providing a major insight into the organization of many real-world systems, unveiling the mechanisms driving the interactions occurring between distinct groups of nodes. One of the most important issues encountered when modeling bipartite networks is devising a way to obtain…

The data in many disciplines such as social networks, web analysis, etc. is link-based, and the link structure can be exploited for many different data mining tasks. In this paper, we consider the problem of temporal link prediction: Given link data for times 1 through T, can we predict the links at time T+1? If our da…

We propose a Bayesian methodology for one-mode projecting a bipartite network that is being observed across a series of discrete time steps. The resulting one mode network captures the uncertainty over the presence/absence of each link and provides a probability distribution over its possible weight values. Additionall…

Estimates treatment effects in bipartite systems with partial eligibility and interference.

We consider matrix completion for recommender systems from the point of view of link prediction on graphs. Interaction data such as movie ratings can be represented by a bipartite user-item graph with labeled edges denoting observed ratings. Building on recent progress in deep learning on graph-structured data, we prop…

Circle graph complexes reveal link properties via Khovanov homology.

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientat…

Typical graph embeddings may not capture type-specific bipartite graph features that arise in such areas as recommender systems, data visualization, and drug discovery. Machine learning methods utilized in these applications would be better served with specialized embedding techniques. We propose two embeddings for bip…

New link polynomials linked to cluster theory.

Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a ch…

Study explores properties of bipartite knots.

Simplified Khovanov-Rozansky operators for link invariants.

Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts, existing approaches to bipartite community detection have drawbacks, including im…

We propose a link prediction algorithm that is based on spring-electrical models. The idea to study these models came from the fact that spring-electrical models have been successfully used for networks visualization. A good network visualization usually implies that nodes similar in terms of network topology, e.g., co…

We introduce an algorithm able to reconstruct the relevant network structure on which the time evolution of country-product bipartite networks takes place. The significant links are obtained by selecting the largest values of the projected matrix. We first perform a number of tests of this filtering procedure on synthe…

New model for detecting communities in weighted bipartite networks.

New research finds six bipartite intrinsically knotted graphs with 23 edges.

In this work we present a complete (no misses, no duplicates) census for closed, connected, orientable and prime 3-manifolds induced by plane graphs with a bipartition of its edge set (blinks) up to $k=9$ edges. Blinks form a universal encoding for such manifolds. In fact, each such a manifold is a subtle class of blin…

Proves Khovanov homology has no torsion for bipartite circle graphs.

New model for detecting communities in weighted bipartite networks.

Bipartite graphs have been used to represent data relationships in many data-mining applications such as in E-commerce recommendation systems. Since learning in graph space is more complicated than in Euclidian space, recent studies have extensively utilized neural nets to effectively and efficiently embed a graph's no…

Moduli space linked to Tait colorings of planar graphs.

PAC learning simplified as bipartite matching.

We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant C_n such that a bipa…

Develops a new variational estimator for node popularity in bipartite networks.