Related papers: Construction of simplicial complexes with prescrib…
Complex structures are typical in machine learning. Tailoring learning algorithms for every structure requires an effort that may be saved by defining a generic learning procedure adaptive to any complex structure. In this paper, we propose…
We solve the graph bi-partitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut-size to scale universally with <k^1/2>. In contrast, earlier results studying the problem in graphs with…
The degree distribution, referred to as the delta-sequence of a network is studied. Using the non-normalized Lorenz curve, we apply a generalized form of the classical majorization partial order. Next, we introduce a new class of small…
When uncertainty meets costly information gathering, a fundamental question emerges: which data points should we probe to unlock near-optimal solutions? Sparsification of stochastic packing problems addresses this trade-off. The existing…
In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \sqcup T_{2}$, such that each $T_{i}$ represents the lines of a copy of the Fano plane $PG(2, \mathbb{F}_{2})$. We generalize this observation by…
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…
We study a number of graph exploration problems in the following natural scenario: an algorithm starts exploring an undirected graph from some seed node; the algorithm, for an arbitrary node $v$ that it is aware of, can ask an oracle to…
We extend the previously observed scaling equation connecting the internode distances and nodes' degrees onto the case of weighted networks. We show that the scaling takes a similar form in the empirical data obtained from networks…
We show that if a system of degree-$k$ polynomial constraints on~$n$ Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most~$s$ many monomials, then it also has one whose degree is of the order of the square…
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
The distinguishing result of this paper is a $\mathbf{P}$-time enumerable partition of all the potential perfect matchings in a bipartite graph. This partition is a set of equivalence classes induced by the missing edges in the potential…
Graphs are ubiquitous to model the irregular (non-Euclidean) structure of complex data, but they are limited to pairwise relationships and fail to model the complexities of the datasets exhibiting higher-order interactions. In that context,…
This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Compared with existing finite-sample results for linear systems and related analyses under…
Topological Signal Processing (TSP) utilizes simplicial complexes to model structures with higher order than vertices and edges. In this paper, we study the transferability of TSP via a generalized higher-order version of graphon, known as…
Many real networks have cliques as their constitutional units. Here we present a family of scale-free network model consist of cliques, which is established by a simple recursive algorithm. We investigate the networks both analytically and…
Structure learning via MCMC sampling is known to be very challenging because of the enormous search space and the existence of Markov equivalent DAGs. Theoretical results on the mixing behavior are lacking. In this work, we prove the rapid…
The purpose of this article is to examine and limit the conditions in which the P complexity class could be equivalent to the NP complexity class. Proof is provided by demonstrating that as the number of clauses in a NP-complete problem…
Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes…
Recent increase in the availability of warped images projected onto a manifold (e.g., omnidirectional spherical images), coupled with the success of higher-order assignment methods, has sparked an interest in the search for improved…
In this paper, we develop the notion of a Morse sequence, which provides an alternative approach to discrete Morse theory, and which is both simple and effective. A Morse sequence on a finite simplicial complex is a sequence composed solely…