Related papers: Cluster Magnification, Root Capacity, Unique Chain…

We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to…

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials…

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a…

We study percolation on nonamenable groups at the uniqueness threshold $p_u$, the critical value that separates the phase in which there are infinitely many infinite clusters from the phase in which there is a unique infinite cluster. The…

Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The…

In this article we study a class of central extensions of $\mathbb{Z}\wr\mathbb{Z}$, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In…

Many systems such as critical infrastructure exhibit a modular structure with many links within the modules and few links between them. One approach to increase the robustness of these systems is to reinforce a fraction of the nodes in each…

Phylogenetic trees (i.e. evolutionary trees, additive trees or X-trees) play a key role in the processes of modeling and representing species evolution. Genome evolution of a given group of species is usually modeled by a species…

We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality. The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second…

The paper establishes a strong correspondence between two important clustering approaches that emerged in the 1970's: clustering by level sets or cluster tree as proposed by Hartigan and clustering by gradient lines or gradient flow as…

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes…

We present a structural clustering algorithm for large-scale datasets of small labeled graphs, utilizing a frequent subgraph sampling strategy. A set of representatives provides an intuitive description of each cluster, supports the…

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a…

Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a…

I review recent progress in the study of the large-scale structure of the Universe through the distribution of clusters of galaxies, concentrating on new results using X-ray selected samples. After discussing the importance of understanding…

The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…

We study constrained clustering, where constraints guide the clustering process. In existing works, two categories of constraints have been widely explored, namely pairwise and cardinality constraints. Pairwise constraints enforce the…

Link prediction problem has increasingly become prominent in many domains such as social network analyses, bioinformatics experiments, transportation networks, criminal investigations and so forth. A variety of techniques has been developed…

One of the central models in distributed computing is Linial's LOCAL model [SIAM J. Comp. 1992]. Over time, researchers have studied distributed graph problems in the LOCAL model under slightly different assumptions, such as whether nodes…