Related papers: A Characterization of Level-k Realizability for Cl…
We provide a constructive proof of the Seymour Second Neighborhood Conjecture (SSNC) by reframing the problem as a set-packing optimization problem. The universal family of oriented graphs $\mathcal{O}$ is classified by their minimum…
Jacob Fox, C. Seshadhri, Tim Roughgarden, Fan Wei, and Nicole Wein introduced the model of $c$-closed graphs--a distribution-free model motivated by triadic closure, one of the most pervasive structural signatures of social networks. While…
This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation. We leverage the fact that, for…
An efficient MCMC algorithm is presented to cluster the nodes of a network such that nodes with similar role in the network are clustered together. This is known as block-modelling or block-clustering. The model is the stochastic blockmodel…
The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new…
Hypergraphs provide a powerful framework for modeling complex systems and networks with higher-order interactions beyond simple pairwise relationships. However, graph-based clustering approaches, which focus primarily on pairwise relations,…
Hierarchical clustering is an effective, interpretable method for analyzing structure in data. It reveals insights at multiple scales without requiring a predefined number of clusters and captures nested patterns and subtle relationships,…
Clustering is typically measured by the ratio of triangles to all triples, open or closed. Generating clustered networks, and how clustering affects dynamics on networks, is reasonably well understood for certain classes of networks…
For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau}…
A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Families $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of sets are cross-$t$-intersecting if, for every $i$ and $j$ in $\{1, 2, \dots, k\}$ with $i…
We study the widely used hierarchical agglomerative clustering (HAC) algorithm on edge-weighted graphs. We define an algorithmic framework for hierarchical agglomerative graph clustering that provides the first efficient $\tilde{O}(m)$ time…
Neighborhood graphs are a critical but often fragile step in spectral clustering of text embeddings. On realistic text datasets, standard $k$-NN graphs can contain many disconnected components at practical sparsity levels (small $k$),…
This paper considers metric spaces where distances between a pair of nodes are represented by distance intervals. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a…
In this paper, a hub refers to a non-terminal vertex of degree at least three. We study the minimum number of hubs needed in a network to guarantee certain flow demand constraints imposed between multiple pairs of sources and sinks. We…
In this paper we consider the clustering coefficient and clustering function in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are…
In this paper, we introduce a neural network framework for semi-supervised clustering (SSC) with pairwise (must-link or cannot-link) constraints. In contrast to existing approaches, we decompose SSC into two simpler classification…
Consider a host hypergraph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that if the edges of $G$ are sampled at the appropriate rate then the spanning structure still appears…
We consider the $k$-Clustering problem, which is for a given multiset of $n$ vectors $X\subset \mathbb{Z}^d$ and a nonnegative number $D$, to decide whether $X$ can be partitioned into $k$ clusters $C_1, \dots, C_k$ such that the cost…
A standard approach to reduce the complexity of very large networks is to group together sets of nodes into clusters according to some criterion which reflects certain structural properties of the network. Beyond the well-known modularity…
An uncertain graph $\mathcal{G} = (V, E, p : E \rightarrow (0,1])$ can be viewed as a probability space whose outcomes (referred to as \emph{possible worlds}) are subgraphs of $\mathcal{G}$ where any edge $e\in E$ occurs with probability…