Related papers: A geometry-induced topological phase transition in…
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…
We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of…
Statistical significance of network clustering has been an unresolved problem since it was observed that community detection algorithms produce false positives even in random graphs. After a phase transition between undetectable and…
We introduce a model for the randomization of complex networks with geometric structure. The geometric randomization (GR) model assumes a homogeneous distribution of the nodes in an underlying similarity space and uses rewirings of the…
Clustering is one of the fundamental tasks in data analytics and machine learning. In many situations, different clusterings of the same data set become relevant. For example, different algorithms for the same clustering task may return…
Previous statistical approaches to hierarchical clustering for social network analysis all construct an "ultrametric" hierarchy. While the assumption of ultrametricity has been discussed and studied in the phylogenetics literature, it has…
Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every $\varepsilon>0,$ there exist (complicate)…
Consider a network consisting of two subnetworks (communities) connected by some external edges. Given the network topology, the community detection problem can be cast as a graph partitioning problem that aims to identify the external…
The Random Geometric Graph (RGG) is a random graph model for network data with an underlying spatial representation. Geometry endows RGGs with a rich dependence structure and often leads to desirable properties of real-world networks such…
Geometric evolution represents a fundamental aspect of many physical phenomena. In this paper we consider the geometric evolution of structures that undergo topological changes. Topological changes occur when the shape of an object evolves…
Real-world networks often exhibit strong transitivity with nontrivial local clustering spectra and degree correlations. Such features are not easily modeled in tractable network models, creating an obstacle to the theoretical understanding…
We show that thermodynamics can be formulated naturally from the intrinsic geometry of phase space alone-without postulating an ensemble, which instead emerges from the geometric structure itself. Within this formulation, phase transitions…
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers…
Based on numerical simulation and local stability analysis we describe the structure of the phase space of the edge/triangle model of random graphs. We support simulation evidence with mathematical proof of continuity and discontinuity for…
We present an algorithm for generating random networks with arbitrary degree distribution and Clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and poisson degree…
This paper studies clustering algorithms for dynamically evolving graphs $\{G_t\}$, in which new edges (and potential new vertices) are added into a graph, and the underlying cluster structure of the graph can gradually change. The paper…
Topological landscape is introduced for networks with functions defined on the nodes. By extending the notion of gradient flows to the network setting, critical nodes of different indices are defined. This leads to a concise and…
Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown,…
The interplay between topology and criticality has been a recent interest of study in condensed matter physics. A unique topological transition between certain critical phases has been observed as a consequence of the edge modes living at…
Two distinct structures of aggregates of atoms connected by anisotropic bonds with a network configuration are discussed from the viewpoint of a point set topology. A specific topological space connects the two types of topological…