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The formation of triangles in complex networks is an important network property that has received tremendous attention. The formation of triangles is often studied through the clustering coefficient. The closure coefficient or transitivity…
Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focuses…
We characterize the geometry and topology of the set of all weight vectors for which a linear neural network computes the same linear transformation $W$. This set of weight vectors is called the fiber of $W$ (under the matrix multiplication…
In this paper we discuss gauging one-form symmetries in two-dimensional theories. The existence of a global one-form symmetry in two dimensions typically signals a violation of cluster decomposition -- an issue resolved by the observation…
Algorithms for node clustering typically focus on finding homophilous structure in graphs. That is, they find sets of similar nodes with many edges within, rather than across, the clusters. However, graphs often also exhibit heterophilous…
A $(\beta,\delta,\Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $\Delta$ such that for every vertex $v\in V$, the probability that…
Node copying is an important mechanism for network formation, yet most models assume uniform copying rules. Motivated by observations of heterogeneous triadic closure in real networks, we introduce the concept of a hidden network model - a…
With the aim of completing the previous study by A. Or{\l}owski and the author concerning intertwining maps between induced representations and conjugation representation, termed here weighted class operators, we compute the latter…
We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity,…
The transport properties on the two-dimensional surface of coupled multilayer heterostructures are studied in the integer quantum Hall states. We emphasize the criticality of the surface state and the phase coherent transport properties in…
The visualization and interactive exploration of geo-referenced networks poses challenges if the network's nodes are not evenly distributed. Our approach proposes new ways of realizing animated transitions for exploring such networks from…
For a continuous semicascade on a metrizable compact set $\Omega $, we consider the weak$^{*}$ convergence of generalized operator ergodic means in ${\rm End}\, \, C^{*} (\Omega)$. We discuss conditions on the dynamical system under which…
For a generic quantum many-body system, the quantum ergodic regime is defined as the limit in which the spectrum of the system resembles that of a random matrix theory (RMT) in the corresponding symmetry class. In this paper we analyse the…
The structure of many complex networks includes edge directionality and weights on top of their topology. Network analysis that can seamlessly consider combination of these properties are desirable. In this paper, we study two important…
We describe and study a transport based procedure called NetOTC (network optimal transition coupling) for the comparison and alignment of two networks. The networks of interest may be directed or undirected, weighted or unweighted, and may…
Symmetry breaking surface fields give rise to nontrivial and long-ranged order parameter profiles for critical systems such as fluids, alloys or magnets confined to wedges. We discuss the properties of the corresponding universal scaling…
Dynamic systems characterized by diversified evolutions are not only more flexible, but also more resilient to attacks, failures and changing conditions. This article addresses the quantification of the diversity of non-linear transient…
A commuting tuple of $n$ operators $(S_1, \dots, S_{n-1}, P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc \[ \Gamma_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots,…
The generic behavior of quantum systems has long been of theoretical and practical interest. Any quantum process is represented by a sequence of quantum channels. We consider general ergodic sequences of stochastic channels with arbitrary…
Node connectivity plays a central role in temporal network analysis. We provide a comprehensive study of various concepts of walks in temporal graphs, that is, graphs with fixed vertex sets but edge sets changing over time. Taking into…