Related papers: Highly irregular separated nets
We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of disjointly strictly singular operators, those not invertible on the span…
We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly's…
Deep neural networks have attained remarkable success across diverse classification tasks. Recent empirical studies have shown that deep networks learn features that are linearly separable across classes. However, these findings often lack…
Recent studies show that in interdependent networks a very small failure in one network may lead to catastrophic consequences. Above a critical fraction of interdependent nodes, even a single node failure can invoke cascading failures that…
Latent Euclidean embedding models a given network by representing each node in a Euclidean space, where the probability of two nodes sharing an edge is a function of the distances between the nodes. This implies that for two nodes to share…
We investigate the Bieri--Neumann--Strebel--Renz (BNSR) invariants of irreducible uniform lattices. In the case of a direct product of a tree and a Euclidean space we show that vanishing of the BNSR invariants for all finite-index subgroups…
In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure…
We use information theory to study recovering sets $\R_L$ and strongly cancellative sets $\C_L$ on different lattices. These sets are special classes of recovering pairs and cancellative sets previously discussed in [1], [3] and [5]. We…
Machine learning is advancing towards a data-science approach, implying a necessity to a line of investigation to divulge the knowledge learnt by deep neuronal networks. Limiting the comparison among networks merely to a predefined…
This research establishes that many real-world networks exhibit bounded expansion, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. We analyze several…
This study investigates the suitability of the annealed approximation in high-dimensional systems characterized by dense networks with quenched link disorder, employing models of coupled oscillators. We demonstrate that dynamic equations…
We provide a new characterisation of the decades old open problem of extending bilipschitz mappings given on a Euclidean separated net. In particular, this allows for the complete positive solution of the open problem in dimension two.…
The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open)…
Networks are useful representations for complex systems. Especially, heterogeneous and asymmetrical relations commonly found in complex systems can be converted to weighted directed edges between nodes. The disparity filter (Serrano et al.,…
We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense,…
We study mechanical structures composed of spatial four-bar linkages that are bistable, that is, they allow for two distinct configurations. They have an interpretation as quad nets in the Study quadric which can be used to prove existence…
Complex systems are large collections of entities that organize themselves into non-trivial structures that can be represented by networks. A key emergent property of such systems is robustness against random failures or targeted attacks…
We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up…
We present a new type of probabilistic model which we call DISsimilarity COefficient Networks (DISCO Nets). DISCO Nets allow us to efficiently sample from a posterior distribution parametrised by a neural network. During training, DISCO…
We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with $(i)$ unconstrained interdependent…