Related papers: Bounds on monotone switching networks for directed…
We study the complexity of small-depth Frege proofs and give the first tradeoffs between the size of each line and the number of lines. Existing lower bounds apply to the overall proof size -- the sum of sizes of all lines -- and do not…
Seeking effective neural networks is a critical and practical field in deep learning. Besides designing the depth, type of convolution, normalization, and nonlinearities, the topological connectivity of neural networks is also important.…
We consider network design problems in which we are given a graph and seek a min-size $2$-connected subgraph that satisfies a prescribed property. $\bullet$ In the 1-Connectivity Augmentation problem the goal is to augment a connected graph…
Diestel and M\"uller showed that the connected tree-width of a graph $G$, i.e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of $G$ and the largest length of a geodesic cycle in…
An important issue in neural network research is how to choose the number of nodes and layers such as to solve a classification problem. We provide new intuitions based on earlier results by An et al. (2015) by deriving an upper bound on…
Distributed data collection is a fundamental task in open systems. In such networks, data is aggregated across a network to produce a single aggregated result at a source device. Though self-stabilizing, algorithms performing data…
In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be…
A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u,w in G, there exists abpath P_{uw} in G that is monotone in some direction l_{uw}. (Namely, the order of the orthogonal projections of…
For any positive integer $k$, there exist neural networks with $\Theta(k^3)$ layers, $\Theta(1)$ nodes per layer, and $\Theta(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are…
We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well…
Findings: We show that the distance distribution in an undirected network Lorenz majorizes the one of a chain. As a consequence, the average and median distances in any such network are smaller than or equal to those of a chain. Research…
In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic…
We prove that the Jones diameter of a link is twice its crossing number whenever the breadth of its Jones polynomial equals the difference between the crossing number and the Turaev genus. This implies that such link is adequate, as per the…
Small-world networks by Watts and Strogatz are a class of networks that are highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. These characteristics result in networks with unique…
A crossing-free straight-line drawing of a graph is monotone if there is a monotone path between any pair of vertices with respect to some direction. We show how to construct a monotone drawing of a tree with $n$ vertices on an $O(n^{1.5})…
The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper…
In this paper we study complexity of an extension of ordered binary decision diagrams (OBDDs) called $c$-OBDDs on CNFs of bounded (primal graph) treewidth. In particular, we show that for each $k$ there is a class of CNFs of treewidth $k…
We describe a synchronous distributed algorithm which identifies the edge-biconnected components of a connected network. It requires a leader, and uses messages of size O(log |V|). The main idea is to preorder a BFS spanning tree, and then…
The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to…
Let $(\Omega, \A, \mu)$ be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $\Omega$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $\Omega$…