Brain EEG Time Series Selection: A Novel Graph-Based Approach for Classification

Brain electroencephalography (EEG), the complex, weak, multivariate, nonlinear, and nonstationary time series, has been recently widely applied in neurocognitive disorder diagnoses and brain-machine interface developments. With its specific features, unlabeled EEG is not well addressed by conventional unsupervised time-series learning methods. In this article, we handle the problem of unlabeled EEG time-series clustering and propose a novel EEG clustering algorithm, that we call mwcEEGc. The idea is to map the EEG clustering to the maximum-weight clique (MWC) searching in an improved Fréchet similarity-weighted EEG graph. The mwcEEGc considers the weights of both vertices and edges in the constructed EEG graph and clusters EEG based on their similarity weights instead of calculating the cluster centroids. To the best of our knowledge, it is the first attempt to cluster unlabeled EEG trials using MWC searching. The mwcEEGc achieves high-quality clusters with respect to intracluster compactness as well as intercluster scatter. We demonstrate the superiority of mwcEEGc over ten state-of-the-art unsupervised learning/clustering approaches by conducting detailed experimentations with the standard clustering validity criteria on 14 real-world brain EEG datasets. We also present that mwcEEGc satisfies the theoretical properties of clustering, such as richness, consistency, and order independence.

Multi-way graph partition by stochastic probe

Exact and heuristic algorithms for the weighted total domination problem

In this paper we propose to generalize Jaccard and related measures, often used as similarity coefficients between two sets. We define Jaccard, Dice-Sorensen and Tversky edge weights on a graph and generalize them to account for vertex weights. We develop an efficient parallel algorithm for computing Jaccard edge and PageRank vertex weights. We highlight that the weights computation can obtain more than 10X speedup on the GPU versus CPU on large realistic data sets. Also, we show that finding a minimum balanced cut for modified weights can be related to minimizing the sum of ratios of the intersection and union of nodes on the boundary of clusters. Finally, we show that the novel weights can improve the quality of the graph clustering by about 15% and 80% for multi-level and spectral graph partitioning and clustering schemes, respectively.

Self-avoiding walks interacting with a surface are a useful model of polymer adsorption and one can choose to weight the walk according to the number of vertices or the number of edges in the surface. We consider a combined model of self-avoiding walks on the d-dimensional hypercubic lattice with weights associated with the numbers of vertices and edges in the surface and investigate its phase diagram. We also derive inequalities relating the critical points in the edge and vertex weighting schemes.

In this paper, we consider different aspects of robust 1‐median problems on a tree network with uncertain or dynamically changing edge lengths and vertex weights which can also take negative values. The dynamic nature of a parameter is modeled by a linear function of time. A linear algorithm is designed for the absolute dynamic robust 1‐median problem on a tree. The dynamic robust deviation 1‐median problem on a tree with n vertices is solved in O(n2 α(n) log n) time, where α(n) is the inverse Ackermann function. Examples show that both problems do not possess the vertex optimality property. The uncertainty is modeled by given intervals, in which each parameter can take a value randomly. The absolute robust 1‐median problem with interval data, where vertex weights might also be negative, can be solved in linear time. The corresponding deviation problem can be solved in O(n2) time. © 2001 John Wiley & Sons, Inc.

We present an algorithm which calculates a minimum cut and its weight in an undirected graph with nonnegative real edge weights, n vertices and m edges, in time O(max(log n, min(m/n,δG/e)) n2), where e is the minimal edge weight, and δG is the minimal weighted degree. For integer edge weights this time is further improved to O(δG n2) and O(λG n2). In both cases these bounds are improvements of the previously known best bounds of deterministic algorithms. These were O(nm + n2 log n) for real edge weights and O(nM + n2) and O(M + λG n2) for integer weights, where M is the sum of all edge weights.

Let G be a simple, connected and undirected graph with vertex set V and edge set E. A total k-labeling is defined as totally irregular total k-labeling if the weights of any two different both vertices and edges are distinct. The weight of vertex x is defined as , while the weight of edge xy is . A minimum k for which G has totally irregular total k-labeling is mentioned as total irregularity strength of G and denoted by ts(G). This paper contains investigation of totally irregular total k-labeling and determination of their total irregularity strengths for caterpillar graphs with each internal vertex between two stars has degree three. The results are and for n > 4:

It is demonstrated how the complexity of a (molecular) graph can be quantified in terms of the walk counts, extremely easily obtained graph invariants that depend on size, branching, cyclicity, and edge and vertex weights (unsaturation, heteroatoms). The influence of symmetry is easily accounted for. The term labyrinthicity is proposed for what is measured by walk counts alone, neglecting symmetry. The total walk count and recently advanced measures of labyrinthicity or complexity are compared with respect to the ordering of structures and to the computational effort required to obtain numerical values.

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).

Approximation algorithms for some optimum communication spanning tree problems

A Polynomial Time Approximation Scheme for Optimal Product-Requirement Communication Spanning Trees

In many domains it is necessary to generate surrogate networks, e.g., for hypothesis testing of different properties of a network. Generating surrogate networks typically requires that different properties of the network are preserved, e.g., edges may not be added or deleted and edge weights may be restricted to certain intervals. In this paper we present an efficient property-preserving Markov chain Monte Carlo method termed CycleSampler for generating surrogate networks in which (1) edge weights are constrained to intervals and vertex strengths are preserved exactly, and (2) edge and vertex strengths are both constrained to intervals. These two types of constraints cover a wide variety of practical use cases. The method is applicable to both undirected and directed graphs. We empirically demonstrate the efficiency of the CycleSampler method on real-world data sets. We provide an implementation of CycleSampler in R, with parts implemented in C.

Graph-based approaches for simulating pedestrian dynamics in building models

Nous montrons des bornes inferieures de $\sqrt{\log n}$ pour l’ordre des fluctuations de trois modeles planaires de croissance (percolation de premier passage, percolation de dernier passage et polymeres diriges) sans autre hypothese sur la loi des poids des sommets ou des aretes que les conditions minimales permettant d’eviter les cas pathologiques. De telles bornes etaient connues auparavant seulement pour certaines classes restreintes de lois. De surcroit, nous montrons que l’exposant des fluctuations autour de la forme limite pour la percolation de premier passage est au moins $1/8$, ce qui etend des resultats precedents a des lois plus generales.