Clustering is a method of allocating data points in various groups, known as clusters, based on similarity. The notion of expressing similarity mathematically and then maximizing it (minimize dissimilarity) can be formulated as an optimization problem. Spectral clustering is an example of such an approach to clustering, and it has been successfully applied to visualization of clustering and mapping of points into clusters in two and three dimensions. Higher dimension problems remained untouched due to complexity and, most importantly, lack of understanding what “similarity” means in higher dimensions. In this paper, we apply spectral clustering to long timeseries EEG (electroencephalogram) data. We developed several models, based on different similarity functions and different approaches for spectral clustering itself. The results of the numerical experiment demonstrate that the created models are accurate and can be used for timeseries classification.

In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

In this paper, we consider the friendship paradox in the context of random walks and paths. Among our results, we give an equality connecting long-range degree correlation, degree variability, and the degree-wise effect of additional steps for a random walk on a graph. Random paths are also considered, as well as applications to acquaintance sampling in the context of core-periphery structure.

Gaussian graphical models are useful tools for conditional independence structure inference of multivariate random variables. Unfortunately, Bayesian inference of latent graph structures is challenging due to exponential growth of $\mathcal{G}_n$, the set of all graphs in n vertices. One approach that has been proposed to tackle this problem is to limit search to subsets of $\mathcal{G}_n$. In this paper we study subsets that are vector subspaces with the cycle space $\mathcal{C}_n$ as the main example. We propose a novel prior on $\mathcal{C}_n$ based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implement a Markov chain Monte Carlo algorithm, and show that (i) posterior edge inclusion estimates computed with our technique are comparable to estimates from the standard technique despite searching a smaller graph space, and (ii) the vector space perspective enables straightforward implementation of MCMC algorithms.

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb{R}^d$. Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.

Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

During the 125th European Study Group with Industry held in Limassol, Cyprus, 5–9 December 2016, one of the participating companies, Engino.net Ltd, posed a very interesting challenge to the members of the study group. Engino.net Ltd is a Cypriot company, founded in 2004, that produces a series of toy sets – the Engino® toy sets – consisting of a number of building blocks, which can be assembled by pupils to compose toy models. Depending on the contents of a particular toy set, the company has developed a number of models that can be built utilizing the blocks present in the set; however, the production of a step-by-step assembly manual for each model could only be done manually. The goal of the challenge posed by the company was to implement a procedure to automatically generate the assembly instructions for a given toy. In the present paper, we propose a graph-theoretic approach to model the problem and provide a series of results to solve it by employing modified versions of well-established algorithms in graph theory. An algorithmic procedure to obtain a hierarchical, physically feasible decomposition of a given toy model, from which a series of step-by-step assembly instructions can be recovered, is proposed.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number #V(G) of the vertices. We give Cohen-Macaulay criteria for such graphs.

In this paper we give an analytic solution for graphs with n nodes and E = cn log n edges for which the probability of obtaining a given graph G is µn (G) = exp (- β ∑i=1ndi2), where di is the degree of node i. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.

The structured coalescent is a continuous-time Markov chain which describes the genealogy of a sample of homologous genes from a subdivided population. Assuming this model, some results are proved relating to the genealogy of a pair of genes and the extent of subpopulation differentiation, which are valid under certain graph-theoretic symmetry and regularity conditions on the structure of the population. We first review and extend earlier results stating conditions under which the mean time since the most recent common ancestor of a pair of genes from any single subpopulation is independent of the migration rate and equal to that of two genes from an unstructured population of the same total size. Assuming the infinite alleles model of neutral mutation with a small mutation rate, we then prove a simple relationship between the migration rate and the value of Wright's coefficient FST for a pair of neighbouring subpopulations, which does not depend on the precise structure of the population provided that this is sufficiently symmetric.

This is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.