Construction of larger singular and nonsingular graphs using a path

As social media becomes more feature-rich and the capability for interactions between users becomes more complex as a result, it may become necessary to expand the models used in data analysis to represent more complex interactions and networks. To that effect, researchers have begun using graphs with different types of vertices or even hyperedges to represent more complex networks. In this chapter, we will explore some of the community detection approaches state-of-the-art research uses to deal with the increasing complexity of social networks, and particularly representing those networks as multi-mode networks (or heterogeneous networks). This chapter will cover the approaches used as well as the graph representations of complex networks. Though the work studied uses social networks as the basis for analysis, the use of multi-mode networks and hyperedges is principled in any analysis task where the complexity of the data calls for multiple types of entities with interactions involving two or more entities in the network.

Real-Time scheduling and analysis of parallel tasks on heterogeneous multi-cores

On strongly asymmetric and controllable primitive graphs

Given a simple graph G, let A(G) be its adjacency matrix. A principal submatrix of A(G) of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A(G) and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue μ if the multiplicity of μ in A(G)−v goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalizeour results to weighted graphs, or equivalently to symmetric matrices.

A nonsingular graph with a singular deck, or NSSD, is an edge-weighted graph with all the diagonal entries of its adjacency matrix and of the inverse equal to zero. We obtain necessary and sufficient conditions for a two-vertex-deleted subgraph of an NSSD G to remain an NSSD by considering triangles in the inverse NSSD whose adjacency matrix is . We also consider the graph obtained by introducing terminal vertices to the NSSD G and show that the conditions for it to be an NSSD are associated with the conditions deduced for the two-vertex deleted subgraph.

We show in this paper that by applying size-effect distortions to a perfect Penrose tiling, on the basis that rhomb-edges which connect different types of vertices assume different lengths, we can obtain a diffraction pattern which shows remarkable similarity to the zero-level (h5=0) section observed in decagonal Al71Co13Ni16. In particular a central clearly delineated decagon is observed, on the inside of which there is reduced intensity, and on the outside of which there is enhanced intensity. Such a transfer of intensity is characteristic of size-effect distortions in crystals but for these systems it is necessary to have disorder involving (at least) two types of atoms. In the present case the effect is observed with all vertices occupied by a single scatterer and the system remains topologically equivalent to the Penrose pattern with long range quasicrystallinity.

On geodetic sets formed by boundary vertices

Non-Singular graphs with a Singular Deck

A graph G is said to be singular if its adjacency matrix is a singular matrix. In this paper, by analyzing the structure of Sachs subgraphs on three types of tricyclic graphs ϖ(a1,a2,b1,b2,c1,c2), σ(a1,a2,b1,b2,c), and τ(a1,a2,a3,a4) and calculating the determinants of their adjacency matrices, we investigate the necessary and sufficient conditions for the singularity of these three types of tricyclic graphs. Furthermore, we derive that the probabilities of singular graphs among tricyclic graphs ϖ(a1,a2,b1,b2,c1,c2), σ(a1,a2,b1,b2,c), and τ(a1,a2,a3,a4) are 217512,113256, and 67128, respectively.

Cospectral Graphs and Regular Orthogonal Matrices of Level 2

Layout classification is an important task used in lithography simulation approaches, such as source optimization (SO), source-mask joint optimization (SMO) and so on. In order to balance the performance and time consumption of optimization, it is necessary to classify a large number of cut layouts with the same key patterns. This paper proposes a new kind of classification method for lithography layout patterns based on graph convolution network (GCN). GCN is an emerging machine learning approach that achieves impressive performance in processing graph signals with nonEuclidean topology structures. The proposed method first transforms the layout patterns into graph signals, where the sum of several adjacent layout pixels is associated with one graph vertex. Next, the adjacent graph vertices are connected by the graph edges, where the edge weights are determined by the correlations between the vertices. Therefore, the layout geometries can be represented by the function values on the graph vertices and the adjacency matrix. Subsequently, the GCN framework is established based on the graph Fourier transform, where the input is the graph signal of the layout, and the output is its classification label. The network parameters of GCN are trained in a supervised manner. The proposed method is compared to the simple convolutional neural network (CNN) with a few layers and VGG-16 network, respectively. Finally, the features of different methods are discussed in terms of classification accuracy and computational efficiency.

The matrix completion problem can be stated as follows: Suppose $X\in \mathbb{R}^{n_{1}\times n_{2}}$ is unknown except for an upper bound $r$ on its rank. By measuring a small number $m\ll n_{1}n_{2}$ of elements of $X$ , is it possible to recover $X$ exactly, or at least, to construct a reasonable approximation of $X$ ? There are two existing approaches to choosing the elements to be measured, namely: choosing them at random or in some deterministic fashion. Practically all of the existing literature focuses on the case where the sampled locations are chosen at random. In contrast, the focus in the present paper is on deterministic methods for choosing the elements to be sampled. Specifically, we choose the measurement matrix to be the adjacency or biadjacency matrix of a so-called Ramanujan graph. Existing explicit techniques for constructing Ramanujan graphs lead to graphs whose degree $d$ is bounded by $n^{1/3}$ where $n$ is the number of vertices. However, for the matrix completion problem, this degree is too small. We point out that the well-known Lubotzky-Phillips-Sarnak construction of Ramanujan graphs can be used to generate graphs of very high degree, and use these for the matrix construction problem. Then we carry out numerical studies of “phase transition” in the matrix completion problem, by comparing the behavior of sampling at random with choosing the sampled locations using a Ramanujan graph. Specifically, we show that for a fixed sampling pattern, there is a maximum rank $\bar{r}$ for which randomly generated matrices of rank $r\leq\overline{r}$ can be recovered with probability one, and the recovery probability drops very sharply to zero if $r$ is increased by just two or three beyond $\bar{r}$ . The technique proposed here is applicable only to the recovery of square matrices. The recovery of rectangular matrices requires so-called asymmetric Ramanujan graphs, and is studied in a separate paper.

We propose to use eigenvector centrality (EVC) as the basis for determining stable paths and arrive at benchmarks for the number of path transitions incurred with stability-based routing in cognitive radio ad hoc networks (CRAHNs) wherein secondary users (SUs) use the available (idle) licensed channels of primary users (PUs). We model a time-variant network graph (at any time instant) of SU nodes whose adjacency matrix indicates the number of common available PU channels in the mutually intersecting neighbourhoods of any two SU nodes at the time instant. Such an adjacency matrix is used as the basis to compute the EVC of the vertices in the network graph of SU nodes and determine an end-to-end source-destination path whose constituent SU nodes have a higher EVC. Simulation results indicate EVC-based paths between any two SU nodes incur significantly fewer transitions without any appreciable increase in the average hop count per path.

Let $G$ be a graph with adjacency matrix $A$, let $H(t)=\exp(itA)$. $G$ is called a periodic graph if there exists a time $\tau$ such that $H(\tau)$ is diagonal. If $u$ and $v$ are distinct vertices in $G$, we say that perfect state transfer occurs from $u$ to $v$ if there exists a time $\tau$ such that $|H(\tau)_{u,v}|=1$. A necessary and sufficient condition for $G$ is periodic is given. We give the existence for the perfect state transfer between antipodal vertices in graphs with extreme diameter.

Constructing cospectral graphs via regular rational orthogonal matrices with level two