Structural Entropy of the Stochastic Block Models

With the rapid expansion of graphs and networks and the growing magnitude of data from all areas of science, effective treatment and compression schemes of context-dependent data is extremely desirable. A particularly interesting direction is to compress the data while keeping the “structural information” only and ignoring the concrete labelings. Under this direction, Choi and Szpankowski introduced the structures (unlabeled graphs) which allowed them to compute the structural entropy of the Erdos–Rényi random graph model. Moreover, they also provided an asymptotically optimal compression algorithm that (asymptotically) achieves this entropy limit and runs in expectation in linear time. In this paper, we consider the stochastic block models with an arbitrary number of parts. Indeed, we define a partitioned structural entropy for stochastic block models, which generalizes the structural entropy for unlabeled graphs and encodes the partition information as well. We then compute the partitioned structural entropy of the stochastic block models, and provide a compression scheme that asymptotically achieves this entropy limit.

Adapting k-means for graph clustering

We propose two new algorithms for clustering graphs and networks. The first, called K‑algorithm, is derived directly from the k-means algorithm. It applies similar iterative local optimization but without the need to calculate the means. It inherits the properties of k-means clustering in terms of both good local optimization capability and the tendency to get stuck at a local optimum. The second algorithm, called the M-algorithm, gradually improves on the results of the K-algorithm to find new and potentially better local optima. It repeatedly merges and splits random clusters and tunes the results with the K-algorithm. Both algorithms are general in the sense that they can be used with different cost functions. We consider the conductance cost function and also introduce two new cost functions, called inverse internal weight and mean internal weight. According to our experiments, the M-algorithm outperforms eight other state-of-the-art methods. We also perform a case study by analyzing clustering results of a disease co-occurrence network, which demonstrate the usefulness of the algorithms in an important real-life application.

Kolmogorov Basic Graphs and Their Application in Network Complexity Analysis

Throughout the years, measuring the complexity of networks and graphs has been of great interest to scientists. The Kolmogorov complexity is known as one of the most important tools to measure the complexity of an object. We formalized a method to calculate an upper bound for the Kolmogorov complexity of graphs and networks. Firstly, the most simple graphs possible, those with O(1) Kolmogorov complexity, were identified. These graphs were then used to develop a method to estimate the complexity of a given graph. The proposed method utilizes the simple structures within a graph to capture its non-randomness. This method is able to capture features that make a network closer to the more non-random end of the spectrum. The resulting algorithm takes a graph as an input and outputs an upper bound to its Kolmogorov complexity. This could be applicable in, for example evaluating the performances of graph compression methods.

Environmental Stratification in Trials of Unbalanced Multiyear Soybean (Glycine max (l.) Merril) via the Integration of GGE Biplot Graphs and Networks of Environmental Similarity

Genotype x enviroment (GE) interaction can difficult soybean breeding programs to atieve the aim of obtain more productive cultivars. Enviroment stratification is a way to circunvent this problem. This work aimed to gather GGE Biplot graphs of a network of trials unbalance multiyear soybean via matrices of coincidence and networks of enviroment to optimize environmental stratification. Data from an experimental network of 43 trials was used, these experiments were implanted during the crop seasons of 2011/12, 2012/13, 2013/14 and 2015/16 in Brazil. The GE interaction were statistically significant for all 43 trials. The step by step of our analses was: GGE Biplots graphs were obtained; the enviroment coincidence matrices were calculated; the values of matrices were used for to obtain the networks of environmental similarity. The study demonstrated that by the method was possible to identify, using unbalanced multiyear data, the formation of four mega-environments. Therefore, integrating GGE Biplot graphs and networks of environmental similarity is an efficient method to optimize a soybean program by environment stratification.

Research of NP-Complete Problems in the Class of Prefractal Graphs

NP-complete problems in graphs, such as enumeration and the selection of subgraphs with given characteristics, become especially relevant for large graphs and networks. Herein, particular statements with constraints are proposed to solve such problems, and subclasses of graphs are distinguished. We propose a class of prefractal graphs and review particular statements of NP-complete problems. As an example, algorithms for searching for spanning trees and packing bipartite graphs are proposed. The developed algorithms are polynomial and based on well-known algorithms and are used in the form of procedures. We propose to use the class of prefractal graphs as a tool for studying NP-complete problems and identifying conditions for their solvability. Using prefractal graphs for the modeling of large graphs and networks, it is possible to obtain approximate solutions, and some exact solutions, for problems on natural objects—social networks, transport networks, etc.

Global Symmetries, Local Symmetries and Groupoids

Local symmetries are primarily defined in the case of spacetime, but several authors have defined them outside this context, sometimes with the help of groupoids. We show that, in many cases, local symmetries can be defined as global symmetries. We also show that groups can be used, rather than groupoids, to handle local symmetries. Examples are given for graphs and networks, color symmetry and tilings. The definition of local symmetry in physics is also discussed.

Optimization Models for Efficient (t, r) Broadcast Domination in Graphs

Known to be NP-complete, domination number problems in graphs and networks arise in many real-life applications, ranging from the design of wireless sensor networks and biological networks to social networks. Initially introduced by Blessing et al., the (t,r) broadcast domination number is a generalization of the distance domination number. While some theoretical approaches have been addressed for small values of t,r in the literature; in this work, we propose an approach from an optimization point of view. First, the (t,r) broadcast domination number is formulated and solved using linear programming. The efficient broadcast, whose wasted signals are minimized, is then found by a genetic algorithm modified for a binary encoding. The developed method is illustrated with several grid graphs: regular, slant, and king’s grid graphs. The obtained computational results show that the method is able to find the exact (t,r) broadcast domination number, and locate an efficient broadcasting configuration for larger values of t,r than what can be provided from a theoretical basis. The proposed optimization approach thus helps overcome the limitations of existing theoretical approaches in graph theory.

ON MOSTAR INDEX OF GRAPHS

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

Graphs in Molecular Epidemiology

Graphs and networks are used in molecular epidemiology to model the evolution of viruses and their spread during outbreaks and epidemics. They are instrumental at different stages of the computational pipelines. This includes the inference of transmission networks using viral sequences sampled from infected individuals, and studies of selection and accumulation of mutations in viral populations and their interactions with hosts' immune systems. This chapter describes some algorithmic and graph-theoretic problems associated with these stages to illustrate the relevance of the concepts of graph theory to molecular epidemiology of viral infections. The chapter will demonstrate how graph-theoretic methods combined with the machinery of differential equations, the Bayesian inference, and computational genomics uncover hidden biological and epidemiological patterns of virus evolution and transmission.