Graph Entropy Research Articles

A Conjecture Regarding the Extremal Values of Graph Entropy Based on Degree Powers

Many graph invariants have been used for the construction of entropy-based measures to characterize the structure of complex networks. The starting point has been always based on assigning a probability distribution to a network when using Shannon’s entropy. In particular, Cao et al. (2014 and 2015) defined special graph entropy measures which are based on degrees powers. In this paper, we obtain some lower and upper bounds for these measures and characterize extremal graphs. Moreover we resolve one part of a conjecture stated by Cao et al.

Trajectories Entropy in Dynamical Graphs with Memory

In this paper we investigate the application of non-local graph entropy to evolving and dynamical graphs. The measure is based upon the notion of Markov diffusion on a graph, and relies on the entropy applied to trajectories originating at a specific node. In particular, we study the model of reinforcement-decay graph dynamics, which leads to scale free graphs. We find that the node entropy characterizes the structure of the network in the two parameter phase-space describing the dynamical evolution of the weighted graph. We then apply an adapted version of the entropy measure to purely memristive circuits. We provide evidence that meanwhile in the case of DC voltage the entropy based on the forward probability is enough to characterize the graph properties, in the case of AC voltage generators one needs to consider both forward and backward based transition probabilities. We provide also evidence that the entropy highlights the self-organizing properties of memristive circuits, which re-organizes itself to satisfy the symmetries of the underlying graph.

Data selection in EEG signals classification.

The alcoholism can be detected by analyzing electroencephalogram (EEG) signals. However, analyzing multi-channel EEG signals is a challenging task, which often requires complicated calculations and long execution time. This paper proposes three data selection methods to extract representative data from the EEG signals of alcoholics. The methods are the principal component analysis based on graph entropy (PCA-GE), the channel selection based on graph entropy (GE) difference, and the mathematic combinations channel selection, respectively. For comparison purposes, the selected data from the three methods are then classified by three classifiers: the J48 decision tree, the K-nearest neighbor and the Kstar, separately. The experimental results show that the proposed methods are successful in selecting data without compromising the classification accuracy in discriminating the EEG signals from alcoholics and non-alcoholics. Among them, the proposed PCA-GE method uses only 29.69% of the whole data and 29.5% of the computation time but achieves a 94.5% classification accuracy. The channel selection method based on the GE difference also gains a 91.67% classification accuracy by using only 29.69% of the full size of the original data. Using as little data as possible without sacrificing the final classification accuracy is useful for online EEG analysis and classification application design.

Application of Graph Entropy in CRISPR and Repeats Detection in DNA Sequences

We analyzed DNA sequences using a new measure of entropy. The general aim was to analyze DNA sequences and find interesting sections of a genome using a new formulation of Shannon like entropy. We developed this new measure of entropy for any non-trivial graph or, more broadly, for any square matrix whose non-zero elements represent probabilistic weights assigned to connections or transitions between pairs of vertices. The new measure is called the graph entropy and it quantifies the aggregate indeterminacy effected by the variety of unique walks that exist between each pair of vertices. The new tool is shown to be uniquely capable of revealing CRISPR regions in bacterial genomes and to identify Tandem repeats and Direct repeats of genome. We have done experiment on 26 species and found many tandem repeats and direct repeats (CRISPR for bacteria or archaea). There are several existing separate CRISPR or Tandem finder tools but our entropy can find both of these features if present in genome.

COUNTING SPANNING TREES IN PRISM AND ANTI-PRISM GRAPHS

In this paper, we calculate the number of spanning trees in prism and antiprism graphs corresponding to the skeleton of a prism and an antiprism. By the electrically equivalent transformations and rules of weighted generating function, we obtain a relationship for the weighted number of spanning trees at the successive two generations. Using the knowledge of difference equations, we derive the analytical expressions for enumeration of spanning trees. In addition, we again calculate the number of spanning trees in Apollonian networks, which shows that this method is simple and effective. Finally we compare the entropy of our networks with other studied networks and find that the entropy of the antiprism graph is larger.

Some New Properties for Degree-Based Graph Entropies

The graph entropies inspired by Shannon’s entropy concept become the information-theoretic quantities for measuring the structural information of graphs and complex networks. In this paper, we continue studying some new properties of the graph entropies based on information functionals involving vertex degrees. We prove the monotonicity of the graph entropies with respect to the power exponent. Considering only the maximum and minimum degrees of the ( n , m ) -graph, we obtain some upper and lower bounds for the degree-based graph entropy. These bounds have different performances to restrict the degree-based graph entropy in different kinds of graphs. Moreover the degree-based graph entropy can be estimated by these bounds.

Influence of the CYP1A1 T3801C Polymorphism on Tobacco and Alcohol-Associated Head and Neck Cancer Susceptibility in Northeast India.

Tobacco and alcohol contain or may generate carcinogenic compounds related to cancers. CYP1A1 enzymes act upon these carcinogens before elimination from the body. The aim of this study was to investigate whether CYP1A1 T3801C polymorphism modulates the relationship between tobacco and alcohol- associated head and neck cancer (HNC) susceptibility among the northeast Indian population. One hundred and seventy histologically confirmed HNC cases and 230 controls were included within the study. The CYP1A1 T3801C polymorphism was determined using PCR-RFLP, and the results were confirmed by DNA sequencing. Logistic regression (LR) and multifactor dimensionality reduction (MDR) approaches were applied for statistical analysis. The CYP1A1 CC genotype was significantly associated with HNC risk (P=0.045). A significantly increased risk of HNC (OR=6.09; P<0.0001) was observed in individuals with combined habits of smoking, alcohol drinking and tobacco-betel quid chewing. Further, gene-environment interactions revealed enhanced risks of HNC among smokers, alcohol drinkers and tobacco-betel quid chewers carrying CYP1A1 TC or CC genotypes. The highest risk of HNC was observed among smokers (OR=7.55; P=0.009) and chewers (OR=10.8; P<0.0001) carrying the CYP1A1 CC genotype. In MDR analysis, the best model for HNC risk was the three-factor model combination of smoking, tobacco-betel quid chewing and the CYP1A1 variant genotype (CVC=99/100; TBA=0.605; P<0.0001); whereas interaction entropy graphs showed synergistic interaction between tobacco habits and CYP1A1. Our results confirm that the CYP1A1 T3801C polymorphism modifies the risk of HNC and further demonstrated importance of gene-environment interaction.

Par@Graph – a parallel toolbox for the construction and analysis of large complex climate networks

Abstract. In this paper, we present Par@Graph, a software toolbox to reconstruct and analyze complex climate networks having a large number of nodes (up to at least 106) and edges (up to at least 1012). The key innovation is an efficient set of parallel software tools designed to leverage the inherited hybrid parallelism in distributed-memory clusters of multi-core machines. The performance of the toolbox is illustrated through networks derived from sea surface height (SSH) data of a global high-resolution ocean model. Less than 8 min are needed on 90 Intel Xeon E5-4650 processors to reconstruct a climate network including the preprocessing and the correlation of 3 × 105 SSH time series, resulting in a weighted graph with the same number of vertices and about 3.2 × 108 edges. In less than 14 min on 30 processors, the resulted graph's degree centrality, strength, connected components, eigenvector centrality, entropy and clustering coefficient metrics were obtained. These results indicate that a complete cycle to construct and analyze a large-scale climate network is available under 22 min Par@Graph therefore facilitates the application of climate network analysis on high-resolution observations and model results, by enabling fast network reconstruct from the calculation of statistical similarities between climate time series. It also enables network analysis at unprecedented scales on a variety of different sizes of input data sets.

Entropy of symmetric graphs

Let FG(P) be a functional defined on the set of all the probability distributions on the vertex set of a graph G. We say that G is symmetric with respect toFG(P) if the distribution P∗ maximizing FG(P) is uniform on V(G). Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we prove that vertex-transitive graphs are symmetric with respect to graph entropy. As the main result of this paper, we prove that a perfect graph is symmetric with respect to graph entropy if and only if its vertices can be covered by disjoint copies of its maximum-size clique. Particularly, this means that a bipartite graph is symmetric with respect to graph entropy if and only if it has a perfect matching.

On the distance based graph entropies

In this note we prove a sharp upper bound for the distance based graph entropy introduced in Chen et al. (2014). Furthermore we resolve one part of the conjecture proposed in the same paper and provide counterexamples for the other part.

A FORMULA FOR THE NUMBER OF SPANNING TREES IN CIRCULANT GRAPHS WITH NONFIXED GENERATORS AND DISCRETE TORI

We consider the number of spanning trees in circulant graphs of ${\it\beta}n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of ${\it\beta}n$ factors, while we derive a formula of ${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators.

Entropy of Weighted Graphs with Randi´c Weights

Shannon entropies for networks have been widely introduced. However, entropies for weighted graphs have been little investigated. Inspired by the work due to Eagle et al., we introduce the concept of graph entropy for special weighted graphs. Furthermore, we prove extremal properties by using elementary methods of classes of weighted graphs, and in particular, the one due to Bollobás and Erdös, which is also called the Randi´c weight. As a result, we derived statements on dendrimers that have been proven useful for applications. Finally, some open problems are presented.

Degree-based entropies of networks revisited

Studies on the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to Shannon and Rashevsky. Various graph parameters have been used for the construction of entropy-based measures to characterize the structure of complex networks. Based on Shannon’s entropy, in Cao et al. (Extremality of degree-based graph entropies, Inform. Sci. 278 (2014) 22–33), we studied graph entropies which are based on vertex degrees by using so-called information functionals. As a matter of fact, there has been very little work to find their extremal values when considered Shannon entropy-based graph measures. We pursue with this line of research by proving further extremal properties of the degree-based graph entropies.

Poset Entropy Versus Number of Linear Extensions: The Width-2 Case

Kahn and Kim (J. Comput. Sci. 51, 3, 390–399, 1995) have shown that for a finite poset P, the entropy of the incomparability graph of P (normalized by multiplying by the order of P) and the base-2 logarithm of the number of linear extensions of P are within constant factors from each other. The tight constant for the upper bound was recently shown to be 2 by Cardinal et al. (Combinatorica 33, 655–697, 2013). Here, we refine this last result in case P has width 2: we show that the constant can be replaced by 2−ε if one also takes into account the number of connected components of size 2 in the incomparability graph of P. Our result leads to a better upper bound for the number of comparisons in algorithms for the problem of sorting under partial information.

Novel inequalities for generalized graph entropies – Graph energies and topological indices

The entropy of a graph is an information-theoretic quantity for measuring the complexity of a graph. After Shannon introduced the entropy to information and communication, many generalizations of the entropy measure have been proposed, such as Rényi entropy and Daróczy entropy. In this article, we prove accurate connections (inequalities) between generalized graph entropies, graph energies, and topological indices. Additionally, we obtain some extremal properties of nine generalized graph entropies by employing graph energies and topological indices.

The Hosoya Entropy of a Graph

This paper demonstrates properties of Hosoya entropy, a quantitative measure of graph complexity based on a decomposition of the vertices linked to partial Hosoya polynomials. Connections between the information content of a graph and Hosoya entropy are established, and the special case of Hosoya entropy of trees is investigated.

What is Heat?

Abstract We define heat as a particular kind of nonwork interaction that involves only energy and entropy transfers, and that is entirely distinguishable from work. The existence of heat interactions is a consequence of the first and second laws of thermodynamics. The requirement that heat be entirely distinguishable from work implies strict conditions on the end states of the interacting systems, and guarantees a definite relation between such states and the energy and entropy transfers. We illustrate these conditions by using energy versus entropy graphs. Many experiences can be represented as heat interactions, including the exchanges between two black bodies at temperatures that differ infinitesimally. We discuss the latter point in a companion paper at this conference.

Analyzing EEG Signals Using Graph Entropy based Principle Component Analysis and J48 Decision Tree

This paper proposed a method using principle component analysis based on graph entropy (PCA-GE) and J48 decision tree on electroencephalogram (EEG) signals to predict whether a person is alcoholic or not. Analysis is performed in two stages: feature extraction and classification. The principle component analysis (PCA) chooses the optimal subset of channels based on graph entropy technique and the selected subset is classified by the J48 decision tree in Weka. K-nearest neighbor (KNN) and support vector machine (SVM) in R package are also used for comparison. Experimental results show that the proposed PCA-GE method is successful in selecting a subset of channels, which contributes to the high accuracy and efficiency in the classification of alcoholics and non-alcoholics.

Adjusting protein graphs based on graph entropy.

Measuring protein structural similarity attempts to establish a relationship of equivalence between polymer structures based on their conformations. In several recent studies, researchers have explored protein-graph remodeling, instead of looking a minimum superimposition for pairwise proteins. When graphs are used to represent structured objects, the problem of measuring object similarity become one of computing the similarity between graphs. Graph theory provides an alternative perspective as well as efficiency. Once a protein graph has been created, its structural stability must be verified. Therefore, a criterion is needed to determine if a protein graph can be used for structural comparison. In this paper, we propose a measurement for protein graph remodeling based on graph entropy. We extend the concept of graph entropy to determine whether a graph is suitable for representing a protein. The experimental results suggest that when applied, graph entropy helps a conformational on protein graph modeling. Furthermore, it indirectly contributes to protein structural comparison if a protein graph is solid.

Singularities in the Entropy of Asymptotically Large Simple Graphs

We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. We also determine the precise modified-bipartite structure of asymptotic graphs with edge density 1/2 and triangle density in the interval $$\displaystyle [0,1/8]$$ .