Search

Porphyrazine is a macrocyclic molecule that has potential uses in biology and materials research. In this work, we investigate the topological characteristics of porphyrazine via topological indices. These indices are important in QSPR and QSAR modeling because they aid in the analysis and prediction of physical, biological, and chemical properties associated with a specific chemical structure. In this paper, we investigate neighborhood M-polynomial and graph index-entropy of porphyrazine graph, deriving several topological indices based on neighborhood degree sum from it, and numerical computation and graphical interpretation are used to explain the results further. This research advances our understanding of the basic principles of physics and chemistry by shedding light on the intricate connections between biological processes, chemical reactivity, and molecular structure. [ABSTRACT FROM AUTHOR]

The porphyrazine structure, known for its high chemical and thermal stability, has become a significant focus in materials science, chemical reactivity, functionalization, and drug design. By utilizing the new Zagreb-type indices to analyze the chemical structure of porphyrazine, we can gather more information about their bonding and connecting patterns. This enables us to construct an entropy measure that helps evaluate the stability of the material and predict its behavior in different scenarios. Furthermore, establishing correlations between these indices and entropy using logarithmic regression models allows for a deeper understanding of complex properties of porphyrazine. This, in turn, opens up new possibilities for the compound’s potential applications across various scientific and technical fields. In our work, we have used the M-polynomial to derive molecular descriptors for degree-based topological indices and determine the entropy of the porphyrazine structure based on these descriptors. [ABSTRACT FROM AUTHOR]

The analysis of algebraic invariants of algebras induced by appropriated multiset systems called Brauer configurations is a Brauer analysis of the data defining the multisets. Giving a complete description of such algebraic invariants (e.g., giving a closed formula for the dimensions of algebras induced by significant classes of Brauer configurations) is generally a tricky problem. Ringel previously proposed an analysis of this type in the case of Dynkin algebras, for which so-called Dynkin functions were used to study the numerical behavior of invariants associated with such algebras. This paper introduces two additional tools (the entropy and the covering graph of a Brauer configuration) for Brauer analysis, which is applied to Dynkin and Euclidean diagrams to define Dynkin functions associated with Brauer configuration algebras. Properties of graph entropies defined by the corresponding covering graphs are given to establish relationships between the theory of Dynkin functions, the Brauer configuration algebras theory, and the topological content information theory. [ABSTRACT FROM AUTHOR]

A nanotube is a nanometer–scale tube-like structure, it is a kind of nanoparticle, and may be large enough to serve as a pipe through which other nanoparticles can be channeled, or, depending on the material, may be used as an electrical conductor or an electrical insulator. For computing the structural information of nanotubes, the graph entropies have become the information theoretic quantities. The graph entropy measure has attracted the research community due to its potential application in discrete mathematics, biology, and chemistry. In this paper, our contribution is to explore graph entropies for structures of some nanotubes based on novel information function, which is the number of different degree vertices along with the number of edges between various degree vertices. More precisely, we computed entropies of some classes of nanotubes such as titania nanotube TNT 3 [ m , n ] , TNT 6 [ m , n ] and carbon nanotubes HAC 5 C 6 C 7 [ m , n ] by making a relation of degree-based topological indices with the help of information function. [ABSTRACT FROM AUTHOR]

Online disinformation has become one of the most severe concerns in today's world. Recognizing disinformation timely and effectively is very hard, because the propagation process of disinformation is dynamic and complicated. The existing newest research leverage uniform time intervals to study the multi-stage propagation features of disinformation. However, uniform time intervals are unrealistic in the real world, cause the process of information propagation is not regular. In light of these facts, we propose a novel and effective framework Multi-stageDynamicDisinformationDetection with Graph Entropy Guidance(MsDD) to better analyze multi-stage propagation patterns. Instead of traditional snapshots, we analyze the dynamic propagation network via graph entropy, which can work effectively in finding the dynamic and variable-length stages. In this way, we can explicitly learn the changing pattern of propagation stages and support timely detection even at the early stages. Based on this effective multi-stage analysis framework, we further propose a novel dynamic analysis model to model both the structural and sequential evolving features. Extensive experiments on two real-world datasets prove the superiority of our model. We open the datasets and source code at https://github.com/researchxr/MsDD. [ABSTRACT FROM AUTHOR]

Anomaly detection is critical in the internet of things (IoT) environment. To this issue, this study provides a novel approach for detecting anomalies in multivariate IoT time series. The proposed approach identified relationships between IoT time series to establish a dynamic graph and estimated the graph entropy to detect anomalies. The presented approach was applied to industrial IoT datasets. The results have shown that the presented method outperformed other models by 0.21 with respect to F1‐score. In addition, we used three distinct algorithms to detect the anomalies from the multivariate IoT time series. According to the results, the local outlier factor approach outperformed the others by 0.18 with respect to F1‐score. [ABSTRACT FROM AUTHOR]

Macromolecules are gaining much attention in various fields today. Dendrimers are artificially synthesized macromolecules by convergent or divergent approach. They are compact regular structures with spherical dimension and has a vast number of application in disparate fields such as drug delivery, material science, and biology, magnetic resonance imaging, an organic light-emitting device, etc. Determining the pharmacological, chemical, and biological characteristics of a substance necessitates a significant amount of effort. From the chemical graph of the dendrimer structure, we can infer those characteristics with the help of numerical descriptors known as the topological index. The Wiener and Szeged indices are two important distance-based topological indices applicable in nanoscience. The degree-based topological indices also have great importance and huge applications in structural chemistry. These indices together with graph entropy are found to be more effective and have found application in different sciences. In this work, the Wiener index, Szeged indices, Mostar indices, and Padmakar Ivan index for cyclen cored dendrimers are evaluated by converting the original graph into quotient graphs using Θ * - classes. This technique is applied since the regular cut method is a lengthy process while moving on to higher generations and also due to the presence of odd cycles in the structure. The degree-based indices and the degree-based graph entropies for the cyclen cored dendrimers are further studied. The comparison graphs with respect to the topological indices as well as graph entropies are also presented. [ABSTRACT FROM AUTHOR]

Determining a globally optimal solution of belief space planning (BSP) in high-dimensional state spaces directly is computationally expensive, as it involves belief propagation and objective function evaluation for each candidate action. However, many problems of interest, such as active SLAM, exhibit structure that can be harnessed to expedite planning. Also, in order to choose an optimal action, an exact value of the objective function is not required as long as the actions can be sorted in the same way. In this paper we leverage these two key aspects and present the topological belief space planning (t-bsp) concept that uses topological signatures to perform this ranking for information-theoretic cost functions, considering only topologies of factor graphs that correspond to future posterior beliefs. In particular, we propose a highly efficient topological signature based on the von Neumann graph entropy that is a function of graph node degrees and supports an incremental update. We analyze it in the context of active pose SLAM and derive error bounds between the proposed topological signature and the original information-theoretic cost function. These bounds are then used to provide performance guarantees for t-bsp with respect to a given solver of the original information-theoretic BSP problem. Realistic and synthetic simulations demonstrate drastic speed-up of the proposed method with respect to the state-of-the-art methods while retaining the ability to select a near-optimal solution. A proof of concept of t-bsp is given in a small-scale real-world active SLAM experiment. [ABSTRACT FROM AUTHOR]

Carbon nanotubes are one of the most extensively studied nanomaterials because of their remarkable mechanical and electrical properties. The Y-junction structures within carbon nanotubes have received significant attention in the field of nanotechnology, primarily due to their immense potential for powering the next generation of multi-terminal nanodevices. Topological indices play a crucial role in exploring the physicochemical properties and structural attributes of chemical compounds, as they are numerical values intricately linked to the molecular structure of these compounds. Moreover, graph-based entropies serve as essential thermophysical parameters used to quantify the heterogeneity and relative stabilities of molecular structures. In this article, we have utilized the NM-polynomial technique to calculate various neighborhood degree sum-based topological indices and graph-based entropies for carbon nanotube Y-junction graphs. [ABSTRACT FROM AUTHOR]

In 1997, Sierpinski graphs, S (n , k) , were obtained by Klavzar and Milutinovic. The graph S (1 , k) represents the complete graph K k and S (n , 3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by S i e (Λ , t) , already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph S i e (Λ , t) . In the second step, we extract some degree-based indices of a generalized Sierpinski graph S i e (Λ , t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph S i e (Λ , t) by using the Randić index. [ABSTRACT FROM AUTHOR]

Quantifying similarities/dissimilarities among different graph models and studying the irregularity (heterogeneity) of graphs in graphs and complex networks are one of the fundamental issues as well as a challenge of scientific and practical importance in many fields of science and engineering. This paper has been motivated by the necessity to establish novel and efficient entropy-based methods to quantify the structural irregularity properties of graphs, measure structural complexity, classify, and compare complex graphs and networks. In particular, we explore how criteria such as Shannon entropy, Von Newman, and generalized graph entropies, already defined for graphs, can be used to evaluate and measure irregularities in complex graphs and networks. To do so, we use some results obtained from graph spectral theory related to the construction of matrices obtained from graphs. We show how to use these irregularity indices based on graph entropies and demonstrate the usefulness and efficiency of each of these complexity measures on both synthetic networks and real-world data sets. [ABSTRACT FROM AUTHOR]

In this study, we analyzed structural changes in financial markets under COVID-19 to support investors' investment decisions. Because an explanation of these changes is necessary to respond appropriately to said changes and prepare for similar major changes in the future, we visualized the financial market as a graph. The hypothesis was based on expertise in the financial market, and the graph was analyzed from a detailed perspective by dividing the graph into domains. We also designed an original change-detection indicator based on the structure of the graph. The results showed that the original indicator was more effective than the comparison method in terms of both the speed of response and accuracy. Explanatory change detection of this method using graphs and domains allowed investors to consider specific strategies. [ABSTRACT FROM AUTHOR]

Objective: This paper applies graph methods to distinguish major depression disorder (MDD) and healthy (H) subjects using the graph features of single-channel electroencephalogram (EEG) signals. Methods: Four network features—graph entropy, mean degree, degree two, and degree three—were extracted from the 19-channel EEG signals of 64 subjects (26 females and 38 males), and then these features were forwarded to a support vector machine to conduct depression classification based on the eyes-open and eyes-closed statuses, respectively. Results: Statistical analysis showed that graph features with degree of two and three, the graph entropy of MDD was significantly lower than that for H (p < 0.0001). Additionally, the accuracy of detecting MDD using single-channel T4 EEG with leave-one-out cross-validation from H was 89.2% and 92.0% for the eyes-open and eyes-closed statuses, respectively. Conclusion: This study shows that the graph features of a short-term EEG can help assess and evaluate MDD. Thus, single-channel EEG signals can be used to detect depression in subjects. Significance: Graph feature analysis discovered that MDD is more related to the temporal lobe than the frontal lobe. [ABSTRACT FROM AUTHOR]

The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. The second law of thermodynamics establishes the concept of entropy as a physical property of a thermodynamic system. Graph entropies defined for various graph invariants have shown applications in different fields of physics, chemistry, statistics, biology, and social sciences. In this paper, we define graph entropies in terms of fifth geometric-arithmetic index, fourth atom-bond connectivity index and Sanskruti index. We computed the bounds of these graph entropies for arbitrary graphs and for molecular graphs of single-walled armchair carbon nanotube Y-junctions and constructed the regression equations between the computed entropies and topological invariants. [ABSTRACT FROM AUTHOR]

One of the fundamental challenges in reticular chemistry is the characterization of the complexity of the underlying molecular network, in which high complexity signifies low symmetry and high diversity. The entropy of a network is one such topological descriptor that serves to characterize the order/disorder structural complexity. In this paper, we obtain the entropy measures by computing systematic expressions of the self-powered degree-based topological indices for three isoreticular metal organic frameworks with a zinc based central unit and stringed binding linkers with varying benzene molecule counts. We have also enumerated the self-returning and nonreturning walks for these isoreticular structures in order to provide potential pathways for charge and ion transport. We have considered other properties such as eccentricities, radius, diameter, vertex and edge equivalence classes that facilitate rapid computations of thermochemistry and hence relative stabilities of isoreticular networks. [ABSTRACT FROM AUTHOR]

In this note we resolve three conjectures from Dehmer et al. (2019) on the comparison of distance measures based on the graph spectra, by constructing families of counterexamples and using computer search. In addition, we prove other results and conditions under which the conjectures hold true. [ABSTRACT FROM AUTHOR]

• We define the entropy of a graph using induced connected subgraphs (called graphlets) of different size. • We embed a graph in a feature space using entropies estimated from graphlets of different sizes. • We also include the higher-order graphlets in the feature vector to provide a more richer representation of a graph. • The higher-order graphlets can be computed in polynomial times and their inclusion can increase the classification accuracy. In this paper, we present a general framework to estimate the network entropy that is represented by means of an undirected graph and subsequently employ this framework for graph classification tasks. The proposed framework is based on local information functionals which are defined using induced connected subgraphs of different sizes. These induced subgraphs are termed graphlets. Specifically, we extract the set of all graphlets of a specific sizes and compute the graph entropy using our proposed framework. To classify the network into different categories, we construct a feature vector whose components are obtained by computing entropies of different graphlet sizes. We apply the proposed framework to two different tasks, namely view-based object recognition and biomedical datasets with binary outcomes classification. Finally, we report and compare the classification accuracies of the proposed method and compare against some of the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Marginal entropy is one of the distances based on the graph entropy. Then, this entropy is computed by the Wiener index of graphs. In this paper, we obtain the marginal entropy of paths, stars, double stars, cycles and vertex-transitive graphs. [ABSTRACT FROM AUTHOR]

Abstract Entropy-based methods are useful tools for investigating various problems in mathematical chemistry, computational physics and pattern recognition. In this paper we introduce a general framework for applying Shannon entropy to fullerene graphs, and used it to investigate their properties. We show that important physical properties of these molecules can be determined by applying Hosoya entropy to their corresponding graphs. [ABSTRACT FROM AUTHOR]

Let H be a hypergraph with n vertices. Suppose that d1,d2,...,dn are degrees of the vertices of H . The t-th graph entropy based on degrees of H is defined as I d t (H) = − ∑ i = 1 n ( d i t ∑ j = 1 n d j t log d i t ∑ j = 1 n d j t ) = log ( ∑ i = 1 n d i t ) − ∑ i = 1 n ( d i t ∑ j = 1 n d j t log d i t) , where t is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of I d t (H) for t = 1, when H is among all uniform supertrees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively. [ABSTRACT FROM AUTHOR]

Abstract Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between subgraph centrality, which is defined using the matrix exponential f (x) = exp ⁡ (x) , and the walk structure of a network. In a walk-regular graph, the number of closed walks of each length must be the same for all nodes, implying uniform f -subgraph centralities for any f (or maximum f - walk entropy). We consider when non-walk-regular graphs can achieve maximum entropy, calling such graphs entropic. For parameterized measures, we are also interested in which values of the parameter witness this uniformity. To date, only one entropic graph has been identified, with only two witnessing parameter values, raising the question of how many such graphs and parameters exist. We resolve these questions by constructing infinite families of entropic graphs, as well as a family of witnessing parameters with a limit point at zero. [ABSTRACT FROM AUTHOR]

In order to measure the amount of different information in a system, entropy concept can be used. Graph entropy measures nodes' contribution to the entropy of the graph. By this way, the influential actors can be identified. Due to this case, a new entropy-based method was proposed to identify the influential actors. Karcı entropy was applied to the social networks first time. The alpha parameter allowed us to combine many different conditions together when measuring in the network. The other important contribution of this paper is to predict the value of alpha parameter of Karcı entropy by using fuzzy logic. After that Karcı and Shannon entropies were compared based on experimental results. Moreover, Karcı entropy was compared to traditional centrality measures. If Karcı entropy definition is considered as a set of entropies, Shannon entropy can be regarded as an element of this set. Accordingly, it can be concluded that Karcı entropy is superior to Shannon entropy. • Applying graph entropy to identify influential nodes instead of traditional centrality measures. • Karcı entropy was applied to graph entropy for the first time. • Karcı entropy's α value was determined by using fuzzy α selection algorithm. • Karcı entropy is a promised method for identifying influential node in social networks. • Karcı entropy is more universal with respect to Shannon entropy. [ABSTRACT FROM AUTHOR]

The advent of borophene sheets and their nanoribbons has inspired significant experimental and theoretical research on the 2D-sheets and nanoribbons of boron, wonder materials of the future with the potential to replace graphene. As a result of the electron deficient nature of boron compounds, borophenes exhibit multiple phases with contrasting network structures. The present study employs novel graph theoretical techniques based on neighborhood degree-sum parameters that provide refined bond partitions to characterize the different phases of borophene nanoribbons. We have applied these techniques to the β 12 and χ 3 phases of borophene nanoribbons. We have also employed computer-assisted combinatorial techniques to generate the boron NMR spectra of different phases of the borophene nanoribbons. • Novel graph theoretical techniques are utilized to compute the topological properties of two phases of borophenes. • Phases of boron nanoribbons, encompassing various boundary configurations, are contrasted by their topological properties. • Degree-sum topological indices and entropies of different phases of borophenes are juxtaposed to show their efficacy. • Machine-generated combinatorial 11B NMR spectra of two phases of borophenes show that 11B NMR can be a powerful tool. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel framework of computing the Quantum-based Entropic Representations (QBER) for un-attributed graphs, through the Continuous-time Quantum Walk (CTQW). To achieve this, we commence by transforming each original graph into a family of k -level neighborhood graphs, where each k -level neighborhood graph encapsulates the connected information between each vertex and its k -hop neighbor vertices, providing a fine representation to reflect the multi-level topological information for the original global graph structure. To further capture the complicated structural characteristics of the original graph through its neighborhood graphs, we propose to characterize the structure of each neighborhood graph with the Average Mixing Matrix (AMM) of the CTQW, that encapsulates the time-averaged behavior of the CTQW evolved on the neighborhood graph. More specifically, we show how the AMM matrix allows us to compute a Quantum Shannon Entropy for each vertex, and thus compute an entropic signature for each neighborhood graph by measuring the averaged value or the Jensen–Shannon Divergence between the entropies of its vertices. For each original graph, the resulting QBER is defined by gauging how the entropic signat ures vary on its k -level neighborhood graphs with increasing k , reflecting the multi-dimensional entropy-based structure information of the original graph. Experiments on standard graph datasets demonstrate the effectiveness of the proposed QBER approach in terms of the classification accuracies. The proposed approach can significantly outperform state-of-the-art entropic complexity measuring methods, graph kernel methods, as well as graph deep learning methods. • We propose a novel framework of computing Quantum-based Entropic Representations (QBER) for un-attributed graphs. • The proposed QBER approaches are based on measuring how the entropic signatures vary on the family of the k-level neighborhood graphs. • The entropic signatures are computed through the Average Mixing Matrix (AMM) of the Continuous-time Quantum Walk (CTQW). • The proposed QBER can simultaneously capture both global and local structural characteristics through the AMM matrix of the CTQW. • Experiments on standard graph datasets demonstrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. The eccentricity of vertex v is the maximum distance between v and other vertices of G . In this paper, we study a new version of graph entropy based on eccentricity of vertices of G . In continuing, we study this graph entropy for some classes of graph operations. Finally, we compute the graph entropy of two classes of fullerene graphs. [ABSTRACT FROM AUTHOR]

A graph is said to be walk-regular if, for each ℓ ≥ 1 , every vertex is contained in the same number of closed walks of length ℓ . We construct a 24-vertex graph H 4 that is not walk-regular yet has maximized walk entropy, S V ( H 4 , β ) = log ⁡ 24 , for some β > 0 . This graph is a counterexample to a conjecture of Benzi (2014) [1, Conjecture 3.1] . We also show that there exist infinitely many temperatures β 0 > 0 so that S V ( G , β 0 ) = log ⁡ n G if and only if a graph G is walk-regular. [ABSTRACT FROM AUTHOR]

Abstract: The entropy of a digraph is a fundamental measure that relates network coding, information theory, and fixed points of finite dynamical systems. In this article, we focus on the entropy of undirected graphs. We prove any bounded interval only contains finitely many possible values of the entropy of an undirected graph. We also determine all the possible values for the entropy of an undirected graph up to the value of four. [ABSTRACT FROM AUTHOR]

Gene networks are beneficial to identify functional genes that are highly relevant to clinical outcomes. Most of the current methods require information about the interaction of genes or proteins to construct genetic network connection. However, the conclusion of these methods may be bias because of the current incompleteness of human interactome. In this paper, we propose an efficient strategy to use gene expression data and gene mutation data for identifying cancer-related key genes based on graph entropy (iKGGE). Firstly, we construct a gene network using only gene expression data based on the sparse inverse covariance matrix, then, cluster genes use the algorithm of parallel maximal cliques for quickly obtaining a series of subgraphs, and at last, we introduce a novel metric that combine graph entropy and the influence of upstream gene mutations information to measure the impact factors of genes. Testing of the three available cancer datasets shows that our strategy can effectively extract key genes that may play distinct roles in tumorigenesis, and the cancer patient risk groups are well predicted based on key genes. [ABSTRACT FROM AUTHOR]

For embedding virtual networks into a large scale substrate network, a massive amount of time is needed to search the resource space even if the scale of the virtual network is small. The complexity of searching the candidate resource will be reduced if candidates in substrate network can be located in a group of particularly matched areas, in which the resource distribution and communication structure of the substrate network exhibit a maximal similarity with the objective virtual network. This work proposes to discover the optimally suitable resource in a substrate network corresponding to the objective virtual network through comparison of their graph entropies. Aiming for this, the substrate network is divided into substructures referring to the importance of nodes in it, and the entropies of these substructures are calculated. The virtual network will be embedded preferentially into the substructure with the closest entropy if the substrate resource satisfies the demand of the virtual network. The experimental results validate that the efficiency of virtual network embedding can be improved through our proposal. Simultaneously, the quality of embedding has been guaranteed without significant degradation. [ABSTRACT FROM AUTHOR]

The computation of a set constituted by few vertices to define a virtual backbone supporting information interchange is a problem that arises in many areas when analysing networks of different natures, like wireless, brain, or social networks. Recent papers propose obtaining such a set of vertices by computing the connected dominating set (CDS) of a graph. In recent works, the CDS has been obtained by considering that all vertices exhibit similar characteristics. However, that assumption is not valid for complex networks in which their vertices can play different roles. Therefore, we propose finding the CDS by taking into account several metrics which measure the importance of each network vertex e.g., error probability, entropy, or entropy variation (EV). [ABSTRACT FROM AUTHOR]

In 1999, De Simone and Körner conjectured that every graph without induced C 5 , C 7 , C ¯ 7 contains a clique cover C and a stable set cover I such that every clique in C and every stable set in I have a vertex in common. This conjecture has roots in information theory and became known as the Normal Graph Conjecture. Here we prove that all C 4 -free graphs of bounded maximum degree and sufficiently large odd girth (linear in the maximum degree) are normal. This is used to prove that for every fixed d , random d -regular graphs are a.a.s. normal. [ABSTRACT FROM AUTHOR]

Entropy-based measures have been used to characterize the structure of complex networks. Various graph entropies based on different graph parameters have been proposed and studied. So, it has been intricate to decide which measure is the right one for solving a particular problem. In this paper, we introduce graph entropy measures based on independent sets and matchings of graphs. The values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures. Based on our findings, we conclude that it should be demanding studying extremal values of the novel entropies. [ABSTRACT FROM AUTHOR]

This paper investigates the design of information theoretic-based fitness function for embedded evolutionary robotics (ERs). Such fitness relies on the assumption that interesting behaviors result in a high sensorimotor (individual) diversity. The current simple entropy as a diversity metric only considers individuals' difference but ignores their spatial relationship. The sensorimotor stream can be analyzed to construct a simple directed graph that has unique entry and exit nodes. This paper proposes a hierarchic entropy as a diversity metric by incorporating the simple entropy and the spatial relationship based graph entropy. Maximizing the hierarchic entropy, achieved by on-board evolutionary algorithm, thus defines a self-driven fitness function enforcing the controller visiting diverse sensorimotor states. The proposed algorithm achieves better performance than the published results of other entropy-based methods only relying on simple entropy, without requiring additional computational resources. [ABSTRACT FROM AUTHOR]

Inspired by the generalized entropies for graphs, a class of generalized degree-based graph entropies is proposed using the known information-theoretic measures to characterize the structure of complex networks. The new entropies depend on assigning a probability distribution about the degrees to a network. In this paper, some extremal properties of the generalized degree-based graph entropies by using the degree powers are proved. Moreover, the relationships among the entropies are studied. Finally, numerical results are presented to illustrate the features of the new entropies. [ABSTRACT FROM AUTHOR]

The estimation of mutual information between graphs has been an elusive problem until the formulation of graph matching in terms of manifold alignment. Then, graphs are mapped to multi-dimensional sets of points through structure preserving embeddings. Point-wise alignment algorithms can be exploited in this context to re-cast graph matching in terms of point matching. Methods based on bypass entropy estimation must be deployed to render the estimation of mutual information computationally tractable. In this paper the novel contribution is to show how manifold alignment can be combined with copula-based entropy estimators to efficiently estimate the mutual information between graphs. We compare the empirical copula with an Archimedean copula (the independent one) in terms of retrieval/recall after graph comparison. Our experiments show that mutual information built in both choices improves significantly state-of-the art divergences. [ABSTRACT FROM AUTHOR]

• The majorization relationships between the continuous PDFs and their discrete area sampling PDFs are analyzed, which provides a theoretical basis to compare the connectivity of the graphs converted from two different distributed random samples, and the probabilities of the graphs being complete. • A novel graph-based confidence verification algorithm to evaluate the analysis results of BPSK signals is proposed, where the block sum of the correlation spectrum is selected as an input fed to the signal-to-graph converter, and the entropy of the quantized samples' histogram is utilized as a novel feature to check the graph's completeness, which results in better performance and lower computational cost than the conventional graph-based detector using the second largest eigenvalue of the Laplacian matrix. • Compared with existing algorithms, the proposed algorithm can obtain superior performance under lower SNRs in the presence of transmission impairments including channel fading, timing offset, Doppler frequency shift, and phase noise. Confidence verification for intercepted signal analysis has important military applications where a priori knowledge of the signal or transmission channel is insufficient or absent. However, existing methods do not perform well under low signal-to-noise ratios (SNRs). In this study, we propose a graph-based verification algorithm for binary phase shift keying (BPSK) signal analysis. The correlation spectrum between the original observed signal and reference signal can be divided into blocks of equal size and the summation of each block (called the block summation (BS) spectrum) obtained. Then, the BS spectrum samples are transformed into a specific simple undirected graph using the signal-to-graph convertor. Accordingly, the verification issue can be simplified to determining whether the graph is fully-connected. The majorization relationships between the probability density functions of the BS and the original correlation spectra are evaluated. This is used to compare the connectivity of the graphs generated from those two types of samples, as well as their probabilities of being complete graphs. Simulation results confirm the superior performance of the proposed algorithm over previous confidence verification algorithms for low SNRs and in the presence of transmission impairments, as well as a lower computational complexity when compared with the conventional graph-based algorithm. [ABSTRACT FROM AUTHOR]

The high incidence of brain tumor has increased significantly in recent years. It is becoming more and more concernful to discover knowledge through mining medical brain image to aid doctors' diagnosis. Clustering medical images for Intelligent Decision Support is an important part in the field of medical image mining because there are several technical aspects which make this problem challenging. In this paper, we propose a medical brain image clustering method to find similar pathology images that can assist doctors to analyze the specific disease, discover its potential cause and make more accurate treatment. Firstly, this method represents medical brain image dataset as a weighted, undirected and complete graph. Secondly, this graph is sparsified so as to describe the similarity of medical images very well. Last but not the least, a graph entropy based clustering method for this sparsified graph is proposed to cluster these medical images. The experimental results show that this method can cluster medical images efficiently and run well in time complexity. The clustering results can better describe the similarity of medical images. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let F G ( 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 to F G ( P ) if the distribution P ∗ maximizing F G ( 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. [ABSTRACT FROM AUTHOR]

The degree-based entropy I d (G) of a graph G on m > 0 edges is obtained from the well-known Shannon entropy − ∑ i = 1 n p (x i) log p (x i) in information theory by replacing the probabilities p (x i) by the fractions d G (v i) 2 m , where { v 1 , v 2 , ... , v n } is the vertex set of G , and d G (v i) is the degree of v i. We continue earlier work on I d (G). Our main results deal with the effect of a number of graph operations on the value of I d (G). We also illustrate the relevance of these results by applying some of these operations to prove a number of extremal results for the degree-based entropy of trees and unicyclic graphs. [ABSTRACT FROM AUTHOR]

• We provide a new graph entropy design to measure the smoothness of graph feature manifold and conclude that motif-based information structures determine this graph entropy. • We propose a novel graph data augmentation strategy that protects not only the integrity of topological structure but also the integrity of motif-based information structures. Our strategy shows advantages in maintaining more graph entropy compared with other methods. • We demonstrate our model on different datasets, and experiments show that our work is superior in real-world node classification tasks. • Our approach significantly enhances the robustness of GCN and could alleviate the over-smoothing phenomenon to a certain extend. The Graph Convolutional Networks (GCN) proposed by Kipf and Welling is an effective model to improve semi-supervised learning of pattern recognition, but faces the obstacle of over-smoothing, which will weaken the representation ability of GCN. Recently some works are proposed to tackle above limitation by randomly perturbing graph topology or feature matrix to generate data augmentations as input for training. However, these operations inevitably do damage to the integrity of information structures and have to sacrifice the smoothness of feature manifold. In this paper, we first introduce a novel graph entropy definition as a measure to quantitatively evaluate the smoothness of a data manifold and then point out that this graph entropy is controlled by triangle motif-based information structures. Considering the preservation of graph entropy, we propose an effective strategy to generate randomly perturbed training data but maintain both graph topology and graph entropy. Extensive experiments have been conducted on real-world datasets and the results verify the effectiveness of our proposed method in improving semi-supervised node classification accuracy compared with a surge of baselines. Beyond that, our proposed approach could significantly enhance the robustness of training process for GCN. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Using a simple relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer–dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients m k are always positive. These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds Δ k ln ( i ! N ( i ) ) ≤ 0 for k ≥ 2 , where N ( i ) is the number of configurations of i dimers on the graph and Δ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients. Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. Moreover, in an exhaustive survey of some classes of regular biconnected graphs with a not too large number v of vertices, we observe only few violations of these bounds. We conjecture that the frequency of the violations vanishes as v → ∞ . Using an inequality by Heilmann and Lieb, we find rigorous upper bounds on N ( i ) valid for arbitrary graphs and for regular graphs. The similarity between this inequality and the one conjectured above suggests that one study the stricter inequality m k ≥ 1 2 k for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we continue studying degree-based entropies for networks. The quantities represent information-theoretic network measures which are based on using information functionals involving vertex degrees. We prove bounds for entropies which are based on information functionals using degree powers and come up with interrelations between different measures. Such interrelations are important to study connections between the measures required to understand the measures in depth. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Randic weight. As a result, we derived statements on dendrimers that have been proven useful for applications. Finally, some open problems are presented. [ABSTRACT FROM AUTHOR]

The entropy of a monomer–dimer system on an infinite regular bipartite lattice can be written as a mean-field part plus a series expansion in the dimer density. In a previous paper it has been conjectured that all coefficients of this series are positive. Analogously on a connected regular graph with v vertices, the “entropy” of the graph ln N ( i ) / v , where N ( i ) is the number of ways of setting down i dimers on the graph, can be written as a part depending only on the number of the dimer configurations over the completed graph plus a Newton series in the dimer density on the graph. In this paper, we investigate for which connected regular graphs all the coefficients of the Newton series are positive (for short, these graphs will be called positive). In the class of connected regular bipartite graphs, up to v = 20 , the only non positive graphs have vertices of degree 3 . From v = 14 to v = 30 , the frequency of the positivity violations in the 3 -regular graphs decreases with increasing v . In the case of connected 4 -regular bipartite graphs, the first violations occur in two out of the 2806490 graphs with v = 22 . We conjecture that for each degree r the frequency of the violations, in the class of the r -regular bipartite graphs, goes to zero as v tends to infinity. This graph-positivity property can be extended to non-regular or non-bipartite graphs. We have examined a large number of rectangular grids of size N x × N y both with open and periodic boundary conditions. We have observed positivity violations only for min ( N x , N y ) = 3 or 4 . [ABSTRACT FROM AUTHOR]