On Entropy of Some Fractal Structures

This paper aims to predict some possible futures for the World Wide Web based on several key network parameters: size, complexity, cost and increasing connection speed thorough the uptake of broadband technology. This is done through the production of a taxonomy specifically evaluating the stability properties of the fully-connected star and complete networks, based on the Jackson and Wolinsky (1996) connections model modified to incorporate complexity concerns. We find that when connection speeds are low neither the star nor complete networks are stable, and when connection speeds are high the star network is usually stable, while the complete network is never stable. For intermediate speed levels much depends upon the other parameters. Under plausible assumptions about the future, we find that the Web may be increasingly dominated by a single intermediate site, perhaps best described as a search engine.

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

Entropic measures of complexity are able to quantify the information encoded in complex network structures. Several entropic measures have been proposed in this respect. Here we study the relation between the Shannon entropy and the von Neumann entropy of networks with given expected degree sequence. We find in different examples of network topologies that when the degree distribution contains some heterogeneity, an intriguing correlation emerges between the two entropic quantities. This results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks, which respectively corresponds to the von Neumann and the Shannon entropy.

this paper proposed the modification of the classic Shannon's formula of information entropy by utilizing a concept that we discovered on April 28, 2016 namely Harmony in Gradation - the Formula for Everything. In more detail, the current formula for calculating information entropy contains the weaknesses because it only contains we call the Harmony, whereas every correct formula that involves the union or random variable which has more than two elements refer to the Harmony in Gradation concept should contain two conflicting elements, namely the Harmony and the Gradation. Then, there are two steps for preparing the modification of Shannon's entropy formula, which the first is making a formula that has at once the Harmony and the Gradation but has the opposite understanding of the Shannon's entropy formula, and we call it the Cavity channel formula, then the second is compiles the entropy modified formula. Furthermore, in this paper, it is proven that by using the formula proposed in this paper, the calculation of entropy will be more accurate compared to the use of the Shannon's formula and will be more in line with the reality of the measurement results. Finally, there are three weaknesses of Shannon's formula which are simply proven here, first is an error in the maximum entropy value, the second is an inaccurate error represents harmony between symbols, and third is not being able to see significant gradation changes in the appearance of symbols.

Entropy has been widely measured the complexity of complex networks. At present, many measures about entropies were defined based on the directed connection of nodes. The similarity of nodes can better represent the relationship among all nodes in complex networks. In the definition of similarity of nodes, the importance of a node in the network depends not only on the degree of the node itself, but also on the extent of dependence of neighboring nodes on the node. In this paper, we proposed a new structure entropy based on nonextensive statistical mechanics and similarity of nodes. In the proposed method, the similarity of nodes and the betweenness of nodes are both quantified. The proposed method considers the extent of dependence between neighbouring nodes. For some complex networks, the proposed structure entropy can distinguish complexity of that while other entropies can not be. Meanwhile, we construct five ER random networks and small-world networks and some real-world complex networks such as the US Air Lines networks, the GD'01-GD Proceedings Self-Citing networks, the Science Theory networks, the Centrality Literature networks and the Yeast networks are measured by the proposed method. The results illustrated our method for quantifying the complexity of complex networks is effective.

The physicochemical properties of molecules are correlated with their chemical structure using degree-based topological indices. The graph entropy zone has captured the imagination of scientists because of its possible application in chemistry. Boron nanotubular structures are high-interest materials with new electronic characteristics due to the presence of multi center bonds. Mechanical and thermal stability are two challenges that these materials have in nanodevice applications. Therefore, they necessitate theoretical research into the other qualities. In this article, study the graph entropy measure by using topological invariants of different versions of atom-bond connectivity index Zagreb indices (including redefined forms and and the forgotten topological index F(G)) which gives a better prediction of the chemical properties of different chemical compounds. We computed these indices for Boron Nanotube because of its significance in nanotechnology. To ensure the correctness of the output, a computational analysis and graphical comparison of generated indices is provided.

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[Formula: see text], TNT[Formula: see text] and carbon nanotubes HAC[Formula: see text] by making a relation of degree-based topological indices with the help of information function.

A computational study of entropy measures and topological indices for Magnesium Nitride using curve fitting techniques.

The Shannon entropies in both coordinate and momentum spaces for some low-lying eigenstates of the Hulthén potential (HP) are investigated over a wide range of screening parameters where bound states exist. The system wave functions are expanded in terms of multiple groups of Slater-type orbitals and the Hamiltonian is solved by employing the Rayleigh–Ritz variational method. The accuracy of the obtained bound-state eigenenergies, Shannon entropies, and radial mean values are validated by comparing with the analytical solutions of the HP in s -wave states. The Bialynicki–Birula and Mycielski inequalities for Shannon entropies in both the coordinate and momentum spaces as well as their sum are examined in both weak and strong screening situations. It is shown that the variation of Shannon entropy sum is governed by its upper bound for the system in a well-defined bound state. In the critical bound region where the system undergoes a bound-continuum transition, the Shannon entropies and entropy sum differ in states by their orbital angular momenta and their asymptotic behavior can be properly shaped by the rigorous upper and lower bounds in terms of the radial and momentum expectation values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>r</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>p</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We finally extended the calculation to a modified HP (MHP) that has been widely discussed in the literature and compared our results with previous predictions. Benchmark Shannon entropies are provided for further investigation of the information measures of MHPs.

We demonstrate that the presence of a single dissimilar chaotic system suppresses chaos in networks of chaotic oscillators, in a diverse set of network topologies, for sufficiently strong coupling. The key property is determined to be the sum of the path lengths between the dissimilar node and all the other nodes (or its maximum, if coupled to unconnected networks), and there exists a linear relation between this quantity and the critical coupling strength for the onset of a spatiotemporal fixed point. This holds true for a chain with the dissimilar node at different locations, a ring and complete network with one embedded dissimilar node, as well as star networks with a dissimilar hub or dissimilar peripheral node. Furthermore, we show that networks with high average degree and high clustering coefficient are more resilient to the influence of an external dissimilar system. These findings will potentially aid in the design of optimally placed dissimilar nodes for controlling chaos in complex networks.

The topic of network quantum nonlocal correlations arises from the large-scale quantum communications. It has been demonstrated that non-m-local correlations (m is the number of sources) in some special kinds of networks will decay under the influence of noises from imperfect devices as these networks extend, including linear and star networks. Furthermore, we observe that under the same noisy level, persistency of non-multi-local correlations in aforementioned different kinds of networks appears obvious discrepancy. Therefore, it becomes a natural and challenging task to check the persistency of non-multi-locality in more complex acyclic network, since star and linear networks can be regarded as acyclic networks. It is worth noting that an acyclic network is one indeterminate forked tree-shaped networks. So the study is focused on the tree-shaped network. We first devote to discussing the determinate forked case. We obtain inequality persistency criteria of determinate k-forked tree-shaped network non-(s n (k))-local correlations (s n (k) is the number of total sources). We show that the larger the fork number k is larger, the better the non-multi-locality of this network persists. In the indeterminate forked case, the inequality persistency criteria are also established. We observe that the larger the number of parties in the last layer of an acyclic network, the better the non-multi-locality of this network is persisted. This theoretical research may guide us how to build quantum networks with the stronger correlation in practical communications.

Understanding the dynamics of microbiome-host interactions under antibiotic-driven selective pressure is critical for addressing microbial dysbiosis and ensuring host health. While compositional and functional analyses have advanced our knowledge of microbial ecosystems, they often fall short in identifying the structural roles of specific microbial taxa. Keystone taxa species that disproportionately influence microbiome stability and host physiology are pivotal in maintaining ecological balance. However, quantifying their systemic influence remains challenging, especially under perturbations such as antibiotic exposure. This study presents a novel framework leveraging network entropy metrics to evaluate the influence of keystone taxa in microbial co-occurrence networks during antibiotic treatment. By applying information-theoretic measures such as Shannon entropy, local node entropy, and global network entropy, we capture how antibiotic-induced disruption alters microbial interaction complexity and the centrality of key taxa. Using longitudinal 16S rRNA sequencing data from murine gut microbiomes subjected to broad-spectrum antibiotics, we reconstructed dynamic interaction networks and tracked entropy changes over time. Our findings reveal that antibiotics reduce overall network entropy, indicating collapse in microbial diversity and connectivity. More importantly, taxa with high local entropy shifts particularly Bacteroides and Lactobacillus species demonstrated outsized influence on network restructuring and host inflammatory markers. These results suggest that entropy-based metrics can identify functional keystone species beyond mere abundance or frequency. This entropy-driven framework offers a scalable, quantitative approach to evaluate microbial resilience, optimize probiotic interventions, and inform precision antimicrobial therapies. Future research can integrate multi-omic and spatial data to refine these insights across different host niches and microbial ecosystems.

A key objective of monitoring networks is to identify potential service threatening outages from events within the network before service is interrupted. Identifying causal events, Root Cause Analysis (RCA), is an active area of research, but current approaches are vulnerable to scaling issues with high event rates. Elimination of noisy events that are not causal is key to ensuring the scalability of RCA. In this paper, we introduce vertex-level measures inspired by Graph Entropy and propose their suitability as a categorization metric to identify nodes that are a priori of more interest as a source of events. We consider a class of measures based on Structural, Chromatic and Von Neumann Entropy. These measures require NP-Hard calculations over the whole graph, an approach which obviously does not scale for large dynamic graphs that characterise modern networks. In this work we identify and justify a local measure of vertex graph entropy, which behaves in a similar fashion to global measures of entropy when summed across the whole graph. We show that such measures are correlated with nodes that generate incidents across a network from a real data set.

Information entropy has played a key role in a wide range of disciplines, for instance, classical and quantum information processing, quantum computing, quantum dynamics and quantum metrology. Here, we develop an information theoretic formalism using Shannon entropy, to investigate the quantum dynamics of Hamiltonian systems with position-dependent mass. Such systems are of fundamental interest in many areas, for instance, condensed matter, mathematical physics and foundations of quantum mechanics. We explore the phenomenon of fractional revivals for the temporal evolution of wave-packet solutions of Schrödinger equation with position-dependent mass by studying, analytically and numerically, the time-development of Shannon information entropy in position and momentum spaces. It is shown by our numerical results that the effect of spatially varying mass on the fractional revivals can not be fully harnessed using conventional measures, for instance, autocorrelation function. However, based on our numerical analysis it is concluded that information entropy is not only more sensitive to identify the fractional revivals but it also better elucidates the effect of position-dependent mass on the structure of fractional revivals in the form of symmetry breaking.