Fractal Box Research Articles

Kinematics, intensity, and spatial arrangement of extensional fractures in the Tuxtla-Malpaso fault system: Chiapas strike-slip fault province, southern Mexico

We describe the orientation, number, size, distribution, and fracture spatial arrangement in the central part of the kilometric Tuxtla-Malpaso fault system, one of the largest in the south of Mexico which cuts Cretaceous calcareous rocks. In this area, there are two sub-vertical extensional fracture sets (T1 and T2). The T1 fractures have ∼ N–S orientation and recorded 4.48 % of extension. The T2 fractures cut the T1 fractures, have NE-SW orientation, and accommodate an amount of extension of 3.87 %. These two fracture sets are parallel to two normal fault groups. The youngest normal faults (F2) have NE-SW strike and cut the N–S normal faults (F1). The F1 faults record a reactivation as strike-slip faults contemporaneous with F2.The T1 and T2 fractures have a fracture intensity between 9.08 and 9.39 (P10) on mesoscopic scale. In the case of the spacing, the value of this parameter (0.28–1) is related to fracture clustering. This behavior agrees with the values of fractal box dimension (0.72–0.81), correlation dimension (0.24–0.6), Lyapunov exponent (1.31–2.42), and coefficient of variation (0.94–1.85) on mesoscopic scale. The cumulative aperture vs. distance shows that T1 and T2 fractures have heterogeneous strain distribution. The fracture spatial arrangement is verified with the normalized correlation count analysis (NCC). The NCC graphs show a clustering behavior for T1 and T2 fractures, which are related to a regulary-spaced fractal clusters pattern. We suggest that this pattern is related to the mechanical stratigraphy and the strain location in the fault zones, while the fracture intensity depends on the degree of fracture clustering and rock type more than the amount of strain.Finally, the T1 fractures were associated with the right strike-slip kinematic for the NW-SE Tuxtla-Malpaso fault. This activity recorded a transtension behavior during the middle Miocene. The T2 fractures were formed during the reactivation (Pliocene) of the Tuxtla-Malpaso fault as the left strike-slip fault behind wrenching-transpression kinematics.

RF fingerprint extraction and device recognition algorithm based on multi-scale fractal features and APWOA-LSSVM

RF fingerprints can be used for device identification and network access authentication. An RF fingerprint extraction and device identification algorithm based on multi-scale fractal features and APWOA-LSSVM is proposed. First, a Hilbert transform is performed on the original RF signal, and differential operations are performed on the I/Q signal. Then, the variational modal decomposition is performed on the differentially processed I/Q signals separately to obtain a number of band-limited Intrinsic Modal Function (IMF) components. The fractal box dimension of each IMF component is calculated separately as the first-dimensional RF fingerprint. The multi-fractal spectrum of the original RF signal is further computed by Multi-fractal Detrended Fluctuation Analysis (MFDFA) as the second-dimensional RF fingerprint. The RF fingerprint feature vector consists of a combination of the first- and second-dimensional RF fingerprints and is used to train the LSSVM model. Since the penalty factor and kernel parameter width directly affect the recognition accuracy and generalization ability of the LSSVM model, APWOA is used to optimize the hyperparameters of the LSSVM model. Finally, the LSSVM model is constructed using the optimal hyperparameters and the performance of the model is verified on a test dataset. The experimental results show that the proposed model achieves an average recognition accuracy of 99.13%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} for 16 Bluetooth devices. Multi-scale fractal features consisting of fractal box dimensions and multiple fractal dimensions are more beneficial in the recognition of wireless devices as compared to single features. In addition, the proposed model possesses superior recognition capabilities when compared to classical machine learning algorithms.

IMPLEMENTASI DIMENSI FRAKTAL BOX COUNTING DAN K-MEANS DALAM KLASIFIKASI JENIS KUPU-KUPU (LEPIDOPTERA) BERDASARKAN BENTUK SAYAP

Butterflies are insects that have aesthetic beauty and are also important assets in the field of nature protection. In nature, butterflies have a crucial role in maintaining ecosystems, especially in the process of pollination of flowering plants which contributes to the natural propagation of plants. Butterfly wings have an important role in identifying butterfly species based on size, shape and color. In mathematics, there is the concept of fractals which is a very suitable method for modeling natural phenomena and used as a tool to represent natural objects. In this study, a classification of butterfly species was carried out based on the shape of the butterfly's wings. A total of 100 butterfly images were used in this study, the first step was to convert the RGB images to grayscale and then segment the butterfly wings. The area is then used to identify butterfly wing type patterns using edge detection using the Canny method. Furthermore, by using the fractal box counting method, the resulting dimensional values are used in the clustering process. The results of the dimension values will be classified using the K-Means grouping method with five groups, namely Adonis Butterfly, Julia Butterfly, Brown Siproeta Butterfly, Metal Marks Butterfly, and African Giant Swallowtail Butterfly, showing an accuracy of 89%.

Analogue study on the classification and identification of landslide dynamics based on spectral features of slidequake signals

Landslides are powerful, destructive events that can generate slidequakes. Monitoring passive slidequake signals can facilitate hazard assessments, dynamic calculations, and early warning. However, there have been few direct observations of landslide dynamics. This hampers the classification and identification of landslide movements based on spectral features of slidequake signals. Here, physical analogue tests were conducted to simulate the landslide movements of three boundary conditions: open flat (OF), gully constrained curve (GC) and barrier blocking (BB). Different slope gradients, movement paths, blocking conditions and slide material structures were employed in 96 experiments to analyse the spectral features of slidequake signals and compensate for the lack of active observation data. Via spectrum identification, we analysed the vibration characteristics of the collected three-component high-frequency slidequake signals of analogue modelling landslides. Then, we modelled the landslide dynamics through feature extraction of the fractal box dimension of the slidequake signals. The OF and BB slidequake signals were identified with a prediction accuracy of 87.5%. Hence, the accurate classification and identification and capability for early warning of remote landslide dynamics based on passive slidequake signal monitoring is presented.

Recognition of rock materials after high-temperature deterioration based on SEM images via deep learning

Accurately identifying the high-temperature history experienced by rocks is essential for understanding their behaviour and predicting properties. However, current approaches are limited by the heterogeneity of rocks, test scale and costs. Here, we proposed an economical, efficient and accurate approach to identifying the rocks after high-temperature deterioration via deep learning. This deep learning-based method exhibited superior abilities in distinguishing the heat-treated rock. Using a scanning electron microscopy (SEM) image covering a size of 14.6 μm × 14.6 μm, the high-temperature deterioration history of rocks can be recognized with an accuracy of 80.2%. Features such as cracks, rock patterns, and cleavage steps in SEM images would further improve the recognition accuracy. For example, SEM images with higher fractal box dimensions show a higher recognization accuracy, especially for temperatures under 600 °C. Besides, using the deep Taylor decomposition algorithm, the high-temperature deterioration regions of the rocks in the microscale were successfully located, extracted, and characterized for the first time. This study highlights the vast potential of the deep learning-based approach in damage deterioration identification of rock after high temperature, which significantly extends the application of deep learning in underground projects.

Waveforms Eavesdropping Prevention Framework: The Case of Classification of EPG Waveforms of Aphid Utilizing Wavelet Kernel Extreme Learning Machine

ABSTRACT Since all information depends solely on the training data, machine learning algorithms typically do not employ external knowledge or other experiences during the learning process. Methods for machine learning have been rigorously tested against novel varieties of highly technical “black box” or “white box” adversarial attacks. By employing attacks, attackers can change systems to serve a harmful end goal. When authorized implementers and eavesdroppers are geographically close together, it is difficult to perform secure beamforming in waveform applications, for instance, leading to erroneous beam forms and, as a result, disastrous beam leakages. As a result, the first move in a prospective black-box offense will be based on the waveform features of a learning signal. By including a non-orthogonality concept into the physical layer signal waveform, the Waveforms Eavesdropping Prevention Framework (WEPF) proposed in this work aims to boost machine learning security to address these difficulties. The implementation scenario is based on a waveforms scenario used to categorize the Electrical Penetration Graph (EPG) for insects, a crucial tool for researching the feeding conduct of piercing-sucking insects and the transition mechanism between viruses and insects. An attribute vector with six dimensions, consisting of low-frequency wavelet energy (LFWE) in the second and third layers of the Wavelet Kernel Extreme Learning Machine, fractal box dimension (FBD), the Hurst exponent (HE), and spectral centroid (SC) in the first two layers of the HHT, was used to test the proposed framework. Two adversarial scenarios were explored. However, the suggested architecture secures all waveform signals, demonstrating the method’s effectiveness in lowering the risk of eavesdropping or tampering with the waveforms used in advanced machine-learning methods.

Spatial Analysis of Structure and Metal Mineralization Based on Fractal Theory and Fry Analysis: A Case Study in Nenjiang−Heihe Metallogenic Belt

Regional tectonics can provide excellent transport channels and precipitation sites for mineralized hydrothermal fluid. Studying the spatial relationship and distribution trends of regional tectonics and metal mineralization has theoretical and practical significance for revealing regional mineralization regularities and guiding mineral exploration. This study considers the Nenjiang-Heihe metallogenic belt, through the fractal box dimension method and Fry analysis, to explore the spatial distribution characteristics and patterns of tectonics. The results were as follows. (1) NE and NW directions are the main tectonic directions in the study area, with high-density areas concentrated in the central-eastern and central-western regions, demonstrating an overall ring-like distribution pattern. (2) Fractal dimensions of the linear structures of the NE and NW directions and the entire study area are 1.543, 1.493, and 1.622, respectively, with a strong coupling relationship between the lineament fractal high-value area and rhombic-grid spatial distribution of known deposits. (3) Gold mineralization shows the NEE and NWW directions as two main mineralization trends; the intersection area is the gold-potential area. The main trend direction of the Cu-Mo metallogenic trend belt is the NNW direction; the intersection area with the NEE direction gold metallogenic trend belt is the Au-Cu-Mo potential mineralization area.

Automatic Identification System for Rock Microseismic Signals Based on Signal Eigenvalues

The microseismic signals of rock fractures indicate that the rock mass in a particular area is changing slowly, and the microseismic signals of rock blasting indicate that the rock mass in a particular area is changing violently. It is of great significance to accurately distinguish rock fracture signals and rock microseismic signals for analyzing the changes in the rock mass in the area where the signal occurs. Considering the microseismic signals of the Dahongshan Iron Mine, the time domain, frequency domain, energy characteristic distribution, and fractal features of each signal were analyzed after noise reduction of the original signal. The results demonstrate that the signal duration and maximum amplitude of the signal could not accurately distinguish the two types of signals. However, the main frequency of the rock fracture signal after noise reduction is distributed above 500 HZ, and the main frequency of the rock blasting signal is mainly distributed below 500 HZ. After the denoised signal is decomposed by the ensemble empirical simulation decomposition, the energy of the IMF1 frequency band of the rock fracture signal occupies an absolute dominant position, and the sum of the energy of the IMF2–IMF4 frequency bands of the rock blasting signal occupies a dominant position. The fractal box dimension of the rock fracture signal is mainly below 1.1, and the fractal box dimension of the rock blasting signal is mainly above 1.25. According to the above research results, an automatic signal recognition system based on the BP neural network is established, and the recognition accuracy of the rock blasting and rock fracture signals reached 93% and 94% respectively, when this system was used.

Comparative investigation on the pyrolysis of crop, woody, and herbaceous biomass: Pyrolytic products, structural characteristics, and CO2 gasification

Pyrolysis and gasification are efficient clean energy conversion technologies used in energy research. Clean utilization of biomass energy can effectively alleviate severe energy challenges in the world. In this study, wheat straw, peach wood, and bamboo were pyrolyzed in a quartz tube furnace, and the pyrolysis gas production and physical and chemical properties of the char products were analyzed systematically. The CO2 gasification performance of the pyrolyzed char was investigated experimentally. The results indicated that the pyrolysis temperature had a significant effect on char properties. A high pyrolysis temperature accelerates the gas release rate and improves the final gas production. As the pyrolysis temperature increased, the quality of the char decreased, and the O/C and H/C molar ratios of the biomass char decreased gradually. The char calorific value was also affected by the temperature. The Fourier Transform Infrared Spectroscopy (FTIR) results demonstrated that the functional groups of the biomass decomposed. Scanning Electron Microscope (SEM) images indicated that as the pyrolysis temperature increased, pore development expanded, and various folds appeared on the surface. The fractal box dimensions first increased and then decreased. The specific surface area and micropore volume of the char first increased and then decreased; however, the temperature at which the peak value occurred was slightly different. In the gasification experiment, the peach wood char prepared by pyrolysis at 600 °C had a superior gasification effect. Biomass with a lower ash content, higher O/C ratio, and more developed pore structure exhibited superior gasification performance. The conclusions of this study have important reference value for large scale, clean, and efficient utilization of biomass resources.

Numerical study of fractal analysis of crack propagation in concrete under different strain rates by meso‑scale particle element modeling

In order to explore the influence of concrete multiple cracks development on dynamic improvement mechanism under higher strain rates, a method to quantitatively describe the relationship between stress and multiple cracks development in the procesds of higher strain rates is proposed. By meso‑scale particle element modeling, SHPB stamping tests of concretes under different strain rates (30s−1≤ε˙≤140s−1) are simulated. The simulation results show that the stress-strain relationship and the law of dynamic strength improvement of concrete are consistent with the experimental results. The crack propagation in concrete under different strain rates is recorded and quantitatively described by the fractal box dimension. The crack propagation in concrete under different higher strain rates is divided into three stages, including micro crack initiation and stability stage, connectivity, extension and widening stage and fragmentation stage. This paper establishes and analyzes the relationships between stress, crack fractal dimension and strain rates under different strain rates. The research shows that at low strain rate the development of multiple cracks and failure of aggregates in concrete have a great impact on the dynamic improvement of concrete. As the strain rate increases, the inertia effect has a greater impact on the dynamic improvement of concrete.

Study on dynamic fracture properties of sandstone under the effect of high-temperature using large-scale sample

High temperatures are a crucial factor in the drilling of rock and the extraction of geothermal resources, to investigate the change rules of the dynamic fracture mechanism and physical parameters of sandstone under high temperature environment, this study conducted a series of physical parameter measurements on the sandstone treated in a range of temperatures (room temperature to 700 °C) using large single cleavage semicircle compression (LSCSC) sample. According to the experimental results, it can be observed that 400 °C is the threshold for the variation of the physical parameters of sandstone. The mass, P-wave velocity, S-wave velocity, elastic modulus and Poisson’ ratio have a significant downward trend with the temperature. The main mineral components of sandstone undergo thermochemical reactions and the variation of quartz is a fundamental cause for the deterioration of sandstone. The dynamic loading test was conducted using a drop hammer impact device and the fractured sample, the dynamic fracture test results indicate that the dynamic crack initiation time (CIT) of the samples is advanced with the temperature, but the crack speed and dynamic fracture toughness decreased with the temperature. Further discussions about SEM images and fractal box dimension of fracture surfaces can be found that with the temperature increases, the proportion of intergranular fractures of sandstone gradually increases, the effect of dynamic loads changes from destroying the grain structure of sandstone to orderly coalescing the micro-cracks in the sandstone.

Detection of Wheel Diameter Difference of Railway Wagon by ACMD-FBD and Optimized MKELM

As a common defect of heavy-haul railway wagons, the wheel diameter difference (WDD) will deteriorate wheel/rail dynamic interaction, which severely threatens the running stability and safety of the wagons. Accordingly, it is of great importance to detect the WDD forms of wagons and take appropriate measures in time. In this paper, a novel method for WDD form detection of the running wagons by analyzing axle box acceleration signals is proposed. To solve the difficulty of weak feature extraction for the vibration signals, a novel feature extraction method combining adaptive chirp mode decomposition (ACMD) with fractal box dimension (FBD) is proposed. Firstly, a 3D feature space is constructed by the FBDs, which is calculated from the extracted chirp modes by ACMD. Then, a multiple kernel extreme learning machine optimized by genetic mutation particle swarm optimization is developed for the classification of the feature vectors. Both simulation and field test results indicate that the proposed detection method is powerful for accurate identification of the wheelsets with standard diameter, in-phase WDD, and anti-phase WDD, and the algorithm efficiency shows practicability in onboard monitoring.

Fractal behaviours of networks induced on infinite tree structures by random walks

Tree graphs such as Cayley trees provide a stage to support the self-organization of fractal networks by the flow of walkers from the root vertex to the outermost shell of the tree graph. This network model is a typical example that demonstrates the ability of a random process on a network to generate fractality. However, the finite scale of the tree structure assumed in the model restricts the size of fractal networks. In this study, we removed the restriction on the size of the trees by introducing a lifetime τ (number of steps of random walks) of walkers. As a result, we successfully induced a size-independent fractal structure on a tree graph without a boundary. Our numerical results show that the mean number of offspring d b of the original tree structure determines the value of the fractal box dimension db through the relation d b — 1 = (n b — 1) -θ . The lifetime τ controls the presence or absence of small-world and scale-free properties. The ideal fractal behaviour can be maintained by selecting an appropriate value of τ. The numerical results contribute to the development of a systematic method for generating fractal small-world and scale-free networks while controlling the value of the fractal box dimension. Unlike other models that use recursive rules to generate self-similar structures, this model specifically produces small-world fractal networks with scale-free properties.

Application of Fractal Analysis in Detecting Trabecular Bone Changes in Periapical Radiograph of Patients with Periodontitis.

Introduction Evaluation of detailed features of the supporting bone is an important step in diagnosis and treatment planning for teeth with clinical attachment loss. Fractal analysis can be used as a method for evaluating the complexity of trabecular bone structures. The aim of this study was to evaluate the trabecular bone changes in periapical radiographs of patients with different stages of periodontitis using fractal analysis. Methods This comparative cross-sectional study was performed on patients with and without clinical attachment loss in mandibular first molars. Teeth with clinical attachment loss were divided into mild, moderate, and severe periodontitis groups. Digital periapical radiographs were obtained from the mandibular first molars using the same exposure parameters. DICOM file of the radiographs was exported to ImageJ software for fractal analysis. Three regions of interest (ROIs) were considered in each radiograph: two proximal ROIs mesial and distal to the mandibular first molar and one apical ROI. Fractal dimension (FD) values were calculated using the fractal box counting approach. Statistical analysis was performed using the chi-square test, Mann–Whitney test, intraclass correlation coefficient, and ANOVA (α = 0.05). Results FD values were significantly different between moderate and severe periodontitis and healthy periodontal bone (P < 0.05), except for the distal ROI for moderate periodontitis cases (P=0.280). However, FD values of the supporting bone in periodontally healthy teeth and teeth with mild periodontitis did not show a statistically significant difference (P > 0.05). Conclusion Fractal analysis is a useful tool for evaluation of bone alterations in moderate and severe periodontitis, but was not able to detect the most initial radiographic bone signs of mild periodontitis.

Application of Drilling Strain Analysis Technology in Mine Dynamic Geological Disaster Monitoring

This article applies the borehole strain observation technology to mine dynamic disaster monitoring for the first time, and uses the fractal box dimension to quantitatively analyse the time history change characteristics of borehole strain data and its correlation with rock bursts. The article uses the drilling multi-parameter test device to test in the mine, and judges the stress distribution by collecting the drilling parameter data obtained from the test hole of the working face of the test point. The study found that the application of borehole strain observation technology to mine dynamic disaster monitoring can directly and accurately record the micro-dynamic change process of rock mass in the area before and after the occurrence of rock burst. The fractal box dimension can quantify the complex changes and evaluation of strain data. Abnormal degree of water tide distortion.

Fractal characteristics of the microstructure of loaded coal based on NMR and μ-CT

ABSTRACT The change in the fracture microstructure of loaded coal affects the gas seepage characteristics of the coal. Using micron CT scanning (μ-CT) and nuclear magnetic resonance (NMR) experiments, the fractal characteristics of coal fracture structure under triaxial compression are studied from two aspects of theoretical and experimental fractal dimension for the first time. VG Studio MAX software was used to reconstruct the three-dimensional (3D) fracture structure of the coal samples, and the 3D fracture structure was modified by the porosity measured by NMR. Through the analysis of the reconstruction model and the T2 relaxation spectrum, the spatial distribution of internal cracks in coal was obtained. The fractal geometry theory, NMR experimental fractal dimension algorithm and 3D box counting method were used to calculate the theoretical and experimental fractal dimensions and then comprehensively analyze the fractal characteristics of cracks in loaded coal. The results show that an external load promotes the development of pores/cracks in coal. With increasing theoretical fractal dimension, CT porosity increases from 0.0630% to 0.101%, and NMR porosity increases from 0.0655% to 0.105%. The relationship between porosity and theoretical fractal dimension can be expressed by y = ax2+ bx+c (fractal dimension is an independent variable). With increasing porosity, the experimental fractal dimension of NMR decreases linearly from 2.975 to 2.951, and the experimental fractal dimension of μ-CT increases linearly from 2.191 to 2.324. The different changing trends of theoretical and experimental fractal dimensions verify the differences in their representational meanings.

Research on FT-IR spectrum data mining of diesel engine lubricating oil

Aiming at the difficulties in quantitatively describing the pollution component information contained in FT-IR spectral data of diesel engine lubricants, wavelet transform and fractal box dimension are used to process FT-IR spectral data and get fractal dimension curve. Multiple peaks can be obtained by analyzing the curve. Two groups of lubricating oil samples with water content of 0.10%, 0.22%, 0.44% and 0.88% respectively and with ethylene glycol content of 0.10%, 0.22%, 0.44% and 0.88% respectively. For the above two groups of oil samples, the corresponding relation between peak position and contaminate ratio can be obtained, and a new method based on FT-IR prediction of contaminate ratio is feasible.

Research on Rapid Evaluation of Rock Mass Quality Based on Ultrasonic Borehole Imaging Technology and Fractal Method

The distribution of discontinuities in the deep stratigraphic rock mass is random and irregular, and the spatial distribution has statistical self‐similarity, which can be analyzed by fractal theory. The borehole wall image obtained based on the ultrasonic borehole imaging technology can clearly reflect the rich structural surface information such as the inclination, density, and crack width of the structural surface in the rock mass. Due to its inherent fractal characteristics, the fractal box dimension of the hole wall image can be used to describe the complexity of the structural surface. In the study, the fractal box dimension of the hole wall image and the RQD value of the rock mass are compared, and it is found that there is a linear correlation between the two. Therefore, the rock quality can be evaluated according to the fractal box dimension value of the hole wall image. Finally, a rapid quantitative evaluation method for rock mass quality classification is established based on ultrasonic borehole imaging technology and fractal method.

Classifying direct normal irradiance 1‑minute temporal variability from spatial characteristics of geostationary satellite-based cloud observations

t Variability of solar surface irradiances in the 1-minute range is of interest especially for solar energy applications. Eight variability classes were previously defined for the 1 min resolved direct normal irradiance (DNI) variability inside an hour. In this study spatial structural parameters derived fromsatellite-based cloud observations are used as classifiers in order to detect the associated direct normal irradiance (DNI) variability class in a supervised classification scheme. A neighbourhood of 3×3 to 29×29 satellite pixels is evaluated to derive classifiers describing the actual cloud field better than just using a single satellite pixel at the location of the irradiance observation. These classifiers include cloud fraction in a window around the location of interest, number of cloud/cloud free changes in a binary cloud mask in this window, number of clouds, and a fractal box dimension of the cloud mask within the window. Furthermore, cloud physical parameters as cloud phase, cloud optical depth, and cloud top temperature are used as pixel-wise classifiers. A classification scheme is set up to search for the DNI variability class with a best agreement between these classifiers and the pre-existing knowledge on the characteristics of the cloud field within each variability class from the reference data base. Up to 55 % of all DNI variability class members are identified in the same class as in the reference data base. And up to 92 % cases are identified correctly if the neighbouring class is counted as success as well – the latter is a common approach in classifying natural structures showing no clear distinction between classes as in our case of temporal variability. Such a DNI variability classification method allows comparisons of different project sites in a statistical and automatic manner e.g. to quantify short-term variability impacts on solar power production. This approach is based on satellite-based cloud observations only and does not require any ground observations of the location of interest.

Fractality and the small-world property of generalised (u, v)-flowers

• We generalise ( u, v )-flowers by introducing probabilities into u and v . • Small-world property provides different values of different fractal dimensions. • Value of fractal cluster dimension reflects the behaviour of the mean path length. • Value of fractal box dimension reflects the behaviour of the diameter of the graph. So-called ( u, v )-flowers are recursive networks which produce self-similar structures with fractality or the small-world property. This paper generalises ( u, v )-flowers by introducing probabilities into the realisations of u and v , which enables the study of intermediate states between small and non-small worlds in fractal networks. We obtain the analytical relation between the diameter of the graph L and the graph size N , L ∼ N 1 / d L , and the degree distribution with a power-law form. We show that the difference between the fractal cluster d c and the fractal box d b dimensions reflects different behaviour of the mean path length 〈 l 〉 and L. There seems to be an apparent contradiction between fractality and the small-world property. However, the small-world property can be reconciled with fractality of the graph by size-dependent fractal dimensions where d b shows a size-dependent increase with an upper limit d L . The invariance and equivalence of d c , d b and d L are maintained only when both 〈 l 〉 and L are subject to the same non-small-world behaviour. Our investigation provides useful information for interpreting empirical fractal data and basic tools for studying the various dynamics that occur in networks.