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Using the discrete version of finite pattern, we obtain the Wiener sum of fractal network modeled by the level-3 Sierpinski gasket.

Fractals are important for studying the physics of water transport in soils. Many authors have assumed a mass fractal structure while others consider a fractal surface approach. Each model needs comparisons on the same data set in terms of goodness-of-fit and physical interpretation of parameters. In this note, it is shown, with some representative data sets, that a pore-solid interface fractal model could fit soil water retention data better than a mass fractal model. In addition to the interfacial fractal dimension, this model predicts the tension at dryness. This value is very close to 10⁶ kPa as theoretically predicted.

In this work, we study the eccentric distance sum of Vicsek networks. To obtain the eccentric distance sum of networks, we investigate the corresponding integral on self-similar measure for Vicsek fractals. We use the self-similarity of distance and measure to solve the integral.

For a class of self-similar networks generated by substitution rules, we investigate them in terms of normalized Laplacian spectra. Accordingly, we obtain the recurrent structure of Markov spectra for these self-similar networks, and also estimate the smallest positive eigenvalue for Laplace operator.

The eigentime identity for random walks on networks is the expected time for a walker going from a node to another node. In this paper, our purpose is to calculate the eigentime identities of flower networks by using the characteristic polynomials of normalized Laplacian and recurrent structure of Markov spectrum.

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

Using the covering theory in fractal geometry, we obtain the fractality of self-similar substitution networks introduced by Li et al. [Scale-free effect of substitution networks, Physica A 492 (2018) 1449–1455].

In this paper, we investigate the average geodesic distance on the Sierpinski hexagon in terms of finite patterns on integrals. Applying this result, we also obtain the asymptotic formula for average distances of Sierpinski hexagon networks.

The substitution network is a deterministic model of evolving self-similar networks. For normalized substitution networks, the limit of metric spaces with respect to networks is a self-similar fractal and the limit of average distances on networks is the integral of geodesic distance of the fractal on the self-similar measure. After some technical handles, we establish the finiteness of integrals and obtain a linear equation set to solve the average distance on the fractal.

Intuitively, edge weight has an effect on the dynamical processes occurring on the networks. However, the theoretical research on the effects of edge weight on network dynamics is still rare. In this paper, we present two weighted network models by adjusting the matching relationship between weights and edges. Both network models are controlled by the weight factor $r\in (0,1]$. They have the same connection structure and weight sequence when $r$ is fixed. Based on their self-similar network structure, we study two types of biased walks with a trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. For both weighted scale-free networks, we obtain exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process in both network models. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule and the spectral dimension of the fractal network, and not all ATT for the weighted networks are affected by the weight factor $r$. In other words, not all weight adjustments can change the trapping efficiency in the network. We hope that the present findings could help us get deeper understanding about the influence factor of biased walk in complex systems.

For any Lalley–Gatzouras self-affine carpet, we find out that the length of any curves from $(x_1,x_2)$ to $(y_1,y_2)$ in the carpet is no less than $|x_1-y_1|+|x_2-y_2|$.

The two-point resistance of fractal network has been studied extensively by mathematicians and physicists. In this paper, for a class of self-similar networks named sailboat networks, we obtain a recursive algorithm for computing resistance between any two nodes, using elimination principle, substitution principle and local sum rules on effective resistance.

In this paper, we discuss a family of non-p.c.f. self-similar networks. Although the boundary of each fractal piece is not a finite set, we obtain the finite geometric patterns for the integral of geodesic distance on the self-similar measure, and then calculate its average distance.

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks.

In this paper, for renormalized Lindström Snowflake networks, we discuss the geodesic metric of their limit space which is quite different from the Euclidean metric on the Lindström Snowflake. Using the dihedral group, we simplify the geodesic patterns and obtain 10,850 patterns and thus calculate the limit of average distances on renormalized Lindström Snowflake networks.

We investigate a class of self-similar networks modeled on the three-dimensional Vicsek fractal, and obtain the asymptotic formula of node-weighted average geodesic distances, which is an integral of geodesic distance in terms of the self-similar measure.

We investigate the $n$-level Sierpinski carpets including the classical one for $n=3$, which are non-p.c.f. self-similar fractals. By using the finitely, many geometric patterns and the self-similar measure, we obtain the mean geodesic distance on the $n$-level Sierpinski carpets and their skeleton networks.

In this paper, the epidemic spreading was investigated from the point of view of fractal. Firstly, the real network was abstracted as a fully connected fractal network. Based on the fractal network, the fractal spreading process was studied. The fractal spreading process was simulated to analyze the change of infection density during transmission. The results showed that the infection density presented an upward trend of the ladder-shaped, and a jumping change in infection density occurred during a certain time. As an illustration, the pandemic of COVID-19 was analyzed, the results indicated that the proposed method was valid. Experiments and analyses have shown that intervention at key jump points in virus propagation can effectively control the spread of the virus.

In this paper, we study the eccentric distance sum of substitution tree networks. Calculation of eccentric distance sum naturally involves calculation of average geodesic distance and it is much more complicated. We obtain the asymptotic formulas of average geodesic distance and eccentric distance sum of both symmetric and asymmetric substitution tree networks. Our result on average geodesic distance generalizes the result of [T. Li, K. Jiang and L. Xi, Average distance of self-similar fractal trees, Fractals26(1) (2018) 1850016.] from symmetric case to asymmetric case. To derive formulas, we investigate the corresponding integrals on self-similar measure and use the self-similarity of distance and measure. For simplicity, we introduce some systematic symbolic assignments and make some assumptions on the graph. We verify that our formulas are correct using the numerical calculation results.