Weight-dependent Walk Research Articles

Exploring trapping problem on simplicial networks encoding higher-order interactions

Biological, social and man-made systems often harbor various types of higher-order interaction, whose presence may substantially impact the dynamics on these systems. As a special random walk, paradigmatic trapping problem is still scarcely explored on networks encoding higher-order interactions. In this paper, applying simplicial networks produced by edge corona product as test bed, we explore how the higher-order interaction impact trapping process. Specifically, an immobile trap is first deployed at an initial node of the networks and average trapping time (ATT) is derived analytically to measure trapping efficiency of standard weight-dependent random walk. Then, delayed random walk incorporating delay phenomenon into random walk process is introduced into the networks and closed form solution of ATT is still obtained for quantifying trapping efficiency. The obtained analytical solutions of ATT in both scenarios show that ATT grows sub-linearly with network size but its leading scaling is quantitatively dependent on the parameter specifying building block of networks. Besides, analytical expression of ATT obtained in the latter scenario shows that the parameter [Formula: see text] manipulating delayed random walk does not affect the leading scaling of ATT but can quantitatively modify its prefactor. Therefore, introduction of delay phenomenon still can modulate trapping efficiency quantitatively in the presence of higher-order interaction. This work may pave the way for modulating trapping process and related dynamical processes on general networks modeling higher-order interactions.

Unveiling the impact of weight heterogeneity on random walks in weighted extended tree-like fractals

In this paper, we put forward a class of weighted extended tree-like fractals and further use them as test bed to unveil the impact of weight heterogeneity on random walks. Specifically, a family of weighted extended tree-like fractals are first proposed, which are parameterized by a growth parameter [Formula: see text] and weight parameter [Formula: see text]. Then, we explore standard weight-dependent walk on the networks by deploying three traps at initial three nodes. To this end, we derive analytically the average trapping time (ATT) to measure the trapping efficiency and the obtained results show that depending on values of [Formula: see text], ATT may grow sub-linearly, linearly and super-linearly with the network size. Besides, it can also quantitatively impact the leading behavior and pre-factor of ATT simultaneously. Finally, more challenging mixed weight-dependent random walk that takes non-nearest-neighbor hopping is addressed. Analytical solutions of ATT derived under this new scenario imply that weight parameter [Formula: see text] still can qualitatively, quantitatively steer leading behavior and quantitatively affect pre-factor of ATT. As to the stochastic parameter [Formula: see text] controlling mixed random walk, it could only impact the pre-factor of ATT and only have negligible effect on the leading behavior of ATT. In summary, this work could further augment our understanding of random walks on networks.

The average trapping time for a weight-dependent walk on a weighted hierarchical graph

A hierarchical graph is a kind of self-similar network, which is widely discussed and has a wide range of applications. In this paper, we introduce a class of weighted hierarchical graphs H (n, k), which is constructed using the hierarchical product of complete graphs. The weighted hierarchical graph depends on two parameters: the weighted factor r $$(0<r\le 1)$$ and the number k $$(k\ge 3)$$ of nodes of the complete graph $$K_k$$ . In addition, we also discuss the trapping problem and the average trapping time (ATT) for a weight-dependent walk on a weighted hierarchical graph. Based on the special structure of the weighted graph, we derived the exact expression of the ATT for a weighted hierarchical graph. The results show that the ATT obeys a power-law function with the exponent $$\frac{\mathrm{ln}(kr)}{\mathrm{ln}k}<1$$ , which is related to the factors k and r and indicate that for $$kr=k(r=1)$$ and $$1<kr<k$$ , the ATT has a linear and sublinear relationship with the network order, respectively. Therefore, the smaller kr is, the more efficient the trapping process of the weighted hierarchical graph is. Compared with the unweighted hierarchical graph, the weighted hierarchical graph has a higher efficiency in the trapping process.

EXACT CALCULATIONS OF FIRST-PASSAGE QUANTITIES ON A CLASS OF WEIGHTED TREE-LIKE FRACTAL NETWORKS

This paper concerns the weight-dependent random walk on a class of weighted tree-like fractal networks controlled by a positive integer parameter [Formula: see text] [Formula: see text] and the weight factor [Formula: see text] [Formula: see text]. We study the first return time (FRT) of a given hub and the global first-passage time (GFPT) to a given hub on the networks. By the probability generating function method, we derive the analytic expressions of the first and second moments of FRT and GFPT. In order to evaluate the fluctuation of FRT and GFPT, we further calculate the variance and the reduced moments of FRT and GFPT.

The average trapping time on the weighted pseudofractal scale-free web

In this paper, we consider the average trapping time (ATT) on the weighted pseudofractal scale-free web (WPSFW) with weight factor r. According to the iteration mechanism of WPSFW, we consider the weight-dependent walk from the node of web subunit to the trap fixed in a hub node. Using the tool of probability generating function, the relationship between the mean first passage time (MFPT) of two generations of the web is discovered, and the expression of the MFPT is obtained. Moreover, we consider the MFPT averaged over all the starting nodes of subunit and finally get the analytical expression of ATT. The result shows that the ATT growth approximates a power-law function of the number of web nodes, and its exponent is .

THE TRAPPING PROBLEM OF WEIGHTED (2,2)-FLOWER NETWORKS WITH THE SAME WEIGHT SEQUENCE

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 [Formula: see text]. They have the same connection structure and weight sequence when [Formula: see text] 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 [Formula: see text]. 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.

Average trapping time of weighted scale-free m-triangulation networks

This paper investigates the average trapping time (ATT) on weighted scale-free m-triangulation networks. ATT is the mean of the first-passage time from any node to the trap fixed at a hub node over the entire network. Based on the structural properties of weighted scale-free m-triangulation networks, we deduce the accurate expression of ATT for weight-dependent walk. The result shows that the scaling expression of ATT with network size obeys a power-law function with exponent characterized by the weight factor r and the growth factor m. Further, we find that the parameters r and m have a significant impact on the diffusion efficiency of the network, namely, the smaller r and m are, the more efficient the trapping process of the network is.

The average weighted receiving time with weight-dependent walk on a family of double-weighted polymer networks

In this paper, a family of the double-weighted polymer networks is introduced depending on the number of copies [Formula: see text] and two weight factors [Formula: see text]. The double-weights represent the selected weights and the consumed weights, respectively. Denote by [Formula: see text] the selected weight connecting the nodes [Formula: see text] and [Formula: see text], and denote by [Formula: see text] the consumed weight connecting the nodes [Formula: see text] and [Formula: see text]. Let [Formula: see text] be related to the weight factor [Formula: see text], and let [Formula: see text] be related to the weight factors [Formula: see text]. Assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the selected weight of edge linking them. The weighted time for two adjacency nodes is the consumed weight connecting the two nodes. The average weighted receiving time (AWRT) is defined on the double-weighted polymer networks. Our results show that in large network, the leading behaviors of AWRT for the double-weighted polymer networks follow distinct scalings, with the trapping efficiency associated with the network size [Formula: see text], the number of copies [Formula: see text], and two weight factors [Formula: see text]. We also found that the scalings of the AWRT with weight-dependent walk in double-weighted polymer networks is due to the use of the weight-dependent walk and the weighted time. The dominant reason is the range of each weight factor. To investigate the reason of the scalings, the AWRT for four cases are discussed.

TRAPPING PROBLEM OF THE WEIGHTED SCALE-FREE TRIANGULATION NETWORKS FOR BIASED WALKS

In this paper, we study the trapping problem in the weighted scale-free triangulation networks with the growth factor [Formula: see text] and the weight factor [Formula: see text]. We propose two biased walks, one is standard weight-dependent walk only including the nearest-neighbor jumps, the other is mixed weight-dependent walk including both the nearest-neighbor and the next-nearest-neighbor jumps. For the weighted scale-free triangulation networks, we derive the exact analytic formulas of the average trapping time (ATT), the average of node-to-trap mean first-passage time over the whole networks, which measures the efficiency of the trapping process. The obtained results display that for two biased walks, in the large network, the ATT grows as a power-law function of the network size [Formula: see text] with the exponent, represented by [Formula: see text] when [Formula: see text]. Especially when the case of [Formula: see text] and [Formula: see text], the ATT grows linear with the network size [Formula: see text]. That is the smaller the value of [Formula: see text], the more efficient the trapping process is. Furthermore, comparing the standard weight-dependent walk with mixed weight-dependent walk, the obtained results show that although the next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT. The smaller the value of probability parameter [Formula: see text], the more efficient the trapping process for the mixed weight-dependent walk is.

Mean first-passage times for two biased walks on the weighted rose networks

Compared with traditional random walk, biased walks have been studied extensively over the past decade especially in the transport and communication networks. In this paper, we first introduce the weighted rose networks. Then, for the weighted rose networks we focus on two biased walks, maximal entropy walk and weight-dependent walk, and obtain the exact expressions of their stationary distributions and mean first-passage times. Finally, we find that the average receiving time for maximal entropy walk is a quadratic function of the weight parameter r while the average receiving time for weighted-dependent walk is a linear function of the weight parameter r. Meanwhile, for the maximal entropy walk, the smaller the value of r is, the more efficient the trapping process is. For the weighted-dependent walk, the larger the value of r(r<r0≈2.6) is, the more efficient for the weighted-dependent walk, the smaller the value of r(r>r0≈2.6) is, the more efficient for the weight-dependent walk.

The trapping problem and the average shortest weighted path of the weighted pseudofractal scale-free networks

In this paper, we study the trapping time in the weighted pseudofractal scale-free networks (WPSFNs) and the average shortest weighted path in the modified weighted pseudofractal scale-free networks (MWPSFNs) with the weight factor [Formula: see text]. At first, for exceptional case with the trap fixed at a hub node for weight-dependent walk, we derive the exact analytic formulas of the trapping time through the structure of WPSFNs. The obtained rigorous solution shows that the trapping time approximately grows as a power-law function of the number of network nodes with the exponent represented by [Formula: see text]. Then, we deduce the scaling expression of the average shortest weighted path through the iterative process of the construction of MWPSFNs. The obtained rigorous solution shows that the scalings of average shortest weighted path with network size obey three laws along with the range of the weight factor. We provide a theoretical study of the trapping time for weight-dependent walk and the average shortest weighted path in a wide range of deterministic weighted networks.

SCALING PROPERTIES OF FIRST RETURN TIME ON WEIGHTED TRANSFRACTALS (1,3)-FLOWERS

Complex networks are omnipresent in science and in our real life, and have been the focus of intense interest. It is vital to research the impact of their characters on the dynamic progress occurring on complex networks for weight-dependent walk. In this paper, we first consider the weight-dependent walk on one kind of transfractal (or fractal) which is named the weighted transfractal [Formula: see text]-flowers. And we pay attention to the first return time (FRT). We mainly calculate the mean and variance of FRT for a prescribed hub (i.e. the most concerned nodes) in virtue of exact probability generating function and its properties. Then, we obtain the mean and the secondary moment of the first return time. Finally, using the relationship among the variance, mean and the secondary moment, we obtain the variance of FRT and the scaling properties of the mean and variance of FRT on weighted transfractals [Formula: see text]-flowers.

The trapping problem of the weighted scale-free treelike networks for two kinds of biased walks.

It has been recently reported that trapping problem can characterize various dynamical processes taking place on complex networks. However, most works focused on the case of binary networks, and dynamical processes on weighted networks are poorly understood. In this paper, we study two kinds of biased walks including standard weight-dependent walk and mixed weight-dependent walk on the weighted scale-free treelike networks with a trap at the central node. Mixed weight-dependent walk including non-nearest neighbor jump appears in many real situations, but related studies are much less. By the construction of studied networks in this paper, we determine all the eigenvalues of the fundamental matrix for two kinds of biased walks and show that the largest eigenvalue has an identical dominant scaling as that of the average trapping time (ATT). Thus, we can obtain the leading scaling of ATT by a more convenient method and avoid the tedious calculation. The obtained results show that the weight factor has a significant effect on the ATT, and the smaller the value of the weight factor, the more efficient the trapping process is. Comparing the standard weight-dependent walk with mixed weight-dependent walk, although next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT.

The Laplacian spectrum and average trapping time for weighted Dyson hierarchical network

In this paper, we consider the weighted Dyson hierarchical network, that is a weighted fully-connected network, where the pattern of weights is ruled by a weight factor. Given that the Laplacian operator is intrinsically implied in the analysis of dynamic processes occurring on complex networks, Laplacian spectrum allows addressing analytically a large class of problems. Exploiting the deterministic recursive structure, we are able to derive explicitly all eigenvalues and their corresponding eigenvectors of Laplacian matrix. Further, we derive exact solutions of the trapping time (TT) and average trapping time (ATT) on the weighted Dyson hierarchical network with weight-dependent walk. The obtained results show that TT and ATT are related to the weight factor of the weighted Dyson hierarchical network.

Eigentime identity of the weighted scale-free triangulation networks for weight-dependent walk

The eigenvalues of the normalized Laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects, in particular for weight-dependent walk. In order to get the eigentime identity for weight-dependent walk, we need to obtain the eigenvalues and their multiplicities of the Laplacian matrix. Firstly, the model of the weighted scale-free triangulation networks is constructed. Then, the eigenvalues and their multiplicities of transition weight matrix are presented, after the recursive relationship of those eigenvalues at two successive generations are given. Consequently, the Laplacian spectrum is obtained. Finally, the analytical expression of the eigentime identity, indicating that the eigentime identity grows sublinearly with the network order, is deduced.

EIGENTIME IDENTITY OF THE WEIGHTED KOCH NETWORKS

The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we present an iterative algorithm for generating the weighted Koch network. Then, we study the eigenvalues for the transition matrix of the weighted Koch networks for weight-dependent walk. We derive explicit expressions for all eigenvalues and their multiplicities. Afterwards, we apply the obtained eigenvalues to determine the eigentime identity, i.e. the sum of reciprocals of each nonzero eigenvalues of normalized Laplacian matrix for the weighted Koch networks. The highlights of this paper are computational methods as follows. Firstly, we obtain two factors from factorization of the characteristic equation of symmetric transition matrix by means of the operation of the block matrix. From the first factor, we can see that the symmetric transition matrix has at least [Formula: see text] eigenvalues of [Formula: see text]. Then we use the definition of eigenvalues and eigenvectors to calculate the other eigenvalues.

Network coherence and eigentime identity on a family of weighted fractal networks

The study on network coherence and eigentime identity has gained much interest. In this paper, the first-order network coherence is characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, while the eigentime identity is quantified by the sum of reciprocals of all nonzero normalized Laplacian eigenvalues. We construct a family of weighted fractal networks with the weight factor r (0 < r ≤ 1). Based on the relationship between the first-order network coherence and the EMFPT, the asymptotic behavior of the first-order network coherence is obtained. The obtained results show that the scalings of first-order coherence with network size obey three laws according to the range of the weight factor. The first law is that the scaling obeys a power-law function of the network size Nn with the exponent, represented by logsr, when 1s<r≤1; The second law is that the scaling obeys (lnNn)2Nn (i.e., the quotient of the square logarithm of the network size and the network size), when r=1s; The third law is that the scaling obeys lnNnNn (i.e., the quotient of the logarithm of the network size and the network size), when 0<r<1s. Thus, the scaling of the first-order coherence of weighted fractal networks decreases with the decreasing of r, when 0 < r ≤ 1.Then, all nonzero normalized Laplacian eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity obey two laws according to the range of the weight factor. The first law is that the scaling obeys ln Nn (i.e., the logarithm of the network size), when 0 < r ≤ 1 and r≠1s; The second law is that the scaling obeys Nnln Nn (i.e., the product of network size and its logarithm), when r=1s.

Two types of weight-dependent walks with a trap in weighted scale-free treelike networks

In this paper, we present the weighted scale-free treelike networks controlled by the weight factor r and the parameter m. Based on the network structure, we study two types of weight-dependent walks with a highest-degree trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. Although some properties have been revealed in weighted networks, studies on mixed weight-dependent walks are still less and remain a challenge. For the weighted scale-free treelike network, we derive exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process. The obtained results show that ATT is related to weight factor r, parameter m and spectral dimension of the weighted network. We find that in different range of the weight factor r, the leading term of ATT grows differently, i.e., superlinearly, linearly and sublinearly with the network size. Furthermore, the obtained results show that changing the walking rule has no effect on the leading scaling of the trapping efficiency. All results in this paper can help us get deeper understanding about the effect of link weight, network structure and the walking rule on the properties and functions of complex networks.

FIRST-ORDER NETWORK COHERENCE AND EIGENTIME IDENTITY ON THE WEIGHTED CAYLEY NETWORKS

In this paper, we first study the first-order network coherence, characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, on the weighted Cayley networks with the weight factor. The analytical formula of the EMFPT is obtained by the definition of the EMFPT. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor. Then, we study eigentime identity quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on the weighted Cayley networks with the weight factor. We show that all their eigenvalues can be obtained by calculating the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity on the weighted Cayley networks obey two laws along with the range of the weight factor.

Scaling of average receiving time on weighted polymer networks with some topological properties

In this paper, a family of the weighted polymer networks is introduced depending on the number of copies f and a weight factor r. The topological properties of weighted polymer networks can be completely analytically characterized in terms of the involved parameters and/or of the fractal dimension. Moreover, assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them, namely weight-dependent walk. Then, we calculate the average receiving time (ART) with weighted-dependent walks, which is the sum of mean first-passage times (MFPTs) for all nodes absorpt at the trap located at the central node as a recursive relation. The obtained remarkable results display that when frac{1}{f+1} < r < 1, the ART grows sublinearly with the network size; when r=frac{1}{f+1}, ART grows with increasing size Ng as {mathrm{ln}}^{2}{N}_{g}; when 0 < r < frac{1}{f+1}, ART grows with increasing size Ng as ln Ng. In the treelike polymer networks, ART grows with linearly with the network size Ng when r = 1. Thus, the weighted polymer networks are more efficient than treelike polymer networks in receiving information.