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

2021 ◽  
pp. 2150369
Author(s):  
Zikai Wu ◽  
Guangyao Xu
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Nearest Neighbor ◽  
Growth Parameter ◽  
Network Size ◽  
Trapping Efficiency ◽  
Test Bed ◽  
Weight Parameter ◽  
The Impact ◽  
Standard Weight

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.

Effect of heterogeneous weights on the average trapping time and two types of random walks in weighted directed networks

2019 ◽  
Vol 33 (05) ◽  
pp. 1950023 ◽  
Author(s):  
Huai Zhang ◽  
Hongjuan Zhang
Keyword(s):  
Random Walks ◽  
Numerical Solutions ◽  
Transition Probability ◽  
Network Size ◽  
Trapping Efficiency ◽  
Directed Networks ◽  
Trapping Time ◽  
Stochastic Parameter ◽  
Weight Parameter ◽  
Trapping Process

In this paper, we define a class of weighted directed networks with the weight of its edge dominated by a weight parameter w. According to the construction of the networks, we study two types of random walks in the weighted directed networks with a trap fixed on the central node, i.e., standard random walks and mixed random walks. For the standard random walks, the trapping process is controlled by a weight parameter w which changes the transition probability of random walks. For the mixed random walks including nonnearest-neighbor hopping, the trapping process is steered by a stochastic parameter [Formula: see text], where [Formula: see text] changes the walking rule. For the above two techniques, we derive both analytically the average trapping time (ATT) as the measure of trapping efficiency, and the obtained analytical expressions are in good agreement with the corresponding numerical solutions for different values of w and [Formula: see text]. The obtained results indicate that ATT scales superlinearly with network size Nn and the weight parameter w changes simultaneously the prefactor and the leading scalings of ATT, while the stochastic parameter [Formula: see text] can only alter the prefactor of ATT and leave the leading scalings of ATT unchanged. This work may help in paving the way for understanding the effects of the link weight and nonnearest-neighbor hopping in real complex systems.

Uncovering the impact of delay phenomenon on random walks in a family of weighted m-triangulation networks

2021 ◽  
pp. 2150234
Author(s):  
Junhao Guo ◽  
Zikai wu
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Analytical Solutions ◽  
Network Models ◽  
Trapping Efficiency ◽  
Trapping Time ◽  
Delay Phenomenon ◽  
The Impact ◽  
Insight Into ◽  
Trapping Process

Uncovering the impact of special phenomena on dynamical processes in more distinct weighted network models is still needed. In this paper, we investigate the impact of delay phenomenon on random walk by introducing delayed random walk into a family of weighted m-triangulation networks. Specifically, we introduce delayed random walk into the networks. Then one and three traps are deployed, respectively, on the networks in two rounds of investigation. In both rounds of investigation, average trapping time (ATT) is applied to measure trapping efficiency and derived analytically by harnessing iteration rule of the networks. The analytical solutions of ATT obtained in both investigations show that ATT increases sub-linearity with the size of the network no matter what value the parameter [Formula: see text] manipulating delayed random walk takes. But [Formula: see text] can quantitatively change both its leading scaling and prefactor. So, introduction of delay phenomenon can control trapping efficiency quantitatively. Besides, parameters [Formula: see text] and [Formula: see text] governing networks’ evolution quantitatively impact both the prefactor and leading scaling of ATT simultaneously. In summary, this work may provide incremental insight into understanding the impact of observed phenomena on special trapping process and general random walks in complex systems.

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

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950112
Author(s):  
CHANGMING XING
Keyword(s):  
Nearest Neighbor ◽  
Influence Factor ◽  
Network Models ◽  
Trapping Efficiency ◽  
Theoretical Research ◽  
Edge Weight ◽  
Weight Factor ◽  
Scale Free ◽  
Weight Sequence ◽  
Standard Weight

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.

scholarly journals Size Matters: The Impact of Training Size in Taxonomically-Enriched Word Embeddings

2019 ◽  
Vol 9 (1) ◽  
pp. 252-267
Author(s):  
Alfredo Maldonado ◽  
Filip Klubička ◽  
John Kelleher
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Public Transport ◽  
Crucial Role ◽  
Word Embeddings ◽  
Performance Gains ◽  
Near Synonyms ◽  
The Impact ◽  
Taxonomic Relations ◽  
The Web

AbstractWord embeddings trained on natural corpora (e.g., newspaper collections, Wikipedia or the Web) excel in capturing thematic similarity (“topical relatedness”) on word pairs such as ‘coffee’ and ‘cup’ or ’bus’ and ‘road’. However, they are less successful on pairs showing taxonomic similarity, like ‘cup’ and ‘mug’ (near synonyms) or ‘bus’ and ‘train’ (types of public transport). Moreover, purely taxonomy-based embeddings (e.g. those trained on a random-walk of WordNet’s structure) outperform natural-corpus embeddings in taxonomic similarity but underperform them in thematic similarity. Previous work suggests that performance gains in both types of similarity can be achieved by enriching natural-corpus embeddings with taxonomic information from taxonomies like Word-Net. This taxonomic enrichment can be done by combining natural-corpus embeddings with taxonomic embeddings (e.g. those trained on a random-walk of WordNet’s structure). This paper conducts a deep analysis of this assumption and shows that both the size of the natural corpus and of the random-walk coverage of the WordNet structure play a crucial role in the performance of combined (enriched) vectors in both similarity tasks. Specifically, we show that embeddings trained on medium-sized natural corpora benefit the most from taxonomic enrichment whilst embeddings trained on large natural corpora only benefit from this enrichment when evaluated on taxonomic similarity tasks. The implication of this is that care has to be taken in controlling the size of the natural corpus and the size of the random-walk used to train vectors. In addition, we find that, whilst the WordNet structure is finite and it is possible to fully traverse it in a single pass, the repetition of well-connected WordNet concepts in extended random-walks effectively reinforces taxonomic relations in the learned embeddings.

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

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950028 ◽  
Author(s):  
MEIFENG DAI ◽  
TINGTING JU ◽  
YUE ZONG ◽  
JIAOJIAO HE ◽  
CHUNYU SHEN ◽  
...  
Keyword(s):  
Nearest Neighbor ◽  
First Passage Time ◽  
Passage Time ◽  
Network Size ◽  
Large Network ◽  
Dominant Term ◽  
Scale Free ◽  
Trapping Problem ◽  
Standard Weight ◽  
Trapping Process

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.

scholarly journals Nonhomogeneous random walks systems on ℤ

2010 ◽  
Vol 47 (02) ◽  
pp. 562-571
Author(s):  
Elcio Lebensztayn ◽  
Fábio Prates Machado ◽  
Mauricio Zuluaga Martinez
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Nearest Neighbor ◽  
The Other ◽  
Active Particle ◽  
Positive Probability ◽  
Small Probability ◽  
Active Particles ◽  
Other Hand ◽  
The Right

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.

scholarly journals Nonhomogeneous random walks systems on ℤ

2010 ◽  
Vol 47 (2) ◽  
pp. 562-571 ◽  
Author(s):  
Elcio Lebensztayn ◽  
Fábio Prates Machado ◽  
Mauricio Zuluaga Martinez
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Nearest Neighbor ◽  
The Other ◽  
Active Particle ◽  
Positive Probability ◽  
Small Probability ◽  
Active Particles ◽  
Other Hand ◽  
The Right

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.

Determining average trapping time of delayed random walks on Apollonian network

2019 ◽  
Vol 33 (20) ◽  
pp. 1950231
Author(s):  
Huai Zhang ◽  
Hongjuan Zhang
Keyword(s):  
Complex Systems ◽  
Random Walks ◽  
Numerical Solutions ◽  
Transition Probability ◽  
Network Size ◽  
Appropriate Technology ◽  
Trapping Efficiency ◽  
Trapping Time ◽  
Stochastic Parameter ◽  
Trapping Process

Designing appropriate technology to effectively control the trapping process in complex systems and achieve the desired trapping efficiency is central in the study of trapping problem in complex systems. In this paper, we study delayed random walks on Apollonian network with a trap fixed at a given initial hub node. In more detail, a stochastic parameter p was introduced in the approach to alter the transition probability of random walks. We further derive analytically the average trapping time (ATT) as the measure of trapping efficiency with the obtained analytical expression being in good agreement with the corresponding numerical solutions. The result indicates that ATT scales sublinearly with network size when [Formula: see text]. Therefore, we introduced stochastic parameter p only alter the prefactor of ATT and left the leading scaling of ATT unchanged. Our work may pave the way for understanding how to control trapping process in real complex systems.

scholarly journals Average trapping time on weighted directed Koch network

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Zikai Wu ◽  
Yu Gao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Numerical Solution ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Trapping Time ◽  
Scale Free ◽  
Weight Parameter ◽  
Scale Free Networks

Abstract Numerous recent studies have focused on random walks on undirected binary scale-free networks. However, random walks with a given target node on weighted directed networks remain less understood. In this paper, we first introduce directed weighted Koch networks, in which any pair of nodes is linked by two edges with opposite directions, and weights of edges are controlled by a parameter θ . Then, to evaluate the transportation efficiency of random walk, we derive an exact solution for the average trapping time (ATT), which agrees well with the corresponding numerical solution. We show that leading behaviour of ATT is function of the weight parameter θ and that the ATT can grow sub-linearly, linearly and super-linearly with varying θ . Finally, we introduce a delay parameter p to modify the transition probability of random walk, and provide a closed-form solution for ATT, which still coincides with numerical solution. We show that in the closed-form solution, the delay parameter p can change the coefficient of ATT, but cannot change the leading behavior. We also show that desired ATT or trapping efficiency can be obtained by setting appropriate weight parameter and delay parameter simultaneously. Thereby, this work advance the understanding of random walks on directed weighted scale-free networks.

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