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.

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.

Delayed random walk on deterministic weighted scale-free small-world network with a deep trap

2020 ◽  
Vol 34 (30) ◽  
pp. 2050333
Author(s):  
Guangyao Xu ◽  
Zikai Wu
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Closed Form Solution ◽  
Small World ◽  
Deep Trap ◽  
Trapping Efficiency ◽  
Small World Network ◽  
Trapping Time ◽  
Scale Free ◽  
Trapping Process

How to effectively control the trapping process in complex systems is of great importance in the study of trapping problem. Recently, the approach of delayed random walk has been introduced into several deterministic network models to steer trapping process. However, exploring delayed random walk on pseudo-fractal web with the co-evolution of topology and weight has remained out of reach. In this paper, we employ delayed random walk to guide trapping process on a salient deterministic weighted scale-free small-world network with the co-evolution of topology and weight. In greater detail, we first place a deep trap at one of initial nodes of the network. Then, a tunable parameter [Formula: see text] is introduced to modulate the transition probability of random walk and dominate the trapping process. Subsequently, trapping efficiency is used as readout of trapping process and average trapping time is employed to measure trapping efficiency. Finally, the closed form solution of average trapping time (ATT) is deduced analytically, which agrees with corresponding numerical solution. The analytical solution of ATT shows that the delayed parameter [Formula: see text] only modifies the prefactor of ATT, and keeps the leading scaling unchanged. In other words, ATT grows sublinearly with network size, whatever values [Formula: see text] takes. In summary, the work may serves as one piece of clues for modulating trapping process toward desired efficiency on more general deterministic networks.

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.

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.

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.

Random walks on real metro systems

2016 ◽  
Vol 27 (10) ◽  
pp. 1650122 ◽  
Author(s):  
Yueying Zhu ◽  
Longfeng Zhao ◽  
Wei Li ◽  
Qiuping A. Wang ◽  
Xu Cai
Keyword(s):  
Random Walks ◽  
Shortest Path ◽  
Path Length ◽  
Trapping Time ◽  
Target Station ◽  
The Mean ◽  
Metro Networks ◽  
Metro Systems ◽  
Trapping Performance ◽  
Trapping Process

In this paper, we investigate the random walks on metro systems in 28 cities from worldwide via the Laplacian spectrum to realize the trapping process on real systems. The average trapping time is a primary description to response the trapping process. Firstly, we calculate the mean trapping time to each target station and to each entire system, respectively. Moreover, we also compare the average trapping time with the strength (the weighted degree) and average shortest path length for each station, separately. It is noted that the average trapping time has a close inverse relation with the station’s strength but rough positive correlation with the average shortest path length. And we also catch the information that the mean trapping time to each metro system approximately positively correlates with the system’s size. Finally, the trapping process on weighted and unweighted metro systems is compared to each other for better understanding the influence of weights on trapping process on metro networks. Numerical results show that the weights have no significant impact on the trapping performance on metro networks.

Controlling the trapping efficiency in a family of scale-free tree networks

2018 ◽  
Vol 32 (21) ◽  
pp. 1850224 ◽  
Author(s):  
Yu Gao ◽  
Zikai Wu
Keyword(s):  
Transition Probability ◽  
Trapping Efficiency ◽  
Edge Weight ◽  
Tree Networks ◽  
Trapping Time ◽  
Scale Free ◽  
The Third ◽  
Weight Parameter ◽  
Free Tree ◽  
Trapping Process

Efficiently controlling the trapping process is very significant in the study of trapping problem in diverse dynamic processes. In this paper, we explore the trapping efficiency on a family of scale-free tree networks with a deep trap positioned at an initial node, which is controlled by three different strategies. In the first technique, the transition probability is modified by an edge weight parameter. In the second method, the transition probability is controlled by a delay parameter. In the third approach, we use the delay parameter and weight parameter simultaneously to control the trapping process. For all the three control methods, the analytical results of average trapping time (ATT) exactly agree with the numerical results. The result of the first control strategy shows that the average trapping time can scale sublinearly, linearly or superlinearly by modifying the weight parameter. The analytic expression of the ATT in the second method shows that the delay parameter can only modify the main coefficient of ATT, but cannot change the dominant behavior of trapping efficiency. The explicit expression of average trapping time when random walk on scale-free tree network is controlled by the third method shows that it is a fine control. We can get desired trapping efficiency by changing the weight parameter and the delay parameter simultaneously. This work provides a better understanding of controlling the trapping process in a family of scale-free tree networks.

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.

Mortal random walks on a family of treelike regular fractals with a deep trap

2020 ◽  
Vol 34 (06) ◽  
pp. 2050031
Author(s):  
Zikai Wu ◽  
Guangyao Xu
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Network Models ◽  
Deep Trap ◽  
Initial Position ◽  
Single Step ◽  
The Mean ◽  
Walk Dimension ◽  
Mean Time ◽  
Temporal Factor

Due to the ubiquitous occurrence of evanescence in many physical, chemical and biological scenarios, mortal random walks that incorporate evanescence explicitly have drawn more and more attention. It has been a hot topic to study mortal random walks on distinct network models. In this paper, we study mortal random walks on T fractal and a family of treelike regular fractals with a trap located at central node (i.e., innermost node). First, with self-similar setting composed of T fractal, initial position of the walker and location of trap, the total trapping probability of the mortal walker reduces to a function of walker’s single-step survival parameter [Formula: see text]. In more detail, the total trapping probability is expressed by the [Formula: see text]th iteration of map (scaling function) of [Formula: see text]. Based on the map, the analytical expression of total trapping probability’s dominant behavior, the mean time to trapping (MFPT) and temporal factor are obtained, which are related to random walk dimension. Last, we extend the analysis to a family of treelike regular fractals. On them, the total trapping probability is still expressed as the [Formula: see text]th iteration of the map scaling [Formula: see text]. Accordingly, dominant behavior of total trapping probability, MFPT and temporal factor are determined analytically. Both analytical results obtained on T fractal and more general treelike regular fractals show that the mean time to trapping and desired random walk dimension can be obtained by tuning the survival probability parameter [Formula: see text]. In summary, the work advances the understanding of mortal random walks on more general deterministic networks.

Experimental Study on the Impact of Slope and Silt-laden inflow Conditions on Vegetation Sediment Trapping Process

2020 ◽  
Author(s):  
Mingjie Luo ◽  
Chengzhong Pan ◽  
Chunlei Liu
Keyword(s):  
Flow Rate ◽  
Influencing Factor ◽  
Stable State ◽  
Sediment Concentration ◽  
Trapping Efficiency ◽  
Sediment Trapping ◽  
Sediment Delivery ◽  
Trapping Effect ◽  
The Impact ◽  
Trapping Process

<p>Vegetation-restored hillslope surfaces not only reduce erosion but they also remove sediment from upslope silt-laden inflow. To investigate the sediment trapping effect of grassland, this study conducted a series of crossed sediment trapping experiments that examined various factors, such as slope (5°–20°), sediment concentration (40–160 g L<sup>−1</sup>), and unit flow rate (7.5–45.0 L min<sup>−1</sup> m<sup>−1</sup>). The duration of each experiment was longer than required to reach the stable state of sediment trapping, so we measured and verified the individual sediment trapping capacity (R<sub>m</sub>) by experiments. The results showed that gentler slopes generated higher instantaneous sediment trapping efficiency (ISTE) and greater R<sub>m</sub>. As the sediment concentration of the silt-laden inflow increased, the impact of slope on R<sub>m</sub> increased. Higher sediment concentration led to lower ISTE but greater R<sub>m</sub>. Similar to the effect of sediment concentration, a larger unit flow rate led to lower ISTE and greater R<sub>m</sub>. Thus, it is evident that interaction among these factors affects sediment trapping process. The experiments revealed the greatest sediment trapping effect of grass strips was concentrated mainly in the first 2-m width, and that 90% of sediment deposition occurred within half the time needed to reach the stable state. Slope and flow rate were found to have an effect on sediment trapping in each section of grass strips, whereas the effect of sediment concentration was concentrated primarily in the first 5-m width. Standard regression coefficients of a comprehensive regression analysis showed that the intensities of the influencing factors on R<sub>m</sub> were as follows: slope (0.736) > grassland width (0.498) > unit flow rate (0.398) > sediment concentration (0.240). It was established that slope is the strongest influencing factor, and that sediment concentration and unit flow rate mainly affect R<sub>m</sub> by changing the rate of sediment delivery. These results will help expand the theoretical basis regarding the effects of vegetation restoration on watersheds in soil erosion research.</p>

Sign in / Sign up

Export Citation Format

Share Document