Mean first return time for random walks on weighted networks

2015 ◽  
Vol 26 (06) ◽  
pp. 1550068 ◽  
Author(s):  
Xing-Li Jing ◽  
Xiang Ling ◽  
Jiancheng Long ◽  
Qing Shi ◽  
Mao-Bin Hu
Keyword(s):  
Random Walk ◽  
Complex Networks ◽  
Random Walks ◽  
Stationary Distribution ◽  
Real World ◽  
Return Time ◽  
Edge Weight ◽  
Weighted Networks ◽  
Strength Based ◽  
First Return Time

Random walks on complex networks are of great importance to understand various types of phenomena in real world. In this paper, two types of biased random walks on nonassortative weighted networks are studied: edge-weight-based random walks and node-strength-based random walks, both of which are extended from the normal random walk model. Exact expressions for stationary distribution and mean first return time (MFRT) are derived and examined by simulation. The results will be helpful for understanding the influences of weights on the behavior of random walks.

Scalings of first-return time for random walks on generalized and weighted transfractal networks

2019 ◽  
Vol 33 (26) ◽  
pp. 1950306
Author(s):  
Qin Liu ◽  
Weigang Sun ◽  
Suyu Liu
Keyword(s):  
Random Walks ◽  
Large Scale ◽  
Probability Generating Function ◽  
Return Time ◽  
Network Size ◽  
Large Network ◽  
Large Scale Network ◽  
Scale Network ◽  
The Mean ◽  
First Return Time

The first-return time (FRT) is an effective measurement of random walks. Presently, it has attracted considerable attention with a focus on its scalings with regard to network size. In this paper, we propose a family of generalized and weighted transfractal networks and obtain the scalings of the FRT for a prescribed initial hub node. By employing the self-similarity of our networks, we calculate the first and second moments of FRT by the probability generating function and obtain the scalings of the mean and variance of FRT with regard to network size. For a large network, the mean FRT scales with the network size at the sublinear rate. Further, the efficiency of random walks relates strongly with the weight factor. The smaller the weight, the better the efficiency bears. Finally, we show that the variance of FRT decreases with more number of initial nodes, implying that our method is more effective for large-scale network size and the estimation of the mean FRT is more reliable.

Exploring activity-driven network with biased walks

2017 ◽  
Vol 28 (09) ◽  
pp. 1750111
Author(s):  
Yan Wang ◽  
Ding Juan Wu ◽  
Fang Lv ◽  
Meng Long Su
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Stationary Distribution ◽  
Significant Role ◽  
Search Strategy ◽  
Transition Probability ◽  
Edge Weighting ◽  
Coverage Function ◽  
Analytical Expressions

We investigate the concurrent dynamics of biased random walks and the activity-driven network, where the preferential transition probability is in terms of the edge-weighting parameter. We also obtain the analytical expressions for stationary distribution and the coverage function in directed and undirected networks, all of which depend on the weight parameter. Appropriately adjusting this parameter, more effective search strategy can be obtained when compared with the unbiased random walk, whether in directed or undirected networks. Since network weights play a significant role in the diffusion process.

scholarly journals Static and dynamic properties of selected stochastic processes on complex networks

2020 ◽  
Author(s):  
Jeremi Ochab
Keyword(s):  
Random Walk ◽  
Complex Networks ◽  
Random Walks ◽  
Dynamic Properties ◽  
Finite Size ◽  
Disease Spread ◽  
Maximal Entropy ◽  
Centrality Measures ◽  
Topological Features ◽  
The Subject

This thesis is concerned with the properties of a number of selected processes taking place on complex networks and the way they are affected by structure and evolution of the networks. What is meant here by 'complex networks' is the graph-theoretical representations and models of various empirical networks (e.g., the Internet network) which contain both random and deterministic structures, and are characterised among others by the small-world phenomenon, power-law vertex degree distributions, or modular and hierarchical structure. The mathematical models of the processes taking place on these networks include percolation and random walks we utilise.The results presented in the thesis are based on five thematically coherent papers. The subject of the first paper is calculating thresholds for epidemic outbreaks on dynamic networks, where the disease spread is modelled by percolation. In the paper, known analytical solutions for the epidemic thresholds were extended to a class of dynamically evolving networks; additionally, the effects of finite size of the network on the magnitude of the epidemic were studied numerically. The subject of the second and third paper is the static and dynamic properties of two diametrically opposed random walks on model highly symmetric deterministic graphs. Specifically, we analytically and numerically find the stationary states and relaxation times of the ordinary, diffusive random walk and the maximal-entropy random walk. The results provide insight into localisation of random walks or their trapping in isolated regions of networks. Finally, in the fourth and fifth paper, we examine the utility of random walks in detecting topological features of complex networks. In particular, we study properties of the centrality measures (roughly speaking, the ranking of vertices) based on random walks, as well as we conduct a systematic comparative study of random-walk based methods of detecting modular structure of networks.These studies thus aimed at specific problems in modelling and analysis of complex networks, including theoretical examination of the ways the behaviour of random processes intertwines with the structure of complex networks.

Enhancing betweenness algorithm for detecting communities in complex networks

2014 ◽  
Vol 28 (09) ◽  
pp. 1450074 ◽  
Author(s):  
Benyan Chen ◽  
Ju Xiang ◽  
Ke Hu ◽  
Yi Tang
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Community Detection ◽  
Real World ◽  
Topological Property ◽  
Edge Weight ◽  
Detection Algorithms ◽  
Improved Algorithm

Community structure is an important topological property common to many social, biological and technological networks. First, by using the concept of the structural weight, we introduced an improved version of the betweenness algorithm of Girvan and Newman to detect communities in networks without (intrinsic) edge weight and then extended it to networks with (intrinsic) edge weight. The improved algorithm was tested on both artificial and real-world networks, and the results show that it can more effectively detect communities in networks both with and without (intrinsic) edge weight. Moreover, the technique for improving the betweenness algorithm in the paper may be directly applied to other community detection algorithms.

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

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850095 ◽  
Author(s):  
MEIFENG DAI ◽  
HUIJIA CHI ◽  
XIANBIN WU ◽  
YUE ZONG ◽  
WENJING FENG ◽  
...  
Keyword(s):  
Complex Networks ◽  
Real Life ◽  
Probability Generating Function ◽  
Return Time ◽  
Exact Probability ◽  
Scaling Properties ◽  
Mean And Variance ◽  
The Mean ◽  
First Return Time ◽  
The Impact

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.

scholarly journals RWNE: A Scalable Random-Walk based Network Embedding Framework with Personalized Higher-order Proximity Preserved

2021 ◽  
Vol 71 ◽  
pp. 237-263
Author(s):  
Jianxin Li ◽  
Cheng Ji ◽  
Hao Peng ◽  
Yu He ◽  
Yangqiu Song ◽  
...  
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real World ◽  
State Of The Art ◽  
Higher Order ◽  
The State ◽  
Random Walk With Restart ◽  
Network Embedding ◽  
Network Proximity ◽  
Embedding Methods

Higher-order proximity preserved network embedding has attracted increasing attention. In particular, due to the superior scalability, random-walk-based network embedding has also been well developed, which could efficiently explore higher-order neighborhoods via multi-hop random walks. However, despite the success of current random-walk-based methods, most of them are usually not expressive enough to preserve the personalized higher-order proximity and lack a straightforward objective to theoretically articulate what and how network proximity is preserved. In this paper, to address the above issues, we present a general scalable random-walk-based network embedding framework, in which random walk is explicitly incorporated into a sound objective designed theoretically to preserve arbitrary higher-order proximity. Further, we introduce the random walk with restart process into the framework to naturally and effectively achieve personalized-weighted preservation of proximities of different orders. We conduct extensive experiments on several real-world networks and demonstrate that our proposed method consistently and substantially outperforms the state-of-the-art network embedding methods.

scholarly journals THE BOW-TIE CENTRALITY: A NOVEL MEASURE FOR DIRECTED AND WEIGHTED NETWORKS WITH AN INTRINSIC NODE PROPERTY

2019 ◽  
Vol 22 (06) ◽  
pp. 1950018
Author(s):  
JAMES B. GLATTFELDER
Keyword(s):  
Complex Networks ◽  
Real World ◽  
Adjacency Matrix ◽  
Intrinsic Property ◽  
Centrality Measures ◽  
The Novel ◽  
Property Value ◽  
Eigenvector Centrality ◽  
Weighted Networks ◽  
Bow Tie

Today, there exist many centrality measures for assessing the importance of nodes in a network as a function of their position and the underlying topology. One class of such measures builds on eigenvector centrality, where the importance of a node is derived from the importance of its neighboring nodes. For directed and weighted complex networks, where the nodes can carry some intrinsic property value, there have been centrality measures proposed that are variants of eigenvector centrality. However, these expressions all suffer from shortcomings. Here, an extension of such centrality measures is presented that remedies all previously encountered issues. While similar improved centrality measures have been proposed as algorithmic recipes, the novel quantity that is presented here is a purely analytical expression, only utilizing the adjacency matrix and the vector of node values. The derivation of the new centrality measure is motivated in detail. Specifically, the centrality itself is ideal for the analysis of directed and weighted networks (with node properties) displaying a bow-tie topology. The novel bow-tie centrality is then computed for a unique and extensive real-world dataset, coming from economics. It is shown how the bow-tie centrality assesses the relevance of nodes similarly to other eigenvector centrality measures, while not being plagued by their drawbacks in the presence of cycles in the network.

scholarly journals Network robustness improvement via long-range links

2019 ◽  
Vol 6 (1) ◽  
Author(s):  
Vincenza Carchiolo ◽  
Marco Grassia ◽  
Alessandro Longheu ◽  
Michele Malgeri ◽  
Giuseppe Mangioni
Keyword(s):  
Complex Networks ◽  
Long Range ◽  
Real World ◽  
Network Robustness ◽  
Scale Free ◽  
Area Of Interest ◽  
Before And After ◽  
Significant Area ◽  
Better Than

AbstractMany systems are today modelled as complex networks, since this representation has been proven being an effective approach for understanding and controlling many real-world phenomena. A significant area of interest and research is that of networks robustness, which aims to explore to what extent a network keeps working when failures occur in its structure and how disruptions can be avoided. In this paper, we introduce the idea of exploiting long-range links to improve the robustness of Scale-Free (SF) networks. Several experiments are carried out by attacking the networks before and after the addition of links between the farthest nodes, and the results show that this approach effectively improves the SF network correct functionalities better than other commonly used strategies.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

An algorithm Walktrap-SPM for detecting overlapping community structure

2017 ◽  
Vol 31 (15) ◽  
pp. 1750121 ◽  
Author(s):  
Fang Hu ◽  
Youze Zhu ◽  
Yuan Shi ◽  
Jianchao Cai ◽  
Luogeng Chen ◽  
...  
Keyword(s):  
Community Structure ◽  
Random Walk ◽  
Community Detection ◽  
Real World ◽  
Overlapping Community Detection ◽  
Overlapping Communities ◽  
Running Time ◽  
Overlapping Community ◽  
Negative Links ◽  
Novel Algorithm

In this paper, based on Walktrap algorithm with the idea of random walk, and by selecting the neighbor communities, introducing improved signed probabilistic mixture (SPM) model and considering the edges within the community as positive links and the edges between the communities as negative links, a novel algorithm Walktrap-SPM for detecting overlapping community is proposed. This algorithm not only can identify the overlapping communities, but also can greatly increase the objectivity and accuracy of the results. In order to verify the accuracy, the performance of this algorithm is tested on several representative real-world networks and a set of computer-generated networks based on LFR benchmark. The experimental results indicate that this algorithm can identify the communities accurately, and it is more suitable for overlapping community detection. Compared with Walktrap, SPM and LMF algorithms, the presented algorithm can acquire higher values of modularity and NMI. Moreover, this new algorithm has faster running time than SPM and LMF algorithms.

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