AN ADAPTIVE RANDOM WALK BASED DISTRIBUTED CLUSTERING ALGORITHM

2012 ◽  
Vol 23 (04) ◽  
pp. 803-830 ◽  
Author(s):  
ALAIN BUI ◽  
ABDURUSUL KUDIRETI ◽  
DEVAN SOHIER
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Adaptive Algorithm ◽  
Clustering Algorithm ◽  
Dynamic Networks ◽  
Maximum Size ◽  
Distributed Clustering ◽  
The Core ◽  
Topological Change ◽  
Bounded Size

In this paper, we present a fully distributed random walk based clustering algorithm intended to work on dynamic networks of arbitrary topologies. A bounded-size core is built through a random walks based procedure. Its neighboring nodes that do not belong to any cluster are recruited by the core as ordinary nodes. Using cores allow us to formulate constraints on the clustering and fulfill them on any topology. The correctness and termination of our algorithm are proven. We also prove that when two clusters are adjacent, at least one of them has a complete core (i.e. a core with the maximum size allowed by the parameter). One of the important advantages of our mobility-adaptive algorithm is that the re-clustering is local: the management of the connections or disconnections of links and reorganization of nodes affect only the clusters in which they are, possibly adjacent clusters, and at worst, the ordinary nodes of the clusters adjacent to the neighboring clusters. This allows us to bound the diameter of the portion of the network that is affected by a topological change.

A distributed clustering algorithm for large-scale dynamic networks

2011 ◽  
Vol 15 (4) ◽  
pp. 335-350 ◽  
Author(s):  
Thibault Bernard ◽  
Alain Bui ◽  
Laurence Pilard ◽  
Devan Sohier
Keyword(s):  
Large Scale ◽  
Clustering Algorithm ◽  
Dynamic Networks ◽  
Distributed Clustering

Mathematics of networks

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Walks ◽  
Adjacency Matrix ◽  
Graph Partitioning ◽  
Dynamic Networks ◽  
Graph Laplacian ◽  
Network Visualization ◽  
Final Part ◽  
Bipartite Networks ◽  
Component Structure ◽  
Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

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.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

A Novel Locality Sensitive K-Means Clustering Algorithm based on Core Clusters

2013 ◽  
Vol 321-324 ◽  
pp. 1939-1942
Author(s):  
Lei Gu
Keyword(s):  
Clustering Algorithm ◽  
Experimental Results ◽  
Clustering Method ◽  
Neighborhood Graph ◽  
The Core ◽  
Random Samples

The locality sensitive k-means clustering method has been presented recently. Although this approach can improve the clustering accuracies, it often gains the unstable clustering results because some random samples are employed for the initial centers. In this paper, an initialization method based on the core clusters is used for the locality sensitive k-means clustering. The core clusters can be formed by constructing the σ-neighborhood graph and their centers are regarded as the initial centers of the locality sensitive k-means clustering. To investigate the effectiveness of our approach, several experiments are done on three datasets. Experimental results show that our proposed method can improve the clustering performance compared to the previous locality sensitive k-means clustering.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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