Random Walk Graph Research Articles

Noise Tolerance of Expanders and Sublinear Expansion Reconstruction

We consider the problem of online sublinear expander reconstruction and its relation to random walks in “noisy expanders. Given access to an adjacency list representation of a bounded-degree graph $G$, we want to convert this graph into a bounded-degree expander $G'$ changing $G$ as little as possible. The graph $G'$ will be output by a distributed filter: this is a sublinear time procedure that, given a query vertex, outputs all its neighbors in $G'$ and can do so even in a distributed manner, ensuring consistency in all the answers. One of the main tools in our analysis is a result on the behavior of random walks in graph that are almost expanders: graphs that are formed by arbitrarily connecting a small unknown graph (the noise) to a large expander. We show that a random walk from almost any vertex in the expander part will have fast mixing properties, in the general setting of irreducible finite Markov chains. We also design sublinear time procedures to distinguish vertices of the expander part from ...

More Benefits of Adding Sparse Random Links to Wireless Networks: Yet Another Case for Hybrid Networks

We theoretically and experimentally analyze the process of adding sparse random links to random wireless networks modeled as a random geometric graph. While this process has been previously proposed, we are the first to prove theoretical bounds on the improvement to the graph diameter and random walk properties of the resulting graph as a function of the frequency of wires used, where this frequency is diminishingly small. In particular, given a parameter k controlling sparsity, any node has a probability of [Formula: see text] for being a wired link station. Amongst the wired link stations, we consider creating a random 3-regular graph superimposed upon the random wireless network to create model [Formula: see text], and alternatively we consider a sparser model [Formula: see text] as well, which is a random 1-out graph of the wired links superimposed upon the random wireless network. We prove that the diameter for [Formula: see text] is [Formula: see text] with high probability and the diameter for [Formula: see text] is [Formula: see text] with high probability, both of which exponentially improve the [Formula: see text] diameter of the random geometric graph around the connectivity threshold, thus also inducing small-world characteristics as the high clustering remains unchanged. Further, we theoretically demonstrate that as long as k is polylogarithmic in the network size, [Formula: see text] has rapidly mixing random walks with high probability, which also exponentially improves upon the mixing time of the purely wireless random geometric graph, which yields direct improvement to the performance of distributed gossip algorithms as well as normalized edge connectivity. Finally, we experimentally confirm that the algebraic connectivities of both [Formula: see text] and [Formula: see text] exhibit significant asymptotic improvement over that of the underlying random geometric graph. These results further motivate future hybrid networks and advances in the use of directional antennas.

Image denoising by random walk with restart Kernel and non-subsampled contourlet transform

To address the drawbacks of continuous partial differential equations, a diffusion method based on spectral graph theory and random walk with restart kernel is proposed, which uses non-subsampled contourlet transform to capture the geometric feature of image. Specifically, a new graph weighting function is constructed based on the geometric feature. Moreover, a second-order random walk with restart kernel was generated. The derivation shows that the proposed method is equivalent to the denoising methods based on partial differential equations. The simulation results demonstrate that the proposed method can effectively reduce Gaussian noise and preserve image edge with superior performance compared with other graph-based partial differential equation methods.

Graph Laplacians and their Convergence on Random Neighborhood Graphs

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.

Systematic discovery of functional modules and context-specific functional annotation of human genome

The rapid accumulation of microarray datasets provides unique opportunities to perform systematic functional characterization of the human genome. We designed a graph-based approach to integrate cross-platform microarray data, and extract recurrent expression patterns. A series of microarray datasets can be modeled as a series of co-expression networks, in which we search for frequently occurring network patterns. The integrative approach provides three major advantages over the commonly used microarray analysis methods: (1) enhance signal to noise separation (2) identify functionally related genes without co-expression and (3) provide a way to predict gene functions in a context-specific way. We integrate 65 human microarray datasets, comprising 1105 experiments and over 11 million expression measurements. We develop a data mining procedure based on frequent itemset mining and biclustering to systematically discover network patterns that recur in at least five datasets. This resulted in 143,401 potential functional modules. Subsequently, we design a network topology statistic based on graph random walk that effectively captures characteristics of a gene's local functional environment. Function annotations based on this statistic are then subject to the assessment using the random forest method, combining six other attributes of the network modules. We assign 1126 functions to 895 genes, 779 known and 116 unknown, with a validation accuracy of 70%. Among our assignments, 20% genes are assigned with multiple functions based on different network environments. http://zhoulab.usc.edu/ContextAnnotation.

Enabling access-privacy for random walk based data analysis applications

Random walk graph and Markov chain based models are used heavily in many data and system analysis domains, including web, bioinformatics, and queuing. These models enable the description and analysis of various behaviors of stochastic systems. If the system being modeled has certain properties, such as if it is irreducible and aperiodic, close form formulations corresponding to its stationary behavior can be used to analyze its behavior. However, if the system does not have these properties or if the user is not interested in the stationary behavior, then an iterative approach needs to be used to determine potential outcomes based on the initial probability distribution inputs to the model. In this paper, we focus on access-privacy enabled outsourced Markov chain based data analysis applications, where a non-trusted service provider takes (hidden) user queries that are described in terms of initial state distributions, and evaluates them iteratively in an oblivious manner. We show that this iterative process can leak information regarding the possible values of the hidden input if the server has a priori knowledge about the underlying Markovian process. Hence as opposed to simple obfuscation mechanisms, we develop an algorithm based on methodical addition of extra states, which guarantees unbounded feasible regions for the inputs, thus preventing a malicious host from having an informed guess regarding the inputs. In particular, we show that if the underlying transition matrix is diagonalizable then we can compute the exact number of states needed for access-privacy, while if the matrix is non-diagonalizable, then only a lower-bound can be computed.

Distributed Routing of Ads and Bids through Random Walks in the IDOS System

We propose a novel highly distributed system called IDOS (Internet Distributed Objects System) for Buy/Sell interactions on the Internet. In IDOS there is no main or single site that handles user interactions. Instead, each new user “joins” a dynamic graph of users that are currently interested in a specific type of Buy/Sell interaction. The objects for the Buy/Sell interactions are moved between the nodes (users) of the dynamic graph. The Buy/Sell interactions are therefore realized by updating the objects whenever they pass through the current node. This should solve the communication bottleneck caused by the common use of central servers for interactions on the Internet. It does, however, pose a distributed routing problem where messages sent by the users must be distributed between all the “interested” users that are currently connected to the graph. The proposed routing algorithm should overcome the dynamic changes that occur in the graph's topology because some of the users leave the graph while others join it. The proposed solution is to use random walks to distribute the messages, so that the ever-changing topology of the dynamic graph can be overlooked. Consequently, there is no notion of addresses in the system; each user knows only how to communicate with her neighbors. The user must wait until the desired object reaches its node. This simplifies the communication layer, since the centralized client/server TCP/IP infrastructure can be replaced by a point-to-point communication system. We use a system of multiple random “postmen” to overcome the potential communications delay caused by the theoretical large cover-time required by random-walks in graphs. Experimental results of the system as well as how it can be employed to realize public auctions are discussed.