Analysis of node2vec random walks on networks.

Random walk-based node embedding algorithms learn vector representations of nodes by optimizing an objective function of embedding vectors and skip-bigram statistics computed from random walks on the network. They have demonstrated state-of-the-art performance in many supervised learning problems such as link prediction and node classification. Yet, their properties remain poorly understood. This paper studies properties of random walk based node embeddings in the unsupervised setting of discovering latent network block structure, i.e., learning node representations whose cluster structure in Euclidean space reflects node adjacencies within the network. We characterize the ergodic limits of the embedding objective, its generalization, and related convex relaxations to derive corresponding non-randomized versions of the node embedding objectives. We also characterize the optimal node embedding Grammians of the non-randomized objectives for the expected graph of a two-community Stochastic Block Model (SBM). We prove that the solution Grammian has rank 1 for a suitable nuclear norm relaxation of the non-randomized objective. Comprehensive experimental results on SBM random networks reveal that our non-randomized ergodic objectives yield node embeddings whose distribution is Gaussian-like, centered at the node embeddings of the expected network within each community, and concentrate in the linear degree-scaling regime as the number of nodes increases.

A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in time that is at most polynomial in the size of the graph. This fundamental property, however, only holds if the graph does not change over time, while on the other hand many distributed networks are inherently dynamic, and their topology is subjected to potentially drastic changes. In this work we study the mixing (i.e., converging) properties of random walks on graphs subjected to random changes over time. Specifically, we consider the edge-Markovian random graph model: for each edge slot, there is a two-state Markov chain with transition probabilities $p$ (add a non-existing edge) and $q$ (remove an existing edge). We derive several positive and negative results that depend on both the density of the graph and the speed by which the graph changes. We show that if $p$ is very small (i.e., below the connectivity threshold of Erd\H{o}s-R\'{e}nyi random graphs), random walks do not mix (fast). When $p$ is larger, instead, we observe the following behavior: if the graph changes slowly over time (i.e., $q$ is small), random walks enjoy strong mixing properties that are comparable to the ones possessed by random walks on static graphs; however, if the graph changes too fast (i.e., $q$ is large), only coarse mixing properties are preserved.

We analyze the Hunter vs. Rabbit game on a graph, which is a model of communication in adhoc mobile networks. Let G be a cycle graph with N nodes. The hunter can move from a vertex to a vertex along an edge. The rabbit can jump from any vertex to any vertex on the graph. We formalize the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over . We classify strategies using the order O(k −β−1) of their Fourier transformation. We investigate lower bounds and upper bounds of the probability that the hunter catches the rabbit. We found a constant lower bound if β∈(0,1) which does not depend on the size N of the graph. We show the order is equivalent to O(1/logN) if β=1 and a lower bound is 1/N (β−1)/β if β∈(1,2]. These results help us to choose the parameter β of a rabbit strategy according to the size N of the given graph. We introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the hunter catches the rabbit, and also show computing simulation results.

We have applied the lattice bond enumeration method to the calculation of the steady-state diffusion in a lattice with fixed traps. We show that, to first order in density of traps, our random walk calculations for the effective diffusion constant in lattices with periodically arrayed traps are in exact agreement with calculations carried out previously for randomly arrayed traps embedded in a three-dimensional continuum medium (fluid). Our lattice random walk results are independent of dimension for d > 1, and we conjecture that this is also true for the continuum diffusion model.

An exhaustive analysis of the shape properties of discrete self-avoiding random walks is presented. The dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self-avoiding walks are studied. By means of high-precision Monte Carlo simulations it is shown that the characteristics of the shape of self-avoiding random walks, when regarded as a function of the walk length, present the expected asymptotic behaviour. The differences with conventional random walks depend upon the observable considered: the probability distributions of the principal inertia eigenvalues of self-avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks, while in the cases of the asphericity or ratios of inertia eigenvalues, the distributions for self-avoiding walks are somewhat more peaked than their counterparts for conventional random walks. Analytical expressions for the probability distributions of inertia moment ratios and the two-dimensional asphericity are given. For common random walks there is an excellent agreement between these analytical distributions and the Monte Carlo data. In this work it is established that this concordance is maintained if the same analytical distributions are applied to the self-avoiding case. This means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self-avoiding walks.

It is a common practice to describe branching random walks in terms of birth, death and walk of particles, which makes it easier to use them in different applications. The main results obtained for the models of symmetric continuous-time heavy-tailed branching random walks on a multidimensional lattice. We will be mainly interested in studying the problems related to the limiting behavior of branching random walks such as the existence of phase transitions under change of various parameters, the properties of the limiting distribution and the survival probability of a particle population. Emphasis is made on the survival analysis. The answers to these and other questions essentially depend on numerous factors which affect the properties of a branching random walk. Therefore, we will try to describe how the properties of a branching random walk depend on such characteristics of the underlying random walk as finiteness or infiniteness of the variance of jumps. The presented results are based on the Green’s function representations of the transition probabilities of a branching random walk.

A random walk is a path that is comprised of random steps taken in random directions. Random walks present themselves in a variety of environments, and for this reason they have been a subject of relevance spanning disciplines ranging from economics to biology. A cycle in a random walk consists of the choice of a random direction, followed by a step forward, occurring for some duration. We investigated isotropic and non-isotropic random walks involving the following four cycle variations: 1. constant time, constant step size 2. constant time, varied step size 3. varied time, constant step size 4. varied time, varied step size. In biological terms, these cases of a random walks are of interest in modeling dispersal of organisms. In this study we have attempted to establish a mechanism to better study cases of non-normal diffusion in a population using a simplistic model.

The goal of network representation learning is to embed nodes so as to encode the proximity structures of a graph into a continuous low-dimensional feature space. In this paper, we propose a novel algorithm called node2hash based on feature hashing for generating node embeddings. This approach follows the encoder-decoder framework. There are two main mapping functions in this framework. The first is an encoder to map each node into high-dimensional vectors. The second is a decoder to hash these vectors into a lower dimensional feature space. More specifically, we firstly derive a proximity measurement called expected distance as target which combines position distribution and co-occurrence statistics of nodes over random walks so as to build a proximity matrix, then introduce a set of T different hash functions into feature hashing to generate uniformly distributed vector representations of nodes from the proximity matrix. Compared with the existing state-of-the-art network representation learning approaches, node2hash shows a competitive performance on multi-class node classification and link prediction tasks on three real-world networks from various domains.

Representing networks in a low dimensional latent space is a crucial task with many interesting applications in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks for sampling context nodes and the traditional Skip-Gram model to capture center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic exponential family graph embedding model, that generalizes random walk-based network representation learning techniques to exponential family conditional distributions. We study three particular instances of this model, analyzing their properties and showing their relationship to existing unsupervised learning models. Our experimental evaluation on real-world datasets demonstrates that the proposed techniques outperform well-known baseline methods in two downstream machine learning tasks.

The concept of pagerank was first started as a way for determining the ranking of Web pages by Web search engines. Based on relations in interconnected networks, pagerank has become a major tool for addressing fundamental problems arising in general graphs, especially for large information networks with hundreds of thousands of nodes. A notable notion of pagerank, introduced by Brin and Page and denoted by PageRank, is based on random walks as a geometric sum. In this paper, we consider a notion of pagerank that is based on the (discrete) heat kernel and can be expressed as an exponential sum of random walks. The heat kernel satisfies the heat equation and can be used to analyze many useful properties of random walks in a graph. A local Cheeger inequality is established, which implies that, by focusing on cuts determined by linear orderings of vertices using the heat kernel pageranks, the resulting partition is within a quadratic factor of the optimum. This is true, even if we restrict the volume of the small part separated by the cut to be close to some specified target value. This leads to a graph partitioning algorithm for which the running time is proportional to the size of the targeted volume (instead of the size of the whole graph).

We discuss some properties of random walks on comb-like structures. These have been used as analogues for the study of anomalous diffusion along a percolation cluster intersected by loopless dead-ends. It is shown that a number of transport properties along a backbone can be found by using an approximation method based on the continuous-time random walk (CTRW). The principal property resulting from this analysis is the asymptotic form of the probability distribution for the location of a random walker at step n. This allows the calculation of such quantities as the mean-squared displacement after n steps, the expected number of distinct sites visited, and changes in properties of the random walk in response to biasing fields. It is shown that when the times between successive visits to the backbone are random, and distributed according to a stable law different critical phenomena can occur in the model.

A new theorem on random walks suggest some possible revisions of the foundations of Quantum Mechanics. This is presented below in the simplified framework of the description of the evolution of a material point in space. Grossly speaking, it is shown that the probabilities generated by normalizing the square modulus of a sum of probability amplitudes, in the setup of Quantum Mechanics, becomes asymptotically close (under the appropriate limiting conditions) to the probabilities generated by the usual causal processes of Classical Mechanics. This limiting coincidence has a series of interesting potential applications. In particular it allows us to reintroduce the concept of causality within the core of Quantum Mechanics. Moreover, it suggests, among other consequences, that gravitational interaction may not even exist. Even though the interpretations of Quantum Mechanics which follow from this mathematical result may seem to bring some unexpected innovations in the context of theoretical physics, there is an obvious necessity to study its theoretical impact on Quantum Mechanics. The first steps toward this aim are taken in the present article.

Network embedding maps the nodes of a network to a continuous vector space, which can then be used as the input to downstream tasks, such as node classification, node clustering, link prediction, and similarity search. To learn network embedding more effectively, many technologies adopt the approach of random walk to obtain the network structure. As the meta-path of heterogeneous networks emerges, network embedding will be equipped with more semantic interpretation. Consequently, various random walks, based on meta-path strategies, have been proposed for network embedding. However, the combination of semantic and structure in a heterogeneous network cannot achieve ideal results. To overcome this challenge, we start from a task-guided issue by combining the timestamps information in the heterogeneous network, and then employing the method of temporal segmentation to decompose the network into a continuous temporal sequence. Finally, the set of context-paths between nodes is calculated in a continuous vector by the depth-first meta-path search algorithm. More precisely, we propose a Temporal Sliding Density Walk (TSDW) algorithm by combining network semantics and structure effectively. Empirical results for network data show that TSDW could significantly outperform the state-of-the-art representation learning models, including DeepWalk, LINE, Node2vec, PTE, Meapath2vec, HIN2vec, HTNE, and CTDNE by 3.02% to 44.9% of Macro-F1, 0.9% to 18.92% of Micro-F1 in multi-class node classification and 21% to 47% of NMI in node clustering.

Stable distributions are special types of probability distributions whose origin is a particular limit regime of other types of distributions. They are closely related to the simple convolution process, which is introduced first for continuous and then for discrete random variables. This leads to the central limit theorem as one of the most important results of probability theory, as well as to its generalized version which is useful in the analysis of random walks. Extreme-value distributions are also presented, as they possess a limit theorem of their own (Fisher–Tippett–Gnedenko). The last part is devoted to the discussion of discrete-time and continuous-time random walks, together with their characteristic diffusion properties.

Link prediction is a key tool for studying the structure and evolution mechanism of complex networks. Recommending new friend relationships through accurate link prediction is one of the important factors in the evolution, development, and popularization of social networks. At present, scholars have proposed many link prediction algorithms based on the similarity of local information and random walks. These algorithms help identify actual missing and false links in various networks. However, the prediction results significantly differ in networks with various structures, and the prediction accuracy is low. This study proposes a method for improving the accuracy of link prediction. Before link prediction, k‐shell decomposition method is used to layer the network, and the nodes that are in 1‐shell and the nodes that are not linked to the high‐shell in the 2‐shell are deleted. The experiments on four real network datasets verify the effectiveness of the proposed method.