The heat kernel as the pagerank of a graph

Random walks have been proven to be useful for constructing various algorithms to gain information on networks. Algorithm node2vec employs biased random walks to realize embeddings of nodes into low-dimensional spaces, which can then be used for tasks such as multi-label classification and link prediction. The performance of the node2vec algorithm in these applications is considered to depend on properties of random walks that the algorithm uses. In the present study, we theoretically and numerically analyse random walks used by the node2vec. Those random walks are second-order Markov chains. We exploit the mapping of its transition rule to a transition probability matrix among directed edges to analyse the stationary probability, relaxation times in terms of the spectral gap of the transition probability matrix, and coalescence time. In particular, we show that node2vec random walk accelerates diffusion when walkers are designed to avoid both backtracking and visiting a neighbour of the previously visited node but do not avoid them completely.

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.

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

Multi-scale local feature detection enables downstream registration and recognition tasks in medical image analysis. This paper articulates a novel robust method for multi-scale local feature extraction on volumetric data. The central idea is the elegant unification of local/global eigen-structures within the powerful framework of anisotropic heat diffusion. First, the local vector field is constructed by way of Hessian matrix and its eigenvectors/eigenvalues. Second, anisotropic heat kernels are computed using the vector field’s global graph Laplacian. Robust local features are manifested as extrema across multiple time scales, serving as volumetric heat kernel signature. To tackle the computational challenge for massive volumetric data, we propose a multiresolution strategy for hierarchical feature extraction based on our feature-preserving down-sampling approach. As a result, heat kernels and local feature identification can be approximated at a coarser level first, and then are pinpointed in a localized region at a finer resolution. Another novelty of this work lies at the initial heat design directly using local eigenvalue for anisotropic heat diffusion across the volumetric domain. We conduct experiments on various medical datasets, and draw comparisons with 3D SIFT method. The diffusion property of our local features, which can be interpreted as random walks in statistics, makes our method robust to noise, and gives rise to intrinsic multi-scale characteristics.

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media.

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.

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.

Supercomputers are today made up of hundreds of thousands of nodes. The interconnection network is responsible for connecting all these nodes to each other. Different interconnection networks have been proposed; high performance topologies have been introduced as a replacement for the conventional topologies of recent decades. A high order, a low degree and a small diameter are the usual properties aimed for by such topologies. However, this is not sufficient to lead to actual hardware implementations. Network scalability and topology simplicity are two critical parameters, and they are two of the reasons why modern supercomputers are often based on torus interconnection networks (e.g., Fujitsu K, IBM Sequoia). In this paper we first describe a new topology, torus-connected cycles (TCCs), realizing a combination of a torus and a ring, thus retaining interesting properties of torus networks in addition to those of hierarchical interconnection networks (HINs). Then, we formally establish the diameter of a TCC, and deduce a point-to-point routing algorithm. Next, we propose routing algorithms solving the Hamiltonian cycle problem, and, in a two dimensional TCC, the Hamiltonian path one. Correctness and complexities are formally proved. The proposed algorithms are time-optimal.

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.

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.

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.

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension DF and a multifractal distribution of probabilities of dimensionsD q (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD' q =D q +D F -d, for a finite interval ofq.

Behavioral thermoregulators leverage environmental temperature to control their body temperature. Habitat thermal quality therefore dictates the difficulty and necessity of precise thermoregulation, and the quality of behavioral thermoregulation in turn impacts organism fitness via the thermal dependence of performance. Comparing the body temperature of a thermoregulator with a null (non-thermoregulating) model allows us to estimate habitat thermal quality and the effect of behavioral thermoregulation on body temperature. We define a null model for behavioral thermoregulation that is a random walk in a temporally and spatially explicit thermal landscape. Predicted body temperature is also integrated through time, so recent body temperature history, environmental temperature, and movement influence current body temperature; there is no particular reliance on an organism's equilibrium temperature. We develop a metric called thermal benefit that equates body temperature to thermally dependent performance as a proxy for fitness. We measure thermal quality of two distinct tropical habitats as a temporally dynamic distribution that is an ergodic property of many random walks, and we compare it with the thermal benefit of real lizards in both habitats. Our simple model focuses on transient body temperature; as such, using it we observe such subtleties as shifts in the thermoregulatory effort and investment of lizards throughout the day, from thermoregulators to thermoconformers.