Entropy Of Random Walks Research Articles

Invariant measures for iterated function systems with inverses

We consider iterated function systems that contain inverses in the overlapping case. We focus on the parameterized families of iterated function systems with inverses, satisfying the transversality condition. We show that the invariant measure is absolutely continuous for a.e. parameter when the random walk entropy is greater than the Lyapunov exponent. We also show that if the random walk entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value.

Generalized entropy for random walks in regular networks and graphs as a super diffusion

In this paper, the entropy of the stochastic processes created by the movement of a walker in a graph is investigated. The Shannon-Khinchin entropy has four axioms that ignore one of them can make the generalized entropy. Here, we investigate the number of different finite paths asymptotically, for determining a generalized entropy. Then, we will study the regular infinite networks and graphs with finite nodes, with two different types of motion.

Dynamic Random Walk for Superpixel Segmentation.

In this paper, we propose a novel random walk model, called Dynamic Random Walk (DRW), which adds a new type of dynamic node to the original RW model and reduces redundant calculation by limiting the walk range. To solve the seed-lacking problem of the proposed DRW, we redefine the energy function of the original RW and use the first arrival probability among each node pair to avoid the interference for each partition. Relaxation of our DRW is performed with the help of a greedy strategy and the Weighted Random Walk Entropy(WRWE) that uses the gradient feature to approximate the stationary distribution. The proposed DRW not only can enhance the boundary adherence but also can run with linear time complexity. To extend our DRW for superpixel segmentation, a seed initialization strategy is proposed. It can evenly distribute seeds in both 2D and 3D space and generate superpixels in only one iteration. The experimental results demonstrate that our DRW is faster than existing RW models and better than the state-of-the-art superpixel segmentation algorithms with respect to both efficiency and segmentation effects.

Random walks on $\mathbb {Z}$ with exponentially increasing step length and Bernoulli convolutions

We establish a correspondence between the limit distribution and the asymptotic entropy of random walks on $\mathbb {Z}$, which have a sequence of step length that is exponentially increasing up to some error, and Bernoulli convolutions.

The application of degree related clustering coefficient in estimating the link predictability and predicting missing links of networks.

Though a lot of valuable algorithms of link prediction have been created, it is still difficult to improve the accuracy of link prediction for some networks. Such difficulties may be due to the intrinsic topological features of these networks. To reveal the correlation between the network topology and the link predictability, we generate a group of artificial networks by keeping some structural features of an initial seed network. Based on these artificial networks and some real networks, we find that five topological measures including clustering coefficient, structural consistency, random walk entropy, network diameter, and average path length significantly show their impact on the link predictability. Then, we define a topological score that combines these important topological features. Specifically, it is an integration of structural consistency with degree-related clustering coefficient defined in this work. This topological score exhibits high correlation with the link predictability. Finally, we propose an algorithm for link prediction based on this topological score. Our experiment on eight real networks verifies good performance of this algorithm in link prediction, which supports the reasonability of the new topological score. This work could be insightful for the study of the link predictability.

On the joint behaviour of speed and entropy of random walks on groups

For every 3/4 \le \delta, \beta< 1 satisfying \delta\leq \beta < \frac{1+\delta}{2} we construct a finitely generated group \Gamma and a (symmetric, finitely supported) random walk X_n on \Gamma so that its expected distance from its starting point satisfies \mathbf E |X_n|\asymp n^{\beta} and its entropy satisfies H(X_n)\asymp n^\delta . In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions f,h up to a constant factor as long as the functions satisfy the relation n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^\gamma for some \gamma<1 .

Erratum to “Topological Entropy of Random Walks on Mapping Class Groups”

Erratum to “Topological Entropy of Random Walks on Mapping Class Groups”

Topological Entropy of Random Walks on Mapping Class Groups

For any pseudo-Anosov diffeomorphism on a closed orientable surface $S$ of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichm\uller space with respect to the Teichm\uller metric. In this paper, we consider random walks on the mapping class group of $S$. The drift of a random walk is defined as the translation distance of the random walk. We define the topological entropy of a random walk and prove that it almost surely agrees with the drift on the Teichm\uller space with respect to the Teichm\uller metric.

Asymptotic entropy of random walks on Fuchsian buildings and Kac–Moody groups

In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac–Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotropic random walk onto an apartment of the building, to prove existence of the asymptotic entropy for this retracted walk, and to ‘lift’ this in order to deduce the existence of the entropy for the random walk on the building.

Asymptotic entropy of random walks on regular languages over a finite alphabet

We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.

Differentiating the entropy of random walks on hyperbolic groups

We show that the asymptotic entropy of a random walk on a nonelementary hyperbolic group, with symmetric and bounded increments, is differentiable and we identify its derivative as a correlation. We also prove similar results for the rate of escape.

Regularity of the entropy for random walks on hyperbolic groups

We consider nondegenerate, finitely supported random walks on a finitely generated Gromov hyperbolic group. We show that the entropy and the escape rate are Lipschitz functions of the probability if the support remains constant.

An Abramov formula for stationary spaces of discrete groups

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

Exponential time improvement for min-wise based algorithms

In this paper we extend the notion of min-wise independent family of hash functions by defining a k-min-wise independent family of hash functions. Informally, under this definition, all subsets of size k of any fixed set X have an equal chance to have the minimal hash values among all the elements in X, when the probability is over the random choice of hash function from the family. This property measures the randomness of the family, as choosing a truly random function, obviously, satisfies the definition for k = ∣ X∣. We define and give an efficient time and space construction of approximately k-min-wise independent family of hash functions by extending Indyk’s construction of approximately min-wise independent. The number of words needed to represent each function is O ( k log log ( 1 ϵ ) + log ( 1 ϵ ) ) , which is only suboptimal by a factor of O ( log log ( 1 ϵ ) ) , where ϵ ∈ (0, 1) is the desired error bound. This construction is the first applicable for sampling bottom- k sketches out of the universe. In addition, we introduce a general and novel technique that utilizes our construction, and can be used to improve many min-wise based algorithms. As an example we show how to apply it for similarity estimation over data streams, and reduce exponentially the run time of the current known result [5]. In addition, we also discuss improvements of known algorithms for estimating rarity and entropy of random walk over graphs.

Asymptotic Entropy of Random Walks on Free Products

Suppose we are given the free product $V$ of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.

Random Walks on Directed Covers of Graphs

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates no longer coincide in general. Furthermore, the behavior of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.

Liouville property for groups and manifolds

We introduce a method to estimate the entropy of random walks on groups. We apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not Liouville. We also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non-trivial Poisson boundary if and only if this group is not virtually nilpotent. This, in particular, shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient Lie group. As another application we get a series of examples of amenable groups such that any finite entropy non-degenerate measure on them has non-trivial Poisson boundary. Since the groups in question are amenable, they do admit measures such that the corresponding random walks have trivial boundary; the above shows that such measures on these groups have infinite entropy.

Entropy of Random Walks on Groups and the Macaev Norm

Entropy of Random Walks on Groups and the Macaev Norm

Entropy of random walks on groups and the Macaev norm

We show that a discrete group on which there is a finitary random walk with positive entropy satisfies a certain condition involving the Macaev norm. This links the entropy of random walks on groups to the author’s work on quasicentral approximate units relative to normed ideals in perturbation theory. On the other hand, the condition we are considering is also an analogue for the Macaev norm of Yamasaki’s hyperbolicity condition.