Deterministic Walks Research Articles

Using deterministic self-avoiding walks as a small-world metric on Watts–Strogatz networks

Using deterministic self-avoiding walks as a small-world metric on Watts–Strogatz networks

Deterministic random walk model in NetLogo and the identification of asymmetric saturation time in random graph

Interactive programming environments are powerful tools for promoting innovative network thinking, teaching science of complexity, and exploring emergent phenomena. This paper reports on our recent development of the deterministic random walk model in NetLogo, a leading platform for computational thinking, eco-system thinking, and multi-agent cross-platform programming environment. The deterministic random walk is foundational to modeling dynamical processes on complex networks. Inspired by the temporal visualizations offered in NetLogo, we investigated the relationship between network topology and diffusion saturation time for the deterministic random walk model. Our analysis uncovers that in Erdős–Rényi graphs, the saturation time exhibits an asymmetric pattern with a considerable probability of occurrence. This behavior occurs when the hubs, defined as nodes with relatively higher number of connections, emerge in Erdős–Rényi graphs. Yet, our analysis yields that the hubs in Barabási–Albert model stabilize the the convergence time of the deterministic random walk model. These findings strongly suggest that depending on the dynamical process running on complex networks, complementing characteristics other than the degree need to be taken into account for considering a node as a hub. We have made our development open-source, available to the public at no cost at https://github.com/bravandi/NetLogo-Dynamical-Processes.

Prime numbers and random walks in a square grid.

In recent years, computer simulations have played a fundamental role in unveiling some of the most intriguing features of prime numbers. In this paper, we define an algorithm for a deterministic walk through a two-dimensional grid, which we refer to as a prime walk. The walk is constructed from a sequence of steps dictated by and dependent on the sequence of the last digits of the primes. Despite the apparent randomness of this generating sequence, the resulting structure-in both two and three dimensions-created by the algorithm presents remarkable properties and regularities in its pattern, which we proceed to analyze in detail.

Deterministic, random, or in between? Inferring the randomness level of wildlife movements

BackgroundWhen assessing connectivity, it is crucial to rely on accurate modeling frameworks that consider species movement preferences and patterns. One important aspect is the level of randomness or unpredictability in the route selection. In this respect, traditional approaches (based on least-cost path or circuit theory) consider species movements unrealistically as totally deterministic or as totally random. A recent approach (randomized shortest path) advocates for choosing intermediate levels of randomness through a single parameter. This parameter may be optimized by validating connectivity surfaces developed from different levels of randomness against observed movement data. However, connectivity models are seldom validated, and it is still unclear how to approach this task. To address this knowledge gap, this paper aims at comparing different validation methods to infer the optimal randomness level in connectivity studies. Additionally, we aimed to disentangle the practical consequences of applying traditional connectivity approaches versus using an optimized level of movement randomness when delineating corridors.MethodsThese objectives were accomplished through the study case of the Iberian lynx, an endangered species whose maintenance and recovery depend on the current connectivity among its population nuclei. We firstly determined a conductance surface based on point selection functions accounting for the behavioral state (territorial or exploratory) of individuals. Secondly, we identified the level of randomness that better fits lynxes’ movements with independent GPS locations and different validation techniques. Lastly, we delineated corridors between lynx population nuclei through a) the randomized shortest path approach and the extreme and optimal levels of randomness of each validation method, and b) the traditional connectivity approaches.ResultsAccording to all used validation methodologies, models with intermediate levels of randomness outperformed those with extreme randomness levels representing totally deterministic or random movements. We found differences in the optimal randomness level among validation methods but similar results in the delineation of corridors. Our results also revealed that models with extreme randomness levels (deterministic and random walk) of the randomized path approach provided equivalent corridor networks to those from traditional approaches. Moreover, these corridor networks calculated with traditional approaches showed notable differences in patterns from the corridor network calculated with an optimized randomness level.ConclusionsHere we presented a connectivity model with a solid biological basis that calibrates the level of movement randomness and is supported by comprehensive validation methods. It is thus a step forward in the search and evaluation of connectivity approaches that lead to improved, efficient, and successful management actions.

Entropy dimension for deterministic walks in random sceneries

AbstractEntropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

THE NEAREST UNVISITED VERTEX WALK ON RANDOM GRAPHS

We revisit an old topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.

Deterministic walks in a random cylinder

Deterministic walks in a random cylinder

Deterministic walks in random environment

Motivated by the random Lorentz gas, we study deterministic walks in random environment and show that (in simple, yet relevant cases) they can be reduced to a class of random walks in random environment where the jump probability depends (weakly) on the past. In addition, we prove few basic results (hopefully, the germ of a general theory) for the latter purely probabilistic model.

Texture classification using deterministic walk and the influence of the neighbor set

Recently, the deterministic partially self-avoiding walk was proposed as an approach for texture characterization. Unfortunately, tourist walk is subjected to different parameters and these parameters may influence its performance. Although the influence of some parameters is well known, there is a lack of information concerning about the influence of the neighbor set used during the walk. As our contribution, we investigated the influence of the neighbor set, thus providing a better understanding on how neighborhood affects the method’s ability to characterize texture samples. To accomplish that, we evaluated three different types of neighborhood and how the radius value used to generate each neighborhood affects the method’s performance.

The cover time of deterministic random walks for general transition probabilities

A deterministic random walk is a deterministic process analogous to a random walk. While there are some results on the cover time of the rotor-router model, which is a deterministic random walk corresponding to a simple random walk, nothing is known about the cover time of deterministic random walks emulating general transition probabilities. This paper is concerned with the shortest remaining time (SRT)-router model with multiple tokens, which is a deterministic process coping with general transition probabilities possibly containing irrational numbers. For the model, we give an upper bound of the cover time, which is the first result on the cover time of deterministic random walks for general transition probabilities. Our upper bound also improves the existing bounds for the rotor-router model in some cases.

Fundamental Law of Memory Recall.

Human memory appears to be fragile and unpredictable. Free recall of random lists of words is a standard paradigm used to probe episodic memory. We proposed an associative search process that can be reduced to a deterministic walk on random graphs defined by the structure of memory representations. The corresponding graph model can be solved analytically, resulting in a novel parameter-free prediction for the average number of memory items recalled (R) out of M items in memory: R=sqrt[3πM/2]. This prediction was verified with a specially designed experimental protocol combining large-scale crowd-sourced free recall and recognition experiments with randomly assembled lists of words or common facts. Our results show that human memory can be described by universal laws derived from first principles.

Dynamic texture analysis using networks generated by deterministic partially self-avoiding walks

Dynamic textures are sequences of images (video) with the concept of texture patterns extended to the spatiotemporal domain. This research field has attracted attention due to the range of applications in different areas of science and the emergence of a large number of multimedia datasets. Unlike the static textures, the methods for dynamic texture analysis also need to deal with the time domain, which increases the challenge for representation. Thus, it is important to obtain features that properly describe the appearance and motion properties of the dynamic texture. In this paper, we propose a new method for dynamic texture analysis based on Deterministic Partially Self-avoiding Walks (DPSWs) and network science theory. Here, each pixel of the video is considered a vertex of the network and the edges are given by the movements of the deterministic walk between the pixels. The feature vector is obtained by calculating network measures from the networks generated by the DPSWs. The modeled networks incorporate important characteristics of the DPSWs transitivity and their spatial arrangement in the video. In this paper, two different strategies are tested to apply the DPSWs in the video and capture appearance and motion characteristics. One strategy applies the DPSWs in three orthogonal planes and the other is based on spatial and temporal descriptors. We validate our proposed method in Dyntex++ and UCLA (and their variants) databases, which are two well-known dynamic texture databases. The results have demonstrated the effectiveness of the proposed approach using a small feature vector for both strategies. Also, the proposed method improved the performance when compared to the previous DPSW-based method and the network-based method.

A discrete shuffled frog-leaping algorithm to identify influential nodes for influence maximization in social networks

Influence maximization problem aims to select a subset of k most influential nodes from a given network such that the spread of influence triggered by the seed set will be maximum. Greedy based algorithms are time-consuming to approximate the expected influence spread of given node set accurately and not well scalable to large-scale networks especially when the propagation probability is large. Conventional heuristics based on network topology or confined diffusion paths tend to suffer from the problem of low solution accuracy or huge memory cost. In this paper an effective discrete shuffled frog-leaping algorithm (DSFLA) is proposed to solve influence maximization problem in a more efficient way. Novel encoding mechanism and discrete evolutionary rules are conceived based on network topology structure for virtual frog population. To facilitate the global exploratory solution, a novel local exploitation mechanism combining deterministic and random walk strategies is put forward to improve the suboptimal meme of each memeplex in the frog population. The experimental results of influence spread in six real-world networks and statistical tests show that DSFLA performs effectively in selecting targeted influential seed nodes for influence maximization and is superior than several state-of-the-art alternatives.

Anomalous Diffusion in Random Dynamical Systems.

Consider a chaotic dynamical system generating diffusionlike Brownian motion. Consider a second, nonchaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this random dynamical system? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line, we find anomalous dynamics characterized by aging, weak ergodicity breaking, breaking of self-averaging, and infinite invariant densities. This result holds for general types of noise and for perturbing nonlinear dynamics in bifurcation scenarios.

Topology-Hiding Computation on All Graphs

A distributed computation in which nodes are connected by a partial communication graph is called topology hiding if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes (Moran–Orlov–Richelson, TCC’15; Hirt et al., Crypto’16) as well as for other graph families, such as cycles, trees, and low circumference graphs (Akavia–Moran, Eurocrypt’17), but the feasibility question for general graphs was open. In this work, we positively resolve the above open problem: we prove that topology-hiding computation is feasible for all graphs under either the decisional Diffie–Hellman or quadratic residuosity assumption. Our techniques employ random or deterministic walks to generate paths covering the graph, upon which we apply the Akavia–Moran topology-hiding broadcast for chain graphs (paths). To prevent topology information revealed by the random walk, we design multiple graph-covering sequences that, together, are locally identical to receiving at each round a message from each neighbor and sending back a processed message from some neighbor (in a randomly permuted order).

A deterministic walk on the randomly oriented Manhattan lattice

Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic walk is then started at the origin and at each step moves diagonally to the nearest vertex in the direction of the horizontal and vertical lines of the present location. This definition can be generalized, in a natural way, to larger dimensions, but we mainly focus on the two dimensional case. In this context the process localizes on two vertices at all large times, almost surely. We also provide estimates for the tail of the length of paths, when the walk is defined on the two dimensional lattice. In particular, the probability of the path to be larger than $n$ decays sub-exponentially in $n$. It is easy to show that higher dimensional paths may not localize on two vertices but will still eventually become periodic, and are therefore bounded.

The entanglement of deterministic aperiodic quantum walks** Project supported by the National Natural Science Foundation of China (Grant Nos. 11575087 and 11175087).

We study the entanglement between the internal (coin) and the external (position) degrees of freedom in the dynamic and the static deterministic aperiodic quantum walks (QWs). For the dynamic (static) aperiodic QWs, the coin depends on the time (position) and takes two coins C(α) and C(β) arranged in the two classes of generalized Fibonacci (GF) and the Thue–Morse (TM) sequences. We found that for the dynamic QWs, the entanglement of three kinds of the aperiodic QWs are close to the maximal value, which are all much larger than that of the homogeneous QWs. Further, the first class of GF (1st GF) QWs can achieve the maximum entangled state, which is similar to that of the dynamic disordered QWs. And the entanglement of 1st GF QWs is greater than that of the TM QWs, being followed closely by the entanglement of the second class of GF (2nd GF) QWs. For the static QWs, the entanglement of three kinds of the aperiodic QWs are also close to the maximal value and 1st GF QWs can achieve the maximum entangled state. The entanglement of the TM QWs is between 1st GF QWs and 2nd GF QWs. However, the entanglement of the static disordered QWs is less than that of three kinds of the aperiodic QWs. This is different from those of the dynamic QWs. From these results, we can conclude that the dynamic and static 1st GF QWs can also be considered as maximal entanglement generators.

A temporal central limit theorem for real-valued cocycles over rotations

On considere des marches aleatoires sur la droite reelle, engendres par des rotations irrationnelles, ou, de maniere equivalente, des produits croises d’une rotation par un nombre reel $\alpha$, dont le cocycle est une fonction constante par morceaux de moyenne nulle admettant un saut de un a une singularite $\beta$. Si $\alpha$ est mal approche par des rationnels et $\beta$ n’est pas bien approche par l’orbite de $\alpha$, nous demontrons une version temporelle du Theoreme de la Limite Centrale (ou un Temporal Central Limit theorem dans la terminologie qui a ete introduite recemment par D. Dolgopyat et O. Sarig). Plus precisement, nous montrons que, pour chaque point initial fixe, les variables aleatoires d’occupation, proprement renormalisees, tendent vers une variable aleatoire de loi normale. Ce resultat generalise un theoreme de J. Beck dans le cas sparticulier ou $\alpha$ est un nombre irrationnel quadratique, $\beta$ est un nombre rationnel et le point initial est l’origine. Ce resultat de Beck a ete montre avec de nouvelles methodes et etendu par Avila–Dolgopyat–Duryev–Sarig (Israel J. Math. 207 (2015) 653–717) et Dolgopyat–Sarig (J. Stat. Phys. 166 (2017) 680–713) a l’aide d’une renormalisation geometrique. Dans ce papier, nous utilisons aussi la renormalisation, mais, au lieu d’avoir recours a un argument geometrique, nous proposons d’utiliser l’algorithme de fraction continue avec une version dynamique de l’expansion de Ostrowski. Cela nous donne un codage symbolique qui nous permet de reduire le resultat principal a un theoreme de la limite centrale pour de chaines de Markov non-homogenes.

The robot crawler graph process

Information gathering by crawlers on the web is of practical interest. We consider a simplified model for crawling complex networks such as the web graph, which is a variation of the robot vacuum edge-cleaning process of Messinger and Nowakowski. In our model, a crawler visits nodes via a deterministic walk determined by their weightings which change during the process deterministically. The minimum, maximum, and average time for the robot crawler to visit all the nodes of a graph is considered on various graph classes such as trees, multi-partite graphs, binomial random graphs, and graphs generated by the preferential attachment model.

Unbounded Discrepancy of Deterministic Random Walks on Grids

Unbounded Discrepancy of Deterministic Random Walks on Grids