Foundations of the 2N-ary Choice Tree Model

Jim Propp's rotor–router model is a deterministic analog of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order.Cooper and Spencer (Comb Probab Comput 15 (2006) 815–822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid ℤd and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors.This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite k‐ary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least exp(Ω(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

The effect of irrelevant preference arguments on consumer choice

How does attention modulate sensory representations? In order to probe the underlying neural mechanisms, we established a simple rodent model of modality-specific attention. Rats were trained to perform distinct auditory two-tone discrimination and olfactory odor discrimination in a two alternative choice (2AC) paradigm. To determine auditory cortex’s role in this frequency discrimination task, we used GABA-A receptor agonist muscimol to transiently and reversibly inactivate auditory cortexes bilaterally in rats performing simple interleaved auditory and olfactory discrimination. With olfactory discrimination performance serving as internal control for motivation and decision making capability, we found only auditory two-tone discrimination was selectively impaired in these rats. This shows the auditory cortex is involved in this two-tone discrimination task.To investigate the neural correlate of modality-specific attention in the auditory cortex, we trained rats to perform interleaved auditory and olfactory blocks (of 50~70 trials each) in a single session. In auditory blocks, pure tones were either presented with or without a neutral odor (caproic acid, n=2 and 3 respectively), and subjects were rewarded for discriminating auditory stimuli. In olfactory blocks, both task odors and pure tones were presented simultaneously, and subjects were rewarded for discriminating olfactory stimuli. We recorded neural responses in primary auditory cortex (area A1) in freely moving rats while subjects performed this behavior. Single unit responses to tones were heterogeneous, and included transient, sustained, and suppressed. We found 205 of 802 units recorded responsive to the stimuli we used. Of these 205 units, 18.5% showed modality-specific attentional modulation of the anticipatory activity before tone onset. In addition, we also observed in smaller proportion of units (11.2%) modality-specific attentional modulation of the tone-evoked responses; in most cases, the responses to a particular auditory stimulus was enhanced in the auditory block (or, equivalently, suppressed in the olfactory block). Attention increased choice probability of the population in the auditory block. We have also observed significant behavior choice probability in small proportions of units. Our results suggest that shifting attention between audition to olfaction tasks can modulate the activity of single neurons in primary auditory cortex.

This study introduces a novel simulation-based approach for addressing the stochastic Dynamic Traffic Assignment (DTA) problem, specifically targeting large, congested networks under dynamic conditions which is a characteristic of urban mobility environment. The proposed methodology leverages an underlying random walk model for route selection, drawing inspiration from the concept of equivalent impedance in electrical networks. This alternative route choice model iteratively condenses non-overlapping subnetworks into virtual links, allowing for the dynamic estimation of equivalent time-dependent virtual travel costs. Consequently, the downstream link choice probabilities for all destinations are computed, and by employing a random walk model, the route choice decision-making process is shifted to nodes. This approach closely aligns with travelers' real-life behavior, supporting a finer temporal segmentation of evolving traffic conditions and improving the precision of performance assessments. Furthermore, the route choice model addresses the limitations of other Markovian route choice models in handling overlapping routes and scaling issues.

We study the neighborhoods of a typical point $$Z_n$$ visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to $$\mu ^{*n}(Z_n)$$ . If such neighborhood contains a ball of radius $$C \sqrt{n}$$ , we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $$\mathbb {Z}$$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa’s criterion for the property $$H_\mathrm{FD}$$ . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $$\mathbb {Z}^2$$ with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

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.

A previously reported positive influence of a distractor alternative on decision accuracy was driven by an unintended covariation between key variables, beyond which the distractor has a modest negative influence on accuracy, possibly mirroring asymmetric multiattribute context effects.

Random walks with similar transition probabilities

A reverse duality between the asymmetric simple exclusion process (ASEP) with open boundary conditions and a biased random walk of a single particle is proved for a special manifold of boundary parameters of the ASEP. The duality function is given by the configuration probabilities of a family of Bernoulli shock measures with a microscopic shock at site x of the lattice. The boundary conditions of the dual random walk, which can be reflecting or absorbing, depend on the choice of the duality function. As a consequence of this duality, the full time-dependent distribution of the open ASEP starting from a fixed shock measure at x is given for any time t>0 by a convex combination of shock measures with shock at position y. The coefficients are the transition probabilities P(x, t|y, 0) at time t of the random walk starting at site y. It is also shown that this family of shock measures arises from a similarity transformation of the two-dimensional representation of the matrix algebra for the stationary matrix product measure of the open ASEP. This implies a further duality between random walks with different boundary conditions.

Spiking Irregularity and Frequency Modulate the Behavioral Report of Single-Neuron Stimulation

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.

In this paper, we define a class of weighted directed networks with the weight of its edge dominated by a weight parameter w. According to the construction of the networks, we study two types of random walks in the weighted directed networks with a trap fixed on the central node, i.e., standard random walks and mixed random walks. For the standard random walks, the trapping process is controlled by a weight parameter w which changes the transition probability of random walks. For the mixed random walks including nonnearest-neighbor hopping, the trapping process is steered by a stochastic parameter [Formula: see text], where [Formula: see text] changes the walking rule. For the above two techniques, we derive both analytically the average trapping time (ATT) as the measure of trapping efficiency, and the obtained analytical expressions are in good agreement with the corresponding numerical solutions for different values of w and [Formula: see text]. The obtained results indicate that ATT scales superlinearly with network size Nn and the weight parameter w changes simultaneously the prefactor and the leading scalings of ATT, while the stochastic parameter [Formula: see text] can only alter the prefactor of ATT and leave the leading scalings of ATT unchanged. This work may help in paving the way for understanding the effects of the link weight and nonnearest-neighbor hopping in real complex systems.

Spontaneous collapse models use non-linear stochastic modifications of the Schrödinger equation to suppress superpositions of eigenstates of the measured observable and drive the state to an eigenstate. It was recently demonstrated that the Born rule for transition probabilities can be modeled using the linear Schrödinger equation with a Hamiltonian represented by a random matrix from the Gaussian unitary ensemble. The matrices representing the Hamiltonian at different time points throughout the observation period are assumed to be independent. Instead of suppressing superpositions, such Schrödinger evolution makes the state perform an isotropic random walk on the projective space of states. The relative frequency of reaching different eigenstates of an arbitrary observable in the random walk is shown to satisfy the Born rule. Here, we apply this methodology to investigate the behavior of a particle in the context of the double-slit experiment with measurement. Our analysis shows that, in this basic case, the evolution of the particle’s state can be effectively captured through a random walk on a two-dimensional submanifold of the state space. This random walk reproduces the Born rule for the probability of finding the particle near the slits, conditioned on its arrival at one of them. To ensure that this condition is satisfied, we introduce a drift term representing a change in the variance of the position observable for the state. It is argued that the drift term accounts for the energy transfer and trapping incurred during the particle’s interaction with the detector. A drift-free model, based on equivalence classes of states indistinguishable by the detector, is also considered. The resulting random walk, with or without drift, serves as a suitable model for describing the transition from the initial state to an eigenstate of the measured observable in the experiment, offering new insights into its potential underlying mechanisms.

Decision Making in Recurrent Neuronal Circuits

(1964). Random Walk with Transition Probabilities That Depend on Direction of Motion. Mathematics Magazine: Vol. 37, No. 4, pp. 248-250.