# RANDOM [WALKER]

The term random walk was first introduced by Karl Pearson in 1905. A random walk is a

**mathematical object**, known as a **stochastic or random process**, that describes a path that consists of a succession of random steps on some **mathematical space** such as the integers. An elementary example of a random walk is the random walk on the integer number line Z which starts at 0 and at each step moves +1 or ?1 with equal probability.

Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology as well as economics. Random walks explain the observed **behaviors** of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of ? can be approximated by the use of random walk in an **agent-based modeling environment**.

The code development is based on the following methodology: **1.**Generate a start point / **2.**Create a random number between -1 and 1 / **3.**Assign each the x,y, and z axis one of random numbers / **4.**Assign each the x,y, and z axis one of random numbers / **5.**Take a step in that direction.

**VARIABLE PARAMETERS:**

– SIZE OF STEP IN X AXIS –

– SIZE OF STEP IN Y AXIS –

– LIMIT OF PROBABILITY IN Z AXIS –

– RANDOM SEED –

– NUMBER OF STEPS TAKEN –

The random walker algorithm is an **algorithm for image segmentation**. In the first description of the algorithm, a user interactively labels a small number of pixels with known labels (called seeds), e.g., “object” and “background”. The unlabeled pixels are each imagined to release a random walker, and the probability is **computed** that each pixel’s random walker first arrives at a seed bearing each label, i.e., if a user places K seeds, each with a different label, then it is necessary to compute, for each pixel, the probability that a random walker leaving the pixel will first arrive at each seed. These probabilities may be determined analytically by solving a **system of linear equations**. After computing these probabilities for each pixel, the pixel is assigned to the label for which it is most likely to send a random walker. The image is modeled as a graph, in which each pixel corresponds to a node which is connected to neighboring pixels by edges, and the edges are weighted to reflect the similarity between the pixels.

**RANDOM [WALKER]** is a project developed at IaaC, Institute for Advanced Architecture of Catalonia, developed at Master in Advanced Architecture 2019/20 by:

Students: Holly Carton, Daria Ciobanu-Enescu, Eszter Olah, Oana Taut

Faculty: Angel Munoz