Random walks are a process to generate random step sequences that can model natural phenomena and be used in algorithms. They involve starting at a location and randomly selecting successive locations, similar to the path of a molecule in a liquid. Random walks can be modeled as Markov chains on graphs and simulated in higher dimensions to create fractal patterns. They are used in applications like recommender systems, stock market modeling, and generating random fractal images. A random walk corresponds to an irreducible, aperiodic Markov chain and will have a unique stationary probability distribution if the graph is well-behaved.

1. Random Walks

2. What is Random Walk?
• Random walk is a process, a model or a rule to generate path
sequence of random motion.
• 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.

3. What is Random Walk?
• Many natural phenomena can be modelled as random walk
• the path traced by a molecule as it travels in a liquid or a gas,
• the search path of a foraging animal,
• the price of a fluctuating stock and
• the financial status of a gambler
• PageRank
• Recommender Systems
• Investment theory of stock market
• Generate fractal images
• Even though they may not be truly random in reality

4. Simulation of Normally Distributed Random
Walk
• We start with initial location 100 and generate the random walk
based on normal probability distribution.

5. Simulation on Higher Dimensions
• In higher dimensions, the set of randomly walked points has
interesting geometric properties. In fact, one gets a discrete fractal.

6. Simulation on
Higher
Dimensions

7. Simple random walks on graphs

13. Simple random walks on graphs

14. Simple random walks on graphs

15. Simple random walks on graphs

16. Random Walk and Markov chain
Correspondence between terminology of random walks and Markov chains

17. Random Walk and Markov chain

18. Random Walk and Markov chain
• A Markov chain describes a stochastic process over a set of states
according to a transition probability matrix
• Markov chains are memoryless
• Random walks correspond to Markov chains:
• The set of states is the set of nodes in the graph.
• The elements of the transition probability matrix are the probabilities to
follow and edge from one node to another.

19. Random Walk and Markov chain
Stochastic matrix

20. Random Walk and Markov chain

21. Random Walk and Markov chain

22. Random Walk and Markov chain
• Since G is connected, the random walk on G corresponds to an
irreducible Markov chain.
Irreducible: There is a path from every node to every other node.
• The Perron-Frobenius theorem for nonnegative matrices implies the
existence of a unique probability distribution π, which is a positive
left eigenvector of P associated to its dominant eigenvalue λ = 1

23. Random Walk and Markov chain

24. Random Walk and Markov chain
Does a stationary distribution always exist? Is it unique?
Yes, if the graph is “well-behaved”.
Irreducible and Aperiodic, a directed graph is said to be aperiodic if
there is no integer k > 1 that divides the length of every cycle of the
graph

25. Several concepts

26. Random Walk with Recommender Systems
A
B
C
D
a
b
c
d
e

27. Random Walk with Recommender Systems
A
B
C
D
a
b
c
d
e
A to c

28. Random Walk with Recommender Systems
A
B
C
D
a
b
c
d
e
A to e
A
B
C
b
c
d
e
a
D

29. Random Walk for Recommendation
TrustWalker: A Random Walk Model for Combining Trust-based and
Item-based Recommendation

30. Random Walk algorithms

31. Random Walk algorithms
A subset of Vc of a set genes V have “a prori” known property C
Can we rank the other genes in the set VVc w.r.t their likelihood to
belong to Vc?

32. Random Walk in DeepWalk
DeepWalk, learning latent representations of vertices in a network

33. Random Walk in DeepWalk
DeepWalk, learning latent representations of vertices in a network

34. Reference
• http://people.revoledu.com/kardi/tutorial/StochasticProcess/Rando
mWalk/index.html
• http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaS
chmidt)/random.htm
• https://www.math.uchicago.edu/~lawler/srwbook.pdf
• https://www.cs.cmu.edu/~avrim/598/chap5only.pdf
• http://www.lirmm.fr/~sau/JCALM/Josep.pdf
• http://homes.di.unimi.it/valentini/MB201213/slide/RandomWalksGr
aphs.pdf
• https://arxiv.org/pdf/1403.6652.pdf

35. Weekly Report
• Released the DeepRec
• Revised the sequential recommendation paper
• I implemented a similar idea for rating prediction-> does not work
• Prepare for the learning group
Next Week:
• Some review work
• Revision