11.4 Applications to random combinatorial structures

Random combinatorial structures are the building blocks of many complex systems. This section explores how these structures behave when formed randomly, focusing on trees, graphs, and permutations.

We'll dive into the fascinating world of stochastic processes and probabilistic models. These tools help us understand and predict the behavior of random systems, from walks and occupancy problems to balls-and-bins and urn models.

Random Structures

Characteristics of Random Trees

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• Random trees represent graph structures with no cycles and random connections between nodes
• Binary search trees emerge from random insertions into initially empty trees
• Cayley trees consist of labeled nodes with random connections, forming tree structures
• Height of random binary search trees typically grows logarithmically with the number of nodes
• Random tree models apply to evolutionary processes (phylogenetic trees) and computer science (decision trees)

Properties of Random Graphs

• Random graphs form by connecting vertices with edges based on probability
• Erdős-Rényi model generates random graphs by independently connecting each pair of vertices with probability p
• Giant component emerges in random graphs when the average degree exceeds 1
• Connectivity threshold occurs when the average degree reaches log n, where n is the number of vertices
• Random graph models describe social networks, neural connections, and internet topology

Analysis of Random Permutations

• Random permutations shuffle elements of a set into random order
• Cycle structure of random permutations follows a Poisson distribution for small cycle lengths
• Expected number of cycles in a random permutation of n elements is approximately log n
• Longest increasing subsequence in a random permutation has an expected length of about 2√n
• Random permutation models apply to cryptography (encryption algorithms) and computational biology (gene rearrangements)

Stochastic Processes

Fundamentals of Random Walks

• Random walks model paths consisting of successive random steps
• Simple random walk on a line moves left or right with equal probability at each step
• Central Limit Theorem applies to random walks, leading to normal distribution of final positions
• First return time to the origin follows a heavy-tailed distribution
• Random walk models describe Brownian motion, stock price fluctuations, and particle diffusion

Analysis of Occupancy Problems

• Occupancy problems study distribution of objects randomly placed into containers
• Birthday problem calculates probability of shared birthdays in a group
• Coupon collector problem analyzes time to collect all items in a set through random sampling
• Load balancing in distributed systems modeled by balls-into-bins occupancy problems
• Occupancy models apply to hash table analysis and network packet distribution

Exploring the Coupon Collector Problem

• Coupon collector problem determines expected time to collect all distinct items
• Expected number of trials to collect all n coupons is n * H_n, where H_n is the nth harmonic number
• Variance of collection time is approximately (π^2 / 6) * n^2
• Tail bounds exist for probability of exceeding expected collection time
• Coupon collector model applies to Pokemon card collecting and testing software with random inputs

Probabilistic Models

Applications of Balls-and-Bins Model

• Balls-and-bins model randomly distributes m balls into n bins
• Maximum load (most balls in any bin) grows as log n / log log n for m = n
• Power of two choices reduces maximum load to log log n / log 2 by selecting least loaded of two random bins
• Balls-and-bins models apply to load balancing in distributed systems and hash table performance
• Generalizations include weighted balls and biased bin selection

Analyzing Urn Models

• Urn models study probability distributions of colored balls drawn from urns
• Pólya urn model adds balls of the drawn color, leading to reinforcement effects
• Ehrenfest urn model transfers balls between urns, modeling particle diffusion
• Hypergeometric distribution arises from sampling without replacement from an urn
• Urn models describe population genetics (allele frequencies) and contagion processes (disease spread)