Markov Chains vs Graph Theory

Markov Chains Compared with Graph Theory

Structural Comparison

Aspect	Markov Chains	Graph Theory
Structure	States and transitions	Nodes and edges
Representation	Transition matrix (probabilities)	Adjacency matrix (connections)
Directionality	Directed transitions between states	Directed or undirected edges
Weights	Probabilities assigned to transitions	Edge weights (e.g., cost or flow)
Memory	Memoryless (Markov property)	Memory via paths or cycles
Analysis Techniques	Steady state, mixing time, absorbing states	Connectivity, shortest paths, centrality
Applications	PageRank, queuing systems, stochastic modeling	Optimization, network design, dependency modeling

Markov Chain as a Graph: Nodes as states, edges as transitions, weights as probabilities
Transition Matrix vs Adjacency Matrix: Probability