This notes is primarily based on Random Graphs and Complex Networks and Stochastic Processes on Random Graphs by Remco van der Hofstad, which can be found  here. Compared to the books I just mentioned, my notes has far less content than the books by Remco, but offers more explanation for every theorem included.  It is the first long notes I wrote as a graduate student, and you can see how the writing quality improves over chapters. I will regularly update the notes, to remove typos and to rephrase awkward sentences.

You can get the pdf version here.

Contents
1 Branching Process
1.1 Introduction
1.2 Survival vs Extinction
1.3 Family Moments
1.4 Supercritical Branching Process
1.5 Hitting-Time Theorem, Total Progeny and Tree-Like Nature
1.6 Poisson Branching Process
1.7 Binomial and Poisson Branching Process
2 Erdős-Rényi Graph
2.1 Construction and the Exploration Algorithm
2.2 Some Preparation Work
2.3 Results in Subcritical Regime
2.3.1 Proofs of the Bounds
2.4 Law of Large Number for Supercritical Cases
2.4.1 Strategy and Overview
2.4.2 Proof of Lemmas and the LLN
3 Critical Erdős-Rényi Graph
3.1 Largest Critical Cluster
3.1.1 The Statement and Lemmas Needed
3.1.2 The Proof of Theorem 3.1
3.1.3 The Proof of Lemmas
3.2 A Refined Cluster Tail
3.2.1 A Modified Exploration Algorithm
3.2.2 The Proof of Theorem 3.2
3.3 The Connectivity Threshold
4 Configuration Model
4.1 Construction and setup
4.2 Erased configuration model
4.3 Repeated configuration model
5 Preferential Attachment Model
5.1 Definition and Construction
5.2 Degree of Fixed Vertices
5.3 Degree Sequences of PAM
6 First Passage Percolation On Finite Graphs
6.1 First Passage Percolation: Concepts
6.2 FPP on Complete Graphs with Exponential Edge Weights
6.2.1 Limits of Minimal Weight and Fluctuation
6.2.2 CLT of Hopcount
6.3 FPP on Configuration Model: First Glimpse and Difficulties
6.3.1 Introduction and Main results
6.3.2 Local Structure of First Passage Percolation on CM
6.3.3 Back to CM
6.4 New Tool: Continuous-Time Branching Process
6.4.1 Introduction
6.4.2 Yule Process
6.4.3 Bellman-Harris Process
6.4.4 General CTBP
6.5 FPP on CM with General Lifetime Distribution
6.5.1 The Setup and Overview
6.5.2 The Execution
7 Percolation On CM
7.1 The Idea of Percolation
7.2 Percolation on CM: Result and Preparation
7.3 Proof of Theorem 7.2
8 Open Problems and Further Remarks
8.1 FPP on the Lattice
8.1.1 Phase Transition for Bond Percolation on the Lattice
8.1.2 The Limit Shape for FPP on the Lattice
8.2 FPP on Complete Graphs: Fluctuation of Hopcount
8.3 FPP on Conservative Configuration Model
8.4 Phase Transition of Barely Subcritical CM with Finite Third Moment
9 Bibliography