What the book covers

1. Weak ties and small worlds. Granovetter’s argument for “the strength of weak ties” and his data; how, in the Watts–Strogatz model, a few “shortcuts” make path lengths collapse while clustering barely changes; and the later tests with LinkedIn and Facebook data, which found that the most useful ties are often of moderate strength, while most help arrives through the very many weak ties.
2. Structural holes. Burt’s network constraint, effective size and betweenness centrality (Brandes’s algorithm in the standard library); what evidence there is for the returns to brokerage, and in which cultural and organisational settings they vanish or turn into a cost.
3. Simple and complex contagion. Why behaviours that need reinforcement from several sources spread better in clustered networks; the width and placement of “wide bridges”; heterogeneous thresholds; and a 2024 theoretical result showing that if thresholds are even slightly noisy, long ties speed spreading up again.
4. Scale-free networks and robustness. The Barabási–Albert preferential-attachment model, the statistical dispute over the claim that “scale-free networks are rare”, and how the largest connected component changes under random failure and targeted attack.
5. Centrality and PageRank. Eigenvector centrality, Bonacich’s beta-centrality (and the power of being depended upon), and the power iteration, damping factor and dangling nodes of PageRank.
6. Information cascades. The sequential Bayesian model of Bikhchandani, Hirshleifer and Welch, the probabilities of a cascade and of a wrong cascade, and why making private signals public and voting independently can break a wrong cascade.

Chapter 7 reads this modern research beside the alliances and counter-alliances of the Stratagems of the Warring States and the Records of the Grand Historian, the “three kinds of friends who benefit” in the Analects, and the passage on “not letting the four kinds of people live mixed together” in the “Discourses of Qi” of the Discourses of the States, and sets out the ethical boundary between Warring States statecraft and modern networks of mutually beneficial cooperation. Chapter 8 offers a four-step topology checklist and a pre-registered trial randomised by team.

This book is related to two other 5D Library volumes. Volume 6, The Economics of Attention, has already implemented a preferential-attachment network (with an exponent of about 2.91), the information cascades of Bikhchandani and colleagues, and the phase transition in Granovetter’s threshold model; this book does not repeat those programs, but builds on them to discuss network robustness and attack, the statistical testing of power laws, and the mechanisms that break wrong cascades. Chapter 5 of volume 13, The Classical and Statistical Mechanics of Action, compared the spread of simple and complex contagions on a clustered ring and on random networks (thresholds 1, 2 and 3); Chapter 3 of this book builds on that to discuss the width and placement of bridges, heterogeneous thresholds and noisy thresholds.

Claims corrected on checking

Several claims in the author’s original framework needed correction or qualification once they were checked. First, the figures in Granovetter’s 1973 paper behind “most people found their jobs through acquaintances they rarely saw” come from a subsample of 54 people who had found jobs through personal contacts, within his study of professional, technical and managerial workers in Newton, a suburb of Boston, not from the whole sample; and later large-scale tests did not simply confirm “the weaker the better”: randomised experiments on LinkedIn found that ties of moderate strength helped most with changing jobs, while Facebook data showed that a single strong tie is more useful but most help comes through the far more numerous weak ties. Second, Burt’s returns to brokerage have clear limits: in a study of Chinese high-tech firms, structural holes were associated with worse career outcomes where the organisational culture emphasised commitment and the collective. Third, “complex contagions spread faster in clustered networks, and long ties are harmful” was derived for deterministic thresholds; if there is a small probability of adopting below the threshold, long ties speed spreading up instead. Fourth, “scale-free networks are everywhere” faces a serious statistical challenge: among nearly a thousand real networks, only about 4 % show the strongest evidence of scale-free structure; and in finite-size simulations the “robustness” advantage of scale-free networks against random failure is small, while their fragility under targeted attack is what really stands out. Fifth, the Records of the Grand Historian places Su Qin before Zhang Yi and makes them fellow students under Master Guigu, but the Mawangdui silk manuscript known as the Letters and Speeches of the Warring States Strategists (Zhanguo zonghengjia shu), together with the work of Tang Lan, Yang Kuan and others, shows that Su Qin’s main activity came after Zhang Yi, and that much of the Records’ account of Su Qin is unreliable. Sixth, the lines the author quoted, “assess the land and grade the levy, and the people will not move away; if government does not lü the old, the people will not be tou”, come from the “Discourses of Qi” in the Discourses of the States; the Guanzi chapter “Xiao kuang” has “assess the land and grade its levies” instead. These lines are about taxing land by grade and keeping the population settled, and reading them as a statement about network structure is strained, so Chapter 7 mentions them only briefly and uses two other passages from the same text (“the four kinds of people must not live mixed together” and “person paired with person”) as its classical sources on closure. Each chapter says so where the point arises, and the Editor’s Notes published with the book files collect them.

How to read the boxes

As in other 5D Library books, different kinds of statement are marked separately.

About this edition

This is version 1.0.0, the first complete edition: a preface and eight chapters, in Chinese and English. The book contains 15 programs and 4 vector figures. The text, boxes and code were drafted with an AI writing tool from the author’s title, framework and notes; the author is responsible for reviewing and revising them and for checking every source. Corrections and missing references are welcome through the 5D Library website.