Chapter 9 · begins on page 343
Networks, Dependence, and Resilience
The models
Each model in the book is stated on one card: who decides, when, with what information and commitment, what they get, what they may do, and the question it answers.
Model K1 · p. 372
How much can pass, and what limits it?
Model K2 · p. 374
Which arcs matter most, and do simple rankings find them?
Model K3 · p. 376
What detection probability can the inspector guarantee, and where should it look?
Model K4 · p. 378
Which chokepoints decide connectivity, and where does added capacity buy resilience?
Model K5 · p. 379
What loss can protection guarantee to hold, and how does the guarantee move with predictability, the number of windows and the threat's strength?
What is proved, in words
The book follows each result with one sentence that says what it means without symbols. The formal statement is a click away; the proofs are in the book.
Proposition 9.3 · Model K1 · p. 373
The stylised supply network carries , and four arcs whose capacities sum to prove that nothing more can pass.
Proposition 9.5 · Exact interdiction values · p. 374
An adversary removing one, two or three links cuts the flow to , and at best, by striking the routes that bypass the central node; protecting any single link does not change these losses (for three links, because no link lies in all eight optimal attacks).
Proposition 9.6 · Rankings are not strategies · p. 375
On this network, removing links in the order of three natural scores — flow, capacity or centrality — ranked once or re-ranked after each removal, does worse than the optimal attack at some budget, and the most central link is not the best single target. Orders that are optimal at every budget exist; the scores do not find them.
Proposition 9.9 · The value is a reciprocal maximum flow · p. 376
In the game of watching links against an evader, the value is the reciprocal of a maximum flow: the inspector watches a minimum cut, and the evader spreads over the routes of a maximum flow.
Proposition 9.10 · Model K3 · p. 377
With certain detection on the transit links the inspector watches the four links of the smallest transit cut a quarter of the time each; with the route-specific detection of the second instance the optimal watch moves to the route-to-market links, where detection is best; in general it sits on a minimum cut of the capacities , not on the single best link. The value is for the maximum flow of the capacities over the watchable arcs; which arcs are watchable is part of the model.
Proposition 9.13 · Ten cables are one corridor · p. 378
In this table, service at the threshold needs the classic corridor, so ten cables through it fail as one; the same four units of capacity buy nothing on the corridor, most on a route sharing none of its chokepoints, and a little less on a bypass that shares one; four routes share no chokepoint and four losses are needed to stop everything, while a single loss already breaks the service threshold.
Proposition 9.15 · Escorts, the threshold and a visible draw · p. 380
Escorts lose nothing only when every window can be held against the threat's whole strength; below that threshold every plan loses something, and what a rotating schedule can guarantee survives its announcement but not the sight of its draw.
Proposition 9.16 · Model K5 at the stylised values · p. 381
In the base case a rotating schedule holds the loss to four thirds of a ship where any fixed or visible schedule loses two, randomising is worth something only at the middle of the three threat strengths compared here, and whether gathering ships into fewer windows or spreading them over more is better depends on the number of escorts.
Theorem 9.2 · Maximum flow equals minimum cut · p. 373
How much a network can carry is set by its narrowest cut, and with whole-number capacities the best flow can be chosen in whole numbers.
Worked cases
From page 353 · sources checked 9 November 2026
Middle East cable corridors
Which chokepoints decide connectivity, and which investments buy resilience rather than capacity?
The mapping, the network and what the mathematics licenses →From page 360 · sources checked
Shipping through the Strait of Hormuz under threat
What loss can protection guarantee, what does predictability cost, and should ships be gathered into fewer windows or spread over more?
The mapping, the network and what the mathematics licenses →Parallaxes in history
1888 · Chapter 9 · p. 359
A passage and a promise of passage
Constantinople, 1888: nine powers declare Suez open to every flag, its fresh-water canal secure.
Europe–Asia cables: a bypass of one chokepoint still crosses another (Section 9.7 (p. 353)).
1984–1988 · Chapter 9 · p. 367
The tanker war and convoy escort
Gulf, 1984–88: attacks on merchant ships counted after the fact; from July 1987 escort for eleven reflagged tankers.
Escorts over transit windows: an allocation seen before a strike is held to the guarantee of a fixed one (Section 9.8 (p. 360)).
The people
Chapter 9 · Networks, Dependence, and Resilience
Lester R. Ford Jr.
1927 – 2017
Mathematician who worked at the RAND Corporation in the 1950s. With D. R. Fulkerson he proved the max-flow min-cut theorem and gave the augmenting-path algorithm for maximum flows; his name is also attached to the Bellman–Ford shortest-path algorithm and, with S. M. Johnson, to the Ford–Johnson merge-insertion sorting method.
Main works
- “Maximal flow through a network”, with D. R. Fulkerson (Canadian J. Math. 8, 1956)
- Flows in Networks, with D. R. Fulkerson (1962)
Chapter 9 · Networks, Dependence, and Resilience
D. Ray Fulkerson
1924 – 1976
Mathematician at the RAND Corporation from 1951 and, from 1971, at Cornell University. He worked on network flows, combinatorial optimisation and the theory of blocking and anti-blocking polyhedra; with George Dantzig and Selmer Johnson he solved a 49-city travelling-salesman problem by the cutting-plane method. A prize in discrete mathematics bears his name.
Main works
- “Solution of a large-scale traveling-salesman problem”, with G. B. Dantzig and S. M. Johnson (Operations Research 2, 1954)
- Flows in Networks, with L. R. Ford Jr. (1962)
- “Blocking and anti-blocking pairs of polyhedra” (Mathematical Programming 1, 1971)
Chapter 9 · Networks, Dependence, and Resilience
Theodore E. Harris
1919 – 2005
Mathematician at the RAND Corporation (1947–66) and then at the University of Southern California. He worked on branching processes and on interacting particle systems, and Harris recurrence of Markov chains is named after him. With F. S. Ross he wrote the RAND report on the capacity of a rail network, whose object of interest was a minimum cut.
Main works
- “Fundamentals of a method for evaluating rail net capacities”, with F. S. Ross (RAND RM-1573, 1955)
- The Theory of Branching Processes (1963)
- “Contact interactions on a lattice” (Annals of Probability 2, 1974)