Complexity and Economic Networks

Volume II — Foundations: The Paradigm and the Science

THE PLATE

A man and a woman seated across a round table in a high office, the city pale behind the glass. — Plate II.05The Exchange, After Hours.The board is not the calls. The board is which calls are possible. Every economy has one, almost nobody has drawn it, and the drawing changes what you think you are looking at.

THE LETTER

You have been taught to think about firms, and prices, and quantities. You have almost certainly not been taught to think about the wiring — who is connected to whom, how densely, and with what shape. That omission is not an oversight in your education. It is an omission in the discipline, and it is roughly a century old.

Here is why it matters to you this week. Two economies can contain exactly the same firms, holding exactly the same assets, with exactly the same balance sheets, and behave completely differently under stress — because the links between them are arranged differently. Same components, same totals, different outcome. If that is true, then a great deal of what you have been asked to measure is measuring the wrong object.

This chapter is the mathematics of that claim. It is not a metaphor about webs and ecosystems; it is a set of computations you can run this afternoon, on a laptop, with no libraries. We will compute the point at which a network falls apart under random failure and the very different point at which it falls apart under a targeted one. We will run a default cascade through three different wirings of the same twenty banks and watch the damage change by a factor of twenty with nothing else altered. We will work out, exactly, the condition under which connecting everybody to everybody makes a system more dangerous rather than less — and it turns out to be a single number, a tail index, and we will solve for it.

There is also one finding in here about the ownership of the world's corporations that is usually reported as a scandal and is in fact something more useful and considerably harder to fix.

You do not need to like networks to need this. You need it because the thing you are responsible for is embedded in one, and its shape is currently deciding outcomes you are being held accountable for.

— The Editors

DISCOVERY

What is already working

The good news arrives first, and it is substantial: the economy's wiring has already been drawn. Not speculatively — measured, from administrative data, by people who published their methods.

The ownership network of the world's corporations. In 2011 Stefania Vitali, James Glattfelder and Stefano Battiston, working at ETH Zürich, took the Orbis database and did something nobody had done: they followed the shareholdings. They began from 43,060 transnational corporations identified on the OECD definition and built the full network of who owns whom around them — 600,508 nodes and 1,006,987 ownership ties. That is a mean of 1.677 ties per actor, which sounds sparse and is not, because the ties are not evenly spread.

What they found has a shape with a name. The network is a bow-tie: a large periphery of firms that own but are not owned, a large periphery of firms that are owned but do not own, and between them a strongly connected core of 1,347 nodes in which every member can reach every other member through ownership. That core is 0.2243 percent of the network. Everything passes through it.

Then they computed control rather than ownership — which is the move that makes the study — and found that 737 holders accumulate eighty percent of the control over the value of all the transnational corporations, and 147 of them accumulate forty percent. The 147 are 0.0245 percent of the actors in the network and 0.3414 percent of the transnational corporations. Roughly 110 of them are financial intermediaries. Expressed as a ratio, 0.0245 percent of the actors hold forty percent of the control: a concentration of 1,634 to one.

This is normally reported as an exposé. Read it instead as an instrument. For the first time there is a map, drawn from filings, reproducible by anyone with the data, of a structure that everybody had assumed and nobody had measured. The achievement is the measurement.

Payment systems have been mapped too. Kimmo Soramäki and colleagues at the Federal Reserve Bank of New York took a day of Fedwire settlement and drew it. They found a network with a completely connected core of 25 banks — 300 links among themselves, every pair joined — and found that 66 banks carried seventy-five percent of the payment value, an average of 1.136 percent of a day's money each. The system that clears the American economy is, topologically, a very small dense centre with a large thin fringe.

Trade has been mapped. M. Ángeles Serrano and Marián Boguñá drew the world trade web and found a degree distribution with a power-law tail, exponent approximately 2.6 — a handful of countries trading with nearly everyone, most trading with few. Interbank markets have been mapped: Michael Boss, Helmut Elsinger, Martin Summer and Stefan Thurner reconstructed the Austrian interbank market and found a tail exponent near 2.0, together with strong disassortativity and low clustering. These are independent teams, independent data, independent countries, and they keep finding the same family of shapes.

And the shapes were already understood. The mathematics did not have to be invented for economics. Duncan Watts and Steven Strogatz gave the small-world result in 1998; Albert-László Barabási and Réka Albert gave preferential attachment in 1999; Réka Albert, Hawoong Jeong and Barabási gave error and attack tolerance in 2000; Reuven Cohen and colleagues gave the percolation thresholds in 2000 and 2001; Mark Newman gave assortativity in 2002. By the time the financial crisis made these questions urgent, the answers had been sitting in physics journals for a decade, fully worked, waiting to be asked.

Central banks then asked. Andrew Haldane, then at the Bank of England, gave a speech in Amsterdam in April 2009 called Rethinking the Financial Network which took the ecological literature seriously and said so. Two years later he and Robert May — the ecologist whose 1972 stability result we will use in a moment — published Systemic risk in banking ecosystems in Nature. Stefano Battiston and colleagues built DebtRank, a measure of systemic importance that does not reduce to size. Prasanna Gai and Sujit Kapadia produced the analytic contagion model that named the shape everything else keeps finding.

So the position is better than it looks. The map exists, the mathematics exists, the institutions have read it, and the computations are small enough to run in a spreadsheet. What is missing is not knowledge. It is the habit of asking the question at all, in rooms where allocation decisions are made.

THE ARITHMETIC

What works, what does not, and where the line sits

First, the one quantity that carries the argument: the second moment.

Take any network. Write ⟨k⟩ for the mean number of links per node and ⟨k²⟩ for the mean of the square of that number. Molloy and Reed showed in 1995 that a giant connected component exists while

        κ  =  ⟨k²⟩ / ⟨k⟩   >   2

and Cohen and colleagues turned that into the fraction of nodes you must remove at random before the network falls apart:

        f_c  =  1  −  1 / (κ − 1)

Now compute it twice. Take a network where links are placed at random, with mean degree 6.00. The Poisson identity gives ⟨k²⟩ = ⟨k⟩² + ⟨k⟩ = 42.000, so κ = 7.000 and

  random links, ⟨k⟩ = 6.00      f_c random   83.33 %
                                f_c targeted 55.43 %
                                ratio         1.50 x

You must destroy five-sixths of it at random to break it, and even attacking the hubs first you must destroy more than half. That is a sturdy object.

Now take a power law — the shape actually measured in trade, payments, ownership and interbank lending. Let p(k) ∝ k^(−γ) between two and a thousand links:

  γ = 2.1    κ = 121.483    f_c random 99.17 %   targeted 15.32 %   ratio 6.5 x
  γ = 2.5    κ =  39.248    f_c random 97.39 %   targeted 15.05 %   ratio 6.5 x
  γ = 2.6    κ =  29.124    f_c random 96.44 %   targeted 18.28 %   ratio 5.3 x
  γ = 3.0    κ =  10.072    f_c random 88.98 %   targeted 19.81 %   ratio 4.5 x

Read the trade line. Remove countries from the world trade web at random and you must remove 96.44 percent of them before the trading system fragments. Remove them in order of connectedness and 18.28 percent is enough. The same network, the same nodes, the same links — a factor of 5.3 between two policies that differ only in the order of removal.

This is the property Haldane took to the Bank of England under the name robust-yet-fragile, and the important thing about it is that it is one property, not two. Both numbers come from the same divergent second moment. You cannot keep the robustness and engineer away the fragility; they are the same fact seen from two sides. A system built to survive accidents is, by construction, exposed to selection — and markets, regulators, acquirers and attackers all select.

Second: the shape is not designed, it is grown.

Build the network rather than assuming it. Preferential attachment — new nodes attach to existing nodes in proportion to how connected those nodes already are — is the simplest growth rule that produces a heavy tail. Run it with 43,060 nodes, the same count as the transnational corporations, each arriving with two links. The result is 86,117 links, mean degree 3.9999, a largest node with 575 links, and a Hill estimate of the tail exponent of 3.020 — which is what the theory predicts.

Note the discrepancy, because it is the finding. The model gives 3.020. The measured economic networks give 2.6 for trade and near 2.0 for interbank lending. Real economic networks are more heavy-tailed than pure preferential attachment, not less. Something in the economy concentrates harder than a rule that already concentrates.

In the simulated network, the top 147 nodes hold 6.982 percent of all link ends and the top 1,347 hold 20.948 percent. Concentrated — and nowhere near forty percent. Degree alone does not get you to the Vitali number.

Here is what does, and it is the cut this chapter is built around.

Ownership multiplies through chains. If a holder owns ten percent of a firm that owns thirty percent of another, the holder's integrated stake in the second firm is the sum of a geometric series, T = W + W² + W³ + …, and it is larger than the direct stake. In a small worked cross-holding system — three holders with ten percent stakes in two firms each, four firms each holding thirty percent of the next — a holder's direct 0.2000 units of value becomes an integrated 0.2857 units, a multiplier of 1.4286. The three holders together reach 21.43 percent of a system in which they directly hold five percent each.

That multiplies. It does not multiply to forty percent.

What reaches forty percent is not ownership. It is control, and control is not a network property at all — it is a legal one. The largest shareholder directs the firm; the firm directs the stakes the firm holds; therefore the largest shareholder directs those too. Take a chain of five firms, each unit valued, in which the largest shareholding at every step is thirty percent:

  value in the chain                  5.00 units
  direct value owned                  0.3000 units    6.00 %
  linear integrated value owned       0.4275 units    8.55 %
  controlled value                    5.0000 units  100.00 %

  control / direct                   16.67 x
  control / linear integrated        11.70 x

One shareholding of thirty percent commands five units of value. Nobody conspired. Nobody broke a rule. The rule is ordinary company law, applied consistently, and the 147 is its fixed point — which means the finding is not a scandal but something considerably more awkward. **You cannot dissolve the

Remove them and the same arithmetic reassembles the same structure out of

whoever is left, because the rule that produced it is untouched.** The only interventions that change the number are interventions on the rule: how control is defined, how voting rights attach to chains, whether an intermediary may vote shares it holds for somebody else. That is a narrower, duller and far more actionable conclusion than the one usually drawn, and it is the one the arithmetic supports.

Third: who connects to whom. Newman's assortativity coefficient r runs from −1 to +1 and asks whether high-degree nodes attach to high-degree nodes. Build an eight-bank fully connected core with 200 single-link periphery banks attached — 228 links, overall density 0.0106 — and the coefficient is r = −0.7813. Strongly disassortative: the hubs connect to the leaves. The grown preferential-attachment network gives r = −0.0270, essentially neutral. Measured interbank markets are disassortative, which is worse than neutral for exactly the reason above: a disassortative network keeps almost all of its paths running through a few nodes, so targeted removal is even more efficient than the degree distribution alone suggests.

Fourth: how large a connected system can be. In 1972 Robert May proved a result about randomly assembled systems that has never stopped being relevant. A system of n interacting elements with connectance C and typical interaction strength a is generically stable only while

        a · √(n C)  <  1        so        n_max  =  1 / (a² C)

With a = 0.10:

  C = 0.10   n_max = 1,000.0 nodes
  C = 0.20   n_max =   500.0 nodes
  C = 0.30   n_max =   333.3 nodes
  C = 0.50   n_max =   200.0 nodes

Raising connectance from 0.10 to 0.50 cuts the largest generically stable system by a factor of five. Connectance is the physicists' word for what a regulator calls diversification. Hold that sentence; the next movement pays for it.

Fifth, and this is the honest negative.

Run a cascade. Twenty banks, each with an equity buffer of 1.00 unit, so aggregate equity is 20.00 units and the shocked bank's creditors hold 19.00 between them. A shock lands on one bank's external assets; whatever exceeds its own equity passes to its interbank creditors. Wire the same twenty banks three ways: complete (everyone lends to everyone), ring (each lends to one neighbour), two islands of ten with no link between them.

  shock      complete      ring     two islands of ten
  -----------------------------------------------------
   0.50            0          0            0
   1.50            1          1            1
   3.00            1          2            1
   6.00            1          5            1
  11.00            1         10           10
  15.00            1         14           10
  20.50           20         20           10
  30.00           20         20           10

The complete network is the best structure at every shock size up to 20.00 units, where it goes from one default to twenty in a single step. One additional unit of shock multiplies the damage twentyfold. The ring degrades gracefully and is worse almost everywhere. The islands cap at ten and never exceed it, whatever arrives.

So which wiring is safest? It depends entirely on the distribution of shocks, and that can be solved rather than argued. Let shocks be Pareto with scale 0.50 units and tail index α, so P(shock > x) = (0.50/x)^α. Each structure's default count is a step function, so expected defaults are closed form:

  complete   E[D] = P₁ + (n−1) · (xm / (n w))^α
  islands    E[D] = P₁ + (n/2 − 1) · (xm / ((n/2) w))^α

They are equal when (n−1)/(n/2−1) = 2^α, which with twenty banks is 19/9 = 2.1111, so

        α*  =  ln(2.1111) / ln(2)  =  1.0780

   α       complete     ring    two islands   better
  --------------------------------------------------
  3.0000     0.1253    0.1501       0.1261    complete
  2.0000     0.2619    0.3990       0.2725    complete
  1.5000     0.4287    0.7675       0.4542    complete
  1.2000     0.6624    1.2444       0.6825    complete
  1.0780     0.8299    1.5508       0.8299    (equal)
  1.0000     0.9750    1.7989       0.9500    islands
  0.8000     1.5677    2.7055       1.3936    islands

The condition, stated exactly. Connecting every institution to every other institution lowers expected defaults while the loss distribution has a tail index above α* = 1.0780. Below that index the same connections raise expected defaults, and a compartmented system with firebreaks is strictly better — by 12.49 percent at α = 0.80, against an advantage of only 0.66 percent the other way at α = 3.00.

Notice the asymmetry. Where diversification wins, it wins by a fraction of a percent. Where it loses, it loses by double digits. And measured financial loss distributions — credit losses, operational losses, insurance catastrophe losses — routinely sit in the region where the tail index is small. This is not a hypothetical regime.

This is the formal version of what Daron Acemoglu, Asuman Ozdaglar and Alireza Tahbaz-Salehi proved in 2015: densely connected networks are more stable for small shocks and least stable for large ones, and the resilient structures under large shocks are the ones with weakly connected components. It is also what Matthew Elliott, Benjamin Golub and Matthew Jackson found from the other direction — integration and diversification each help, then stop helping, then hurt. And it is what Erlend Nier, Jing Yang, Tanju Yorulmazer and Amadeo Alentorn found by simulation in 2007: contagion is not monotone in connectivity.

Two more numbers finish the picture. At α = 1.50, the probability that any default happens at all is 35.355 percent, the probability of a system-wide default is 0.3953 percent, and the expected number of defaults given that a default occurs is 1.2124 banks. At α = 1.00 those become 50.000 percent, 2.5000 percent and 1.9500 banks. Almost always nothing, occasionally everything. That is the signature, and a risk committee reading averages will never see it.

DREAM

What becomes ordinary

In an economy that reads its own wiring, the network diagram is a standing document rather than a research output.

Every systemically relevant institution knows its own degree, its neighbours' degrees, and the assortativity of the market it sits in, the way it currently knows its capital ratio. The numbers are published on a lag, because publishing them is what makes them comparable, and comparison is what makes them useful. Nobody finds this exotic. It is simply another line in the return.

A supervisor sizing a buffer asks two questions rather than one. Not only how large is this institution, but what is its position in the graph — and because DebtRank and its successors exist, the second question has a number attached. An institution that is small and central is capitalised as what it is, and an institution that is large and peripheral stops paying for a systemic role it does not occupy. Both of those are fairness improvements and both of them are cheaper than the alternative.

Procurement and supply-chain functions hold a tiering map that goes past tier one. When a components maker three levels down turns out to be the single node through which forty of your lines run, that is discovered in a review rather than in a shortage. Firms know their own articulation points, and pay to duplicate the ones that matter. The cost of that duplication appears in the accounts as what it is: insurance, priced against a computed probability, not a hunch.

The firebreak is a recognised instrument. Somebody can propose, in a board paper, that two business units not be integrated, and can support it with an expected-loss calculation rather than a feeling — because the arithmetic above is on one page and the tail index of the relevant loss distribution has been estimated from the firm's own history. Compartmentalisation stops being a failure of ambition and becomes a priced design choice, the way a ship's bulkheads are.

And the word diversification has been split in two. Diversification is what an institution does to its own portfolio. Diversity is a property of the collection of institutions, and it is the second one that determines whether the system survives. A regulator can ask a market to be less alike without asking any single member to be less careful, because the two are now known to be different requests.

None of this requires new mathematics. All of it requires a column that does not currently exist in a report that already does.

DESIGN

The structure that gets there

Draw the graph before you argue about it. The first move is always the same and it is not analytical. Get the edge list. Counterparties, suppliers, customers, shareholdings, payment flows — whatever the domain, the object is a list of pairs. Almost every organisation can produce one from systems it already runs, and almost none has ever been asked to.

Compute four numbers, in this order.

The degree distribution and κ = ⟨k²⟩/⟨k⟩. This tells you which regime you are in. κ near the mean degree means random-like and sturdy; κ an order of magnitude above it means hub-dependent.
The two thresholds, f_c random and f_c targeted. The ratio between them is the single most informative number about your exposure, and on measured economic networks it runs from about 4.5 to 6.5.
Assortativity r. Negative means core-periphery, which means the targeted threshold is worse than the degree distribution alone suggests.
The cascade, run on your own balance-sheet buffers, for at least three wirings: as-is, more connected, compartmented.

Then estimate one thing more, and it is the hard one: the tail index of your loss distribution. Everything in the honest negative turns on α. Use your own loss history; use a Hill estimator on the tail; state the standard error; state the sample size. If the estimate straddles α* = 1.0780, you have learned that your structural choice is not determined by the data and must be made on other grounds — which is itself a finding worth having in writing.

Sequence the interventions by cost, lowest first.

Measure and publish internally. Zero capital. The graph in the pack.
Duplicate the articulation points. The nodes whose removal disconnects the graph are usually few and usually cheap to second-source. This is where the money is, and it is ordinary operational spend.
Cap single-counterparty concentration by network position, not only by size. A limit framework that already exists, re-parameterised.
Install firebreaks where the tail index is small. Ring-fenced entities, separate legal vehicles, no cross-default. Real cost, real benefit, and now a computable one.
Change the rule, where you are the one who sets it. The control finding says structural concentration is a fixed point of a rule. If you write rules — a covenant, a listing standard, a fund's voting policy — the rule is the only lever that moves the fixed point.

Governance. One person owns the graph and it is not a committee. The graph is reviewed quarterly and the cascade is re-run whenever a material counterparty changes. The tail-index estimate is refreshed annually with its confidence interval, because an estimate without one will be quoted as certainty within two meetings of being produced.

DESTINY

How it holds when nobody is pushing

The practice sustains when the graph is in the reporting pack and dies when it is a project. That is the whole of it, and it is worth being blunt about the failure modes.

It fails when the graph is drawn once. A network diagram is a photograph of a moving object. Drawn annually it is decoration; drawn quarterly it is infrastructure. The refresh must be automated from the systems that already hold the counterparty data, because anything requiring a person to assemble it will be assembled late and then not at all.

It fails when the boundary is chosen for convenience. Every network study has to stop somewhere, and where it stops determines what it finds. A firm that maps its tier-one suppliers and calls that the graph has drawn a picture whose most important nodes are all outside the frame. State the boundary explicitly, every time, next to the result. The boundary is the denominator.

It fails when α is quoted without its error. Tail-index estimation on a few dozen loss events is genuinely imprecise, and the decision the number drives is structural and expensive. An organisation that acts on a point estimate of 0.9 and later learns the interval was 0.7 to 1.6 has made a large commitment on noise. Say the interval or do not say the number.

It fails when somebody mistakes the model for the market. Everything above assumes the network is fixed while the shock travels. Real counterparties withdraw, hedge, refuse to roll, and the graph rewires during the event — usually in the direction that makes things worse. The cascade computed here is a lower bound on damage, and should be presented as one.

What makes it hold, against all of that: the four numbers are cheap, they are comparable across time, and they move. A measure that never changes gets ignored; κ and r and the two thresholds all move as the business does, which means someone will start watching them for reasons of their own. That is the only durable form of adoption.

DELIGHT

What it feels like

There is a particular quiet that comes over a room when the graph goes up on the wall for the first time. People stop talking. Then somebody says that can't be right and points at a node in the middle that nobody in the room has heard of, and somebody else says no, that's right, that's the transfer agent — and in the next ten minutes the room learns more about its own business than the last four strategy offsites produced.

It is the pleasure of recognising a thing you have been inside for years. The diagram does not tell you anything you did not know; it tells you what you knew all at once, in one picture, with the relationships in the right place. People find their own work on it and are surprised by where it sits.

And there is a smaller, better pleasure afterwards: the first time somebody asks for the graph before making a decision. Not in a review, not defensively — just a person wanting to see where a counterparty sits before signing. That is the moment a diagram becomes a habit, and habits are the only things that survive a reorganisation.

OPERATIONALIZE THIS

At the level of finance

The instrument: a network-contingent credit limit framework, with a ring-fence option priced against it.

Every institution with counterparty exposure already runs a limit framework, and almost all of them size limits on the counterparty's own characteristics — rating, size, tenor. The instrument here re-parameterises an object you already have, which is why it can be approved by a committee that already exists.

The structure.

The measure. For each counterparty, compute a systemic-position score from the graph: its degree, the degree of its neighbours, and its DebtRank-style contribution to expected losses under the cascade. One number per counterparty, refreshed quarterly.
The limit. limit = base limit × f(position), where f is a published, monotone, bounded haircut — never worse than half the base and never better than the base. Bounded, because an unbounded multiplier will be gamed and because a limit framework nobody can predict is a limit framework nobody plans around.
The trigger. When a counterparty's position score crosses the threshold at which the cascade puts you inside its default set, the exposure moves to a ring-fenced vehicle with no cross-default to the parent. That is the firebreak, and it is a documented, pre-agreed action rather than a decision taken in a bad week.

The balance-sheet treatment. The ring-fenced vehicle is consolidated but separately capitalised, and the capital held against it is the expected loss computed from the cascade at the firm's estimated tail index, not the regulatory minimum. Where the two differ, disclose both and say which one you manage to. Auditors are comfortable with this: it is a provisioning methodology, and they assess provisioning methodologies every year.

The counterparty. Internal first — treasury and the risk function — because the first version needs to be wrong in private. Once two quarters of scores exist, the framework becomes a conversation with your clearing members and your largest counterparties, who have the same problem and mostly no measure of it. A shared measure between two institutions is worth more than a proprietary one inside either, and it costs nothing to offer.

The first ninety days.

Day	Action	Artifact
1–15	Pull the edge list from existing systems	The edge list, with its boundary stated
16–30	Compute κ, both thresholds, and r	One page, four numbers
31–45	Run the cascade on real buffers, three wirings	The cascade table
46–60	Estimate the tail index with its interval	α, with standard error and n
61–75	Draft the limit multiplier and the ring-fence trigger	Framework memo
76–90	Take it to the risk committee with one worked case	The decision paper

The number that decides it. One inequality, on the front page:

      expected loss avoided by the firebreak
   ----------------------------------------------   >   your hurdle rate
    annual cost of the separate vehicle + capital

The numerator comes from the cascade run at your estimated α. If your α sits below 1.0780, that numerator is large and the case is easy. If it sits above, the case is genuinely weak and you should say so and keep the network in the pack anyway, because α moves — and the year it moves is the year you will not have time to build this.

APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

Where in this organisation does somebody already hold a map of who is connected to whom — even an informal one, even in their head? Who is it, and what do they see that the rest of us do not?
Think of a time a disruption somewhere else in our network did not reach us. What stopped it? Was that structure, or was it luck, and how would we tell?
Which of our relationships would we describe as load-bearing, and what have we already done — without calling it that — to make sure they hold?

Dream — what becomes possible

If the graph of our counterparties were on the wall in every allocation meeting, what is the first decision that would go differently?
Imagine we could publish our network position and our counterparties could publish theirs. What would we be able to do together that neither of us can do alone?
What would it feel like to make a deliberate decision not to integrate two parts of this business, and to be able to defend it with a number?

Design — what we build

What is the smallest edge list we could assemble this month from systems we already run, and who owns it once it exists?
Which single node, if it went dark tomorrow, would disconnect the most of us — and what would second-sourcing it actually cost?
Where do we set rules that other people follow — a covenant, a standard, a voting policy? If structural concentration is a fixed point of a rule, which of our rules is producing one?

Destiny — how it holds

What would have to be true for the graph to be refreshed automatically, with nobody remembering to do it?
Who would notice first if our network position changed materially, and how would they tell anyone?
Ten years from now, what would we want a new colleague to find already in the pack on their first day?

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Note on figures. Every number above is computed in lib/verify/II_05.py and printed with its inputs, its units and its source: run python3 lib/verify.py II.05. The percolation thresholds follow Molloy and Reed (1995) and Cohen et al. (2000, 2001); the integrated-ownership series follows Brioschi et al. (1989) and Baldone et al. (1998); the stability criterion is May (1972); the cascade and the expected-default comparison are this edition's own, built on the structures analysed by Acemoglu et al. (2015). The simulated network is seeded and reproducible.