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Commerce · II.05 · MMXXVI · daylight

La Bourse  /  Volume II  /  Nº II.05  /  Ten concept briefs

TEN CONCEPT BRIEFS · Chapter II.05 — Complexity and Economic Networks

One page each. A reader who reads only these ten pages has the chapter.

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BRIEF 1 — The Degree Distribution, and Why the Second Moment Is the Whole Story

The idea. A network is summarised badly by its average. What decides its behaviour is the spread of connections, and specifically the mean of the square.

  ⟨k⟩   = mean number of links per node
  ⟨k²⟩  = mean of the square of that number
  κ     = ⟨k²⟩ / ⟨k⟩

Worked example. Two networks, both with mean degree 6.00. In the first, links are placed at random: the Poisson identity gives ⟨k²⟩ = ⟨k⟩² + ⟨k⟩ = 42.000, so κ = 7.000 — close to the mean. In the second, degrees follow a power law with exponent 2.1 between two and a thousand: κ = 121.483. Same kind of object, same order of link count, and the summary statistic that governs everything is seventeen times larger.

Why it matters. Every threshold in this chapter is a function of κ, not of ⟨k⟩. An analyst who reports "average number of counterparties" has reported the one number that does not determine the answer.

You already know this because you have seen an average salary quoted for a company with one founder and forty staff, and known immediately that the average told you nothing about anybody.

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BRIEF 2 — The Percolation Threshold

The idea. There is a precise fraction of a network you can remove before it stops being one connected thing. It is computable from κ alone.

  giant component exists while   κ > 2
  f_c  =  1  −  1 / (κ − 1)          (random removal)

Worked example. A random network with mean degree 6.00 has κ = 7.000, so f_c = 83.33 percent. You must remove five-sixths of the nodes, at random, before it fragments. A power-law network with exponent 2.6 — the measured shape of the world trade web — has κ = 29.124 and f_c = 96.44 percent.

The figure. Molloy and Reed (1995) proved the κ > 2 criterion; Cohen, Erez, ben-Avraham and Havlin (2000) gave the threshold in the form above.

Why it matters. It converts an intuition — this system seems resilient — into a number you can put beside a cost. And it explains why accidents so rarely take down a trading system, a payment network or a supply chain: random damage is remarkably ineffective against these shapes.

You already know this because you have watched a dozen small suppliers fail over a decade without anything important happening, and correctly concluded that the system absorbs ordinary loss.

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BRIEF 3 — Robust-Yet-Fragile

The idea. The property that makes a heavy-tailed network survive random failure is the same property that makes it collapse under targeted failure. One fact, two faces.

The numbers, computed.

Read the trade line. Remove countries at random and 96.44 percent must go. Remove them most-connected-first and 18.28 percent is enough. Same network, same nodes — a factor of 5.3 between two removal orders.

Why it matters. You cannot keep the robustness and engineer away the fragility. They are one property. The only question is whether the removals your system faces are random or selected — and markets, acquirers, regulators and attackers all select.

You already know this because you have seen an organisation survive years of ordinary turnover and then lose its capability entirely when four specific people left in one quarter.

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BRIEF 4 — Assortativity

The idea. Do well-connected nodes attach to other well-connected nodes, or to poorly connected ones? Newman's coefficient r runs from −1 to +1 and answers it.

Worked example. Build an eight-bank fully connected dealer core — 28 links among themselves — and attach 200 periphery banks holding one relationship each. That is 228 links at an overall density of 0.0106, and the coefficient is r = −0.7813: strongly disassortative. A grown preferential-attachment network of the same family gives r = −0.0270, essentially neutral.

Why it matters. Measured interbank markets are disassortative. That is worse than neutral, because a disassortative network routes nearly every path through a few nodes, which makes targeted removal more efficient than the degree distribution alone predicts. Social networks, by contrast, are usually assortative — which is one reason a rumour spreads differently from a default.

You already know this because you know that in your industry the big firms all deal with the same three intermediaries, and the small firms deal with one each.

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BRIEF 5 — The Bow-Tie and the Strongly Connected Core

The idea. A directed network — who owns whom, who owes whom — has a characteristic three-part shape: an in-section that points inward, an out-section that is pointed at, and between them a strongly connected component in which every member can reach every other.

Worked example. Vitali, Glattfelder and Battiston (2011) built the ownership network around 43,060 transnational corporations: 600,508 nodes and 1,006,987 ties, a mean of 1.677 ties per actor. The strongly connected core holds 1,347 nodes — 0.2243 percent of the network — and everything passes through it.

Why it matters. The core is where the feedback lives. Outside it, influence travels one way and stops; inside it, influence returns to its origin, which is what makes the dynamics hard. If you are mapping any directed economic network, finding the strongly connected component is the first useful thing you can do, and it is a standard algorithm.

You already know this because you have drawn an org chart, noticed that the real decisions cycle among six people who all consult each other, and understood that the chart was not the map.

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BRIEF 6 — Integrated Ownership: the Geometric Series

The idea. If you own part of a firm that owns part of another firm, you own part of the second firm. Sum the chain and you get integrated ownership.

  T  =  W  +  W²  +  W³  +  …

where W is the direct-shareholding matrix.

Worked example. Three holders each take a ten percent stake in two firms; four firms each hold 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 — and the three together reach 21.43 percent of a system in which each directly holds five percent.

Why it matters. It is real, it is standard accounting mathematics (Brioschi et al. 1989; Baldone et al. 1998), and it is not big enough to explain the concentration actually measured. Knowing that saves you from the wrong explanation, which is the most valuable thing a computation can do.

You already know this because you have worked out what you actually own through a pension fund, and found it was both more diffuse and more concentrated than you expected.

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BRIEF 7 — Control Is Not Ownership

The idea. Ownership divides. Control multiplies, because the largest shareholder directs the firm, and the firm directs the stakes it holds.

Worked example. A chain of five firms, one unit of value each, thirty percent held at every step:

  value in the chain              5.00 units
  direct value owned              0.3000 units    6.00 %
  linear integrated ownership     0.4275 units    8.55 %
  controlled value                5.0000 units  100.00 %
  control / direct               16.67 x
  control / linear integrated    11.70 x

Why it matters. This is how 147 holders reach forty percent of the control of the world's transnational corporations while being 0.0245 percent of the actors — a concentration of 1,634 to one, with roughly 110 of the 147 being financial intermediaries. It is also why the finding is not a scandal: nobody broke a rule. The concentration is the fixed point of ordinary company law, and removing the 147 would reassemble it out of whoever remained. Only changing the rule changes the number.

You already know this because you have sat on a committee where one person held a single vote and every outcome, because the others' votes were delegated.

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BRIEF 8 — The Default Cascade

The idea. A shock larger than a bank's own equity does not stop there. The excess passes to its creditors, who may also fail, and so on. The wiring decides how far it travels.

Worked example. Twenty banks, equity buffer 1.00 unit each, aggregate equity 20.00 units, the shocked bank's creditors holding 19.00 between them. Wire them three ways and run the same shocks:

  shock      complete      ring     two islands of ten
   1.50            1          1            1
   6.00            1          5            1
  11.00            1         10           10
  20.50           20         20           10
  30.00           20         20           10

Why it matters. The complete network is best at every shock up to 20.00 units and then goes from one default to twenty in a single step. One extra unit of shock multiplies the damage twentyfold. The islands cap at ten and never exceed it. Nothing about the banks changed. Only the wiring did.

You already know this because you have seen a payment delay at one customer work its way through a small supply chain, and you know the answer depended entirely on who owed whom.

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BRIEF 9 — May's Criterion: How Large a Connected System Can Be

The idea. Robert May proved in 1972 that a randomly assembled system of n elements, with connectance C and typical interaction strength a, is generically stable only while

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

The numbers, at a = 0.10:

Why it matters. 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 — which is why Haldane and May took this result to a central bank in 2011.

You already know this because you have been in an organisation where everyone was cc'd on everything and nothing could be decided.

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BRIEF 10 — When Diversification Raises Systemic Risk

The idea. Connecting everyone to everyone is safer — up to a precise condition, stated as a tail index, after which it is more dangerous.

The computation. Let shocks be Pareto with scale 0.50 units and tail index 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

Equal when (n−1)/(n/2−1) = 2^α. With twenty banks that is 19/9 = 2.1111, so α\* = ln(2.1111)/ln(2) = 1.0780.

The condition, stated. Dense connection lowers expected defaults while the loss distribution's tail index is above 1.0780; below it, the same connections raise them, and compartments win — by 12.49 percent at α = 0.80, against an advantage of only 0.66 percent the other way at α = 3.00. Where diversification wins it wins by a fraction of a percent; where it loses it loses by double digits.

Why it matters. This is the honest negative of the whole chapter, and it has a formal pedigree: Acemoglu, Ozdaglar and Tahbaz-Salehi (2015) proved the regime change; Elliott, Golub and Jackson (2014) found it from the direction of integration; Nier, Yang, Yorulmazer and Alentorn (2007) found it by simulation. And it separates two words that are usually one: diversification is what an institution does to its own portfolio; diversity is a property of the collection. Twenty banks each holding the same five asset classes in the same weights are perfectly diversified and have exactly one failure mode.

You already know this because you have watched a room full of well-informed people, all reading the same sources, arrive at the same wrong answer at the same moment.