The shape of fame

In any creator economy, a few accounts have most of the followers and most of the views. This is not peculiar to social media. Vilfredo Pareto noticed in the 1890s that incomes in several countries fell off as a power of income: the number of people earning more than $x$ was roughly proportional to $x^{-\alpha}$ (Pareto 1896). The same shape appears in citations of scientific papers, the sizes of cities, the sales of books and records, and the links between web pages (Newman 2005).

A power law has a simple signature. If the probability that a creator has $k$ followers falls as

$$P(k) \propto k^{-\gamma},$$

then doubling $k$ always divides the probability by the same factor, $2^{\gamma}$, however large $k$ is. There is no typical size. The tail is long enough that the largest few accounts hold a large share of the total.

How much do the few hold?

For a continuous Pareto distribution with tail exponent $\alpha = \gamma – 1$, the share of the total held by the top fraction $p$ of the population is $p^{1 – 1/\alpha}$, and the Gini coefficient, the standard measure of inequality between 0 (all equal) and 1 (one holds everything), is $1/(2\alpha – 1)$.

# Pareto distribution P(X > x) = (x_min / x) ** alpha. The top fraction p holds p ** (1 -
# 1/alpha) of the total,
# and the Gini coefficient is 1 / (2 * alpha - 1). (alpha = gamma - 1 for a density P(k) ~ k **
# -gamma.)
print(" alpha  gamma   top 1%   top 20%   Gini")
for alpha in (1.16, 1.5, 2.0, 3.0):
    share = lambda p: p ** (1 - 1 / alpha)
    print(f"{alpha:6.2f} {alpha + 1:6.2f} {share(0.01):8.1%} {share(0.20):9.1%} {1 / (2 * alpha - 1):6.2f}")
#  alpha  gamma   top 1%   top 20%   Gini
#   1.16   2.16    53.0%     80.1%   0.76
#   1.50   2.50    21.5%     58.5%   0.50
#   2.00   3.00    10.0%     44.7%   0.33
#   3.00   4.00     4.6%     34.2%   0.20

The smaller the exponent, the more extreme the concentration. With $\gamma$ near 2, the top 1% holds about half of everything.

Where power laws come from: the rich get richer

In 1999 Albert-László Barabási and Réka Albert proposed a mechanism for networks such as the web. Networks grow: new nodes keep arriving. And new nodes link preferentially to nodes that already have many links. Together, growth and preferential attachment produce a degree distribution with $\gamma = 3$, whatever the details (Barabási and Albert 1999). The idea had earlier forms: Derek de Solla Price’s “cumulative advantage” in citations (Price 1976) and Robert Merton’s “Matthew effect” in scientific reputation, named after the verse “to every one that hath shall be given” (Merton 1968).

import random
from math import log

def grow(n, m, preferential, seed=1):
    """Grow a network to n nodes; each new node links to m existing nodes.
    preferential=True: chosen in proportion to current degree (Barabasi-Albert); False: uniformly."""
    rng = random.Random(seed)
    deg = [m] * (m + 1)                      # start from a small complete graph
    ends = [i for i in range(m + 1) for _ in range(m)]   # each node listed once per link end
    for new in range(m + 1, n):
        targets = set()
        while len(targets) < m:
            targets.add(rng.choice(ends) if preferential else rng.randrange(new))
        deg.append(m)
        for t in targets:
            deg[t] += 1
            ends += [t, new]
    return deg

def gini(x):
    x = sorted(x); n = len(x)
    return sum((2 * i - n - 1) * v for i, v in enumerate(x, 1)) / (n * sum(x))

def gamma_mle(deg, kmin):
    tail = [k for k in deg if k >= kmin]     # discrete approximation, Clauset et al. (2009)
    return 1 + len(tail) / sum(log(k / (kmin - 0.5)) for k in tail)

print("growth rule      max degree  top 1% share  Gini   tail exponent")
for name, pref in (("preferential", True), ("uniform", False)):
    deg = grow(20000, 2, pref)
    top = sorted(deg, reverse=True)[: len(deg) // 100]
    print(f"{name:14s} {max(deg):10d} {sum(top) / sum(deg):12.1%} {gini(deg):6.3f} {gamma_mle(deg, 10):8.2f}")
# growth rule      max degree  top 1% share  Gini   tail exponent
# preferential          356        12.1%  0.391     2.91
# uniform                24         3.9%  0.300     5.51

Under preferential attachment the best-connected node has 356 links, against 24 under uniform attachment, and the estimated exponent is close to the theoretical 3. Under uniform attachment the distribution is not a power law at all; its “exponent” in the table is only what the estimator returns when applied to the wrong shape.

In creator commerce, the analogue of preferential attachment is easy to see. A platform’s recommendation system shows more people the accounts and videos that already perform well; viewers follow accounts that others follow; brands book creators with large audiences. Each of these feeds visibility back into visibility.

Two cautions about power laws

The cautions matter for practice. “The distribution is a power law” is a strong claim that needs data and a test; “the distribution is heavy-tailed and highly concentrated” is usually what can be shown, and it is enough for most of the conclusions in this book.

Superstars and luck

Sherwin Rosen explained why small differences in talent can produce very large differences in income: when one performer can serve a huge market at little extra cost, buyers prefer the best available, and rewards concentrate on the top (Rosen 1981). Digital distribution is the extreme case: a video can be watched by ten people or ten million at almost the same cost.

For a creator or a brand, the practical lesson of this chapter is a sober one. In a market shaped by preferential attachment, the average outcome is a poor guide to the typical one: most accounts have far fewer followers than the mean, and early advantages compound. Chapter 7 returns to what the old Chinese merchants made of crowded and neglected markets.