Prediction and Prevention of Disproportionally Dominant Agents in Complex Networks
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Prediction and Prevention of Disproportionally Dominant Agents in Complex Networks
Sandro Claudio Leraa,b , Alex ‘Sandy’ Pentlanda , Didier Sornetteb,c,d
a Massachusetts Institute of Technology, Massachusetts, USA
b
Southern University of Science and Technology, Shenzhen, China
c
ETH Zurich, Zurich, Switzerland
d
Tokyo Tech World Research Hub Initiative (WRHI), Tokyo, Japan
Abstract
arXiv:1711.09890v6 [physics.soc-ph] 16 Sep 2020
We develop an early warning system and subsequent optimal intervention policy to avoid the formation of disproportional dom-
inance (‘winner-takes-all’) in growing complex networks. This is modeled as a system of interacting agents, whereby the rate at
which an agent establishes connections to others is proportional to its already existing number of connections and its intrinsic fitness.
We derive an exact 4-dimensional phase diagram that separates the growing system into two regimes: one where the ‘fit-get-richer’
(FGR) and one where, eventually, the ‘winner-takes-all’ (WTA). By calibrating the system’s parameters with maximum likelihood,
its distance from the WTA regime can be monitored in real time. This is demonstrated by applying the theory to the eToro social
trading platform where users mimic each others trades. If the system state is within or close to the WTA regime, we show how to
efficiently control the system back into a more stable state along a geodesic path in the space of fitness distributions. It turns out that
the common measure of penalizing the most dominant agents does not solve sustainably the problem of drastic inequity. Instead,
interventions that first create a critical mass of high-fitness individuals followed by pushing the relatively low-fitness individuals
upward is the best way to avoid swelling inequity and escalating fragility.
In just about a decade, a handful of companies have con- one should foster more balanced growth and competition by
tributed to an alarmingly centralized world wide web. This improving the relative fitness of under-represented agents.
hardly resembles the Internet’s founding credo of net neutral- The applications of our framework are manifold, since
ity. Dominance on the Internet is not self-contained, but tran- fitness-adjusted preferential attachment has been shown to ap-
scends the economy as a whole, such as digital advertising and proximate the growth dynamics of many systems at a phe-
e-commerce. Beyond economic concerns, such unprecedented nomenological level. Examples include the evolution of the
digital monopolies have important implications for politics, and world-wide-web [3], citation networks [4], trust relationships
society [1]. But how can one tell that a company is too large? At [5], social media [6], streaming services [7], photon emission
what point should regulators interfere and how? We present a rate [8], supply chains [9], Escherichia coli metabolic networks
framework to answer these and similar questions. At its heart is [5], market investments [10], and transactions on the bitcoin
the law of preferential attachment, stating that the growth rate is blockchain [11]. A theoretical justification for fitness-based be-
proportional to size. We consider a system of interacting agents, havioral preferential attachment has recently been provided as
whereby the rate at which any given agent establishes connec- the result of strategic minimization of maximum exposure to
tions to others is proportional to its already existing number of least fit nodes [12]. In this article, we apply the method on
connections and the agent’s intrinsic fitness in attracting con- the eToro trading platform, where participants observe others’
nections. This representation is known to carry a risk of cen- trades and can chose to either engage in their own strategies
tralization [2], in the sense that removal of a few dominating or copy others. We find that the platform does not give rise
agents would collapse the entire system. We derive a system- to“winners that take it all”, which we attribute to the inherent
atic classification of the system’s phase space and show un- component of luck that underlies trading performance. We also
der what circumstances such unfavorable centralization occurs. discuss the implications of theses insights for a universal base-
This allows us to anticipate the emergence of overly dominant income, progressive tax, anti-trust laws and social-networks.
agents ex ante and construct methods for early intervention. In
socio-economic systems, typical countermeasures against cen- 1. Model Definition & Regime Characterization
tralization are progressive taxes, anti-trust laws and similar leg-
islation. However, our analysis reveals that such an approach In this section, we propose an exact criterion specifying the
may be ineffective because it addresses only the symptoms - conditions for which the dominance of an agent (firm, city, web-
disproportionally dominant agents - rather than the underlying site, individual, etc.) becomes overwhelming.
cause - a fundamentally imbalanced system that catalyses such
dominance. Instead of punishing the most competitive agents, 1.1. Fitness-Based Preferential Attachment & Deletion
Aiming at a generic description, preferential attachment is a
Email address: slera@mit.edu (Sandro Claudio Lera) natural first building block [13–15]. Assuming some hetero-geneity among agents, it is appropriate to add to it a fitness- in socio-economic [22] and complex systems [23]. The no-
based proportional attachment [16], which has been shown [12] tion of disproportional dominance manifests itself in the num-
to be particularly suitable to describe growth dynamics in com- ber of connections of those agents (representing their influ-
plex networks. In practice, the appropriateness of this form may ence/importance/wealth/. . . ). This is made rigorous as follows:
be tested by means of specific Bayesian statistical methods [6]. there exists at least one agent aDK whose ratio of its degree kDK
We thus consider an undirected network that is growing by to the total number of connections does not decay to zero in
attaching one agent and m edges per unit time. Edges may the limit of infinite system size. In other words, even as the
also be established between already existing nodes. Each agent system grows to infinity, a non-zero fraction of connections is
ai has an intrinsic ‘attractiveness’ or, from hereon, ‘fitness’ controlled by aDK . The influence of such an agent is felt in
ηi ∈ (0, 1) sampled from some fitnesses distribution ρ. Agent the entire system no matter what the total size of the network
ai ’s number of connections, i.e. its degree, is denoted by ki . (Figure 1(a)). From a stability point of view, the system is then
At each time-step, agent ai establishes a new connection with strongly reliant on a few agents, and might collapse if those
probability pi proportional to the product of its fitness and de- fail. From an economic point of view, any action of such an
gree, pi = ηi ki / j (η j p j ). The fitter and the more connected the
P
agent affects the entire economy, reminiscent of the companies
agent, the more attractive it is. To add one more realistic, yet mentioned in the introduction.
generic assumption [17], we allow agents to be removed from The WTA regime occurs if and only if
the system (failure, death, etc.) with probability proportional to
cηωi . Here, c > 0 is the base rate of removal and ηωi accounts Z1
ρ(η)
for the dependence of the probability of failure on the fitness of
∗
I ≡ dη 0, the fit agents are more likely to fail, for 0 ξω +hηiω + 1−c ω η + 1
η1−ω
−1
c hη i
instance due to targeted attacks. For ω < 0, weak agents are
more likely to fail, i.e. η itself is a measure of robustness with where ξ is a constant that depends on ρ, c and ω, but is typi-
respect to failure. cally close to 1. Expectations h·i are calculated with respect to
Assuming that the i-th agent is still alive at time t, the mean the distribution ρ from which η is sampled. Importantly, (2) de-
field rate at which its degree ki changes reads pends on the fitness landscape ρ as a whole, and not just on the
fitness of a few agents. This suggests that acting on the domi-
∂ki ηi ki c hηiω ki nant agents by removing many of their connections or targeting
=m − (1)
∂t S (t) 1 − c hηω i t their growth specifically will not be effective: this may tran-
siently decrease their influence but is short-lived as the growth
where S is a normalization factor and expectation values h·i dynamics leads to the resurgence of new agents that dominate.
are taken over the fitness distribution ρ. The first term in the The problem is systemic: as we shall see, in order to prevent the
right hand side of (1) is a direct consequence of the attachment occurrence of unwanted winners-takes-all agents, one should
rule described above. The second factor is a product of two act on the fitness landscape ρ as a whole.
independent probabilities: the probability ∼ c hηiω that an agent Finally, it is important to note that the FGR regime is not to
is deleted, and the probability ∼ ki /t that agent ai is connected be mistaken for a state of ‘equality’. The asymptotic size distri-
to that failed agent. A detailed derivation of this equation and bution is still heavy-tailed with a power law exponent controlled
subsequent results is found in the supplementary information by the shape of ρ [16]. However, the important difference is that
(SI). 1 the influence of any given agent is more localized and negligi-
While (1) captures the growth dynamics of a large class of ble within a large enough system, in contrast to the WTA regime
systems at least to first order, it may also be extended to take where some agents dominate system-wide.
into consideration for instance preferential deletion [17], sub-
linear or super-linear preferential attachment [18], assortative 1.3. Parametric Regime Classification
mixing [19], multi-dimensional fitness [20], or different bound- (2) shows that the appearance of WTA agents depends on
ary conditions such as minimal lower-bounds or inflow of new the deletion parameters c and ω, as well as the entire distri-
agents at different rates [14]. bution of fitnesses ρ. In order to classify the possible different
regimes, we propose to explore the space of fitness distributions
1.2. The fit-get-richer (FGR) and the winner-takes-all (WTA) parameterized by the Beta-distribution whose density is given
regimes by fα, β (η) ∝ ηα−1 (1 − η)β−1 where varying α, β > 0 samples all
As shown in the SI, (1) gives rise to two distinct asymp- shapes of practical interest, see Figure 1(b).
totic regimes: the fit-get-richer (FGR) and the winner-takes- The winner-takes-all (WTA) condition (2) can now be ex-
all (WTA). In the later, the system is largely dependent and pressed as a function of four parameters {α, β, ω, c}, via the
controlled by just a few agents. This regime is analogous to function I ∗ (α, β, ω, c) and the two corresponding regimes are
a Bose-Einstein condensate in statistical quantum mechanics FGR (I ∗ > 1) and WTA (I ∗ < 1). In general, the integration
[2, 21] and the dominant agents are known as ‘dragon-kings’ in (2) with ρ has to be solved numerically. Figure 1(c) shows
the domain of existence for the WTA and FGR regimes, for dif-
ferent preferential deletion parameters c and ω in the space of
1 The SI Appendix is available from the authors upon request. the parameters (α, β). Comparing with Figure 1(b), we see that
22.5 5.5
(a) (b) α = 12 , β = 1
(c)
fraction of links captured
by most connected agent
2 (α0 , β0 ) WTA regime
2.0 α = 1, β = 1
10−1 5.0
α = 1, β = 3
1.5
α = 2, β = 2
fα,β
α = 2, β = 5 4.5
10 −2 1.0
α = 5, β = 1
winner-takes-all 0.5 4.0
fit-get-richer ω =-1, c = 0.01
10−3 0.0
104 106 0.0 0.2 0.4 0.6 0.8 1.0
3.5 ω =-1, c = 0.10
ω = 0, c = 0.10
β
number of agents η
3.0 ω = 1, c = 0.01 FGR regime
Eu
3.0 (d) 3.0 (e) ω = 1, c = 0.10
cli
de
2.5 2.5 2.5
an
ge
2.0 2.0
od
es
2.0
ρ
ρ
1.5 1.5
ic
1.0 1.0
1.5 (α1 , β1 )
0.5 0.5 Wasserstein geodesic
0.0 0.0 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
η η α
Figure 1: (a) Fraction of links connected to the most dominant agent nDK in the system, as a function of the total number of agents. The mean values and ± one
standard deviation bands are obtained over a sample of 100 realizations. We have fixed c = 1% and ω = 0. Two different fitness distributions are used, one in the
WTA and one in the FGR regime (red and blue cross in panel (c)). (b) Beta distribution density for different parameters α and β used to parametrize the class of
fitness distributions on [0, 1]. As α and β vary, all typically relevant shapes of the fitness distribution are sampled. The WTA regime is characterized by the bulk of
the probability mass being concentrated on agents of relatively low fitness, with only few fit agents that end up dominating the system (e.g. α = 2, β = 5). (c) Phase
diagram showing the domains of existence for the fit-get-richer (FGR) and the winner-takes-all (WTA) regimes, for different preferential deletion parameters c and
ω in the space of the parameters (α, β) of the Beta-fitness distribution. The WTA regime corresponds to values where I ∗ < 1 ((2)). This WTA regimes lies above a
line parametrized by (c, ω). (d) Geodesic with respect to Wasserstein-2 metric, i.e. cost minimizing interpolation from a Beta distribution in the WTA regime to a
Beta distribution in the FGR regime. The (approximate) path in the space of Beta-distributions is shown in panel (c). (e) Distributions along the shortest path with
respect to the Euclidean distance in the (α, β)-plane (cf. dashed line in panel (c)). This intervention policy is more expensive than the one along the Wasserstein
geodesic.
the WTA regime corresponds to fitness landscapes in which the plied a Bayesian statistical method for the joint estimation of
bulk of the agents have relatively low fitness. Smaller values of preferential attachment and node fitness without imposing func-
ω favor the failure of relatively low fitness agents, and hence in- tional constraints [6]. See SI for details.
crease the area of the WTA regime. In contrast, for large ω, the Translating to our model, each trader is an agent, and the
fit-get-richer (FGR) regime increases in importance, as more fit number of traders followings a given trader ai defines its num-
agents tend to be removed by targeted attacks. The larger c is, ber of connections (degree) ki . The rate at which a trader at-
the stronger is this effect. tracts new connections is a measure of the attachment probabil-
ity pi . In this way, by measuring agent ai ’s popularity ki as well
as its current rate of attraction pi , one can infer an effective
2. Empirical Calibration on the eToro Social Trading Plat-
fitness ηi ∝ pi /ki without detailed knowledge of the underly-
form
ing system dynamics. The access to all the active accounts and
We apply the above theory to the social trading dynamics on traders allows us a faithful reconstruction of the underlying net-
the multi-asset brokerage platform eToro. On eToro, traders work. We measure pi and ki over rolling windows at different
have access to different assets, in particular foreign exchange times t from July 2012 to November 2013, such that the evo-
markets, and can leverage their position up to 400 times for lution of the fitness can be tracked: t 7→ ηi (t). After having
both long and short positions. The feature that most distin- inferred the fitness of every agent at time t, we construct an em-
guishes eToro from other trading platforms is its ‘OpenBook’ pirical fitness distribution ρ̂(t) as a histogram over all fitnesses.
social investment function. Instead of making their own trad- The distribution of fitnesses turns out to be fairly stationary and
ing decisions, users can chose to ‘mimic’ one or several other unimodal, with most traders having a fitness around ηi ≈ 0.4
traders and automatically execute the same trades as they do. (Figure 2(a)). The histogram ρ̂(t) is fitted fitted by the Beta-
Since more popular traders are more prominently displayed on distribution via maximum likelihood to obtain the parameters
the platform, they are more likely to be noticed and hence fol- α̂(t) and β̂(t). In with the stationarity of ρ̂, the evo-
accordance
lowed. This suggests that a preferential attachment model is lution of t 7→ α̂(t), β̂(t) is confined to a small radius around
suitable to model the imitation tendency. The decision to follow α̂ ≈ 1.25 and β̂ ≈ 1.25 (Figure 2(b)). Similarly, the deletion
a specific trader also depends on idiosyncratic, heterogeneous parameters ĉ(t) and ω̂(t) are inferred from direct observation
properties (e.g. recent performance or risk level), thus adding at different times t and are found to be stationary and fluctuat-
a fitness component seems appropriate. To confirm the appli- ing within limiting bounds (Figure 2(c)). See SI for details on
cability of fitness-adjusted preferential attachment, we have ap- methods. Here, we have defined node removal as the process
3(b)
taken, resulting in large losses and explaining why particularly
3.5
(a)
fit traders are more likely to vanish from the platform.
1.2
These insights highlight a secondary benefit of our approach:
1.0
it complements and guides a more system specific understand-
ρ
3.0 0.8
ing of the problem at hand. As long as the system allows for
0.6
0.4
a representation of the form (1) (or a related variant thereof),
0.2 0.4 0.6 0.8 one can absorb the intrinsic system dynamics into η. The phe-
2.5
η nomenological model then allows one to determine the state of
β
0
< the system. System specific knowledge is again required to un-
0
ω
>
derstand how the fitness distribution is to be modified in prac-
ω
WTA regime FGR regime
2.0
tice (cf. also below).
We now continue with the main purpose of the method: as-
1.5 sessing the state of disproportional dominance. Given the esti-
mates of α, β, c and ω, we can now calculate t 7→ I α̂, β̂, ω̂, ĉ
∗
from (2) and track its distance from the transition threshold
0.50 0.75 1.00 1.25 1.50
α
1.75 2.00 2.25 2.50
I ∗ = 1. For the case of eToro, we find that the system dynam-
0.004 (c)
0.4
ics is confined in the stable FGR regime I ∗ > 1 (Figure 2(d)),
ω
suggesting that the competitive environment is enough to avoid
c
0.2
0.003
the emergence of disproportionally
dominant traders. This is
alternatively visualized in the α̂, β̂ -plane (Figure 2(b)), where
(d) we can see that the dynamics is well below the separation line
I = 10
to the WTA regime. If the dynamics was such that unfit traders
I=5 are more likely to exit (ω < 0) at higher rates (1 > c 0),
I∗ = 1
2012-09-01 2012-11-01 2013-01-01 2013-03-01 2013-05-01 2013-07-01
the system would however lie closer to the critical threshold at
I ∗ = 1.
Figure 2: Analysis of eToro trading network at different times t from July We note that the strength of the above methodology is that
2012 to November 2013. (a): Time-averaged distribution of trader fitnesses it can also be implemented to incomplete or noisy data, as the
ρ̂ (dashed black line) as well as 80% confidence interval as determined over regime classification depends only on the empirical distribution
all times t. (b) Evolution of the eToro network in the (α, β) plane obtained
by the procedure explained in the text. The straight lines separate the WTA
of fitnesses ρ̂ as a whole. A current limitation is that the fit-
regime from the FGR regime. The green, dashed line corresponds to the regime nesses are assumed stationary, or at least that they change over
separation with time-averaged ω̂ ≈ 0.3 and ĉ ≈ 0.003 value. The purple, dotted time-scales that are larger than for changes in the number of
line highlights the hypothetical effect of removing relatively low fitness traders established connections. The SI of this article confirms this is
at a high rate (ω = −0.3, c = 0.5). (c) Time evolution of parameters c (multi
colors an left axis) and ω (green and right axis). (d) Time evolution of the
the case for eToro. Relaxing this assumption is part of future
parameter I ∗ compared with the horizontal line at I ∗ = 1 that separates the two research. Realizing that a large part of the traders’ performance
regimes (FGR for I ∗ > 1 and WTA for I ∗ < 1). observed on eToro is likely driven by luck, it seems particularly
interesting to take into consideration fitness values that decay
with time [26].
whereby a trader is no more being mimicked by anyone after
having been mimicked previously. Interestingly, ω̂ is consis-
tently positive, suggesting that particularly fit traders are more 3. Optimal Control of Disproportional Dominance
likely to disappear. Our method is phenomenological in that it
captures idiosyncratic, system specific details into the fitness The analysis of theToro social trading network in the previ-
term. To interpret this result, one must thus apply domain- ous section showed an instance where the network was confined
specific knowledge. On eToro, a trader’s recent performance in the FGR regime. In less competitive, large-scale economic
is displayed to others by means of several metrics (percentage systems, this may not always be the case. In such situations,
gain, portfolio volatility etc.) We can then expect η to be a an external regulator (e.g. the state) may wish to intervene to
(generally complex) function of these variables. Performance ensure greater opportunities for new entrants and less inequity.
on financial markets is known to have a large contribution of Interventions only targeting the most dominant agents may ap-
luck [24] and presumably even more so for the primarily retail- pear ‘unfair’, as it punishes arguably highly fit agents. Perhaps
traders that are active on eToro. more important, such intervention is inefficient in light of (2),
In the SI, we show that the most mimicked (i.e. high η) which shows that the emergence of the WTA regime is a conse-
traders indeed correspond to the ones who seem to outperform quence of the distribution of fitnesses as a whole. Rather than
the market on short time scales, but that this performance is just acting on the right tail of the distribution in form of punish-
not sustained over longer times. Since, on short time-scales, ments, more holistic, distribution-wide interventions are called
luck is easily mistaken as skill [25], the mimicking function- for. In particular, as inferred from panel (b) and (c) in Figure
ality of eToro gives rise to imitation of noise traders. Implied 1, it is the high population of weak agents that needs to be ad-
over-confidence may then be the reason that more risk is being dressed.
4Relying again on our parametrization of the fitness space in cient intervention (Wasserstein geodesic) focuses first on the
terms of the Beta-distribution, we formulate the regulator’s in- high fitness part of the distribution, by decreasing β which de-
tervention as an action that aims to modify the distribution of scribes the part of the fitness population close to 1 while keep-
fitnesses. Mathematically, this amounts to the problem of shift- ing α approximately constant. Then, the intervention shifts to
ing optimally the distribution from an initial shape character- increasing α at quasi-fixed β, thus working on the part of the
ized by (α0 , β0 ) for instance inside or uncomfortably close to fitness population close to 0. In contrast, the straight line (Eu-
the WTA regime to another shape represented by (α1 , β1 ) inside clidean geodesic) works on both α and β simultaneously, which
the FGR domain (see Figure 1(c)). Practically, this amounts corresponds to first building a large pool of agents with inter-
to increasing the concentration of high-fitness agents relative to mediate fitnesses (Figure 1(e)). The optimal intervention thus
relatively low-fitness agents (e.g. in form of subsidies, start- amounts to decreasing the number of agents of relatively low
up promotion, professional education and so forth). Techni- fitness while increasing the number of relative fit ones, while
cally, we formulate the intervention technique as an optimal avoiding a population that peaks at intermediate fitness levels.
transport problem, seeking the most cost efficient way of trans-
forming one distribution into another. We assume that increas- 4. Implications & Applications
ing an agent’s fitness from η1 to η2 is associated with a cost
c(η1 , η2 ). The problem then amounts to finding a transport map Since the growth dynamics of many systems may be captured
τ : [0, 1] → [0, 1] that pushes fα0 , β0 (·) onto fα1 , β1 (·) while min- at least to first order by fitness-based preferential attachment
R1 [3–5, 5–11], the insights from this article can be applied to a
imizing the total cost dη c(η, τ(η)) fα0 , β0 (η). The solution of large class of systems. Let us discuss a few examples.
0
the problem depends on the functional shape of c(·, ·). Here, Already in the early 20th century, Gibrat had recognized that
we provide a specific example by assuming a quadratic cost the growth dynamics of firms may be described with propor-
c(η1 , η2 ) = |η1 − η2 |2 . This is a reasonable assumption for sys- tional growth [28]. This model has ever since been extended nu-
tems where fitness can be controlled directly or indirectly via merous times to take into consideration effects such as firm het-
external manipulation. Whether a quadratic cost is appropriate erogeneity [29–32], minimum firm size [33], births and deaths
generally depends on the specific system under scrutiny, but our of firms [34–36], merger and acquisitions [37], or bankruptcy
method generalizes to other functional shapes for c. In full gen- [38]. In the context of our model, the firm dynamics can be de-
erality, different cost functions will yield different intervention scribed for instance via preferential attachment of the supply-
protocols. However, as discussed in the SI Appendix, our con- chain network [9]. The number of customers of a firm is de-
clusions remain similar when extended to c(η1 , η2 ) = |η1 − η2 |δ noted by k and η is the rate at which its customer base grows.
for any δ > 1. By varying δ, a large class of cost functions may Our results then have implications for instance for the formu-
be approximated reasonably well. lation of anti-trust laws. Regulators are concerned with the
merger of two large firms. The impact of a merger is often
Denote by Fα, β (·) the cumulative distribution function of the assessed in terms of concentration of market power [39]. Firm
−1
Beta distribution, and by Qα, β (·) ≡ Fα, β (·) its inverse (quantile) size may be one important aspect, especially in sectors with
function. It can be shown [27] that the cost minimizing trans- only a few firms. However, our results suggest that a poten-
port map τ is of the form τ(η) = Qα1 , β1 (Fα0 , β0 (η)). The interpo- tially greater harm stems from the less monitored acquisitions
lation from fα0 , β0 (·) to fα1 , β1 (·) is equivalent to a geodesic path of successful smaller firms (such as start-ups) that gradually de-
with respect to the Wasserstein-2 metric. It is parametrized by plete the pool of high fitness agents.
ft (η) = ∂τ
∂η fα1 , β1 (τt (η)) with τt = t η+(1−t) τ(η) and t is running
t
At the level of personal wealth, the concept of fitness based
from 0 to 1. Hence, ft (η) interpolates from f0 (η) = fα0 , β0 (η) to preferential attachment is also relevant. The richer an agent, the
f1 (η) = fα1 , β1 (η) through the space of fitness distributions along more likely they are to attract more wealth by being exposed to
a geodesic path with respect to the Wasserstein-2 metric (Fig- more opportunities and transactions [10, 11, 40]. Denote by k
ure 1(d)). We can again approximate any intermediate distribu- the wealth of an agent. Then, it holds dk ∝ ηk where the fitness
tions ft (η) as Beta-distribution with parameters (αt , βt ) (see SI η is the growth rate in this context. From such a representa-
for mathematical details). This traces out a path t 7→ (αt , βt ) tion, the effect of a universal base income is to put a floor on
that can be interpreted as the most cost-efficient intervention the minimum wealth ki that any agent may have. It thereby acts
plan. In Figure 1(c), such a geodesic path is shown, starting as a reflecting lower boundary and this boundary condition can
from the red cross deep in the WTA domain and ending on the even make the wealth distribution more unequal, transforming
blue cross in the FGR domain. Panel (d) of Figure 1 presents it from log-normal to power law [33]. However, to sustain-
a sequence of snapshots for the distribution of fitnesses, as it ably address the problem of income and wealth inequality, it
changes along the optimal path. This geodesic in the distri- is the growth rates (i.e. fitnesses) themselves that need to be
bution space deviates very strongly from the naive solution of addressed.
taking a straight path in the 2-dimensional (α, β) plane (which While our model sheds light on what should be done and
would be the geodesic with Euclidean distances). As already where resources should be targeted, system specific domain
mentioned, the WTA regime corresponds to a relatively low- knowledge is required to put these insights into best practice.
fitness population with a few high fitness individuals, who even- A progressive tax system that takes from the rich and gives to
tually become winners that ‘take it all’. The most cost effi- the poor may free up the necessary resources, but does not solve
5the problem sustainably by itself. While it is expected that the [8] M. Golosovsky. Mechanisms of complex network growth: Synthesis
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97(6):062310, 2018.
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growth opportunities for low-income individuals e.g. by en- [10] D. Garlaschelli, S. Battiston, M. Castri, V. Servedio, and G. Caldarelli.
The scale-free topology of market investments. Physica A: Statistical
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