Scale-free networks are rare

Abstract

A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree k follows a power law, decaying like k−αk^{-\alpha}, often with 2<α<32 < \alpha < 3. However, empirical evidence for this belief derives from a relatively small number of real-world networks. We test the universality of scale-free structure by applying state-of-the-art statistical tools to a large corpus of nearly 1000 network data sets drawn from social, biological, technological, and informational sources. We fit the power-law model to each degree distribution, test its statistical plausibility, and compare it via a likelihood ratio test to alternative, non-scale-free models, e.g., the log-normal. Across domains, we find that scale-free networks are rare, with only 4% exhibiting the strongest-possible evidence of scale-free structure and 52% exhibiting the weakest-possible evidence. Furthermore, evidence of scale-free structure is not uniformly distributed across sources: social networks are at best weakly scale free, while a handful of technological and biological networks can be called strongly scale free. These results undermine the universality of scale-free networks and reveal that real-world networks exhibit a rich structural diversity that will likely require new ideas and mechanisms to explain.

Full text

ARTICLE
Scale-free networks are rare
Anna D. Broido1& Aaron Clauset 2,3,4
Real-world networks are often claimed to be scale free, meaning that the fraction of nodes
with degree kfollows a power law k−α, a pattern with broad implications for the structure and
dynamics of complex systems. However, the universality of scale-free networks remains
controversial. Here, we organize different definitions of scale-free networks and construct a
severe test of their empirical prevalence using state-of-the-art statistical tools applied to
nearly 1000 social, biological, technological, transportation, and information networks.
Across these networks, we find robust evidence that strongly scale-free structure is
empirically rare, while for most networks, log-normal distributions fit the data as well or
better than power laws. Furthermore, social networks are at best weakly scale free, while a
handful of technological and biological networks appear strongly scale free. These findings
highlight the structural diversity of real-world networks and the need for new theoretical
explanations of these non-scale-free patterns.
https://doi.org/10.1038/s41467-019-08746-5 OPEN
1Department of Applied Mathematics, University of Colorado, 526 UCB, Boulder, CO 80309, USA. 2Department of Computer Science, University of
Colorado, 430 UCB, Boulder, CO 80309, USA. 3BioFrontiers Institute, University of Colorado, 596 UCB, Boulder, CO 80309, USA. 4Santa Fe Institute, 1399
Hyde Park Road, Santa Fe, NM 87501, USA. Correspondence and requests for materials should be addressed to A.D.B. (email: anna.broido@colorado.edu)
or to A.C. (email: aaron.clauset@colorado.edu)
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Networks are a powerful way to represent and study the
structure of complex systems. Examples today are plen-
tiful and include social interactions among individuals,
protein or gene interactions in biological organisms, commu-
nication between digital computers, and various transportation
systems. Across scientific domains and classes of networks, it is
common to encounter the claim that most or all real-world
networks are scale free. The precise details of this claim vary1–7,
but generally a network is deemed scale free if the fraction of
nodes with degree kfollows a power-law distribution k−α, where
α> 1. Some versions of this “scale-free hypothesis”have stronger
requirements, e.g., requiring 2 < α< 3 or that node degrees evolve
by the preferential attachment mechanism8,9. Other versions
make them weaker, e.g., the power law need only hold in the
upper tail10, it can exhibit an exponential cutoff11, or it is merely
more plausible than a thin-tailed distribution like an exponential
or normal12.
The study and use of scale-free networks is widespread
throughout network science1,9,13–15. Many studies investigate
how the presence of scale-free structure shapes dynamics running
over a network6,7,14,16–22. For example, under the Kuramoto
oscillator model, a transition to global synchronization is well-
known to occur at a precise threshold K
c
, whose value depends on
the power-law parameter αof the degree distribution23–27. Scale-
free networks are also widely used as a substrate for network-
based numerical simulations and experiments, and the study of
specific generating mechanisms for scale-free networks has been
framed as providing a common basis for understanding all net-
work assembly3,8,9,28–32.
The universality of scale-free networks, however, remains con-
troversial. Many studies find support for their ubiquity4,5,16,17,33–35,
while others challenge it on statistical or theoretical grounds2,3,10,36–44.
This conflict in perspective has persisted because past work has
typically relied upon small, often domain-specific data sets, less rig-
orous statistical methods, differing definitions of “scale-free”structure,
and unclear standards of what counts as evidence for or against the
scale-free hypothesis4–7,16,17,45–48. Additionally, few studies have
performed statistically rigorous comparisons of fitted power-law dis-
tributions to alternative, non-scale-free distributions, e.g., the log-
normal or stretched exponential, which can imitate a power-law form
in realistic sample sizes49. These issues raise a natural question of the
pervasiveness of strong empirical evidence for scale-free structures in
real-world networks.
Central to this debate are the ambiguities induced by the
diversity of uses of the term “scale-free network.”The classic
definition1,21,35,37 states that a network is scale free if its degree
distribution Pr(k) has a power law k−αform. A power law is the
only normalizable density function f(k) for node degrees in a
network that is invariant under rescaling, i.e., fðckÞ¼gðcÞfðkÞ
for any constant c14, and thus “free”of a natural scale. For a
network’s degree distribution, being scale free implies a power-
law pattern, and vice versa. Scale invariance can also refer to non-
degree-based aspects of network structure, e.g., its subgraphs may
be structurally self-similar50,51, and sometimes these networks are
also called scale free.
Scale-free networks are commonly discussed in the literature
on network assembly mechanisms, particularly in the context of
preferential attachment1,28,29, in which the probability that a
node gains a connection is proportional to its current degree k.
Although preferential attachment is the most famous mechanism
that produces scale-free networks, there exist other mechanisms
that can also produce them13–15. And, some variations of pre-
ferential attachment do not produce power-law degree distribu-
tions35, although those networks are still sometimes, confusingly,
called scale free. Because the shape of a degree distribution
imposes only modest constraints on overall network structure52,
it represents relatively weak evidence when trying to distinguish
generating mechanisms53–56, even when the distribution’s func-
tional form is clear. However, identifying that form from
empirical data can be non-trivial, e.g., because log-normals often
fit degree distributions as well or better than power laws49,56,57.
Across this broad literature, the term “scale-free network”may
mean a precise or approximate statistical pattern in the degree
distribution, an emergent behavior in an asymptotic limit, or a
property of all networks assembled in part or in whole by a
particular family of mechanisms. This imprecision has con-
tributed to the controversy around the scale-free hypothesis.
Here, we focus narrowly on the traditional degree-based defi-
nition of a scale-free network, which has the advantage of being
directly testable using empirical data. Even within this scope, the
definition is often modified by introducing auxiliary hypoth-
eses58. For instance, the scale-free pattern may only hold for the
largest degrees, implying Pr(k)∝k−αfor k≥k
min
> 1, so that the
power law governs the distribution’s upper tail, while the lower
tail or “body”follows some non-power-law pattern. In other
settings, finite-size effects may suppress the frequency of nodes
with degrees close to the underlying system’s size, implying Pr(k)
∝k−αe−λk, where λgoverns the transition between a power law
and an exponential cutoff in the extreme upper tail. Or, extreme
heterogeneity among degrees may be of primary interest,
implying a restriction like 2 < α< 3, where the distribution’s mean
is finite while its variance is infinite, asymptotically. Finally, the
power law may not even be meant to be a good model of the data
itself, but rather simply a better model than some alternatives,
e.g., an exponential or log-normal distribution, or just a generic
stand-in for a “heavy-tailed”distribution, i.e., one that decays
more slowly than an exponential.
A consequence of these varied uses of the term scale-free
network is that different researchers can use the same term to
refer to slightly different concepts, and this ambiguity complicates
efforts to empirically evaluate the basic hypothesis. Here, we
construct a severe test58 of the ubiquity of scale-free networks by
applying state-of-the-art statistical methods to a large and diverse
corpus of real-world networks. To explicitly cover the variations
in how scale-free networks have been defined in the literature, we
formalize a set of quantitative criteria that represent differing
strengths and types of evidence for scale-free structure in a par-
ticular network. This set of criteria unifies the common varia-
tions, and their combinations, and allows us to assess different
types and degrees of evidence of scale-free degree distributions.
For each network data set in the corpus, we estimate the best-
fitting power-law model, test its statistical plausibility, and com-
pare it to alternative non-scale-free distributions. We analyze
these results collectively, consider how the evidence for scale-free
structure varies across domains, and quantitatively evaluate their
robustness under several alternative criteria. We conclude with a
forward-looking discussion of the empirical relevance of the
scale-free hypothesis and offer suggestions for future research on
the structure of networks.
Results
Preliminaries. A key component of our evaluation of the scale-free
hypothesis is the use of a large and diverse corpus of real-world
networks. This corpus is composed of 928 network data sets drawn
from the Index of Complex Networks (ICON), a comprehensive
online index of research-quality network data, spanning all fields of
science59. It includes networks from biological, information, social,
technological, and transportation domains that range in size from
hundreds to millions of nodes (Fig. 1). These networks also exhibit
a wide variety of graph properties, such as being simple, directed,
weighted, multiplex, temporal, or bipartite.
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The scale-free hypothesis is defined most clearly for simple
graphs, which have only one degree distribution. More compli-
cated networks, e.g., a directed, weighted, multiplex network, can
have multiple degree distributions, which complicates testing
whether it is scale free; we must determine which degree
distributions count as evidence and which do not. We address
this problem in two ways. First, we apply a sequence of graph
transformations that convert a given network data set, defined as
a network with multiple graph properties, into a set of simple
graphs, each of which can be tested unambiguously for scale-free
structure (Supplementary Figs. 1 and 2). In this process, we
discard any resulting simple graph that is either too dense or too
sparse, under pre-specified thresholds, to be plausibly scale free.
(See Supplementary Note 1 for complete details.)
Then, for each simple graph associated with a network data set,
we apply standard statistical methods49 to identify the best-fitting
power law in the degree distribution’s upper tail, evaluate its
statistical plausibility using a goodness-of-fit test, and compare it
to four alternative distributions fitted to the same part of the
upper tail using a likelihood-ratio test. The outputs of these
fitting, testing, and comparison procedures for a given simple
graph encode in a vector the statistical evidence for its scale-free
structure. We then evaluate the set of these vectors for a given
network data set under criteria that formalize the different
definitions of a scale-free network.
For a given degree distribution, a key step in this process is the
selection of a value k
min
, above which the degrees are most closely
modeled by a scale-free distribution (see Methods). Hence, the
fitting procedure truncates non-power-law behavior among low-
degree nodes, enabling a more clear evaluation of potentially
scale-free patterns in the upper tail. For technical reasons, all
model tests and comparisons must then be made only on the
degrees k≥k
min
in the upper tail49. Although our primary
evaluation uses a normalized likelihood ratio test60 that has been
specifically shown valid for comparing the distributions con-
sidered here49, we also present results based on using standard
information criteria to compare distributional models61.
This approach for evaluating evidence for scale-free structure
has several advantages. It provides a systematic procedure
applicable to any network data set, and treats every data set
equivalently. It provides an evaluation of the scale-free hypothesis
over a maximally broad variety of networks, which facilitates the
characterization of their empirical ubiquity. And, it provides a
means to assess different kinds of evidence for scale-free
structure, by combining results from multiple degree distribu-
tions, if available in a network data set. The graph-simplification
process or the particular evidence criteria used may also
introduce biases into the results. We control for these possibilities
by considering alternative criteria under multiple robustness
analyses.
Definitions of a scale-free network. The different notions of
evidence for scale-free structure found in the literature can be
organized into a nearly nested set of categories (Fig. 2) and
assessed by applying standard statistical tools to each graph
associated with a network data set. Evidence for scale-free
structure typically comes in two types: (i) a power-law distribu-
tion is not necessarily a good model of the degrees, but it is a
relatively better model than alternatives, or (ii) a power law is
itself a good model of the degrees.
The first type represents indirect evidence of scale-free
structure, because the observed degree distribution is not itself
required to be plausibly scale free, only that a scale-free pattern is
more believable than some non-scale-free patterns. A network
data set that exhibits this kind of evidence is placed into a
category called
●Super-Weak: For at least 50% of graphs, no alternative
distribution is favored over the power law.
The second type represents direct evidence of scale-free
structure, and the various modifications of a purely scale-free
pattern can be organized in a set of nested categories that
represent increasing levels of evidence:
●Weakest: For at least 50% of graphs, a power-law distribution
cannot be rejected (p≥0.1).
●Weak: Requirements of Weakest, and the power-law region
contains at least 50 nodes (n
tail
≥50).
150
Number of
networks
Number of
networks
Number of nodes n
Mean degree 〈k〉
50
102
101
100
101102103104105106200 450
Fig. 1 Mean degree hkias a function of the number of nodes n. The 928
network data sets in the corpus studied here vary broadly size and density.
For data sets with more than one degree sequence (see text), we plot the
median of the corresponding set of mean degrees
Not Scale Free
Strongest
Super-Weak
Strong
Weak
Weakest
Fig. 2 Taxonomy of scale-free network definitions. Super-Weak meaning
that a power law is not necessarily a statistically plausible model of a
network’s degree distribution but it is less implausible than alternatives;
Weakest, meaning a degree distribution that is plausibly power-law
distributed; Weak, adds a requirement that the distribution’s scale-free
portion cover at least 50 nodes; Strong, adds a requirement that 2 <^
α<3
and the Super-Weak constraints; and, Strongest, meaning that almost every
associated simple graph can meet the Strong constraints. The Super-Weak
overlaps with the Weak definitions and contains the Strong definitions as
special cases. Networks that fail to meet any of these criteria are deemed
Not Scale Free
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●Strong: Requirements of Weak and Super-Weak, and
2<^
α<3 for at least 50% of graphs.
●Strongest: Requirements of Strong for at least 90% of graphs,
and requirements of Super-Weak for at least 95% of graphs.
The progression from Weakest to Strongest categories
represents the addition of more specific properties of the
power-law degree distribution, all found in the literature on
scale-free networks or distributions. We define a sixth category of
networks that includes all networks that do not fall into any of the
above categories:
●Not Scale Free: Networks that are neither Super-Weak nor
Weakest.
This evaluation scheme is parameterized by the different
fractions of simple graphs required by each evidence category.
The particular thresholds given above are statistically motivated
in order to control for false positives and overfitting, and to
provide a consistent treatment across all networks (see Methods).
A more permissive parameterization of the scheme is also
considered as a robustness check. The above scheme favors
finding evidence for scale-free structure in three ways: (i) graphs
identified as being too dense or too sparse to be plausibly scale
free are excluded from all analyses, (ii) the estimation procedure
selects, by choosing k
min
, the subset of data in the upper tail that
best-fits a power law, and (iii) the comparisons to alternatives are
performed only on the data selected by the power law.
Scaling parameters. Across the corpus, the distribution of med-
ian estimated scaling parameters parameters ^
αis concentrated
around a value of ^
α¼2, but with a long right-tail such that 32%
of data sets exhibit ^
α3 (Fig. 3). The range α2ð2;3Þis
sometimes identified as including the most emblematic of scale-
free networks8,9, and we find that 39% of network data sets have
median estimated parameters in this range. We also find that 34%
of network data sets exhibit a median parameter ^
α<2, which is a
relatively unusual value in the scale-free network literature.
Because every network produces some ^
α, regardless of the
statistical plausibility of the network being scale free, the shape of
the distribution of ^
αis not necessarily evidence for or against the
ubiquity of scale-free networks. It does, however, enable a check
of whether the estimation methods are biased by network size n.
Comparing ^
αand n,wefind little evidence of strong systematic
bias (r2=0.24, p=1.82 × 10−13; Supplementary Fig. 3).
Across the five categories of evidence for scale-free structure,
the distribution of median ^
αparameters varies considerably
(Fig. 3, insets). For networks that fall into the Super-Weak
category, the distribution has a similar breadth as the overall
distribution, with a long right-tail and many networks with ^
α>3.
Most of the networks with ^
α<2 are spatial networks, represent-
ing mycelial fungal or slime mold growth patterns62. However,
few of these exhibit even Super-Weak or Weakest evidence of
scale-free structure, indicating that they are not particularly
plausible scale-free networks. Among the Weakest and Weak
categories, the distribution of median ^
αremains broad, with a
substantial fraction exhibiting ^
α>3. The Strong and Strongest
categories require that ^
α2ð2;3Þ, and the few network data sets in
these categories are somewhat concentrated near ^
α¼2.
Alternative distributions. Independent of whether the power-law
model is a statistically good model of a network’s degree
sequence, it may nevertheless be a better model than non-power-
law alternatives.
Across the corpus, likelihood ratio tests find only modest
support for the power-law distribution over four alternatives
(Table 1). In fact, the exponential distribution, which exhibits a
thin tail and relatively low variance, is favored over the power law
(41%) more often than vice-versa (33%). This outcome accords
with the broad distribution of scaling parameters, as when α>3
(32% of data sets; Fig. 3), the degree distribution must have a
relatively thin tail.
The log-normal is a broad distribution that can exhibit heavy
tails, but which is nevertheless not scale free. Empirically, the log-
normal is favored more than three times as often (48%) over the
power law, as vice versa (12%), and the comparison is
inconclusive in a large number of cases (40%). In other words,
the log-normal is at least as good a fit as the power law for the
vast majority of degree distributions (88%), suggesting that many
previously identified scale-free networks may in fact be log-
normal networks.
160
160
Number of data sets
140
120
100
80
60
40
20
0234567
Power-law parameter, 
180
All data sets
Super-Weak scale-free
Weakest scale-free
Weak scale-free
Strong scale-free
Strongest scale-free
Super-Weak
scale-free
Weakest
scale-free
Weak
scale-free
Strong
scale-free
Strongest
scale-free
80
0
160
80
0
160
80
0
160
80
0
160
80
0
234567 234567 234567
234567
234567
Fig. 3 Distribution of ^
αby scale-free evidence category. For networks with more than one degree sequence, the median estimate is used, and for visual
clarity the 8% of networks with a median ^
α7 are omitted
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The Weibull or stretched exponential distribution can produce
thin or heavy tails, and is a generalization of the exponential
distribution. Compared to the power law, the Weibull is more
often the better statistical model (47%) than vice versa (33%).
Finally, the power-law distribution with an exponential cut-off
requires special consideration, as it contains the pure power-law
model as a special case. As a result, the likelihood of the power
law can never exceed that of the cutoff model, and the interesting
outcome is the degree to which the test is inconclusive between
the two. In this case, a majority of networks (56%) favor the
power law with cutoff model, indicating that finite-size effects
may be common.
The above findings are corroborated by replacing the likelihood
ratio test with information criteria to perform the model
comparisons, which yield qualitatively similar conclusions
(Supplementary Table II).
Assessing the scale-free hypothesis. Given the results of fitting,
testing, and comparing the power-law distribution across net-
works, we now classify each according to the six categories
described above.
Across the corpus, fully 49% of networks fall into the Not Scale
Free category (Fig. 4). Slightly less than half (46%) fall into the
Super-Weak category, in which a scale-free pattern among the
degrees is not necessarily statistically plausible itself, but remains
no less plausible than alternative distributions. The Weakest and
Weak categories represent networks in which the power-law
distribution is at least a statistically plausible model of the
networks’degree distributions. In the Weak case, this power-law
scaling covers at least 50 nodes, a relatively modest requirement.
These two categories account for only 29 and 19% of networks,
respectively, indicating that it is uncommon for a network to
exhibit direct statistical evidence of scale-free degree distributions.
Finally, only 10 and 4% of network data sets can be classified as
belonging to the Strong or Strongest categories, respectively, in
which the power-law distribution is not only statistically
plausible, but the exponent falls within the special α2ð2;3Þ
range and the power law is a better model of the degrees than
alternatives. Taken together, these results indicate that genuinely
scale-free networks are far less common than suggested by the
literature, and that scale-free structure is not an empirically
universal pattern.
The balance of evidence for or against scale-free structure does
vary by network domain (Fig. 5). These variations provide a
means to check the robustness of our results, and can inform
future efforts to develop new structural mechanisms. We focus
our domain-specific analysis on networks from biological, social,
and technological sources (91% of the corpus).
Among biological networks, a majority lack any direct or
indirect evidence of scale-free structure (63% Not Scale Free;
Fig. 5a), in agreement with past work on smaller corpora of
biological networks42. The aforementioned fungal networks
represent a large share of these Not Scale Free networks, but
this group also includes some protein interaction networks and
some food webs. Among the remaining networks, one third
exhibit only indirect evidence (33% Super-Weak), and a modest
fraction exhibit the weakest form of direct evidence (19%
Weakest). This latter group includes cat and rat brain
connectomes. Compared to the corpus as a whole, biological
networks are slightly more likely to exhibit the strongest level of
direct evidence of scale-free structure (6% Strongest), and these
are primarily metabolic networks.
We note that the fungal networks comprise 28% of the corpus
and our analysis places 100% of them in the Not Scale Free
category. Given their spatially embedded nature, it could be
argued that these networks were unlikely to be scale-free in the
first place. Because we know a posteriori that these networks
are Not Scale Free, omitting them will necessarily increase the
fraction of networks in at least some of the other categories.
We find that these increases occur primarily in the weaker
evidence categories: 5% of non-fungal networks fall into the
Strongest category (up from 4%), 13% in Strong (from 10%), 27%
in Weak (from 19%), 40% in Weakest (from 29%), and 65%
Super-Weak (from 46%). Hence, the qualitative conclusions from
our primary analysis are robust to the inclusion of this particular
subset of networks.
In contrast, social networks present a different picture. Like the
corpus overall, half of social networks lack any direct or indirect
evidence of scale-free structure (50% Not Scale Free; Fig. 5b),
while indirect evidence is slightly less prevalent (41% Super-
Weak). The former group includes the Facebook100 online social
networks, and the latter includes many Norwegian board of
director networks.
However, among the categories representing direct evidence of
scale-free structure, more networks fall into the Weakest (48%)
and Weak (31%) categories, but not a single network falls into the
Strong or Strongest categories. Hence social networks are at best
only weakly scale free, and even in cases where the power-law
distribution is plausible, non-scale-free distributions are often a
better description of the data. The social networks exhibiting
weak evidence include many scientific collaboration networks and
roughly half of the Norwegian board of director networks.
Technological networks exhibit the smallest share of networks
for which there is no evidence, direct or indirect, of scale-free
structure (8% Not Scale Free; Fig. 5c), and the largest share
exhibiting indirect evidence (90% Super-Weak). The former
group includes some digital circuit networks and various water
Not
Scale Free
Super-Weak
All data sets
Weakest
Weak
Strong
Strongest 36 (0.04)
89 (0.10)
177 (0.19)
268 (0.29)
431 (0.46)
456 (0.49)
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 4 Proportion of networks by scale-free evidence category. Bars
separate the Super-Weak category from the nested definitions, and from
the Not Scale Free category, defined as networks that are neither Weakest
or Super-Weak
Table 1 Comparison of scale-free and alternative
distributions
Test outcome
Alternative p(x)∝f(x)M
PL
Inconclusive M
Alt
Exponential e−λx33% 26% 41%
Log-normal 1
xelogxμ
ðÞ
2
2σ212% 40% 48%
Weibull ex
b
ðÞ
a
33% 20% 47%
Power law with
cutoff
x−αe−λx–44% 56%
The percentage of network data sets that favor the power-law model M
PL
, alternative model M
Alt
,
or neither, under a likelihood-ratio test, along with the form of the alternative distribution f(x)
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distribution networks. Among the categories representing direct
evidence, less than half exhibit the weakest form of direct
evidence (42% Weakest). This group includes roughly half of
CAIDA’s networks of autonomous systems, several digital circuit
networks, and several peer-to-peer networks. In contrast to
biological or social networks, however, technological networks
exhibit a modest fraction with strong direct evidence of scale-free
structure (28% Strong). Networks in this category include the
other half of the CAIDA graphs. But, almost none of the
technological networks exhibit the strongest level of direct
evidence (1% Strongest).
Transportation networks do not represent a large enough
fraction of the corpus for a similar statistical analysis, but do offer
some useful insights for future work. Most of these networks
exhibit little evidence of scale-free structure. For example, all
three airport networks and 46 of 49 road networks fall into the
Not Scale Free category, while two of the remaining three road
networks fall into the Weak category and one into Super-Weak.
All of the subway networks fall into the Super-Weak category,
and nearly all fall into the Weakest category. These results suggest
that scale-free networks may represent poor models of many
transportation systems.
Robustness analysis. In order to assess the dependence of these
results on the evaluation scheme itself, we conduct a series of
robustness tests.
Specifically, we test whether the above results hold qualitatively
when (i) we consider only network data sets that are naturally
simple (unweighted, undirected, monoplex, and no multi-edges);
(ii) we remove the power-law with cutoff from the set of alternative
distributions; (iii) we lower the percentage thresholds for all
categories to allow admission if any one constituent simple graph
satisfies the requirements; and (iv) we analyze the scaling behavior
ofthedegreedistribution’sfirst and second moment ratio. Details
for each of these tests, and two others, are given in Supplementary
Note 5. We also test whether the evaluation scheme correctly
classifies four different types of synthetic networks with known
structure, both scale free and non-scale free. Details and results for
these tests are given in Supplementary Note 6.
The first test evaluates whether the extension of the scale-free
hypothesis to non-simple networks and the corresponding graph-
simplification procedure biases the results. The second evaluates
whether the presence of finite-size effects drives the lack of
evidence for scale-free distributions. Applied to the corpus, each
test produces qualitatively similar results as the primary
evaluation scheme (see Supplementary Note 5, and Supplemen-
tary Fig. 4), indicating that the lack of empirical evidence for
scale-free networks is not driven by these particular choices in the
evaluation scheme itself.
The third considers a “most permissive”parameterization,
which evaluates the impact of our requirements that a minimum
percentage of degree sequences satisfy the constraints of a
category. Under this test, we specifically examine how the
evidence changes if we instead require that only one degree
sequence satisfies the given requirements. That is, this test lowers
the threshold for each category to be maximally permissive: if
scale-free structure exists in any associated degree sequence, the
network data set is counted as falling into the corresponding
category.
Under this modification, the Strong and Strongest categories
become equivalent, and 18% of network data sets fall into this
combined category (Fig. 6). We note that under this modified
evaluation, synthetic directed networks assembled by preferential
attachment should and do fall into the Strongest category of
evidence. The most permissive category, Super-Weak, only
changes slightly from 46 to 49%. And finally, performing this
test on only the directed networks within the corpus produces
similar results (see Supplementary Note 5 and Supplementary
Fig. 5). These tests demonstrate that the percentage requirements
used in the category definitions of the primary evaluation scheme
are not overly restrictive, and our qualitative conclusions are
robust to variations in the precise thresholds the evaluation uses.
The fourth test provides a model-independent evaluation of a
key prediction of the scale-free hypothesis. Scale-free distribu-
tions are mathematically unusual because only the moments hkmi
for m<α–1 are finite, and all higher moments diverge14,
asymptotically. Hence, in the most widely analyzed range of α2
ð2;3Þfor scale-free networks, the moment ratio hk2i=hki2
diverges as the network size nincreases. This behavior underpins
the practical relevance of many theoretical analyses of scale-free
Not
Scale Free
Not
Scale Free
Super-Weak
Super-Weak
Strong
Strongest
Weakest
Weak
Strong
Strongest
Weakest
Weak
Not
Scale Free
Super-Weak
Technological
Strong
Strongest
Weakest
Weak
Social Biological
0.0 0.2 0.4 0.6 0.8 1.0
3 (0.01)
56 (0.28)
76 (0.37)
85 (0.42)
183 (0.90)
0 (0.00)
45 (0.31)
71 (0.48)
61 (0.41)
74 (0.50)
30 (0.06)
30 (0.06)
48 (0.10)
94 (0.19)
163 (0.33)
310 (0.63)
0 (0.00)
17 (0.08)
+ 0.13
– 0.14
– 0.10
– 0.09
– 0.04
+ 0.02
+ 0.01
– 0.05
– 0.10
+ 0.12
+ 0.19
– 0.04
– 0.41
– 0.02
+ 0.18
+ 0.18
+ 0.13
+ 0.44
a
b
c
Fig. 5 Proportion of networks by scale-free evidence category and by
domain. aBiological networks, bsocial networks, and ctechnological
networks. Tickers show change in percent from the pattern in all of the data
sets
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networks. Of course, diverging moments cannot be identified
from finite-sized networks, and no real-world network can
validate this prediction of the scale-free hypothesis. However, if
most networks are scale free in this way, the scaling behavior of
their moment ratios should exhibit a strongly diverging trend.
Across the corpus as a whole, we find little evidence of a general
pattern of diverging moment ratios (Fig. 7). Instead, we find
enormous variation in ratios across networks, domains, and
scales, such that networks with 102n103often have larger
ratios than networks several orders of magnitude larger, and even
those moments that do appear to increase with ndo not increase
fast enough to be consistent with scale-free behavior (Supple-
mentary Fig. 8). We leave a more detailed investigation of these
variations for future work.
Overall, the results of these tests corroborate our primary
findings of relatively little empirical evidence for the ubiquity of
scale-free networks, and suggest that empirical degree distribu-
tions exhibit a richer variety of patterns, many of which are lower
variance, than predicted by the scale-free hypothesis.
Discussion
By evaluating the degree distributions of nearly 1000 real-world
networks from a wide range of scientific domains, we find that
scale-free networks are not ubiquitous. Fewer than 36 networks
(4%) exhibit the strongest level of evidence for scale-free struc-
ture, in which every degree distribution associated with a network
is convincingly scale free. Only 29% of networks exhibit the
weakest form, in which a power law is simply a statistically
plausible model of some portion of the degree distribution’s
upper tail. And, for 46% of networks, the power-law form is not
necessarily itself a good model of the degree distribution, but is
simply a statistically better model than alternatives. Nearly half
(49%) of networks show no evidence, direct or indirect, of scale-
free structure, and in 88% of networks, a log-normal fits the
degree distribution as well as or better than a power law. These
results demonstrate that scale-free networks are not a ubiquitous
phenomenon, and suggest that their use as a starting point for
modeling and analyzing the structure of real networks is not
empirically well grounded.
Across different scientific domains, the evidence for scale-free
structure is generally weak, but varies somewhat in interesting
ways. These differences provide hints as to where scale-free
structure may genuinely occur. For instance, our evidence indi-
cates that scale-free patterns are more likely to be found in certain
kinds of biological and technological networks. These findings
corroborate theoretical work on domain-specific mechanisms for
generating scale-free structure, e.g., in biological networks via the
well-established duplication-mutation model for molecular
networks3,30,54 or in certain kinds of technological networks via
highly optimized tolerance13,63.
In contrast, we find that social networks are at best weakly scale
free, and although a power-law distribution can be a statistically
plausible model for these networks, it is often not a better model
than a non-scale-free distribution. Class imbalance in the corpus
precludes broad conclusions about the prevalence of scale-free
structure in information or transportation networks. However,
the few of these in the corpus provide little indication that they
would exhibit strongly different structural patterns than the better
represented domains.
The variation of evidence across social, biological, and tech-
nological domains (Fig. 5) is consistent with a general conclusion
that no single universal mechanism explains the wide diversity of
degree structures found in real-world networks. The failure to
find broad evidence for scale-free patterns in the degree dis-
tributions of networks indicates that much remains unknown
about how network structure varies across different domains64
and what kinds of structural patterns are common across them.
We look forward to new investigations of statistical differences
and commonalities, which seem likely to generate new insights
about the structure of complex systems.
The statistical evaluation here considers only the degree dis-
tributions of networks, and hence says relatively little about other
structural patterns or the underlying processes that govern the
form of any particular network. However, the finding that scale-
free networks are empirically uncommon does imply a generally
limited role for any mechanism that necessarily produces power-
law degree distributions9,15,32,56, especially in domains where the
evidence for strongly scale-free networks is weak, e.g., social
networks. The mechanisms that govern the shape of a particular
network generally cannot be determined from a static network’s
degree distribution alone, as it is both a weak constraint on
network structure52 and a weak discriminator between mechan-
isms54. For some networks, there is strong evidence that
mechanisms like preferential attachment apply, e.g., scientific
citation networks28,29,55,56. However, the results described here
imply that if such mechanisms apply more broadly, they are
heavily modified or even dominated by other, perhaps domain-
specific mechanisms. A claim that some network is scale free
should thus be established using a severe statistical test58 that goes
beyond static degree distributions.
Number of nodes n
101
10–1
100
101
102
103
102103104105106
〈k2〉 / 〈k〉2
Smoothed mean
Biological data set
Informational data set
Social data set
Technological data set
Transportation data set
Fig. 7 Moment ratio scaling. For 3662 degree sequences, the empirical ratio
of the second to first moments hk2i=hki2as a function of network size n,
showing substantial variation across networks and domains, little evidence
of the divergence pattern expected for scale-free distributions, and perhaps
a roughly sublinear scaling relationship (smoothed mean via exponential
kernel, with smoothed standard deviations)
All data sets
Not
Scale Free
Super-Weak
Strong
Strongest
Weakest
Weak
– 0.02
+ 0.02
+ 0.09
+ 0.09
+ 0.08
+ 0.14
165 (0.18)
165 (0.18)
258 (0.28)
354 (0.38)
452 (0.49)
441 (0.48)
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 6 Proportions of networks in each scale-free evidence category with
removed degree percentage requirements
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In theoretical network science, assuming a power law for a
random graph’s degree distribution can simplify mathematical
analyses, and a power law can be a useful conceptual model for
building intuition about the impact of extreme degree hetero-
geneity. And, for some types of calculations, e.g., the location of
the epidemic threshold, scale-free networks can be useful models,
even when real-world degree distributions are simply heavy
tailed, rather than scale free65–67. On the other hand, if a math-
ematical result depends strongly on the asymptotic behavior of a
scale-free degree distribution, the results’practical relevance will
necessarily depend on the empirical prevalence of scale-free
structures, which we show to be uncommon or rare, depending
on the kind of scale-free structure of interest. Mathematical
results based on extreme degree heterogeneity may, in fact, have
more narrow applicability than previously believed, given the lack
of evidence that empirical moment ratios diverge as quickly as
those results typically assume (Fig. 7and Supplementary Fig. 8).
The structural diversity of real-world networks uncovered here
presents both a puzzle and an opportunity. The strong focus in
the scientific literature on explaining and exploiting scale-free
patterns has meant relatively less is known about mechanisms
that produce non-scale-free structural patterns, e.g., those with
degree distributions better fitted by a log-normal. Two important
directions of future work will be the development and validation
of novel mechanisms for generating more realistic degree struc-
ture in networks, and novel statistical techniques for identifying
or untangling them given empirical data. Similarly, theoretical
results concerning the behavior of dynamical processes running
on top of networks, including spreading processes like epide-
miological models, social influence models, or models of syn-
chronization, may need to be reassessed in light of the genuine
structural diversity of real-world networks.
The statistical methods and evidence categories developed and
used in our evaluation of the scale-free hypothesis provide a
quantitatively rigorous means by which to assess the degree to
which some network exhibits scale-free structure. Their applica-
tion to a novel network data set should enable future researchers
to determine whether assuming scale-free structure is empirically
justified.
Furthermore, large corpora of real-world networks, like the one
used here, represent a powerful, data-driven resource by which to
investigate the structural variability of real-world networks64.
Such corpora could be used to evaluate the empirical status of
many other broad claims in the networks literature, including the
tendency of social networks to exhibit high clustering coefficients
and positive degree assortativity68, the prevalence of the small-
world phenomena69, the prevalence of “rich clubs”in networks70,
the ubiquity of community71 or hierarchical structure72, and the
existence of “super-families”of networks73. We look forward to
these investigations and the new insights they will bring to our
understanding of the structure and function of networks.
Methods
Network data sets. Network data sets were obtained through the ICON59,an
online index of real-world network data sets from all domains of science. The
composition of the corpus is roughly half biological networks, a third social or
technological networks, and a sixth information or transportation networks
(Supplementary Table 1). The 928 networks included span five orders of magni-
tude in size, are generally sparse with a mean degree of hki3 (Fig. 1), and possess
a range of graph properties, e.g., simple, directed, weighted, multiplex, temporal, or
bipartite.
Prior to analysis, each network data set is transformed into one or more graphs,
whose degree sequences can be unambiguously tested for a scale-free pattern (for
example, Supplementary Fig. 1). For each non-simple graph property of a network,
a specific transformation is applied that increases the number of graphs in the data
set while removing the given graph property. Full details of this process are given in
Supplementary Note 1, and Supplementary Fig. 2. Complicated network data sets
can produce a combinatoric number of simple graphs under this process. Treating
every simplified degree sequence independently could lead to skewed results, e.g., if
a few non-scale-free data sets account for a large fraction of the total extracted
simple graphs. To avoid this bias, results are reported at the level of network data
sets. Additionally, we require that simplified graphs are neither too sparse nor too
dense to be potentially scale free and thus retain for analysis only simplified graphs
with mean degree 2 <hki<
ffiffiffi
n
p.
Simplifying the 928 network data sets produced 18,448 simple graphs, of which
14,415 were excluded for being too sparse and 371 excluded for being too dense
(about 80.4% of derived simple graphs). Results in the main text are reported only
in terms of the remaining 3662 simple graphs (about 3.9 per network data set). Of
the 928 network data sets, 735 (79%) produced no graphs that were excluded for
being too sparse. More than 90% of graphs excluded for being too sparse were
produced by simplifying three network data sets (<1% of the corpus). Similarly, 874
(94%) of the network data sets produced no graphs that were excluded for being
too dense. More than 70% of graphs excluded for being too dense were produced
by simplifying three network data sets. Finally, 782 (84%) of the data sets generated
at most three degree sequences prior to applying the too-sparse and too-dense
filters. Hence, the vast majority of data sets were uninvolved in the production of
many excluded graphs.
Modeling degree distributions. For the degree sequence fkig¼k1;k2;¼;knof a
given network data set, we estimate the best-fitting power-law distribution of the
form
PrðkÞ¼Ck
αα>1;kkmin 1;ð1Þ
where αis the scaling exponent, Cis the normalization constant, and kis integer
valued. This specification models only the distribution’s upper tail, i.e., degree
values k≥k
min
, and discards data from any non-power-law portion in the lower
distribution.
Fitting this model to an empirical degree sequence requires first choosing the
location
^
kmin at which the upper tail begins, and then estimating the scaling
exponent ^
αon the truncated data k
^
kmin. Because the choice of k
min
changes the
sample size, it cannot be directly estimated using likelihood or Bayesian techniques.
Here, the standard KS-minimization approach is used to choose
^
kmin and the
discrete maximum likelihood estimator is used to choose ^
α49. Technical details of
the estimation procedure are given in Supplementary Note 2.
Fitting the power-law distribution always returns some parameters
^
θ¼ð
^
kmin;^
αÞ. However, parameters alone give no indication of the quality of the
fitted model. A standard goodness-of-fit test is used to assess the statistical
plausibility of the fitted model, which returns a standard p-value (see
Supplementary Note 2). Following standard practice in this setting49,ifp≥0.1,
then the degree sequence is deemed plausibly scale free, while if p< 0.1, the scale-
free hypothesis is rejected. Hence, if the underlying data generating process is
indeed scale free, this test has a false negative rate of 0.1. The results of this test
provide direct evidence for or against a network exhibiting scale-free structure.
Each power-law model
^
θis compared to four non-scale-free alternative models,
estimated via maximum likelihood on the same degrees k
^
kmin, using a standard
Vuong normalized likelihood ratio test (LRT)49,60 (see Supplementary Notes 3, 4).
The restriction to k
^
kmin is necessary to make the model likelihoods directly
comparable, and slightly biases the test in favor of the power law, as the best choice
of
^
kmin for an alternative may not be the same as the best choice for the power
law49. The results of this test provide indirect evidence about the scale-free
hypothesis, as a power-law model can be favored over some alternative even if the
power law itself is not a statistically plausible model of the data. The non-scale free
alternatives used here are the (i) exponential, (ii) log-normal, (iii) power-law with
exponential cutoff, and (iv) stretched exponential or Weibull distributions
(Table 1), all of which have been used previously as models of degree
distributions74–78, and for which the validity of the LRT used here has specifically
been previously established49. Results from an alternative comparison based on
information criteria61 are given in Supplementary Table II and in Supplementary
Figs. 6 and 7.
The fitted power law and each alternative are compared using a likelihood ratio
test (see Supplementary Note 4), with the test statistic R¼L
PL L
Alt;where LPL
is the log-likelihood of the power-law model and LAlt is the log-likelihood of a
particular alternative model. The sign of Rindicates which model is a better fitto
the data: the power law R>0ðÞ, the alternative ðR <0Þ, or neither R¼0ðÞ.
The test statistic Ris derived from data, meaning that it is itself a random
variable subject to statistical fluctuations49,60. As a result, the sign of Ris
meaningful only if its magnitude jRj is statistically distinguishable from 0. This
determination is made by a standard two-tailed test against a null hypothesis of
R¼0, which yields a standard p-value. If p≥0.1, then jRj is statistically
indistinguishable from 0 and neither model is a better explanation of the data than
the other. If p< 0.1, then the data provide a clear conclusion in favor of one model
or the other, depending on the sign of R. This threshold sets the false positive rate
for the alternative distribution at 0.05. Corrections for multiple tests, e.g., a family-
wise error rate method like Bonferroni or a false discovery correction like
Benjamini-Hochberg, are not employed. Such corrections would simply lower the
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obtained p-values without changing the overall conclusions, while introducing
additional assumptions into the analysis.
To report results at the level of a network data set, we apply the LRTs to all the
associated simple graphs and then aggregate the results. For each alternative
distribution, we count the number of simple graphs associated with a particular
network data set in which the outcome favored the alternative, favored the power
law, or had an inconclusive result. Normalizing these counts across outcome
categories provides a continuous measure of the relative evidence that the data set
falls into each of category.
Parameters for defining scale-free network. Threshold parameters for the pri-
mary evaluation criteria were selected to balance false positive and false negative
rates, and to provide a consistent evaluation of evidence independent of the
associated graph properties or source of data. For the Super-Weak and Weakest
categories, a threshold of 50% ensures that the given property is present in a
majority of simple graphs associated with a network data set. For the Weak
category, a threshold of at least 50 nodes covered by the best-fitting power law in
the upper tail follows standard practices49 to reduce the likelihood of false positive
errors due to low statistical power. For the Strong category, α2ð2;3Þcovers the
full parameter range for which scale-free distributions have an infinite second
moment but a finite first moment. For the Strongest category, the thresholds of
90% for the goodness-of-fit test and 95% for likelihood ratio tests against alter-
natives match the expected error rates for both tests under the null hypothesis. If
every graph associated with a network data set is scale free, the goodness-of-fit test
is expected to incorrectly reject the power-law model 0.1 of the time, and the
likelihood ratio test will falsely favor the alternative 0.05 of the time. In the “most
permissive”parameterization of the scheme (see Supplementary Note 5), we relax
the threshold requirements so that if at least one graph meets the given criteria, the
network is placed in this category. In this permissive parameterization, a directed
network with a power-law distribution in the in-degrees should be and is classified
as Strongest.
For specific networks, domain knowledge may suggest that some degree
sequences are potentially scale free while others are likely not. A non-uniform
weighting scheme on the set of associated degree sequences would allow such prior
knowledge to be incorporated in a Bayesian fashion. However, no fixed non-
uniform scheme can apply universally correctly to networks as different as, for
example, directed trade networks, directed social networks, and directed biological
networks. To provide a consistent treatment across all networks, regardless of their
properties or source, we employ an uninformative (uniform) prior, which assigns
equal weight to each associated degree sequence. In future work on specific
subgroups of networks, a domain-specific weight scheme could be used with the
evaluation criteria described here.
Results for synthetic networks. The accuracy of the fitting, comparing, and
testing methods, and the overall evaluation scheme itself, were evaluated using four
classes of synthetic data with known structure. Three of these generated networks
that contain power-law degree distributions: a directed version of preferential
attachment79, a directed vertex copy model21, and a simple temporal power-law
random graph. One generated networks that do not: simple Erdös-Rényi random
graphs. Applied to synthetic networks generated by these models, our evaluation
scheme correctly classified each of the synthetic network data sets according to the
scale-free categories suitable for their generating parameters (see Supplementary
Note 6).
Data availability
The network data sets used are available via https://icon.colorado.edu. Code for graph-
simplification functions and power-law evaluations, and data for replication are available
at https://github.com/adbroido/SFAnalysis.
Received: 23 January 2018 Accepted: 23 January 2019
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Acknowledgements
The authors thank Eric Kightley, Johan Ugander, Cristopher Moore, Mark Newman,
Cosma Shalizi, Alessandro Vespignani, Marc Barthelemy, Juan Restrepo, Petter Holme,
and Albert-László Barabási for helpful conversations, and acknowledge the BioFrontiers
Computing Core at the University of Colorado Boulder for providing high performance
computing resources (NIH 1S10OD012300) supported by BioFrontiers IT. This work
was supported in part by Grant No. IIS-1452718 (A.C.) from the National Science
Foundation. Publication of this article was funded by the University of Colorado Boulder
Libraries Open Access Fund.
Author contributions
A.D.B. and A.C. conceived the research, designed the analyzes, and wrote the manu-
script. A.D.B. conducted the analyzes.
Additional information
Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467-
019-08746-5.
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