Emergent scale-free networks

Abstract Many complex systems—from the Internet to social, biological, and communication networks—are thought to exhibit scale-free structure. However, prevailing explanations require that networks grow over time, an assumption that fails in some real-world settings. Here, we explain how scale-free structure can emerge without growth through network self-organization. Beginning with an arbitrary network, we allow connections to detach from random nodes and then reconnect under a mixture of preferential and random attachment. While the numbers of nodes and edges remain fixed, the degree distribution evolves toward a power-law with an exponent γ=1+1p that depends only on the proportion p of preferential (rather than random) attachment. Applying our model to several real networks, we infer p directly from data and predict the relationship between network size and degree heterogeneity. Together, these results establish how scale-free structure can arise in networks of constant size and density, with broad implications for the structure and function of complex systems.


Introduction
Scale-free structure is a hallmark feature of many complex networks, with the probability of a node having k links (or degree k) following a power law k −γ .First studied in networks of scientific citations, 1,2 scale-free structure has now been reported across a staggering array of complex systems, from social networks (of romantic relationships, 3 scientific collaborations, 4 and online friendships 5 ); to biological networks (of connections in the brain, 6 metabolic interactions, 7 and food webs 8 ); to the online and physical wiring of the Internet; [9][10][11][12] to language, 13 transportation, 14 and communication networks. 15Although empirically measuring power laws in real networks poses important technical challenges, 16,17 the study of scale-free structure continues to provide deep insights into the nature of complex systems.
Scale-free networks are highly heterogeneous (or heavy-tailed), with a small number of well-connected hub nodes dominating in a sea of low-degree nodes. 18This heterogeneity has critical implications for the function and dynamics of such systems. 19[27] Despite extensive investigations, there remains a basic limitation in our understanding of how scale-free structure emerges in real systems.Prevailing explanations primarily rely on two mechanisms: growth (wherein nodes are constantly added to the network) and preferential attachment (such that well-connected nodes are more likely to gain new connections). 2,18 hile alternatives have been proposed to preferential attachment (such as random attachment to edges, 28 random copying of neighbors, 29 and deterministic attachment rules 30 ), the dependence on growth remains widespread. 19,31,32 I many real-world contexts, however, this dependence on constant growth is unrealistic. 31,33,34 I biological networks, for example, brains do not grow without bound, 35 and just as animals or species are added to a population, others die out. 8,36 n these systems, rather than relying on growth, scale-free structure emerges organically at relatively constant size.
8][39][40] However, these explanations rely on global choices for the optimization or fitness functions, and therefore do not address the self-organization of network structure.Meanwhile, there exist models for the selforganization of power-law degree distributions, 33,34 but these yield unrealistic exponents γ ≤ 1, whereas most real-world exponents lie in the range 2 ≤ γ ≤ 3. Thus, understanding whether, and how, realistic scale-free structure self-organizes remains a central open question.
Here, we begin by analyzing the dynamics of real networks, demonstrating empirically that systems can maintain scale-free structure even without growth.To explain this observation, we propose an intuitive model in which nodes die at random, and the disconnected edges reattach to new nodes either preferentially (with probability p) or randomly (with probability 1 − p).Under these simple dynamics, the number of edges is held constant, and the network quickly approaches a steady-state size.Importantly, we show (both analytically and numerically) that scale-free structure emerges naturally, with a realistic power-law exponent γ = 1 + 1 p ≥ 2 that depends only on the proportion p of preferential attachment.

Emergent scale-free structure in real networks
In some complex systems-including many biological, language, and real-life social networkstopological properties (such as scale-free structure) arise without constant growth. 33,34 eanwhile, other systems-particularly online social and communication networks, scientific collaborations, and the Internet-are often viewed as growing by accumulating new nodes and edges over time. 19t even for these networks, we will see that scale-free structure can arise without growth.
The dynamics of a network are defined by a sequence of connections (i t , j t ), ordered by the time t at which they occur.Letting these edges accumulate over time, we arrive at a single growing network.Alternatively, one can divide the connections into groups of equal size E, thus defining a sequence of independent snapshots, each representing the structure of the network within a specific window of time (Fig. 1a).For clarity, we let N denote the total number of nodes in the sequence, while n reflects the size of a single snapshot (Fig. 1a).Consider, for example, the social network of friendships on Flickr (Fig. 1b,c). 41Dividing the sequence of connections into groups of size E = 10 3 , we can study the evolution of different network properties.In particular, we find that the Flickr network fluctuates around a constant size n (Fig. 1b).Yet even without growing, we see that the network maintains a clear power-law degree distribution (Fig. 1c), and we verify that this scale-free structure remains consistent over time (see Supplementary Information).By contrast, if we randomize the edges in each snapshot, then the degrees drop off super-exponentially as a Poisson distribution, and the scale-free structure vanishes (Fig. 1c).
We can repeat the above procedure for any time-evolving network, such as links between pages on Wikipedia (Fig. 1d,e) or email correspondence among scientists (Fig. 1f,g). 42,43 cross a number of different social, web, communication, and transportation networks (see Table 1 and Methods for details on network selection), we divide the dynamics into snapshots with E = 10 3 edges each, the largest number that can be applied to all systems.While some networks grow slowly in time (such as Wikipedia in Fig. 1d), all of the networks approach a steady-state size (see Supplementary Information).In fact, the snapshots are limited to n ≤ 2E by definition, and therefore cannot grow without bound.Even still, many of the networks exhibit scale-free structure (such as Wikipedia in Fig. 1e).We note that some of the networks are not scale-free (such as the emails in Fig. 1g), but even these still display heavy-tailed degree distributions with many of the same structural properties. 44In what follows, we will develop a simple dynamical model capable of describing all of these networks.

Model of emergent scale-free networks
The above results demonstrate that scale-free structure can arise without growth in real networks.We divide the sequence of edges into groups of equal size E, thus forming a series of network snapshots.
Each snapshot contains n ≤ N nodes, where N is the total number of nodes in the dataset.b, Trajectory of system size n over time for the network of friendships on Flickr. 41c, Degree distribution of the Flickr network (green) computed across all network snapshots.Randomizing each snapshot (grey) yields a Poisson distribution (solid line).Dashed line illustrates a power law for comparison.d-e, Trajectory of network size (d) and degree distribution (e) for the hyperlinks between pages on English Wikipedia. 42f-g, Trajectory of network size (f) and degree distribution (g) for emails between scientists at a research institution. 43t how can we explain this observation?Here we present a simple model in which scale-free structure emerges through self-organization, with connections rearranging under a mixture of preferential and random attachment.We begin with an arbitrary network of N nodes and E edges (for simplicity, we always begin with a random network).At each time step, one node dies at random, losing all of its connections (Fig. 2a, center).Each of these connections then reattaches in one of two ways: (i) with probability p, it connects to a node via preferential attachment (that is, it attaches to node i with probability proportional to its degree k i ; Fig. 2a, bottom left), or (ii) with probability 1 − p, it connects to a random node (Fig. 2a, bottom right).In this way, the total numbers of nodes N and edges E remain constant, with the wiring between nodes simply rearranging over time.Notably, besides N and E, the model only contains a single parameter p, representing the proportion of preferential (rather than random) attachment.Do the above dynamics produce scale-free structure?To answer this question, we can write down a master equation describing the evolution of the degree distribution P t (k) from one time step t to the next.At each step, the death of a random node (Fig. 2a, center) yields an average decrease in probability of − 1 N P t (k).On average, killing a node produces k = 2E/N disconnected edges that must be reattached.With probability p, each edge attaches preferentially (Fig. 2a, bottom left), connecting to a node of degree k with probability k 2E ; on average, this preferential attachment yields an increase in probability of kp k−1 2E P t (k − 1) and a decrease of − kp k 2E P t (k).Alternatively, with probability 1−p, each disconnected edge reattaches randomly (Fig. 2a, bottom right), yielding an increase in probability of k(1 − p) 1 N P t (k − 1) and a decrease of − k(1 − p) 1 N P t (k).Combining these contributions and simplifying, we arrive at the master equation, We are now prepared to study the evolution of the degree distribution.To compare against the real networks (for which N E), we begin by randomly placing E = 10 3 edges among N = 2 × 10 4 nodes, for an average degree k = 0.1.Running the dynamics with equal amounts of preferential and random attachment (such that p = 0.5), we find that the master equation [Eq.
Eventually, the distribution develops a clear power law P (k) ∼ k −γ in the high-degree limit k 1, with a realistic exponent γ = 3 (Fig. 2b, right).We therefore find that scale-free structure emerges naturally from our simple dynamics (Fig. 2a).
The emergence of scale-free structure leaves an imprint on network properties beyond just the degree distribution.Consider, for example, the size of the network n, which (for consistency with the real networks) is defined as the number of nodes with at least one connection.As the dynamics unfold, edges tend to collect around a small number of high-degree hubs, thus decreasing the size of the network n (Fig. 2c).These hubs comprise the heavy tail of the degree distribution.
To quantify this heavy tail, rather than using the variance of the degrees (which diverges for powerlaw distributions with γ ≤ 3), we instead compute the heterogeneity 1 2 |k i − k j | / k , which is normalized to lie between zero and one (where • represents an average over degrees k ≥ 1 and |k i − k j | measures the average absolute difference in degrees). 25As the network evolves, and scale-free structure emerges (Fig. 2b), we see that the degree heterogeneity increases (Fig. 2d).
Notably, both the network size and degree heterogeneity approach steady-state values, with larger proportions p of preferential attachment yielding networks that are smaller (Fig. 2c), yet more heterogeneous (Fig. 2d).

Steady-state scale-free structure
Thus far, we have explored the network dynamics numerically (using the master equation) and through simulations.To make analytic progress, we must solve for the steady-state degree distribution.Setting P t (k) = P t+1 (k) = P (k), the master equation reduces to the recursion relation In the thermodynamic limit N, E → ∞ (holding fixed the average degree k = 2E/N ), one can then solve for the steady-state distribution where C is the normalization constant and Γ(•) is Euler's gamma function.In what follows, we normalize P (k) to run over positive degrees k ≥ 1, ) .In the high-degree limit k k/p, the above distribution falls off as a power law P (k) ∼ k −γ with scale-free exponent γ = 1 + 1 p (see Methods).We therefore find that the network dynamics produce a wide range of exponents γ ≥ 2 observed in real-world systems.Moreover, this scalefree structure depends only on the proportion p of preferential attachment (independent from the average degree k).
For equal amounts of preferential and random attachment (p = 0.5), the model generates a scalefree exponent γ = 3 (Fig. 3a, center), as observed previously in Fig. 2b.For larger proportions p of preferential attachment, high-degree hubs become more prevalent, strengthening the heavy tail in P (k) and decreasing the exponent γ (Fig. 3a, right).Indeed, as p increases, the dynamics produce networks that are smaller (Fig. 3b) and more heterogeneous (Fig. 3c; see Methods for analytic predictions).Our model thus predicts a specific inverse relationship between network size and heterogeneity (Fig. 3d), which we will be able to test in real networks.Together, these results establish analytically that our simple network dynamics give rise to scale-free structure with realistic exponents γ.

Modeling real networks
Ultimately, we would like to use our model to study real-world systems.To compare against real networks (such as those in Fig. 1), we fix the number of edges (here, E = 10 3 ) and approximate the number of nodes in the model N by the total number that appear in a given dataset.This leaves one free parameter, the proportion p of preferential attachment, which we can fit to the degree distribution of a given network (see Methods).For example, the networks in Fig. 1 are best described as arising from nearly equal amounts of preferential and random attachment (p ≈ 0.5; Fig. 4a-c).
In fact, despite only fitting one parameter, our simple model provides a surprisingly good descrip- with E = 10 3 , N set to the number of nodes in a given dataset, and p fit to the observed degree distributions (data points) for friendships on Flickr (a), 41 hyperlinks on English Wikipedia (b), 42 and emails between researchers (c). 43 tion of nearly all the networks in Table 1 (see Supplementary Information)-even those that are merely heavy-tailed and not obviously scale-free (such as the emails in Fig. 4c).Across these networks, the proportion of preferential attachment ranges from 20% to 100%, slightly outpacing random attachment on average (Fig. 4d).
As we sweep over p, adjusting the ratio of preferential to random attachment, the model predicts a specific tradeoff between the size of a network and its degree heterogeneity (Fig. 3d).
Computing the average properties (over different snapshots) for each of the real networks (Table 1), we find a similar inverse relationship between network size and heterogeneity (Fig. 4e).If we instead hold p fixed and sweep over the number of nodes N , the model also predicts the drop in degree heterogeneity observed in small networks (Fig. 4e, inset).Moreover, even at the level of individual networks, we discover similar tradeoffs between size and heterogeneity across different snapshots (see Supplementary Information).We therefore find that our model not only captures the degree distributions observed in real-world systems (Fig. 4a-c; Supplementary Information), but it also predicts the relationships between different network properties (Fig. 4e).

Extensions and robustness
In designing the model (Fig. 2a), we sought the simplest dynamics that would self-organize to produce scale-free structure.Given this simplicity, there are a number of natural extensions one could explore.To investigate the impact of model extensions on the degree distribution P (k), we again consider the heterogeneity of degrees.In the original model (with the number of edges E held fixed), as we sweep over the proportion p of preferential attachment and the number of nodes N (or, equivalently, the average degree k = 2E/N ), we arrive at a phase diagram for the network structure (Fig. 5a).As p and k increase, the network dynamics produce degree distributions with heavier tails, thus increasing the degree heterogeneity (Fig. 5a).
When performing preferential attachment, we note that these simple dynamics (Fig. 2a) rely on global information about the degrees of all the nodes in a network.In some scenarios, however, a node may only have access to local information about the degrees of nodes in its own neighborhood (for example, its neighbors and their neighbors; Fig. 5b). 45,46 estricting to local information, we find that the degree heterogeneity is significantly reduced for large p (when preferential attachment dominates) and small k (when connections are sparse, and therefore local information becomes severely restrictive; Fig. 5c and d, top).By contrast, for k 1, the networks are dense enough that local information is sufficient to generate heterogeneous degrees (Fig. 5c) and, indeed, scale-free structure (Fig. 5d, bottom).
Beyond global information, the original model also allows multi-edges (where two nodes are connected by multiple edges; Fig. 5e, top) and self-loops (where a node connects to itself; Fig. 5e, bottom).If we disallow multi-edges, the network dynamics still produce scale-free structure for all of parameter space besides p ≥ 0.9 and k 1 (when networks are both highly heterogeneous and dense; Fig. 2f, top).Similarly, if we disallow self-loops, the degree distribution is almost entirely unaffected (Fig. 2f, bottom).As a final extension, when a node dies, rather than losing all of its connections, one could imagine that it only loses some fraction f (Fig. 5g).In the limit f = 0, the dynamics halt and the network becomes static.As f increases, so too does the degree heterogeneity, until at f = 1, we recover the original model (Fig. 5h).Indeed, as long as dying nodes lose a fraction f 0.5 of their edges, the model still produces power-law degree distributions (Fig. 5i, bottom), which we confirm for different average degrees k (see Supplementary Information).In combination, these results demonstrate specific ways that the network dynamics can be extended, restricted, and generalized, while still giving rise to scale-free structure.

Discussion
Understanding how scale-free structure arises from fine-scale mechanisms is central to the study of complex systems.However, existing mechanisms typically require constant growth, an assumption that fails dramatically in many real-world networks.Even in networks that are usually viewed as growing, we show that individual snapshots (which cannot grow without bound by definition) can still exhibit scale-free structure (Fig. 1).Here, we propose a simple model in which scalefree structure emerges naturally through the self-organization of nodes and edges.By allowing nodes to die, and letting connections rearrange under a mixture of preferential and random attachment, we show (both analytically and through simulations) that the degree distribution develops a power-law tail P (k) ∼ k −γ (Fig. 2).Moreover, the scale-free exponent (which takes the simple form γ = 1 + 1 p ) only depends on the proportion p of preferential attachment and captures a wide range of values γ ≥ 2 observed in real systems (Fig. 3).In fact, despite containing only one free parameter, the model provides a surprisingly good description of many real-world networks (Fig.

4; Supplementary Information).
Given the simplicity of the model, one can immediately begin to extend the network dynamics to include additional features and mechanisms.For example, here we investigate the effects of local information, multi-edges, self-loops, and fractional edge removal (Fig. 5).Future work can build upon this progress to develop new models for the emergence of scale-free networks.Beyond node degrees, we note that power-law distributions also arise in many other contexts, from the strengths of connections in the brain and the frequencies of words in language to the populations of cities and the net worths of individuals. 16,47 o these power laws rely on the constant growth of a system?Or, instead, can scale-free distributions arise through the self-organization of existing resources?The framework presented here may provide fundamental insights to these questions.For each network, we list its type, name, total number of nodes N , and total number of edges T .Dividing each network into snapshots of E = 10 3 edges each, we list the average degree k = 2E/N , average network size n (that is, the average number of nodes in each snapshot), the average degree heterogeneity, and the proportion p of preferential attachment that best describes the degree distribution.For descriptions of the networks, distributions of the above quantities, and references, see Supplementary Information.
distribution in Eq. ( 3) to the network's measured degree distribution.Specifically, we begin by setting E = 10 3 and letting N (the total number of nodes in our model) equal the total number of nodes in a given dataset, which results in an estimate for the average degree k = 2E/N (see Table 1).This leaves one free parameter p, which we compute by minimizing the root-mean-square error of the log probabilities.The degree distributions and model fits for all of the networks are displayed and discussed in the Supplementary Information.The gradient descent algorithm used to compute p has been made openly available (see Code Availability).

Fig. 1 |
Fig. 1 | Degree distributions of real dynamical networks.a, Procedure for measuring network dynamics.

3 Fig. 2 |
Fig. 2 | Modeling the emergence of scale-free structure.a, Illustration of network dynamics.Beginning with an arbitrary network (top), at each step in time a random node dies, losing all of its connections (center).Each disconnected edge attaches to another node either preferentially (with probability p; bottom left) or randomly (with probability 1 − p; bottom right).b, Degree distributions for initially random networks (left), after two full network updates (that is, after 2N steps of the dynamics; center), and after ten network updates (right).Distributions are computed over 100 simulations, each containing E = 10 3 edges, N = 2 × 10 4 nodes (for average degree k = 0.1), and equal amounts of preferential and random attachment (p = 0.5).Solid lines depict predictions from the master equation [Eq.(1)], and dashed line illustrates a power law for comparison.c-d, Trajectories of network size n (c) and degree heterogeneity (d) over the course of ten network updates (10N steps of the dynamics) for different values of p. Thick lines reflect individual simulations (beginning with random networks), and thin lines represent master equation predictions.See Methods for a detailed description of simulations.

1 Fig. 3 |
Fig. 3 | Analytic predictions in steady-state.a, Steady-state degree distributions for increasing proportions p of preferential attachment in networks with E = 10 3 edges and N = 2 × 10 4 nodes (for average degree k = 0.1).Data points depict simulations (see Methods), solid lines reflect the analytic prediction in Eq. (3), and dashed lines illustrate power laws with the predicted exponents γ = 1 + 1 p .b-c, Network size n (b) and degree heterogeneity (c) as functions of p. d, Degree heterogeneity versus network size while sweeping over p.In panels b-d, data points are computed using simulations, dashed lines are calculated numerically using Eq.(3), and solid lines are analytic predictions in the limit of sparse connectivity k → 0 (see Methods).

Fig. 5 |
Fig. 5 | Extending the original model.a, Degree heterogeneity of the original model (Fig. 2a) while sweeping over the preferential attachment proportion p and average degree k for networks with E = 10 3 edges.b, Constraining to local information, each node can preferentially attach only to its neighbors and their neighbors.c, Relative change in degree heterogeneity after restricting to local information while sweeping over p and k. d, Degree distributions for the original model [Eq.(3); solid lines] and with local information (data points) for parameters p and k indicated in panel c. e, Multi-edges and self-loops are allowed in the original model.f, Relative change in heterogeneity when disallowing multi-edges (top) or self-loops (bottom).g,When a node dies, rather than removing all of its edges, one could remove only a fraction f .h, Relative change in heterogeneity with fractional edge removal while sweeping over p and f (for networks of average degree k = 1).i, Degree distributions for the original model [Eq.(3); solid lines] and with fractional edge removal (data points) for parameters p and f indicated in panel h.In all panels, values are computed using simulated networks with E = 10 3 edges (see Methods).

Table 1 |
Real temporal networks and their properties