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Emergence of Scaling in Random Networks
Open accessAbstract
Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
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“Other topologically constrained networks such as transportation networks and internet routers face similar issues.6”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [6]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Like most social networks, the in-degree distribution in the #Ferguson network follows roughly a powerlaw, with a small number of nodes receiving a disproportionately high number of retweets and mentions, and most receiving none at all (Barabási & Albert, 1999).”
From #Ferguson is everywhere: initiators in emerging counterpublic networks (2016) · cites this paper as Barabási & Albert, 1999
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“A network with the power-law distribution of degrees has the property of scale-freeness, showing that only few linguistic units have the strong general syntactic capacity to combine with other linguistic units (words), while most other linguistic units have weaker combining ability (Barabási and Albert, 1999).”
From Syntactic Networks of Interlanguage Across L2 Modalities and Proficiency Levels (2021) · cites this paper as Barabási and Albert, 1999
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The distribution of degrees of vertices in a complex network and a random network generally follows the power-law distribution and Poisson distribution, respectively (Barabási and Albert, 1999; Newman, 2003, 2010).”
From Syntactic Networks of Interlanguage Across L2 Modalities and Proficiency Levels (2021) · cites this paper as Barabási and Albert, 1999
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In the citing paper
“complex network theory has demonstrated that many (non-quantum) systems exhibit similarities in their complex features 14 -18 , in the organization of their structure and dynamics 19 -24 , the controllability of their constituents 25 , and their resilience to structural and dynamical perturbations 26 -31 .”
From Complex networks from classical to quantum (2019) · cites this paper as [15]
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“Several models, based on mechanisms like preferential attachment 15 , fi tness 94 , or constrained random wiring 54 , to mention some of them, were developed to reproduce degree distributions commonly observed in empirical systems.”
From Complex networks from classical to quantum (2019) · cites this paper as [15]
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“Concurrent with the formulation of small-world networks, another feature found to be common in many realworld networks is the existence of a small number of highly connected nodes (called “hubs”).6”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [6]
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In the citing paper
“For instance, when new nodes are connected preferentially to a highly connected “rich club,” the degree distribution matches that of other real-world networks, a scenario known as “the rich get richer.”6”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [6]
The cited source says what the claim attributes to it.
In the citing paper
“This network model could be generated using a simple process of sequentially adding nodes to a graph and preferentially attaching them to nodes that already had many connections (e.g., everyone wants to be friends with the popular person).The resulting degree distribution follows a power law.”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [6]
The cited source says what the claim attributes to it.
In the citing paper
“A network with the power-law distribution of degrees has the property of scale-freeness, showing that only few linguistic units have the strong general syntactic capacity to combine with other linguistic units (words), while most other linguistic units have weaker combining ability (Barabási and Albert, 1999).”
From Syntactic Networks of Interlanguage Across L2 Modalities and Proficiency Levels (2021) · cites this paper as Barabási and Albert, 1999
The cited source says what the claim attributes to it.
In the citing paper
“The distribution of degrees of vertices in a complex network and a random network generally follows the power-law distribution and Poisson distribution, respectively (Barabási and Albert, 1999; Newman, 2003, 2010).”
From Syntactic Networks of Interlanguage Across L2 Modalities and Proficiency Levels (2021) · cites this paper as Barabási and Albert, 1999
The cited source says what the claim attributes to it.
In the citing paper
“Traditionally complex network theory explains the evolution of scale-free phenomenon by Bgrowth^and Bpreferential attachment^ (Xie and Levinson 2009;Barabasi and Albert 1999).”
From A Network Approach to Link Visibility and Urban Activity Location (2018) · cites this paper as Barabasi and Albert 1999
In the citing paper
“A power law connectivity degree distribution is the defining feature of a scale-free network46,47”
From Descending networks transform command signals into population motor control (2024) · cites this paper as [46]
In the citing paper
“It has been found that despite the diversity of complex networks, ranging from the Internet to the protein interaction networks in the cell, most networks obey universal properties: they are small world [8], they are scale free [9], and they have a nontrivial community structure [7].”
From Interdisciplinary and physics challenges of network theory (2015) · cites this paper as [9]
In the citing paper
“Therefore, as long as δ < d − 2, the generalized degree increases dynamically due to an effective “linear preferential attachment24”, according to which the generalized degree of a δ-face increases at each time by one, with a probability increasing linearly with the current value of its generalized degree.”
From Complex Quantum Network Manifolds in Dimension d > 2 are Scale-Free (2015) · cites this paper as [24]
In the citing paper
“As our societies are increasingly instantiated in digital form, once-mathematical abstractions of social processes—networks are one prominent example—become very real parts of daily life”
From Stewardship of global collective behavior (2021) · cites this paper as [23]
In the citing paper
“Here, a better fit is achieved with a scale-free network32; however, neither model simultaneously fits both distributions.”
From A mesoscale connectome of the mouse brain (2014) · cites this paper as [32]
In the citing paper
“suggesting relatively heavier tails and higher inequality, indicating a greater tendency for the emergence of super-hubs-nodes with degrees larger than expected in the standard preferential attachment model (Barabási and Albert 1999).”
From "Rich-Get-Richer"? Platform Attention and Earnings Inequality using Patreon Earnings Data (2025) · cites this paper as Barabási and Albert 1999
In the citing paper
“The preferential attachment model (Barabási and Albert 1999) is based on two mechanisms: growth, where a new node is added at each time step, and preferential attachment, where each new node connects to m existing nodes with a probability proportional to their degrees.”
From "Rich-Get-Richer"? Platform Attention and Earnings Inequality using Patreon Earnings Data (2025) · cites this paper as Barabási and Albert 1999
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Functional and topological characterization of protein interaction networks
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The brainstem reticular formation is a small-world, not scale-free, network
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Small-world properties of the Indian railway network
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Metadata
- Authors
- Barabási, A.-L., & Albert, R.
- Affiliations
- Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA.
- Venue
- Science
- Published
- 1999-10-15
- Volume
- 286
- Issue
- 5439
- Pages
- 509-512
- ISSN
- 0036-8075 (ISSN-L)
- 1095-9203
- OpenAlex
- W2008620264
- Cited by
- 36,135
- References
- 19
- Citations by year
- 2026: 1122 · 2025: 1428 · 2024: 1382 · 2023: 1410 · 2022: 1628 · 2021: 1838 · 2020: 1757 · 2019: 1878 · 2018: 1959 · 2017: 1885 · 2016: 1886 · 2015: 1997 · 2014: 2037 · 2013: 1914 · 2012: 1848
- FWCI
- 104.59
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Science (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Complex Systems and Time Series Analysis
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Keywords
- scale-free networks, power-law degree distribution, random networks, preferential attachment, network expansion, self-organization, complex network topology, vertex connectivity
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