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.
“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]
The passage labeled Abstract (page unavailable) names transportation networks among complex networks that may share growth and preferential attachment. It discusses the World Wide Web, but not internet routers, and does not establish that either example faces the unspecified “similar issues” as a topologically constrained network. (LLM, conf=0.83)
Experimental — accuracy not yet measuredThe 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.
“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
The paper’s network discussion (page unavailable in our copy) reports a power-law tail for an actor collaboration network and says highly connected nodes can dominate connectivity. It does not examine the #Ferguson network, retweets or mentions, or establish that most nodes in that network receive none. (LLM, conf=0.88)
Experimental — accuracy not yet measuredIn a scale-free network, 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.
“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 passage labeled Abstract (page not provided) describes scale-free networks with power-law connectivity and highly connected vertices, supporting the general idea that connectivity is uneven. It does not discuss linguistic units or establish that a word’s number of connections measures its syntactic capacity to combine with other words. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe distribution of degrees of vertices in a random network generally follows the Poisson distribution.
“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 passage on the Erdős–Rényi (ER) random graph model says its vertex degrees follow a Poisson distribution. It does not establish this for random networks generally: the paper also describes many large random networks, including its own growth model, whose degree distributions follow a power law. The supplied copy labels these passages as Abstract and provides no page number. (LLM, conf=0.87)
Experimental — accuracy not yet measuredComplex network theory has demonstrated that many non-quantum systems exhibit similarities in their complex features, in the organization of their structure and dynamics, the controllability of their constituents, and their resilience to structural and dynamical perturbations.
“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]
Several models based on mechanisms such as preferential attachment, fitness, or constrained random wiring were developed to reproduce degree distributions commonly observed in empirical systems.
“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]
A common feature in many real-world networks is the existence of a small number of highly connected nodes (hubs).
“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]
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.
“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]
A network model with hubs can be generated by sequentially adding nodes and preferentially attaching them to nodes that already had many connections, and the resulting degree distribution follows a power law.
“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]
A network with the power-law distribution of degrees has the property of scale-freeness.
“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 distribution of degrees of vertices in a complex network generally follows the power-law distribution.
“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
Traditionally complex network theory explains the evolution of scale-free phenomenon by 'growth' and 'preferential attachment'.
“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
A power law connectivity degree distribution is the defining feature of a scale-free network.
“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]
Despite their diversity, most complex networks obey universal structural properties: they are small world, they are scale free, and they have a nontrivial community structure.
“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]
For β = 0 and δ < d − 2, the generalized degree increases dynamically due to an effective linear preferential attachment.
“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]
As our societies are increasingly instantiated in digital form, once-mathematical abstractions of social processes-networks-become very real parts of daily life.
“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]
A better fit is achieved with a scale-free network; however, neither model simultaneously fits both distributions.
“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]
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Scale-Free Networks: A Decade and Beyond
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- 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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