# Rank-Based Attachment Leads to Power Law Graphs

@article{Janssen2010RankBasedAL, title={Rank-Based Attachment Leads to Power Law Graphs}, author={Jeannette C. M. Janssen and Paweł Prałat}, journal={SIAM J. Discret. Math.}, year={2010}, volume={24}, pages={420-440} }

We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power $-\alpha$, for some $\alpha\in(0,1)$. Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent $1+1/\alpha$ whenever vertices are ranked…

## 9 Citations

### Large deviations for the degree structure in preferential attachment schemes

- Mathematics
- 2011

Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path…

### Rank-Based Models of Network Structure and the Discovery of Content

- Computer ScienceWAW
- 2011

A rank model of how content may change amongst agents over time within a stochastic system is proposed and a model of network self-organization based on this rank model is proposed.

### Discovery of Nodal Attributes through a Rank-Based Model of Network Structure

- Computer ScienceInternet Math.
- 2013

It is found that node ranks may be reliably estimated by examining node degree in conjunction with the average degree of first- and higher-order neighbors.

### Connectivity threshold and recovery time in rank-based models for complex networks

- Computer ScienceDiscret. Math.
- 2011

### CONNECTIVITY THRESHOLD AND RECOVERY TIME IN RANK-BASED MODELS FOR COMPLEX NETWORKS

- Computer Science
- 2011

A generalized version of the protean graph with a power law degree distribution, in which the degree of a vertex depends on its age as well as its rank, is studied.

### Winner does not take all: contrasting centrality in adversarial networks

- Computer ScienceArXiv
- 2022

A novel hypothesis that low-key leaders are ubiquitous in adversarial networks is presented and evidence is provided by considering data from real-world networks, including dominance networks in 172 animal populations, trading networks between G20 nations, and Bitcoin trust networks.

### Relative Value and Customer Choice in Loan Decisions: An Application of the Wallet Allocation Rule

- Business
- 2016

Abstract Purpose – This research applies the Wallet Allocation Rule (WAR) to provide marketing scholars and practitioners with a deeper understanding of the key drivers of loan selection and to…

## References

SHOWING 1-10 OF 22 REFERENCES

### First to market is not everything: an analysis of preferential attachment with fitness

- Computer ScienceSTOC '07
- 2007

A rigorous analysis of preferential attachment with fitness, suggested by Bianconi and Barabási and studied by Motwani and Xu, in which the degree of a vertex is scaled by its quality to determine its attractiveness.

### Scale-free network growth by ranking.

- Computer SciencePhysical review letters
- 2006

A criterion of network growth that explicitly relies on the ranking of the nodes according to any prestige measure, be it topological or not is proposed, which may explain the frequency and robustness of scale-free degree distributions in real networks, as illustrated by the special case of the Web graph.

### The degree sequence of a scale‐free random graph process

- MathematicsRandom Struct. Algorithms
- 2001

Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.

### Random evolution in massive graphs

- Mathematics, Computer ScienceProceedings 2001 IEEE International Conference on Cluster Computing
- 2001

This paper gives three increasingly general directed graph models and one general undirected graph model for generating power law graphs by adding at most one node and possibly one or more edges at a time and describes a method for scaling the time in the evolution model such that the power law of the degree sequences remains invariant.

### A NOTE ON THE DIAMETER OF PROTEAN GRAPHS

- Mathematics
- 2007

The web graph is a real-world self-organizing network whose vertices correspond to web pages, and whose edges correspond to links between pages. Many stochastic models for the web graph have been…

### Scale-free networks from varying vertex intrinsic fitness.

- BiologyPhysical review letters
- 2002

A new mechanism leading to scale-free networks is proposed, which is called a good-get-richer mechanism, in which sites with larger fitness are more likely to become hubs (i.e., to be highly connected).

### Universal behavior of load distribution in scale-free networks.

- Computer SciencePhysical review letters
- 2001

It is conjecture that the load exponent is a universal quantity to characterize scale-free networks and valid for both undirected and directed cases.

### Growing Protean Graphs

- Computer Science, MathematicsInternet Math.
- 2007

This paper proposes an extended version of a new random model of the web graph in which the degree of a vertex depends on its age, and uses the differential equation method to obtain basic results on the probability of edges being present.

### Sudden Emergence of a Giantk-Core in a Random Graph

- MathematicsJ. Comb. Theory, Ser. B
- 1996

These proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one, and demonstrate that, unlike the 2-core, when ak- core appears for the first time it is very likely to be giant, of size ?pk(?k)n.

### Statistics of changes in lead node in connectivity-driven networks.

- Computer SciencePhysical review letters
- 2002

The number of lead changes increases logarithmically with network size N, independent of the details of the growth mechanism, and the probability that the first node retains the lead approaches a finite constant for popularity-driven growth.