Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degrees in Preferential Attachment Networks with an Anomaly

Qiu Liang, Remco van der Hofstad, Nelly Litvak

TL;DR

The work analyzes a preferential attachment network augmented by an anomalous vertex that appears at time $\tau$ and attracts new edges with fixed probability $p$ (approximately $\frac{\beta}{2m+\beta+\delta}$). It derives an explicit expression for the anomaly's expected degree and establishes convergence results for ordinary vertices' degrees under the altered dynamics, using martingale arguments and gamma-function-based formulas. Through three regime analyses (late, mid-way, early anomaly), it provides heuristic limiting-degree distributions that reproduce standard PA exponents in some cases while altering tail behavior in others. The results reveal that early anomalies significantly modify the degree tail and that late anomalies largely preserve PA-like behavior, offering insights into anomaly effects and potential detection in evolving networks.

Abstract

We consider a preferential attachment model that incorporates an anomaly. Our goal is to understand the evolution of the network before and after the occurrence of the anomaly by studying the influence of the anomaly on the structural properties of the network. The anomaly is such that after its arrival it attracts newly added edges with fixed probability. We investigate the growth of degrees in the network, finding that the anomaly's degree increases almost linearly. We also provide a heuristic derivation for the exponent of the limiting degree distributions of ordinary vertices, and study the degree growth of the oldest vertex. We show that when the anomaly enters early, the degree distribution is altered significantly, while a late anomaly has minimal impact. Our analysis provides deeper insights into the evolution of preferential attachment networks with an anomalous vertex.

Degrees in Preferential Attachment Networks with an Anomaly

TL;DR

The work analyzes a preferential attachment network augmented by an anomalous vertex that appears at time and attracts new edges with fixed probability (approximately ). It derives an explicit expression for the anomaly's expected degree and establishes convergence results for ordinary vertices' degrees under the altered dynamics, using martingale arguments and gamma-function-based formulas. Through three regime analyses (late, mid-way, early anomaly), it provides heuristic limiting-degree distributions that reproduce standard PA exponents in some cases while altering tail behavior in others. The results reveal that early anomalies significantly modify the degree tail and that late anomalies largely preserve PA-like behavior, offering insights into anomaly effects and potential detection in evolving networks.

Abstract

We consider a preferential attachment model that incorporates an anomaly. Our goal is to understand the evolution of the network before and after the occurrence of the anomaly by studying the influence of the anomaly on the structural properties of the network. The anomaly is such that after its arrival it attracts newly added edges with fixed probability. We investigate the growth of degrees in the network, finding that the anomaly's degree increases almost linearly. We also provide a heuristic derivation for the exponent of the limiting degree distributions of ordinary vertices, and study the degree growth of the oldest vertex. We show that when the anomaly enters early, the degree distribution is altered significantly, while a late anomaly has minimal impact. Our analysis provides deeper insights into the evolution of preferential attachment networks with an anomalous vertex.
Paper Structure (13 sections, 1 theorem, 34 equations, 8 figures)

This paper contains 13 sections, 1 theorem, 34 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. The growth of the degrees in $G_t$
  4. Expected degree of the anomaly
  5. Expected degree of the ordinary vertices
  6. Convergence of degrees for ordinary vertices
  7. Heuristic derivation of the limiting degree distribution for ordinary vertices
  8. Late anomaly: $\tau=t-t^\gamma, \gamma \in (0,1)$.
  9. Mid-way anomaly: $\tau=\alpha t$, $\alpha\in(0,1)$
  10. Early anomaly: $\tau = t^\gamma, \gamma \in (0,1)$.
  11. Comparison to the empirical degree distribution
  12. Degree growth of the oldest vertex
  13. Conclusion and Further Research

Key Result

proposition 1

Consider an anomaly that occurs at time $\tau$, where $1< \tau < t$, and follows the attachment rule rule-2 with given $\delta > -m$, and $\beta > 0$. Then where $c_0 = m+\delta - \frac{m \beta \tau }{m+\beta+\delta}$. In particular,

Figures (8)

  • Figure 1: $\beta = 0.5$
  • Figure 2: $\beta = 2.0$
  • Figure 4: The degree of the anomaly as a function of time $t$. The parameters used are $\tau = 200, \delta = 0, \beta = 2.0, m = 1.$
  • Figure 5: $m =1, \beta = 5.0, \delta = 0$.
  • Figure 6: $m =4, \beta = 10.0, \delta = 0$.
  • ...and 3 more figures

Theorems & Definitions (2)

  • proposition 1: Degree of the anomaly
  • proof