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Anomalous scaling in redirection networks

Harrison Hartle, P. L. Krapivsky, S. Redner, Yuanzhao Zhang

Abstract

In networks that grow by isotropic redirection (IR), a new node selects an initial target node uniformly at random and attaches to a randomly chosen neighbor of the target. The emerging networks exhibit leaf proliferation, in which the number of nonleaves scales sublinearly as $N^μ$ and the degree distribution has an algebraic tail with exponent $1+μ$. To understand these mysterious properties, we introduce a class of models with redirection to leaves whenever possible. The resulting networks exhibit qualitatively similar phenomenology to IR networks, but avoid the inherent non-locality of the IR growth rule. These networks admit an analytical description of the leaf degree distribution, from which we extract the exponent $μ$.

Anomalous scaling in redirection networks

Abstract

In networks that grow by isotropic redirection (IR), a new node selects an initial target node uniformly at random and attaches to a randomly chosen neighbor of the target. The emerging networks exhibit leaf proliferation, in which the number of nonleaves scales sublinearly as and the degree distribution has an algebraic tail with exponent . To understand these mysterious properties, we introduce a class of models with redirection to leaves whenever possible. The resulting networks exhibit qualitatively similar phenomenology to IR networks, but avoid the inherent non-locality of the IR growth rule. These networks admit an analytical description of the leaf degree distribution, from which we extract the exponent .

Paper Structure

This paper contains 13 sections, 81 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Theoretical analysis
  4. DAN model
  5. PAN model
  6. VAN$(\omega)$ model
  7. VAN($\infty$) model
  8. Discussion
  9. Star and line graphs and the scaled core-size distribution $\mathcal{P}(z)$
  10. VAN$(\infty)$ model
  11. Two-population model
  12. The average and the variance
  13. The probability distribution $P_N(C)$

Figures (11)

  • Figure 1: Illustration of isotropic redirection (IR). A new node (purple circle) selects a pre-existing target node (blue) uniformly at random (dashed line). The new node attaches (solid line) to one of the neighbors of the selected node (orange circles) uniformly at random.
  • Figure 2: Schematic structure a tree consisting of leaves (green dots), rank-$1$ nodes (blue dots), and rank $>1$ nodes (red dots). The rank $>1$ nodes form the nucleus, and combined with rank-$1$ nodes they form the core.
  • Figure 3: Random trees grown by: (a) the IR model, (b) the DAN model, and (c) the PAN model. Node colors are as in Fig. \ref{['fig:schematic']}.
  • Figure 4: Illustration of self-averaging for the nucleus to core ratio. The scaled probability distribution $P(\mathcal{N}/\mathcal{C})$ for the ratio of nucleus to core size for a given network realization, $\mathcal{N}/\mathcal{C}$, as a function of the number of nodes $N$. The distribution converges to a delta function that is located at the deterministic value $q_0$. Shown are data for the DAN model (upper panel) and the PAN model (lower panel).
  • Figure 5: The four processes that contribute to the change in $M_1$. The new node, necessarily a leaf, is indicated by the open green circle. The dashed green line indicates the initially selected node and the solid green line indicates the attachment event. The blue node is rank-$1$ and the red node is protected (other neighbors not displayed). In (a) and (b) the new attachment turns a leaf into a rank-$1$ node.
  • ...and 6 more figures