The Topological Behavior of Preferential Attachment Graphs

TL;DR

This work analyzes the higher-order topology of preferential attachment clique complexes in scale-free networks by combining algebraic topology with a Polya urn formulation of the model. It establishes the asymptotic almost-sure scaling of Betti numbers, proves that infinite preferential attachment clique complexes become $q$-homotopy-connected above a critical threshold, and shows the threshold is tight, implying non-contractibility in general. The results reveal two phase transitions per dimension and provide counter-evidence to Weinberger's contractibility conjecture for these infinite complexes, while highlighting subtle limits between topological properties and analytical limits. Numerical simulations suggest a power-law behavior for mean-normalized Betti numbers, motivating conjectures about scaling limits and broader universality across related random models.

Abstract

We investigate the higher-order connectivity of scale-free networks using algebraic topology. We model scale-free networks as preferential attachment graphs, and we study the algebraic-topological properties of their clique complexes. We focus on the Betti numbers and the homotopy-connectedness of these complexes. We determine the asymptotic almost sure orders of magnitude of the Betti numbers. We also establish the occurence of homotopical phase transitions for the infinite complexes, and we determine the critical thresholds at which the homotopy-connectivity changes. This partially verifies Weinberger's conjecture on the homotopy type of the infinite complexes. We conjecture that the mean-normalized Betti numbers converge to power-law distributions, and we present numerical evidence. Our results also highlight the subtlety of the scaling limit of topology, which arises from the tension between topological operations and analytical limiting process. We discuss such tension at the end of the Introduction.

Figures (3)

• Figure 1: (Left; Figure 1 of siu23_PrefAtt_betti) An illustration of the preferential attachment and clique building mechanism. When new nodes (drawn as people) in the left column are added to the network, they are more likely to attach to already popular nodes (who have high degrees), like the light blue person in the figure. Fully connected subsets of nodes form triangles, tetrahedra and their higher-dimensional analogues in the clique complex. Note that in order to have triangles, each new node must connect to at least 2 nodes, but we only drawn one connection for each new node to keep the illustration simple. See \ref{['sec:setup']} for the precise definitions. (Right) The log-log plot of the evolution of the complementary cumulative distribution functions ($\log (1-F(w))$ against $\log w$) of the mean-normalized Betti numbers at dimension 2. Green curves correspond to the distributions for complexes with fewer nodes, and blue ones, larger complexes. The dotted red line is the line of best-fit for the largest complex. Its slope is $-2.51$. Model parameters are detailed in \ref{['sec:simulations']}.
• Figure 2: Phase transitions at different dimensions for infinite preferential attachment clique complexes for moderate $m$. The symbols $\delta$ and $m$ were introduced in the first paragraph of the Introduction and they are precisely defined in \ref{['def:preferential_attachment']}. The symbol $x$, defined in \ref{['eqn:x']}, is a monotone function of $\delta/m$. The strength of preferential attachment increases from left to right. The annotated conditions hold almost surely in their respective regions. Dotted lines indicate that the condition does not hold at the corresponding endpoint. See \ref{['sec:main_results']} for the precise statements.
• Figure 3: (Left) The log-log plot of the evolution of the Kolmogorov-Smirnov norm between the distributions of normalized Betti numbers at time $t$ and time $10000$ (log of the KS norm against $\log t$). The orange dashed line is the least-square line of best-fit. Its slope is $-0.651$. (Right) The plot of the fitted exponents of the tails of the complementary cumulative distribution functions ($1 - \text{cdf}$) of the Betti numbers of preferential attachment complexes with $10000$ nodes but different choices of $\delta$ and $m$ (exponent against $-\delta/m$). Throughout all simulations, $m$ is either $8$ or $10$. This simulation and the generation of this plot were done by Avhan Misra.

Theorems & Definitions (64)

• Definition 1.1: Homotopy and Homotopy-Connectedness
• Definition 1.2: Homology Group, Betti Number, Cycle and Boundary; Section 5 of munkres84algtopo
• Remark
• Example 1.3: Increasingly Large Cycles
• Example 1.4: Preferential Attachment Graphs
• Example 1.5: Subtleties of Categorical Limits
• Example 1.6: Eden Model
• Definition 3.1: Affine Preferential Attachment Graphs; Definition 4.3.1 of garavaglia19_preferentialAttachment_thesis
• Remark
• Remark