[Paper Review] Diagonal degree correlations vs. epidemic threshold in scale-free networks

Maria Letizia Bertotti, Giovanni Modanese|arXiv (Cornell University)|Aug 24, 2021

Complex Network Analysis Techniques18 references3 citations

TL;DR

This paper demonstrates that even weak diagonal assortative degree correlations in scale-free networks drastically reduce the epidemic threshold, causing it to vanish rapidly as network size increases. Using the Vazquez-Weigt correlation matrix and Porto-Weber reconstruction, it shows that identical average nearest neighbor degree functions (knn) can yield vastly different epidemic thresholds due to distinct correlation matrix structures and eigenvalue spectra.

ABSTRACT

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is $P(h|k)=(1-r)P^U_{hk}+r\delta_{hk}$, where $P^U$ is uncorrelated and $r$ (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent $\gamma$, if the network size is measured by the largest degree $n$. We also prove that it is possible to construct, via the Porto-Weber method, correlation matrices which have the same $k_{nn}$ as the $P(h|k)$ above, but very different elements and spectrum, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form $P(h|k) o P(h|k)+\Phi(h,k)$ with $\Phi(h,k)$ depending on a parameter which leave $k_{nn}$ invariant. Such transformations affect in general the epidemic threshold. We find however that this does not happen when they act between networks with constant $k_{nn}$, i.e. networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).

Motivation & Objective

• To investigate how diagonal assortative degree correlations affect the epidemic threshold in large-scale-free networks.
• To examine the ambiguity in reconstructing correlation matrices from the same knn function, showing that different matrices yield different epidemic dynamics.
• To analyze the impact of admissible transformations preserving knn on the epidemic threshold, particularly in networks with constant knn.
• To establish that networks with constant knn (independent of degree) are invariant under such transformations, preserving the epidemic threshold despite correlation changes.

Proposed method

• Uses the Vazquez-Weigt correlation matrix P(h|k) = (1−r)hP(h)/⟨k⟩ + rδhk, which combines uncorrelated and perfectly assortative components.
• Applies the Porto-Weber method to reconstruct correlation matrices from a given knn function, enabling comparison with the original Vazquez-Weigt matrix.
• Analyzes the eigenvalue spectrum of the connectivity matrix Ckh = kP(h|k), showing that for the Vazquez-Weigt matrix, eigenvalues are Λ(i) = ri, with i = 1,…,n.
• Introduces a family of transformations P(h|k) → P(h|k) + Φ(h,k) that preserve knn(k), using symmetric perturbations Φ(h,k) parameterized by φ1,1.
• Computes the largest eigenvalue of Ckh under these transformations to assess changes in the epidemic threshold λc = 1/Λmax.
• Compares results across scale-free networks with varying γ (2.1, 2.5, 2.9), evaluating threshold sensitivity under different correlation structures.

Experimental results

Research questions

• RQ1 How does the presence of even a small amount of diagonal assortative degree correlation affect the epidemic threshold in large-scale-free networks?
• RQ2 To what extent is the epidemic threshold determined by knn(k) alone, or does the full correlation matrix P(h|k) play a critical role?
• RQ3 Can different correlation matrices produce the same knn(k) but lead to significantly different epidemic thresholds?
• RQ4 Under what conditions do transformations preserving knn(k) fail to alter the epidemic threshold?
• RQ5 What is the role of the eigenvalue spectrum of the connectivity matrix in determining the epidemic threshold?

Key findings

• Even a small diagonal assortative correlation (r > 0) causes the epidemic threshold λc to scale as n⁻¹, leading to a rapid vanishing of the threshold as network size n increases.
• The eigenvalues of the Vazquez-Weigt connectivity matrix are exactly Λ(i) = ri for i = 1,…,n, resulting in a maximal eigenvalue Λmax = r, so λc = 1/r, which decreases rapidly with r.
• For γ = 2.1, 2.5, and 2.9, the transformation P(h|k) → P(h|k) + Φ(h,k) with φ1,1 > 0 increases the largest eigenvalue of Ckh and thus decreases the epidemic threshold.
• In contrast, for strictly uncorrelated networks (P(h|k) = hP(h)/⟨k⟩), the same transformation preserves the largest eigenvalue at ⟨k²⟩/⟨k⟩, so λc remains unchanged despite altered P(h|k).
• Networks with constant knn(k) are invariant under such transformations: the epidemic threshold remains unchanged even when P(h|k) is modified, despite non-zero correlations.
• The study confirms that knn(k) alone is insufficient to predict the epidemic threshold; the full structure of P(h|k) and its spectral properties are essential.