[Paper Review] Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks

Tiandong Wang, Sidney I. Resnick|arXiv (Cornell University)|Jan 10, 2016

Complex Network Analysis Techniques16 references3 citations

TL;DR

This paper establishes conditions under which discrete multivariate mass functions, such as those in preferential attachment networks, are regularly varying by embedding them into continuous densities. It proves that the joint in- and out-degree distribution in such networks is nonstandard regularly varying, extending multivariate regular variation theory to discrete settings with applications to power-law behavior in complex networks.

ABSTRACT

Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity conditions. We extend these arguments to discrete mass functions and their associated measures using the concept that the the mass function can be embedded in a continuous density function. We give two different conditions, monotonicity and convergence on the unit sphere, both of which can make the discrete function embeddable. Our results are then applied to the preferential attachment network model, and we conclude that the joint mass function of in- and out-degree is embeddable and thus regularly varying.

Motivation & Objective

• To extend multivariate regular variation theory from continuous densities to discrete mass functions in multivariate settings.
• To identify sufficient conditions—monotonicity and convergence on the unit sphere—under which a discrete mass function can be embedded into a continuous function.
• To apply the theory to preferential attachment networks, where in- and out-degrees exhibit power-law behavior.
• To demonstrate that the joint in- and out-degree mass function is regularly varying under nonstandard scaling when attachment rates differ.
• To bridge the gap between regular variation of the associated measure and the discrete mass function in network models.

Proposed method

• Uses the concept of embedding a discrete mass function into a continuous density function to transfer regular variation properties.
• Applies two embedding conditions: monotonicity of the mass function and convergence of the normalized function on the unit sphere.
• Employs Karamata’s representation and dominated convergence to analyze asymptotic behavior of integrals involving gamma functions.
• Applies Stirling’s formula variant (Lemma 4.1) to establish uniform convergence of gamma function ratios on compact sets.
• Transforms integrals via change of variables to analyze convergence to a limiting integral involving exponential and power-law terms.
• Uses Corollary 4.1 to link regular variation of the embedded function to the original discrete mass function, ensuring the limit function is positive and bounded.

Experimental results

Research questions

• RQ1 Under what conditions can a discrete multivariate mass function be embedded into a continuous function to preserve regular variation?
• RQ2 How does the regular variation of a discrete mass function relate to the regular variation of its associated measure in multivariate settings?
• RQ3 Is the joint in- and out-degree distribution in a preferential attachment network regularly varying when the attachment rates for in- and out-degrees differ?
• RQ4 Can the asymptotic behavior of the joint degree distribution in such networks be characterized using multivariate regular variation?
• RQ5 What role does the embedding of the mass function play in establishing regular variation in discrete, heavy-tailed network models?

Key findings

• The joint in- and out-degree mass function $ p(i,j) $ in a preferential attachment model with $ c_1 eq c_2 $ is nonstandard regularly varying with scaling functions $ b_1(t) = t^{c_1} $, $ b_2(t) = t^{c_2} $.
• The limit function of the normalized mass function converges uniformly on compact sets away from the origin, as shown in equation (4.9) and (4.10).
• The limit function is positive and bounded on the set $ ilde{f E}_0 $, satisfying the conditions for regular variation in the discrete case.
• The convergence of the integral in (4.8) to a limiting integral involving $ e^{-(x/z + y/z^a)} $ confirms the asymptotic form of the tail behavior.
• The results confirm that regular variation of the discrete mass function implies regular variation of the associated measure, consistent with findings in [25, 26].
• The application of Lemma 4.1 ensures uniform convergence of gamma function ratios, which is essential for establishing the limit function’s regularity.