Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks

Maria Letizia Bertotti,Giovanni Modanese

doi:10.3390/sym13010141

Abstract

It is well known that dynamical processes on complex networks are influenced by the degree correlations. A common way to take these into account in a mean-field approach is to consider the function knn(k) (average nearest neighbors degree). We re-examine the standard choices of knn for scale-free networks and a new family of functions which is independent from the simple ansatz knn∝kα but still displays a remarkable scale invariance. A rewiring procedure is then used to explicitely construct synthetic networks using the full correlation P(h∣k) from which knn is derived. We consistently find that the knn functions of concrete synthetic networks deviate from ideal assortativity or disassortativity at large k. The consequences of this deviation on a diffusion process (the network Bass diffusion and its peak time) are numerically computed and discussed for some low-dimensional samples. Finally, we check that although the knn functions of the new family have an asymptotic behavior for large networks different from previous estimates, they satisfy the general criterium for the absence of an epidemic threshold.

Highlights

As is nowadays well known, a large variety of real-world networks display a scale-free property, see, e.g., in [1,2,3,4]
A further feature typically found to be present for example in social networks is represented by assortative mixing, see, e.g., in [2,5]
Degree assortativity—which, at least in the mathematical and physical literature, provides the most studied example of assortativity—suggests that high degree nodes are preferentially linked to other high degree nodes and low degree nodes to other low degree nodes

Summary

Introduction

As is nowadays well known, a large variety of real-world networks display a scale-free property, see, e.g., in [1,2,3,4]. The main purpose of this paper is to contribute to a better comprehension of theoretical and foundational aspects concerning assortative scale-free networks and their role for the mathematical modeling of systems with social interaction Current work in this direction objectively displays some research gaps, because on one side quantities like the correlation coefficient r and the average nearest neighbor degree function knn(k) (see below the definitions) can be measured for real networks without any ambiguity; on the other hand, formal models based on a mean-field approach move from strong assumptions on the form of knn(k) without questioning whether such assumptions can be realized on assortative scale-free networks at least at the level of construction algorithms. Symmetry 2021, 13, 141 and β; and a qualitative agreement between the estimates of diffusion peak times obtained with the two different methods employed

Measuring Assortative and Disassortative Mixing

The Function knn in the Mean-Field Approach and for Concrete Networks

The Peak Time in Innovation Diffusion Dynamics

Further Remarks on the Function knn and Its Average

Conclusions