Probabilistic Growth and Vari-linear Preferential Attachment in Random Networks

TL;DR

This work proposes a new network formation model: the vari-linear network, which includes two core mechanisms: exponential probabilistic growth and vari-linear preferential attachment, which overcomes the limitation of traditional growth mechanism in characterising low-degree distributions.

Abstract

Random networks are convenient foundational platforms widely employed in network experiments. Generating networks that more accurately reflect real-world patterns is a significant topic within complex network research. This work propose a new network formation model: the vari-linear network, which includes two core mechanisms: exponential probabilistic growth and vari-linear preferential attachment. It overcomes the limitation of traditional growth mechanism in characterising low-degree distributions. And confirms that controlling the extent of non-linear in preferential attachment is key to achieving a better fit to the real network's degree distribution pattern. The results show that the vari-linear network model maintains high fitting accuracy across multiple real-world networks of varying types and scales. And exhibits several-fold performance advantages over traditional methods. Meanwhile, it provides a unified theoretical explanation for classic topological characteristics such as small-world networks and scale-free networks. It not only provides a more quality foundational network framework for network research, but also serve as the brand new paradigm for bridging the conceptual divide between various classical network models.

Figures (6)

• Figure 1: Conceptual diagram of vari-linear network model based on (i) exponential probabilistic growth and (ii) vari-linear preferential attachment.
• Figure 2: Degree distribution of vari-linear network generated for typical parameter values. (a) Comparison of resulting network degree distributions for different number of nodes parameter $n$. (b) Comparison of the resulting network degree distributions for different average degree parameters $k$. (c) Comparison of the resulting network degree distributions for different vari-linear parameters $r$.
• Figure 3: Comparison of degree distributions between vari-linear network and traditional networks. (a) Comparison of vari-linear network with the BA scale-free network and the NPA (nonlinear preferential attachment) network. Vari-linear network $\#1$ parameters: $n=2000$, $k=5.22$, $r=1$; BA network parameters: $n=2000$, $m=3$; NPA network parameters: $n=2000$, $m=3$, $\alpha=1$. (b) Comparison of vari-linear network with ER network. Vari-linear network $\#2$ parameters: $n=2000$, $k=5.22$, $r=-1.5$; ER network parameters: $n=2000$, $p=3.03\times10^{-3}$. (c) Comparison of vari-linear network with the WS network. Vari-linear network $\#3$ parameters: $n=2000$, $k=5.22$, $r=-10$; WS network parameters: $n=2000$, $k=6$, $p=0.2$. Note that the purpose of setting the relevant parameter values here is to ensure that the node and edge (average degree $\langle k \rangle$) values of the result networks of each method remain basically consistent.
• Figure 4: Comparison of degree distribution probabilities between the optimal result network of vari-linear network model and the real network (with EMD metric as the optimization objective). In all subfigures, the blue scatters represent the degree distribution of vari-linear network in the optimal case, and the scatters in other colors are the degree distributions of the corresponding real networks. Types of authentic networks include (a-k) social networks (red); (i-p) scholarly co-authorship networks (orange); (q-s) scholarly citation networks (yellow); (t-v) communication networks (purple); (w-ab) biological networks (green); and (ac-af) literary and artistic networks (pink).
• Figure 5: Comparison of degree distribution probabilities between the optimal result network of vari-linear network model and the real network (with KS metric as the optimization objective). In all subfigures, the blue scatters represent the degree distribution of vari-linear network in the optimal case, and the scatters in other colors are the degree distributions of the corresponding real networks. Types of authentic networks include (a-k) social networks (red); (i-p) scholarly co-authorship networks (orange); (q-s) scholarly citation networks (yellow); (t-v) communication networks (purple); (w-ab) biological networks (green); and (ac-af) literary and artistic networks (pink).