Evolving Network Modeling Driven by the Degree Increase and Decrease Mechanism

TL;DR

The paper develops a novel evolving-network model in which vertex-degree dynamics drive network growth and reduction via a degree-driven queueing system. The input rate $λ(k)$ is made positively correlated with degree, with two concrete forms $λ(k)=k$ and $λ(k)=\ln(1+k)$, and the decrease rate is a constant $μ$, yielding a non-homogeneous Poisson process that embodies a new preferential-attachment mechanism. Using birth–death (Markov) analysis, the authors derive stationary degree distributions for both forms (the $λ(k)=k$ case requires $μ>1$, while $λ(k)=\ln(1+k)$ admits a stationary distribution for all $μ>0$ with $P_1$ computed numerically via a convergent series $S(μ)$); simulations validate these results and demonstrate long-tail, non-power-law behavior in some regimes. They further show the model can fit real networks (co-authorship, email, AS) with high fidelity, often outperforming a contemporary evolving-network model, indicating practical applicability for modeling networks with simultaneous growth and reduction and suggesting potential integration with dynamic graph representations and GNNs.

Abstract

Ever since the Barabási-Albert (BA) scale-free network has been proposed, network modeling has been studied intensively in light of the network growth and the preferential attachment (PA). However, numerous real systems are featured with a dynamic evolution including network reduction in addition to network growth. In this paper, we propose a novel mechanism for evolving networks from the perspective of vertex degree. We construct a queueing system to describe the increase and decrease of vertex degree, which drives the network evolution. In our mechanism, the degree increase rate is regarded as a function positively correlated to the degree of a vertex, ensuring the preferential attachment in a new way. Degree distributions are investigated under two expressions of the degree increase rate, one of which manifests a ``long tail'', and another one varies with different values of parameters. In simulations, we compare our theoretical distributions with simulation results and also apply them to real networks, which presents the validity and applicability of our model.

Figures (10)

• Figure 1: An illustration of the degree queueing system. A snapshot during a short time $t$ is displayed (The degree queueing system evolves with time). The purple larger circle indicates vertex $i$ which is the objective of this system. Blue circles indicate vertices that are connecting or are going to connect to vertex $i$, and those with blue lines in the middle square indicate edges connected during time $t$, while black lines indicate edges connected before time $t$. Red dot lines and red circles indicate edges and vertices disconnected to vertex $i$. Yellow wide lines indicate that there are edges connected to other vertices in the network, not denoting specific edges.
• Figure 2: An illustration of the state transition of a vertex (denoted by $i$)'s degree: As we can see above the arrow $f(k)$$(k=1,2,...)$ indicates the degree increase rate relevant to the degree $k$, and beneath the arrow, $k\mu$ indicates the degree decrease rate. The figures in circles indicate the degree value.
• Figure 3: Illustration of networks under two degree increase rates and different decrease rates: The initial network is a scale-free network with 10 vertices. New vertices come with two edges connected to existing vertices. Sub-figures (a), (b), and (c) respectively show snapshots of networks with $\mu$=1.01, 1.21, and 1.41 under the first situation that $\lambda(k)=k$, while sub-figures (d), (e), and (f) respectively present snapshots of networks with $\mu$=0.005, 0.01, and 0.05 under the second situation that $\lambda(k)=ln(1+k)$. The size and the color of nodes symbolize their degree value, where big and yellow circles indicate a large degree while small and purple circles indicate a small degree value. Vertices with a medium degree value are middle-size and dark blue.
• Figure 4: The degree distribution with different decrease rates $\mu$ and $\lambda(k)$=$k$: Sub-figures (a)-(f) are respectively set with $\mu$=1.5, 2.0, 3.5, 3.0, 4.0, and 5.0. Time is set large enough for the distribution to be stationary. Red lines indicate theoretical distributions, and the blue circle plots indicate distributions obtained by simulations. Distributions are all heterogeneous, featured with "long tail".
• Figure 5: The degree distribution with different decrease rates $\mu$ and $\lambda(k)$=$ln(1+k)$: Sub-figures (a)-(f) are respectively set with $\mu$=0.075, 0.15, 0.75, 1.5, 2.0, and 2.5. Red lines indicate theoretical distributions by numerical calculation, and the blue triangle plots indicate distributions obtained by simulations. Distributions with $\mu$=0.075 and 0.15 different from the rest distributions are skew distributions, while others are heterogeneously featured with "long tail".

Theorems & Definitions (9)

• Lemma 1
• proof
• Lemma 2
• Proof
• Theorem 1
• Proof
• Theorem 2
• Lemma 3
• Proof