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Complex networks are powerful tools for characterizing and analyzing complex systems, with wide applications in fields such as physics, sociology, technology, biology, and port terminal management. One of the core issues in complex networks is the mechanism behind the emergence of scaling laws. In real-world networks, the mechanisms underlying the emergence of scaling laws may be highly complex, making it difficult to design network evolution mechanisms that fully align with reality. Explaining real networks through simple mechanisms is a meaningful research topic. Since Barabási and Albert discovered that growth and linear preferential attachment are mechanisms that generate power-law distributions, scholars have identified various forms of preferential attachment that produce power-law degree distributions. However, the most famous and useful one remains the linear preferential attachment in the BA model. Can scale-free behavior also emerge from random attachment and growth? In traditional network analysis, nodes are assumed to join the system at discrete, equally spaced time intervals, often based on the unfounded assumption that interarrival times follow a uniform distribution. In reality, nodes arrive randomly, and their interarrival times do not necessarily follow a uniform distribution. Although complex networks have flourished over the past two decades, they still cannot fully describe real systems with multiple interactions. Hypernetworks, which capture interactions involving more than two nodes, have become an important subject of study, and the mechanisms underlying the emergence of scaling in hypernetworks are a key research focus. The paper first introduces the concept of cliques in hypernetworks. A 1-element clique is a node, a 2-element clique is an edge in a complex network, a 3-element clique represents a triangle in higher-order networks, and a 4-element clique corresponds to a tetrahedron in higher-order networks. Secondly, we propose a clique-driven random hypernetwork evolution model. By combining stochastic processes, nodes arrive in continuous time, which better reflects real-world scenarios and provides a justified distribution for node interarrival times. Using Poisson process theory, we analyze the clique-driven random hypernetwork evolution model, avoiding arbitrary assumptions about node interarrival time distributions commonly made in traditional network analysis, thereby making the network analysis more rigorous. We derive an analytical expression for the cumulative degree distribution and the power-law exponent of the node degree distribution. Finally, we validate the theoretical predictions through computer simulations and empirical analysis of collected real-world data. The results show that the clique-driven random hypernetwork evolution model employs a simple connection mechanism, and that scale-free behavior emerges from growth and random attachment in higher-order structural networks. In our model, not only do nodes join the network in continuous time, but new nodes also randomly select d-element cliques, resulting in a power-law degree distribution. When d = 2, the power-law exponent of the node degree distribution in our model matches that of the BA model. When d > 2, the power-law exponent of the degree distribution depends on the number of elements of the driving clique (simplex dimension). We can directly estimate the power-law exponent of the model's degree distribution using the number of elements of the driving clique.
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Keywords:
- complex network /
- hypernetwork /
- higher-order network /
- hypergraph /
- simplicial complex
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图 1 模型前两步演化示意图 (a) $ N\left(t\right)=0; $ (b) $ N\left(t\right)=1; $ (c) $ N\left(t\right)=2 $
Figure 1. Schematic illustration of the first two steps of the evolution for the model: (a) $ N\left(t\right)=0; $ (b) $ \text{N}\left(t\right)=1; $ (c) $ \text{N}\left(t\right)=2 $.
图 2 模型的累积度分布与回归曲线, 横轴是度, 纵轴是累积度分布$ {P}_{\mathrm{c}\mathrm{u}\mathrm{m}}\left(k\right) $ (a) $ d=2, m=1 $时模型的累积度分布与回归曲线, 回归曲线的相关系数$ R=-0.9825 $, 幂律指数$ \tau =2.013 $; (b) $ d=3, m=1 $时模型的累积度分布与回归曲线, 回归曲线的相关系数$ R=-0.9934 $, 幂律指数$ \tau =1.565 $
Figure 2. The cumulative degree distribution and regression curve of the model, the horizontal axis represents degrees and the vertical axis represents the cumulative degree distribution $ {P}_{\mathrm{c}\mathrm{u}\mathrm{m}}\left(k\right) $: (a) The cumulative degree distribution and regression curve of the model when $ d=2, m=1 $, the correlation coefficient of the regression curve is $ R=-0.9825 $, and the power-law exponent is $ \tau =2.013 $; (b) the cumulative degree distribution and regression curve of the model when $ d=3, m=2 $, the correlation coefficient of the regression curve is:$ R=-0.9934 $, and the power-law exponent is: $ \tau =1.565 $.
图 3 $ d=2 $时模型的累积度分布与理论预测曲线, 横轴是度, 纵轴是累积度分布$ {P}_{\mathrm{c}\mathrm{u}\mathrm{m}}\left(k\right) $, 蓝色+表示模型的模拟值, 红色实线对应于方程(5)给出的理论预测值, 幂律指数为$ \tau =2 $ (a) $ m=1 $时模型的累积度分布模拟值与理论预测曲线; (b) $ m=2 $时模型的累积度分布模拟值与理论预测曲线
Figure 3. The cumulative degree distribution and theoretical prediction curve of the model with $ d=2 $, the horizontal axis represents degrees and the vertical axis represents the cumulative degree distribution $ {P}_{\mathrm{c}\mathrm{u}\mathrm{m}}\left(k\right), $ the blue + denotes the simulation value of the model. The red solid line corresponds to the theoretical predicted value given by Eq. (5), and the power-law exponent is $ \tau =2 $: (a) The cumulative degree distribution and theoretical prediction curve of the model when $ m=1 $; (b) the cumulative degree distribution and theoretical prediction curve of the model when $ m=2 $.
图 4 $ d=3 $时模型的累积度分布与理论预测曲线, 横轴是度, 纵轴是累积度分布$ {P}_{\mathrm{c}\mathrm{u}\mathrm{m}}\left(k\right) $, 蓝色+表示模型的模拟值, 红色实线对应于方程(5)给出的理论预测值, 幂律指数为$ \tau =1.5 $, 其余部分与图3相同
Figure 4. The cumulative degree distribution and theoretical prediction curve of the model with $ d=3, $ the horizontal axis represents degrees and the vertical axis represents the cumulative degree distribution $ {P}_{\mathrm{c}\mathrm{u}\mathrm{m}}\left(k\right) $, the blue + denotes the simulation value of the model, the red solid line corresponds to the theoretical predicted value given by Eq. (5), and the power-law exponent is $ \tau =1.5 $, the rest of the caption is the same as that in Fig. 3.
图 5 新能源汽车专利合作超网络的累积度分布, 网络规模的是N = 1582, 蓝色叉表示新能源汽车专利合作超网络的累积度分布, 红色实线表示回归拟合曲线, 相关系数$ R=-0.9916 $, 累积度分布的幂律指数$ \tau =1.962 $
Figure 5. The cumulative degree distribution of the new energy vehicle patent cooperation hypernetwork, the network scale is N = 1582, the blue cross denotes the cumulative degree distribution of the new energy vehicle patent cooperation hypernetwork, the red solid line represents the regression fitting curve, with a correlation coefficient $ R= $$ -0.9916 $, and the power-law exponent of the cumulative degree distribution is $ \tau =1.962 $.
图 6 下秩3的科学家合作超网络累积度分布, 网络规模的是N = 1830, 蓝色叉表示下秩3的科学家合作超网络累积度分布. 红色实线表示回归拟合曲线, 相关系数$ R= $$ -0.9766 $, 累积度分布的幂律指数$ \tau =1.88 $
Figure 6. The accumulation degree distribution of the scientist cooperative hypernetwork with the lower rank 3, the network scale is N = 1830, the blue cross denotes the cumulative degree distribution of the hypernetwork. The red solid line represents the regression fitting curve, with a correlation coefficient $ R=-0.9766 $, and the power-law exponent of the cumulative degree distribution is $ \tau =1.88 $.
图 7 下秩4的科学家合作超网络累积度分布, 网络规模的是N = 1342, 蓝色叉表示下秩4的科学家合作超网络累积度分布, 红色实线表示回归拟合曲线, 相关系数$ R= $$ -0.9833 $, 累积度分布的幂律指数$ \tau =1.524 $
Figure 7. The accumulation degree distribution of the scientist cooperative hypernetwork with the lower rank 4, the network scale is N = 1342, the blue cross denotes the cumulative degree distribution of the hypernetwork, the red solid line represents the regression fitting curve, with a correlation coefficient $ R=-0.9833 $, and the power-law exponent of the cumulative degree distribution is $ \tau =1.542 $.
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