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 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 behind 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 incorporating 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.