Emergent Topological Complexity in the Barabasi-Albert Model with Higher-Order Interactions
University of Mohaghegh Ardabili（莫哈格赫阿尔达比勒大学）；Shahid Beheshti University（沙希德贝赫什蒂大学）；Swiss Federal Institute for Forest, Snow and Landscape Research, WSL（瑞士联邦森林、雪地和景观研究所）
We examine the homological structure of the Barabasi-Albert model, focusing on the time evolution of $Δ$-dimensional simplices and topological holes as functions of time $t$ and the attachment parameter $m$ (the number of edges added by each incoming node). Numerical simulations reveal a non-trivial topological transition (TT) in the $(Δ, m)$ plane, marking a change from a topologically trivial regime to non-trivial topology. This transition signals the emergence of topological complexity in the model, where higher-order structures develop self-similarly across scales. Beyond this transition, the network exhibits self-similar topological growth, evidenced by a power-law decay in the increments of $Δ$-simplices with $m$-dependent exponents. An analogous transition occurs in the Betti numbers, which display self-similar behavior near the TT and an arctangent dependence farther from it. Based on simulation data, we propose explicit scaling relations describing the behavior of both $Δ$-simplices and Betti numbers near the TT. Overall, the analysis reveals a rich, gapful topological transition structure, where topological quantities exhibit discrete jumps at the transition point.
Comment to "Emergence of Scaling in Random Networks" (cond-mat/9910332)
Barabasi和Albert最近的一篇论文“Emergence of scaling in random networks”（cond-mat/9910332）提出了一种增长机制，可以在大型网络（如Web）中产生每个节点的边数的平稳无标度分布。Barabasi-Albert模型预测，较老的顶点会以牺牲较年轻的顶点为代价获得新的边，从而在引入顶点的时间与其拥有的边数之间存在很强的相关性。我们目前的数据显示，没有这样的相关性，并指出一个替代的增长模型，产生正确的网站的年龄和链接的数量之间的关系。
