Emergent Topological Complexity in the Barabasi-Albert Model with Higher-Order Interactions

We examine the homological structure of the Barabasi-Albert model, focusing on the time evolution of $\Delta$-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 $(\Delta, m)$ 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 $\Delta$-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 $\Delta$-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.