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

Abstract

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.

Figures (11)

• Figure 1: (a) The increments in the number of $\Delta$-dimensional simplexes, $\delta\Sigma_{\Delta,m}$, display a power-law scaling with time $t$ for various dimensions at $m = 29$. (b) and (c) present the corresponding intercepts ($F$) and slopes ($\psi$) of this scaling behavior as functions of $m$ and $\Delta$. For a fixed $\Delta$, the values of $\psi$ approach constant asymptotic limits as $m$ increases.
• Figure 2: (a) The increments in the number of $\Delta$-dimensional simplexes, $\delta\Sigma_{\Delta,m}$, exhibit a power-law scaling with time $t$ for various dimensions at $\Delta = 2$. (b) To verify the time-independence of the function $F$ defined in Eq. \ref{['eqt:sigma_delta_t_m']}, both sides of the equation are multiplied by $t^{\psi}$, confirming that $F$ remains constant over time. (c) The dependence of the scaling exponent $\psi$ on $(m, \Delta)$ is illustrated. For clarity, the plots are shown for only four representative values of $m$.
• Figure 3: Simplicial phase transition: The total number of $\Delta$-dimensional simplexes at $t=N$ is shown for various combinations of $(m, \Delta)$. Colored disks represent parameter pairs where $\Delta$-simplexes are present, while empty disks correspond to $(m, \Delta)$ values for which such structures do not form. The boundary separating these regions, $m_{\Delta}^S=\Delta$, marks the critical parameters governing the emergence of higher-dimensional connectivity in the growing network.
• Figure 4: (a) The number of $\Delta$-dimensional simplexes at $t = t_{\mathrm{max}} = 10^4$ is shown for various combinations of $m$ and $\Delta$. The dashed lines represent the fitting curves described by Eq. \ref{['eq:sigma']}. (b) The corresponding fitting parameters, $b_{\Sigma}$ (inset) and $c_{\Sigma}$ (main panel), are plotted as functions of the simplex dimension $\Delta$. Error bars are smaller than the symbol size. The parameter $c_{\Sigma}$ exhibits a linear dependence on $\Delta$, with a slope of $1.02 \pm 0.02$ and an intercept of $0.72 \pm 0.09$. (c) When the number of $\Delta$-dimensional simplexes is plotted against $m - b_{\Sigma}$ in log–log scale, the data collapse onto straight lines, confirming the power-law behavior and supporting the validity of the proposed fitting function.
• Figure 5: The number of $\Delta$-dimensional simplexes at $t=t_{max}=10^4$ for different $m$ and $\Delta$ values normalized by $Nm$. The range of $m$ is from 1 (the line at the bottom, for $\Delta>1$) to 29 (the line at the top, for $\Delta>1$).