Global hierarchy vs local structure: Spurious self-feedback in scale-free networks

Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a nontrivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barabási-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.

Figure 1

Energy $E$ (a) and magnetization $M$ (b) as functions of temperature for a BAI network with $N={10}^4$ and $m_0=4$. Results from local mean-field theory (MF) and TAP are compared to Monte Carlo simulations (MC). All data are obtained for the same adjacency matrix $\mathbf{A}$. The insets in both panels show the differences of MF and TAP results to MC.

Figure 2

(a) Degree-wise magnetization $m\left(k\right)$ as a function of the degree $k$ obtained by Monte Carlo simulation for $N={10}^4$ and $m_0=4$ at the effective transition temperature $T_T^{\mathrm{MC}}=16.12\pm 0.05$, as extracted from the position of the maximum in ${\chi }_{\mathrm{MC}}$. The dashed line is parallel to the identity, to show the approximate linear relationship. (b) Local structure. Green curves: $m\left(k\right)$, for $k\in \{4,20\}$ as a function of temperature. Blue dashed curves: Degree-wise averages over the nearest neighbors of the hub $u$ (24).

Figure 3

(a) Leading eigenvalue ${\lambda }_{\mathbf{B}\left(\beta \right),\max }$ of $\mathbf{B}\left(\beta \right)$ (31) as a function of temperature $T$ for different system sizes $N$ and $m_0=4$. The black solid line shows the identity and the vertical dashed lines are predictions for the point where ${\lambda }_{\mathbf{B}\left(\beta \right),\max }=T$ from global TAP theory (32). (b) Projection $p_{\mathbf{B}\left(\beta \right)}$ (33) of the magnetization obtained from solving the TAP equations onto the leading eigenvector of $\mathbf{B}\left(\beta \right)$ for different system sizes $N$.

Figure 4

(a) Hub $u$, the node with the largest degree, shown as central dark dot. Subset of the first 25 nodes $i$ of the network, colored according to the minimal number $d_{i,u}$ of edges between $i$ and hub $u$. (b) Average magnetization $M$ (42), magnetization $m_{\mathrm{NN},u}$ (44) averaged over the nearest neighbors of $u$, and magnetization $m\left(k_u\right)$ (40) of the hub as functions of the temperature $T$ within the TAP approach for a BAI model. Symbols denote the corresponding Monte Carlo data. (c) Eigenvector of the largest eigenvalue of $\mathbf{B}\left(\beta \right)$ at $T=\frac{m_0}{2}log\left(N\right)$. Dots: Entries of the eigenvector, sorted by their degree $k_i$, and colored according to the distance $d_{i,u}$ to the central hub $u$. Red curve: Average of the entries with nodes of equal degree. Dotted orange curve: Linear fit of the averaged entry per degree. Dashed yellow curve: Scaling (43) of entry of a nearest neighbor of the central hub $u$. The common prefactor is calculated as ${\beta }_{\mathrm{TAP}}S=k_u^{-1}{v}_{\mathbf{B}\left({\beta }_{\mathrm{TAP}}\right),u}$. All three panels show data from the same BAI model with $N={10}^4$, $m_0=4$.

Figure 5

Specific heat $C$ (a) and susceptibility $\chi$ (b) as functions of temperature $T$ for a BAI with $N={10}^4$ and $m_0=4$. ${\chi }_{\mathrm{MC}}$ (10) is shown as a full line below (overlaid by TAP) and as a dotted line above the Monte Carlo transition temperature. Monte Carlo data for ${\chi }_0$ (8) above the effective transition temperature, which is the appropriate estimator for the susceptibility in the paramagnetic regime. All data obtained for the same realization of the adjacency matrix $\mathbf{A}$ of the network, where $N={10}^4$ and $m_0=4$.

Figure 6

Effective transition temperature as extracted from the maximum of the susceptibility as a function of system size $N$ for $m_0=4$. Red dots: Estimated from Monte Carlo simulations using Eq. (10). Yellow dots: Solutions of the local mean-field equation Eq. (16). Solutions of the TAP Eq. (29). Violet curve: Global mean-field (22) or, equivalently, TAP prediction (32). Black triangles: Leading eigenvalue of the adjacency matrix ${\lambda }_{\mathbf{A},\max }$. Orange curve: Fit of ${\lambda }_{\mathbf{A},\max }$ to $a{N}^{1/4}+b$ to illustrate the $N^{1/4}$ scaling.

Figure 7

Average energy $E$ (a) and magnetization $M$ (b) from 10 different realizations of the BAI model, as functions of temperature $T$ for different system sizes $N$ and $m_0=4$ obtained from TAP approximation. Dashed lines are drawn one standard deviation above and below the mean, calculated according to Eq. (C1). Lines seem to align where the standard deviations are very small. Insets show the standard deviations alone.