Synchronizing individual activities is essential for the stable functioning of diverse complex systems. Understanding the relation between dynamic fluctuations and the connection topology of substrates is therefore important, but it remains restricted to regular lattices. Here we investigate the fluctuation of loads, assigned to the locally least-loaded nodes, in the largest-connected components of heterogeneous networks while varying their link density and degree exponents. The load fluctuation becomes finite when the link density exceeds a finite threshold in weakly heterogeneous substrates, which coincides with the spectral dimension becoming larger than 2 as in the linear diffusion model. The fluctuation, however, diverges also in strongly heterogeneous networks with the spectral dimension larger than 2. This anomalous divergence is shown to be driven by large local fluctuations at hubs and their neighbors, scaling linearly with degree, which can give rise to diverging fluctuations at small-degree nodes. Our analysis framework can be useful for understanding and controlling fluctuations in real-world systems.

We generalize an algorithm used widely in the configuration model such that power-law degree sequences with the degree exponent λ and the number of links per node K controllable independently may be generated. It yields the degree distribution in a different form from that of the static model or under random removal of links while sharing the same λ and K. With this generalized power-law degree distribution, the critical point K c for the appearance of the giant component remains zero not only for λ ≤ 3 but also for 3 < λ < λ l ≃ 3.81. This is contrasted with K c = 0 only for λ ≤ 3 in the static model and under random link removal. The critical exponents and the cluster-size distribution for λ < λ l are also different from known results. By analyzing the moments and the generating function of the degree distribution and comparison with those of other models, we show that the asymptotic behavior and the degree exponent may not be the only properties of the degree distribution relevant to the critical phenomena but that its whole functional form can be relevant. These results can be useful in designing and assessing the structure and robustness of networked systems.

The fluctuation of dynamic variables in complex networks is known to depend on the dimension and the heterogeneity of the substrate networks. Previous studies, however, have reported inconsistent results for the scaling behavior of fluctuation in strongly heterogeneous networks. To understand the origin of this conflict, we study the dynamic fluctuation on scale-free networks with a common small degree exponent but different mean degrees and minimum degrees constructed by using the configuration model and the static model. It turns out that the global fluctuation of dynamic variables diverges algebraically and logarithmically with the system size when the minimum degree is one and two, respectively. Such different global fluctuations are traced back to different, linear and sub-linear, growth of local fluctuation at individual nodes with their degrees, implying a crucial role of degree-one nodes in controlling correlation between distinct hubs.

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