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Nonlinear growth: an origin of hub organization in complex networks

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ABSTRACT

Many real-world networks contain highly connected nodes called hubs. Hubs are often crucial for network function and spreading dynamics. However, classical models of how hubs originate during network development unrealistically assume that new nodes attain information about the connectivity (for example the degree) of existing nodes. Here, we introduce hub formation through nonlinear growth where the number of nodes generated at each stage increases over time and new nodes form connections independent of target node features. Our model reproduces variation in number of connections, hub occurrence time, and rich-club organization of networks ranging from protein–protein, neuronal and fibre tract brain networks to airline networks. Moreover, nonlinear growth gives a more generic representation of these networks compared with previous preferential attachment or duplication–divergence models. Overall, hub creation through nonlinear network expansion can serve as a benchmark model for studying the development of many real-world networks.

No MeSH data available.


Normalized rich-club indices of control networks (red) and simulated, very preterm brain networks (blue). Shaded areas indicate standard deviation, as obtained from 20 curves per scenario. The x-axis indicates the rich club degree, i.e. the cut-off degree defining the rich-club. For the normalization of rich-club indices (y-axis), 100 reference networks preserving degree distribution were generated (hence, the number of nodes and node degree distribution do not differ in control, preterm and reference networks). These results demonstrate that the nonlinear growth model is in accordance with observations of increased rich-club organization in the structural connectome of very preterm-born adults (see also fig. 4 of [24]).
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RSOS160691F5: Normalized rich-club indices of control networks (red) and simulated, very preterm brain networks (blue). Shaded areas indicate standard deviation, as obtained from 20 curves per scenario. The x-axis indicates the rich club degree, i.e. the cut-off degree defining the rich-club. For the normalization of rich-club indices (y-axis), 100 reference networks preserving degree distribution were generated (hence, the number of nodes and node degree distribution do not differ in control, preterm and reference networks). These results demonstrate that the nonlinear growth model is in accordance with observations of increased rich-club organization in the structural connectome of very preterm-born adults (see also fig. 4 of [24]).

Mentions: For a valid comparison of the two scenarios, the normalized rich-club indices are computed by dividing the rich-club coefficient by the mean rich-club coefficient of 100 reference networks: , where ϕ(k) and ϕr(k) denote rich-club coefficients for grown and reference networks, respectively, for a given rich-club degree k. As shown in figure 5, the pathological networks indeed exhibit a stronger rich-club organization than control networks. The reason is that the simulated premature brain networks have comparably enhanced strength within rich clubs, as the weight factor is decreased for the connections that are formed late during development (i.e. connections of the last 25% nodes). Hence, our model is in agreement with the finding of Karolis et al. [24] that there are no topological connectivity differences between the two groups. More information on the growth model for the weighted networks is available in the electronic supplementary material.Figure 5.


Nonlinear growth: an origin of hub organization in complex networks
Normalized rich-club indices of control networks (red) and simulated, very preterm brain networks (blue). Shaded areas indicate standard deviation, as obtained from 20 curves per scenario. The x-axis indicates the rich club degree, i.e. the cut-off degree defining the rich-club. For the normalization of rich-club indices (y-axis), 100 reference networks preserving degree distribution were generated (hence, the number of nodes and node degree distribution do not differ in control, preterm and reference networks). These results demonstrate that the nonlinear growth model is in accordance with observations of increased rich-club organization in the structural connectome of very preterm-born adults (see also fig. 4 of [24]).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5383813&req=5

RSOS160691F5: Normalized rich-club indices of control networks (red) and simulated, very preterm brain networks (blue). Shaded areas indicate standard deviation, as obtained from 20 curves per scenario. The x-axis indicates the rich club degree, i.e. the cut-off degree defining the rich-club. For the normalization of rich-club indices (y-axis), 100 reference networks preserving degree distribution were generated (hence, the number of nodes and node degree distribution do not differ in control, preterm and reference networks). These results demonstrate that the nonlinear growth model is in accordance with observations of increased rich-club organization in the structural connectome of very preterm-born adults (see also fig. 4 of [24]).
Mentions: For a valid comparison of the two scenarios, the normalized rich-club indices are computed by dividing the rich-club coefficient by the mean rich-club coefficient of 100 reference networks: , where ϕ(k) and ϕr(k) denote rich-club coefficients for grown and reference networks, respectively, for a given rich-club degree k. As shown in figure 5, the pathological networks indeed exhibit a stronger rich-club organization than control networks. The reason is that the simulated premature brain networks have comparably enhanced strength within rich clubs, as the weight factor is decreased for the connections that are formed late during development (i.e. connections of the last 25% nodes). Hence, our model is in agreement with the finding of Karolis et al. [24] that there are no topological connectivity differences between the two groups. More information on the growth model for the weighted networks is available in the electronic supplementary material.Figure 5.

View Article: PubMed Central - PubMed

ABSTRACT

Many real-world networks contain highly connected nodes called hubs. Hubs are often crucial for network function and spreading dynamics. However, classical models of how hubs originate during network development unrealistically assume that new nodes attain information about the connectivity (for example the degree) of existing nodes. Here, we introduce hub formation through nonlinear growth where the number of nodes generated at each stage increases over time and new nodes form connections independent of target node features. Our model reproduces variation in number of connections, hub occurrence time, and rich-club organization of networks ranging from protein–protein, neuronal and fibre tract brain networks to airline networks. Moreover, nonlinear growth gives a more generic representation of these networks compared with previous preferential attachment or duplication–divergence models. Overall, hub creation through nonlinear network expansion can serve as a benchmark model for studying the development of many real-world networks.

No MeSH data available.