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Physical Networks

Physical networks are networks of tangible objects, i.e., nodes and edges are embedded in a physical space under physical constraints (e.g., volume exclusion) [(Dehmamy, Milanlouei, and Barabási 2018), (Liu, Dehmamy, and Barabási 2021), (Pósfai et al. 2024), (Pete et al. 2024), (Blagojević and Pósfai 2024), (Glover and Barabási 2024)]. Examples of physical networks include biological neural networks, vascular networks, porous media, granular materials, to name a few. The recent advances of imaging techniques and machine learning methods for image segmentation have significantly improved the availablity of complex three-dimensional data of physical systems, which drives the study of physical networks.

Over the recent years, the physicality of the network has been shown to result in the emergent properties such as entanglement [(Liu, Dehmamy, and Barabási 2021), (Glover and Barabási 2024)], bundling (Bonamassa et al. 2024), jamming (Pósfai et al. 2024), and degree-volume correlations (Pete et al. 2024).

For my PhD thesis, I am investigating the spectral and topological properties of physical networks, as well as dynamical processes on physical networks.

Localization

Synchronization

Magnetization

Entanglement

Higher-Order Networks

Diffusion dynamics on networks with triadic interactions

Triadic interactions are interactions in which a node can regulate an edge between two other nodes either positively or negatively. Such interactions are ubiquitous in real-world networks, such as glia cells in the brain that regulate the connectivity between neurons.

In order to understand the role of triadic interactions in diffusion processes, we investigated the diffusion dynamics on networks with triadic interactions and Gaussian noise and we employed this model as a null model to detect the presence of triadic interactions in gene regulatory networks. To learn more, refer to (Baptista et al. 2024).

Fractal Scale-Free Networks

A network is fractal if the minimum number of subgraphs (boxes) of a given diameter l_{\textup{B}} required to cover the network scales as \begin{equation} N_{\textup{B}}(l_{\textup{B}}) \sim l_{\textup{B}}^{-D_{\textup{f}}} \end{equation} where D_{\textup{f}} is the fractal dimension of the network (Song, Havlin, and Makse 2005). In fractal networks, the average path lengths also scales as the power-law of the network size, i.e., \begin{equation} \langle l \rangle \sim N^{1/D_{\textup{f}}}. \end{equation} While many empirical networks are shown to be small-world, in which the average path lengths scales as logarithm of the number of nodes, i.e., D_{\textup{f}} \to \infty, the fractality can still be valid in length scale shorter than the characteristic length scale of the network (although it is difficult to observe the scaling in terms of shortest path lengths, as most of the networks do not have long enough diameter).

Putting aside the issue of observability of network fractality in real-world (finite) networks, networks with fractal structure exhibit intrigueing properties, such as the long-range degree correlations in the form of hub repulsion (Rozenfeld et al. 2008).

Bifractality

We showed that, in the coexistence of scale-free and fractal properties, a broad class of networks exhibits bifractality, characterized by the coexistence of two different local fractal dimensions, d_{\textup{f}}^{\min} and d_{\textup{f}}^{\max} depending on the position in the network (Yamamoto and Yakubo 2023). In particular, if a fractal scale-free network satisfies the condition that the number \nu_{\textup{B}} of nodes in a supernode (subgraph) of diameter l_{\textup{B}} is proportional to the degree k_{\textup{B}} of the supernode in the renormalization process, i.e., \begin{equation} \nu_{\textup{B}} \propto k_{\textup{B}}, \end{equation} then the network contains the local fractal dimension (equivalent to H"{o}lder exponent) \begin{equation} d_{\textup{f}}^{\max} = D_{\textup{f}}, \quad d_{\textup{f}}^{\min} = D_{\textup{f}} \left(\frac{\gamma-1}{\gamma-2}\right), \end{equation} where \gamma is the exponent of the degree distribution and D_{\textup{f}} is the (global) fractal dimension of the network.

From our extensive considerations of deterministic and stochastic models, the above condition seems to stand for any fractal scale-free networks and we conjectured that the bifractality is a generic property of fractal scale-free networks.

To learn more about the bifractality, see (Yamamoto and Yakubo 2023).

Random walks

Following the bifractality conjecture, we investigated the the walk and spectral dimensions of random walks on fractal scale-free networks. We found that the walk dimension d_{\textup{w}} of random walks on fractal scale-free networks (which characterizes the scaling of the mean topological displacement of random walkers) is constant and does not depend on the position in the network, while the spectral dimension (that determines the scaling of the return probability of a random walker) takes two distinct values, d_{\textup{s}}^{\min} and d_{\textup{s}}^{\max}, depending on the position in the network due to the bifractality of the network.

To learn more, see (Yakubo, Shimojo, and Yamamoto 2024).

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References

Baptista, Anthony, Marta Niedostatek, Jun Yamamoto, Ben MacArthur, Jürgen Kurths, Ruben Sanchez-Garcia, and Ginestra Bianconi. 2024. “Mining Higher-Order Triadic Interactions.” arXiv Preprint arXiv:2404.14997 [Nlin.AO].
Blagojević, Luka, and Márton Pósfai. 2024. “Three-Dimensional Shape and Connectivity of Physical Networks.” Scientific Reports 14 (1): 16874.
Bonamassa, I, B Ráth, M Pósfai, M Abért, D Keliger, B Szegedy, J Kertész, L Lovász, and A-L Barabási. 2024. “Logarithmic Kinetics and Bundling in Physical Networks.” arXiv e-Prints, arXiv–2401.
Dehmamy, Nima, Soodabeh Milanlouei, and Albert-László Barabási. 2018. “A Structural Transition in Physical Networks.” Nature 563 (7733): 676–80.
Glover, Cory, and Albert-László Barabási. 2024. “Measuring Entanglement in Physical Networks.” Physical Review Letters 133 (7): 077401.
Liu, Yanchen, Nima Dehmamy, and Albert-László Barabási. 2021. “Isotopy and Energy of Physical Networks.” Nature Physics 17 (2): 216–22.
Pete, Gábor, Ádám Timár, Sigurdur Örn Stefánsson, Ivan Bonamassa, and Márton Pósfai. 2024. “Physical Networks as Network-of-Networks.” Nature Communications 15 (1): 4882.
Pósfai, Márton, Balázs Szegedy, Iva Bačić, Luka Blagojević, Miklós Abért, János Kertész, László Lovász, and Albert-László Barabási. 2024. “Impact of Physicality on Network Structure.” Nature Physics 20 (1): 142–49.
Rozenfeld, Hernán D, Lazaros K Gallos, Chaoming Song, and Hernán A Makse. 2008. “Fractal and Transfractal Scale-Free Networks.” arXiv Preprint arXiv:0808.2206.
Song, Chaoming, Shlomo Havlin, and Hernan A Makse. 2005. “Self-Similarity of Complex Networks.” Nature 433 (7024): 392–95.
Yakubo, Kousuke, Gentaro Shimojo, and Jun Yamamoto. 2024. “Random Walks on Bifractal Networks.” Phys. Rev. E 110 (December): 064318. https://doi.org/10.1103/PhysRevE.110.064318.
Yamamoto, Jun, and Kousuke Yakubo. 2023. “Bifractality of Fractal Scale-Free Networks.” Phys. Rev. E 108: 024302. https://doi.org/10.1103/PhysRevE.108.024302.