Projects
Under construction…
Physical Networks
Physical networks are networks of tangible objects, i.e., nodes and edges are embedded in a physical space under physical constraints including but not limited to 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 just a few. The recent advances of imaging techniques and machine learning methods for image segmentation have significantly improved the availablity of complex three-dimensional map of physical systems, which motivates 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).
Networks with fractal structure exhibit intriguing 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 (the number of adjacent subgraphs) in the renormalization process (the number of adjacent subgraphs), 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-2}{\gamma-1}\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).