Projects
Physical networks
Physical networks are networks of tangible components, in which nodes and edges are embedded in physical space and subject to constraints such as volume exclusion [(Dehmamy et al. 2018)]. Examples include biological neural networks, vascular systems, porous media, and granular materials. Advances in imaging technologies and accelerated 3D reconstruction by ML/AI have enabled to obtain detailed three-dimensional maps of such complex systems, providing new opportunities to study general organizing principles for tangible complex systems.
While the physical layout of networks has been shown to give rise to emergent structural properties, such as entanglement [Glover and Barabási (2024)], bundling (Bonamassa et al. 2025), jamming (Pósfai et al. 2024), and correlations between node degree and node volume (Pete et al. 2024), it remains elusive how these would impact dynamical processes that unfold on physical networks.
For my PhD thesis, I am exploring the spectral and topological properties of physical networks and their dynamical implications through physical Laplacian, a vertex-weighted graph Laplacian for physical networks derived in (Pete et al. 2024) as \begin{align} \mathbf{Q}_{\textrm{P}} = \mathbf{V}^{-1/2} \mathbf{Q}_{\textrm{G}} \mathbf{V}^{-1/2} \end{align} where \mathbf{G_{\textrm{G}}} is the combinatorial graph Laplacian and \mathbf{V}=\mathrm{diag}(v_1, \dots, v_{N}) is the diagonal volume matrix.
Higher-order networks with triadic interactions
Triadic interactions are higher-order, three-body interactions in which a node modulates edge between two other nodes. Such interactions are reported in various real-world systems; for example, glia cells in the brain regulate synaptic connectivity between two neurons, either inhibiting or exciting synaptic signaling. To model such interactions, we defined triadic interactions on network G=(V,E) as an interaction between a node i \in V and an edge \ell \in E encoded by incidence matrix \mathbf{K} \in \{-1, 0, 1\}^{|E| \times |V|} with entries \begin{align} K_{\ell i} = \begin{cases} -1 & \text{if node $i$ inhibits edge $\ell$}, \\ 1 & \text{if node $i$ excites edge $\ell$}, \\ 0 & \text{otherwise}. \end{cases} \end{align} Signed triadic interactions turn percolation processes into dynamical ones, with phenomena such as a period doubling and chaos [(Sun et al. 2023), (Sun and Bianconi 2024), (Millán et al. 2024)], and dynamical implications of triadic interactions remain an important but open question.
Fractal scale-free networks
A network is fractal with respect to shortest-path lengths, if the minimum number of subgraphs (boxes) of 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 et al. 2005). In fractal networks, the average path lengths also scales as the power of the network size, \begin{equation} \langle l \rangle \sim N^{1/D_{\textup{f}}}. \end{equation} While many empirical networks exhibit the small-world property, where the average path lengths scales logarithmically with the number of nodes, the fractal property can still hold at length scales shorter than the network’s characteristic size. Detecting this scaling through shortest paths, however, is often challenging as most real networks do not have a sufficiently large diameter.
Fractal networks are of particular interest due to their well-defined structural scale invariance, which gives rise to distinctive features such as long-range degree correlations manifested as hub repulsion (Rozenfeld et al. 2008). Fractal network models therefore provide a natural analytical framework for studying network coarse-graining and renormalization.