Research
My research explores how the structures of complex networks impact dynamical processes on them.
Physical networks Hierarchical networks Critical networks
Network structure
How network architecture is shaped by physical constraints, hierarchy, scale-invariance, and heterogeneity.
Random walks Diffusion Synchronization
Network dynamics
How structural properties of networks impact diffusion, synchronization, and stochastic dynamics.
Network structure
The first pillar of my research concerns the non-trivial structural organization of complex networks. I am particularly interested in (i) complex networks where the abstract graph representation is incomplete due to the physicality of nodes and edges, (ii) hierarchical and critical networks with scale-invariant properties, and (iii) the coarse-graining of complex networks.
Physical networks
Physical networks are networks of tangible components, in which nodes and edges are embedded in physical space and subject to volume exclusion, geometry, and material constraints (Dehmamy, Milanlouei, and Barabási 2018). Examples include biological neural networks, vascular systems, porous media, and granular materials, to list a few.
Advances in imaging technologies and accelerated three-dimensional reconstruction by AI/machine learning have made detailed spatial maps of these systems increasingly accessible. This creates an opportunity to develop theoretical frameworks that incorporate physicality into abstract network models.
The physical layout of networks has been shown to give rise to structural features such as entanglement (Liu, Dehmamy, and Barabási 2021; 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; Piazza et al. 2025). These observations motivate a broader question: how should network theory change when nodes and edges are not dimensionless objects, but physical entities occupying space?
One approach is to represent a full-scale physical network as a network-of-networks, in which each subnetwork is a network representation of the (spatial) layout of a physical node while the inter-subnetwork edges represent interactions between subnetworks (Pete et al. 2024). In the limit of weak inter-subnetwork coupling, this leads to a dynamical reduction with node-weighted Laplacian operator that explicitly incorporates node volumes into combinatorial graph Laplacians. This framework provides a route to studying how physical constraints reshape structural properties and dynamical processes on networks.
Fractal scale-free networks
Fractal networks emerge in various contexts, including the incipient percolation cluster at the percolation critical point (Cohen and Havlin 2004) and hierarchical network models (Rozenfeld et al. 2008; Song, Havlin, and Makse 2006; Yakubo and Fujiki 2022). Fractal scale-free networks provide analytically tractable models for understanding scale-invariance and long-range degree correlations in networks.
A network is fractal with respect to shortest-path lengths if the minimum number of boxes (subgraphs) of diameter l_{\mathrm{B}} required to cover the network scales as
\begin{equation} N_{\mathrm{B}}(l_{\mathrm{B}})\sim l_{\mathrm{B}}^{-D_{\mathrm{f}}}, \end{equation}
where D_{\mathrm{f}} is the fractal dimension (Song, Havlin, and Makse 2005). In fractal networks, average path lengths scale as a power of the network size,
\begin{equation} \langle l\rangle\sim N^{1/D_{\mathrm{f}}}, \end{equation}
in contrast to logarithmic scaling in small-world networks.
Fractal network models are useful because they posess well-defined structural scale-invariance and renormalization. They are also known to possess non-trivial structural properties, including hub repulsions, a manifestion of long-range degree correlations beyond nearest neighbors.
Bifractality
An interesting property of fractal scale-free networks is bifractality (Yamamoto and Yakubo 2023), which implies the distinct local dimensionality of the neighborhood of hub nodes versus non-hub nodes. For a class of fractal scale-free networks in which, under renormalization, the number of nodes \nu_{\mathrm{B}} in a supernode of diameter l_{\mathrm{B}} is proportional to the degree k_{\mathrm{B}} of that supernode,
\begin{equation} \nu_{\mathrm{B}}\propto k_{\mathrm{B}}, \end{equation}
the network exhibits two distinct local fractal dimensions,
\begin{equation} d_{\mathrm{f}}^{\max}=D_{\mathrm{f}},\qquad d_{\mathrm{f}}^{\min}=D_{\mathrm{f}}\left(\frac{\gamma-2}{\gamma-1}\right), \end{equation}
where \gamma is the exponent of the degree distribution and D_{\mathrm{f}} is the box-covering fractal dimension.
The minimum local fractal dimension d_{\mathrm{f}}^{\min} is associated with the neighborhood of hub nodes, while the maximum local fractal dimension d_{\mathrm{f}}^{\max} is associated with the neighborhood of non-hub nodes. Analytical and numerical calculations for deterministic and stochastic hierarchical fractal scale-free networks, together with fractal scale-free random graphs, suggest that bifractality is a general property of fractal scale-free networks (Yamamoto and Yakubo 2023).
Network dynamics
The second pillar of my research concerns dynamical processes on networks. I am interested in how non-trivial structural properties of complex networks modify dynamical processes such as diffusion and synchronization.
Dynamics on physical networks
A central question is how physical constraints should enter the dynamical operators used to model diffusion and synchronization. As an analytical model, we study multiscale diffusion on a network-of-networks representation of physical networks (Pete et al. 2024).
In the weak inter-subnetwork coupling limit relative to the intra-subnetwork coupling, the relevant operator reduces to a node-weighted Laplacian,
\begin{equation} \mathbf{Q}_{\mathrm{P}} = \mathbf{V}^{-1/2} \mathbf{Q}_{\mathrm{G}} \mathbf{V}^{-1/2}, \end{equation}
where \mathbf{Q}_{\mathrm{G}} is the combinatorial graph Laplacian of the inter-subnetwork edges and \mathbf{V}=\mathrm{diag}(v_1,\dots,v_N) is the diagonal matrix of node volumes or subnetwork sizes.
This formulation makes node volume an explicit part of the network operator and provides a route to studying diffusion, localization, and synchronization in physical networks.
Laplacian localization
We study eigenmode localization of the node-weighted Laplacian and ask how node physicality reshapes localization phenomena in empirical and synthetic physical networks.
This project aims to identify when physical constraints suppress, enhance, or reorganize Laplacian localization, and how this affects dynamical observables such as diffusion timescales and eigenmode shapes.
More coming soon.
Modular synchronization
Motivated by the coarse-grained Laplacians that arise in the weak-subnetwork coupling limit, we study synchronization dynamics in heterogeneous modular networks.
As a concrete physical setting, we are working in collaboration with an experimental group to design and test modular synchronization in a physical network of coupled spin-Hall nano-oscillators. The goal is to understand how local synchronization within modules and global synchronization across modules are controlled by topological and physical heterogeneity of physical networks.
More coming soon.
Random walks in bifractal networks
Building on the bifractality conjecture (Yamamoto and Yakubo 2023), we studied random walks on bifractal networks and found that the walk dimension remains position-independent, whereas the spectral dimension splits into two distinct local values. This splitting reflects the underlying bifractal structure and shows how local structural scaling can appear in dynamical observables (Yakubo, Shimojo, and Yamamoto 2024).
This project connects the structural pillar of bifractality to dynamical questions about diffusion and transport on fractal scale-free networks.
Higher-order network dynamics with triadic interactions
Triadic interactions are higher-order, three-body interactions in which a node modulates an edge between two other nodes. Such interactions are reported in real systems: for example, glial cells can regulate synaptic interactions between neurons, and a third species can interfere with pairwise ecological interactions (Wang et al. 2009; Cho, Barcelon, and Lee 2016; Grilli et al. 2017; Li et al. 2026).
To model such interactions, we define a triadic interaction on a network G=(V,E) as an interaction between a node i\in V and an edge \ell\in E, encoded by an incidence matrix \mathbf{K}\in\{-1,0,1\}^{|E|\times |V|} with entries
\begin{equation} K_{\ell i} = \begin{cases} -1, & \text{if node } i \text{ inhibits edge } \ell,\\ 1, & \text{if node } i \text{ activates edge } \ell,\\ 0, & \text{otherwise}. \end{cases} \end{equation}
Signed triadic interactions turn percolation processes into dynamical ones and can induce phenomena such as period doubling and chaos (Sun et al. 2023; Sun and Bianconi 2024; Millán et al. 2024).
In my MSc dissertation and subsequent collaborative work, I studied Langevin dynamics on networks with triadic interactions, where the interaction matrix coevolves with node states through a triadic Laplacian. The model takes the form
\begin{equation} d\mathbf{X}=-(\mathbf{L}^{(\mathrm{T})}+\alpha\mathbf{I})\mathbf{X}\,dt+\mathbf{\Gamma}\,d\mathbf{W}_t, \end{equation}
where \mathbf{L}^{(\mathrm{T})} is a state-dependent triadic Laplacian. The pairwise baseline without triadic interactions is analytically tractable, while triadic interactions produce conditional statistical signatures.
We show that conditional correlations and conditional mutual information can reveal the existence, sign, and strength of triadic interactions. This framework was used as a null model to identify node tuples with triadic interactions in gene regulatory networks (Niedostatek et al. 2025).