arXiv subjects

Takehisa Hasegawa

Publications and source records attributed to Takehisa Hasegawa.

At least 19 recordsLinked to original sources

arXiv2026

Berezinskii-Kosterlitz-Thouless-type Transition in Site Percolation on the Diamond Hierarchical Lattice

We study site percolation on the diamond hierarchical lattice, a finite-dimensional fractal network, using an exact generating-function analysis. In contrast to bond percolation, site percolation on this lattice does not undergo a transition from a nonpercolating phase to a percolating phase. Instead, the system exhibits a nonpercolating phase for $p p_{\rm c}$. In the critical phase, the size of the largest cluster remains subextensive, scaling as $N^{ψ(p)}$, where the fractal exponent $ψ(p)$ varies continuously with $p$. By analyzing the renormalization-group recursion relation in the vicinity of $p_{\rm c}$, we show that the correlation length exhibits a Berezinskii-Kosterlitz-Thouless-type essential singularity, $ξ(p)\sim \exp \left({\rm const}/\sqrt{p_{\rm c}-p}\right)$ for $p \to p_{\rm c}^-$, which is further confirmed by finite-size scaling analyses showing excellent data collapse. These results demonstrate that critical phases in percolation can emerge even on finite-dimensional networks and that exponential volume growth is not necessary for such phases to appear. We argue that the critical phase on the diamond hierarchical lattice stems from site dilution remaining relevant under renormalization.

cond-mat.stat-mech↗

arXiv2026

Analytical Framework for the Approximate Master Equation

The approximate master equation (AME) provides a highly accurate description of dynamical processes on networks, yet its steady states are generally analytically intractable. In this study, we develop an analytical framework to derive the steady states of the AME by introducing a controlled approximation that enables closure of the moment equations. This framework reproduces the steady state of the pair approximation by achieving closure with the minimum required order of moments, and can be systematically refined to approach the exact steady states of the AME. We apply this to the SIS model, the voter model, and evolutionary games, demonstrating that the steady states can be derived. In particular, for evolutionary games, we show that combining our framework with the singular perturbation method enables the analytical derivation of the time evolution.

physics.soc-ph↗

arXiv2025

Critical and Nonpercolating Phases in Bond Percolation on the Song-Havlin-Makse Network

We investigate bond percolation on the Song-Havlin-Makse (SHM) network, a scale-free tree with a tunable degree exponent and dimensionality. Using a generating function approach, we analytically derive the average size and the fractal exponent of the root cluster for deterministic cases. Our analysis reveals that bond percolation on the SHM network remains in a nonpercolating phase for all $p < 1$ when the network is fractal (i.e., finite-dimensional), whereas it exhibits a critical phase, where the cluster size distribution follows a power-law with a $p$-dependent exponent, throughout the entire range of $p$ when the network is small-world (i.e., infinite-dimensional), regardless of the specific dimensionality or degree exponent. The analytical results are in excellent agreement with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2024

Influence of initiators on the tipping point in the extended Watts model

In this paper, we study how the influence of initiators (seeds) affects the tipping point of information cascades in networks. We consider an extended version of the Watts model, in which each node is either active (i.e., having adopted an innovation) or inactive. In this extended model, the adoption threshold, defined as the fraction of active neighbors required for an inactive node to become active, depends on whether the node is a seed neighbor (i.e., connected to one or more initiators) or an ordinary node (i.e., not connected to any initiators). Using the tree approximation on random graphs, we determine the tipping point, at which the fraction of active nodes in the final state increases discontinuously with an increasing seed fraction. The occurrence of a tipping point and the scale of cascades depend on two factors: whether a giant component of seed neighbors is formed when the seed fraction is large enough to trigger cascades among seed neighbors, and whether the giant component of ordinary nodes is maintained when newly activated nodes trigger further activations among ordinary nodes. The coexistence of two giant components suggests that a tipping point can appear twice. We present an example demonstrating the existence of two tipping points when there is a gap between the adoption thresholds of seed neighbors and ordinary nodes. Monte Carlo simulations clearly show that the first cascade, occurring at a small tipping point, occurs in the giant component of seed neighbors, while the second cascade, occurring at a larger tipping point, extends into the giant component of ordinary nodes.

physics.soc-ph↗

arXiv2024

Robustness of Random Networks with Selective Reinforcement against Attacks

We investigate the robustness of random networks reinforced by adding hidden edges against targeted attacks. This study focuses on two types of reinforcement: uniform reinforcement, where edges are randomly added to all nodes, and selective reinforcement, where edges are randomly added only to the minimum degree nodes of the given network. We use generating functions to derive the giant component size and the critical threshold for the targeted attacks on reinforced networks. Applying our analysis and Monte Carlo simulations to the targeted attacks on scale-free networks, it becomes clear that selective reinforcement significantly improves the robustness of networks against the targeted attacks.

physics.soc-ph↗

arXiv2023

Optimal Network Robustness Against Attacks in Varying Degree Distributions

In varying degree distributions, we investigate the optimally robust networks against targeted attacks to nodes with higher degrees. In considering that a network tends to have more robustness with a smaller variance of degree distributions, we clarify the optimal robustness at random regular graphs in their comprehensive discrete or random perturbations. By comparing robustness measurements on them, we find that random regular graphs have the optimal robustness against attacks in varying degree distributions.

physics.soc-ph↗

arXiv2022

Cascading behavior of an extended Watts model on networks

In this study, we propose an extended Watts model to examine the effect of initiators on information cascades. The extended Watts model assumes that nodes with connections to initiators have low adoption thresholds than other nodes, due to the significant influence of initiators. We develop a tree approximation to describe the active node fraction for the extended Watts model in random networks and derive the cascade condition for a global cascade to occur with a small fraction of initiators. By analyzing the active node fraction and the cascade window of the extended Watts model on the Erdos-Renyi random graph, we find that increasing the influence of initiators facilitates the possibility of global cascades, i.e., how many nodes eventually become active is significantly affected by the fraction of initiators and the threshold of nodes directly connected to initiators, which determine cascade dynamics at early stages.

physics.soc-ph↗

arXiv2022

Synergistic epidemic spreading in correlated networks

We investigate the effect of degree correlation on a susceptible-infected-susceptible (SIS) model with a nonlinear cooperative effect (synergy) in infectious transmissions. In a mean-field treatment of the synergistic SIS model on a bimodal network with tunable degree correlation, we identify a discontinuous transition that is independent of the degree correlation strength unless the synergy is absent or extremely weak. Regardless of synergy (absent or present), a positive and negative degree correlation in the model reduces and raises the epidemic threshold, respectively. For networks with a strongly positive degree correlation, the mean-field treatment predicts the emergence of two discontinuous jumps in the steady-state infected density. To test the mean-field treatment, we provide approximate master equations of the present model. We quantitatively confirm that the approximate master equations agree with not only all qualitative predictions of the mean-field treatment but also corresponding Monte-Carlo simulations.

physics.soc-ph↗

arXiv2022

Impact of assortative mixing by mask-wearing on the propagation of epidemics in networks

In this study, we discuss the impacts of assortative mixing by mask-wearing on the effectiveness of mask use in suppressing the propagation of epidemics. We employ the mask model, which is an epidemic model involving mask wearers and non-mask wearers. We derive the occurrence probability and mean size of large outbreaks, epidemic threshold, and average epidemic size for the mask model in an assortatively mixed random network that follows an arbitrary degree distribution. Applying our analysis to the Poisson random networks, we find that the assortative (disassortative) mixing by mask-wearing decreases (increases) the epidemic threshold. Assortative mixing, the tendency for (non-)mask wearers to prefer to connect with (non-)mask wearers, is not effective in containing epidemics in that the transmissibility required for large outbreaks to occur is small. On the other hand, in high-transmissibility cases, mask use is most effective in decreasing the occurrence probability and mean size of large outbreaks, as well as the average epidemic size, when the mixing pattern is strongly assortative. Strongly assortative mixing, resulting in the separation of mask wearers and non-mask wearers, reduces the probability and degree of a large outbreak in high-transmissibility cases, although it allows a large outbreak to occur even in low-transmissibility cases. In scale-free networks, mask use is most effective when the mixing pattern is strongly assortative and when it is maximally disassortative, provided that the mask coverage is not low. Both for the Poisson random and scale-free networks, all analytical treatments are in good agreement with simulation results.

physics.soc-ph↗

arXiv2021

Revisiting finite size effect of percolation in degree correlated networks

In this study, we investigate bond percolation in networks that have the Poisson degree distribution and a nearest-neighbor degree-degree correlation. Previous numerical studies on percolation critical behaviors of degree-correlated networks remain controversial. We perform finite-size scaling for the peak values of the second-largest cluster size and the mean cluster size and find a large finite-size effect when a network has a strong degree-degree correlation. Evaluating the size dependence of estimated critical exponents carefully, we demonstrate that the bond percolation in the networks exhibits the mean-field critical behavior, independent of the strength of their nearest-neighbor degree correlations.

physics.soc-ph↗

arXiv2021

Observability transitions in clustered networks

We investigate the effect of clustering on network observability transitions. In the observability model introduced by Yang, Wang, and Motter [Phys. Rev. Lett. 109, 258701 (2012)], a given fraction of nodes are chosen randomly, and they and those neighbors are considered to be observable, while the other nodes are unobservable. For the observability model on random clustered networks, we derive the normalized sizes of the largest observable component (LOC) and largest unobservable component (LUC). Considering the case where the numbers of edges and triangles of each node are given by the Poisson distribution, we find that both LOC and LUC are affected by the network's clustering: more highly-clustered networks have lower critical node fractions for forming macroscopic LOC and LUC, but this effect is small, becoming almost negligible unless the average degree is small. We also evaluate bounds for these critical points to confirm clustering's weak or negligible effect on the network observability transition. The accuracy of our analytical treatment is confirmed by Monte Carlo simulations.

physics.soc-ph↗

arXiv2020

Structure of percolating clusters in random clustered networks

We examine the structure of the percolating cluster (PC) formed by site percolation on a random clustered network (RCN) model. Using the generating functions, we formulate the clustering coefficient and assortative coefficient of the PC. We analytically and numerically show that the PC in the highly clustered networks is clustered even at the percolation threshold. The assortativity of the PC depends on the details of the RCN. The PC at the percolation threshold is disassortative when the numbers of edges and triangles of each node are assigned by Poisson distributions, but assortative when each node in an RCN has the same small number of edges, most of which form triangles. This result seemingly contradicts the disassortativity of fractal networks, although the renormalization scheme unveils the disassortative nature of a fractal PC.

physics.soc-ph↗

arXiv2020

Emergence of Long-Range Correlations in Random Networks

We perform an analytical analysis of the long-range degree correlation of the giant component in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the giant component is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the giant component becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of the $l$-distant node pairs, which behaves as an exponentially decreasing function of distance in the off-critical region. The correlation function obeys a power-law with an exponential cutoff near the critical point. The Erdős-Rényi random graph is employed to confirm this critical behavior.

physics.soc-ph↗

arXiv2020

Percolation on a maximally disassortative network

We propose a maximally disassortative (MD) network model which realizes a maximally negative degree-degree correlation, and study its percolation transition to discuss the effect of a strong degree-degree correlation on the percolation critical behaviors. Using the generating function method for bipartite networks, we analytically derive the percolation threshold and the order parameter critical exponent, $β$. For the MD scale-free networks, whose degree distribution is $P(k) \sim k^{-γ}$, we show that the exponent, $β$, for the MD networks and corresponding uncorrelated networks are same for $γ>3$ but are different for $2<γ<3$. A strong degree-degree correlation significantly affects the percolation critical behavior in heavy-tailed scale-free networks. Our analytical results for the critical exponents are numerically confirmed by a finite-size scaling argument.

physics.soc-ph↗

arXiv2018

Disassortativity of percolating clusters in random networks

We provide arguments for the property of the degree-degree correlations of giant components formed by the percolation process on uncorrelated random networks. Using the generating functions, we derive a general expression for the assortativity of a giant component, $r$, which is defined as Pearson's correlation coefficient for degrees of directly connected nodes. For uncorrelated random networks in which the third moment for the degree distribution is finite, we prove the following two points. (1) Assortativity $r$ satisfies the relation $r\le 0$ for $p\ge p_{\rm c}$. (2) The average degree of nodes adjacent to degree-$k$ nodes at the percolation threshold is proportional to $k^{-1}$ independently of the degree distribution function. These results claim that disassortativity emerges in giant components near the percolation threshold. The accuracy of the analytical treatment is confirmed by extensive Monte Carlo simulations.

physics.soc-ph↗

arXiv2018

Sudden spreading of infections in an epidemic model with a finite seed fraction

We study a simple case of the susceptible-weakened-infected-removed model in regular random graphs in a situation where an epidemic starts from a finite fraction of initially infected nodes (seeds). Previous studies have shown that, assuming a single seed, this model exhibits a kind of discontinuous transition at a certain value of infection rate. Performing Monte Carlo simulations and evaluating approximate master equations, we find that the present model has two critical infection rates for the case with a finite seed fraction. At the first critical rate the system shows a percolation transition of clusters composed of removed nodes, and at the second critical rate, which is larger than the first one, a giant cluster suddenly grows and the order parameter jumps even though it has been already rising. Numerical evaluation of the master equations shows that such sudden epidemic spreading does occur if the degree of the underlying network is large and the seed fraction is small.

physics.soc-ph↗

arXiv2017

Efficiency of prompt quarantine measures on a susceptible-infected-removed model in networks

This study focuses on investigating the manner in which a prompt quarantine measure suppresses epidemics in networks. A simple and ideal quarantine measure is considered in which an individual is detected with a probability immediately after it becomes infected and the detected one and its neighbors are promptly isolated. The efficiency of this quarantine in suppressing a susceptible-infected-removed (SIR) model is tested in random graphs and uncorrelated scale-free networks. Monte Carlo simulations are used to show that the prompt quarantine measure outperforms random and acquaintance preventive vaccination schemes in terms of reducing the number of infected individuals. The epidemic threshold for the SIR model is analytically derived under the quarantine measure, and the theoretical findings indicate that prompt executions of quarantines are highly effective in containing epidemics. Even if infected individuals are detected with a very low probability, the SIR model under a prompt quarantine measure has finite epidemic thresholds in fat-tailed scale-free networks in which an infected individual can always cause an outbreak of a finite relative size without any measure. The numerical simulations also demonstrate that the present quarantine measure is effective in suppressing epidemics in real networks.

physics.soc-ph↗

arXiv2016

Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree

We study bond percolation of the Cayley tree (CT) by focusing on the probability distribution function (PDF) of a local variable, namely, the size of the cluster including a selected vertex. Because the CT does not have a dominant bulk region, which is free from the boundary effect, even in the large-size limit, the phase of the system on it is not well defined. We herein show that local observation is useful to define the phase of such a system in association with the well-defined phase of the system on the Bethe lattice, that is, an infinite regular tree without boundary. Above the percolation threshold, the PDFs of the vertex at the center of the CT (the origin) and of the vertices near the boundary of the CT (the leaves) have different forms, which are also dissimilar to the PDF observed in the ordinary percolating phase of a Euclidean lattice. The PDF for the origin of the CT is bimodal: a decaying exponential function and a system-size-dependent asymmetric peak, which obeys a finite-size-scaling law with a fractal exponent. These modes are respectively related to the PDFs of the finite and infinite clusters in the nonuniqueness phase of the Bethe lattice. On the other hand, the PDF for the leaf of the CT is a decaying power function. This is similar to the PDF observed at a critical point of a Euclidean lattice but is attributed to the nesting structure of the CT around the boundary.

cond-mat.stat-mech↗