Properties of the connected components in projections of random bipartite networks: effects of clique size fluctuations
Yuka Fujiki; Shogo Mizutaka, Last, Projection is a helpful description for treating bipartite networks as (monopartite) networks with pairwise interactions. Projections induce correlation spontaneously, avoiding negative degree correlation, even if bipartite networks are entirely random. In this study, we examined the structure of projections of random bipartite networks characterized by the degree distribution of individual and group nodes through the generating function method. We decomposed a projection into two subgraphs, the giant component, and finite components and analyzed their degree correlation. We demonstrate that positive degree correlations in projections originating from the clique size fluctuation remain after the decomposition at the set of finite components, although the values of their clustering coefficient are still finite. The giant component can exhibit either positive or negative degree correlations based on the structure of the projection. However, they are positively correlated in most cases. In addition, we found that a projection removed the giant component coincides with one in the subcritical phase, i.e., the discrete duality relation, when the degree distributions for group and individual are of Poisson.
Applied Network Science, Dec. 2024, [Reviewed]

Crossover phenomenon in adversarial attacks on voter model
Shogo Mizutaka, Lead, A recent study (Chiyomaru and Takemoto 2022 Phys. Rev. E 106 014301) considered adversarial attacks conducted to distort voter model dynamics in networks. This method intervenes in the interaction patterns of individuals and induces them to be in a target opinion state through a small perturbation ϵ. In this study, we investigate adversarial attacks on voter dynamics in random networks of finite size n. The exit probability P +1 to reach the target absorbing state and the mean time τ n to reach consensus are analyzed in the mean-field approximation. Given ϵ > 0, the exit probability P +1 converges asymptotically to unity as n increases. The mean time τ n to reach consensus scales as ( ln ϵ n ) / ϵ for homogeneous networks with a large finite n. By contrast, it scales as ( ln ( ϵ μ 1 2 n / μ 2 ) ) / ϵ for heterogeneous networks with a large finite n, where µ 1 and µ 2 represent the first and second moments of the degree distribution, respectively. Moreover, we observe the crossover phenomenon of τ n from a linear scale to a logarithmic scale and find n c o ∼ ϵ − 1 / α above which the state of all nodes becomes the target state in logarithmic time. Here, α = 1 for homogeneous networks and α = ( γ − 1 ) / 2 for scale-free networks with a degree exponent 2 < γ < 3 .
Journal of Physics: Complexity, 01 Sep. 2023, [Reviewed]

Synergistic epidemic spreading in correlated networks
Shogo Mizutaka; Kizashi Mori; Takehisa Hasegawa, Lead, 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., AMER PHYSICAL SOC
Physical Review E, Sep. 2022, [Reviewed]

Revisiting Finite Size Effect of Percolation in Degree Correlated Networks
Shogo Mizutaka; Takehisa Hasegwa, Lead, 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., PHYSICAL SOC JAPAN
Journal of the Physical Society of Japan, Apr. 2022, [Reviewed]

Emergence of long-range correlations in random networks
Shogo Mizutaka; Takehisa Hasegawa, Lead, We perform an analysis of the long-range degree correlation of the giant component (GC) in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the GC is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the GC becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of two nodes separated by the shortest path length l, 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 Erdös-Rényi random graph is employed to confirm this critical behavior.
Journal of Physics: Complexity, Dec. 2020, [Reviewed]

Structure of percolating clusters in random clustered networks
Takehisa Hasegawa; Shogo Mizutaka, 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., AMER PHYSICAL SOC
Physical Review E, Jun. 2020, [Reviewed]

Pregnancy parturition scars in the preauricular area and the association with the total number of pregnancies and parturitions
Yuriko Igarashi; Kunio Shimizu; Shogo Mizutaka; Kotaro Kagawa, Objectives: The aim of the present study was to clarify the association between the degree of development of pregnancy parturition scars (PPSs) and the total number of pregnancies and parturitions (TNPPs) on the basis of new identification standards for PPS in the preauricular area. Materials and Methods: Preauricular grooves were macroscopically observed on the pelves of 103 early modern males and 295 females (62 early modern females; 233 present-day females). Three categories of PPS in the preauricular area were defined. The association between the degree of development of PPS in the preauricular area and the TNPP was analyzed in 90 present-day females with detailed lifetime data. Results: PPS could not estimate the exact TNPP. However, it was shown that no PPS indicated no TNPP, weak PPS indicated a lower TNPP, and developed PPS indicated a higher TNPP. Discussion: Even though the possibility remains that some PPS indicate no TNPP, the results showed that the percentage of each PPS category indicated fertility in the population, suggesting that the strength of the association between the degree of development of PPS and the TNPP was affected by the classification system, the reliability of lifetime data, and the statistical methods used for analysis., WILEY
American Journal of Physical Anthropology, 01 Feb. 2020, [Reviewed]

Simple model of fractal networks formed by self-organized critical dynamics
Shogo Mizutaka, Lead, In this paper, a simple dynamical model in which fractal networks are formed by self-organized critical (SOC) dynamics is proposed; the proposed model consists of growth and collapse processes. It has been shown that SOC dynamics are realized by the combined processes in the model. Thus, the distributions of the cluster size and collapse size follow a power-law function in the stationary state. Moreover, through SOC dynamics, the networks become fractal in nature. The criticality of SOC dynamics is the same as the universality class of mean-field theory. The model explains the possibility that the fractal nature in complex networks emerges by SOC dynamics in a manner similar to the case with fractal objects embedded in a Euclidean space.
Journal of the Physical Society of Japan, 2019, [Reviewed]

Percolation on a maximally disassortative network
Shogo Mizutaka; Takehisa Hasegawa, Lead, 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, we show that the exponent, β, for the MD networks and corresponding uncorrelated networks are the same for &IMG ALIGN="MIDDLE" ALT="$\gamma>3$ " SRC="epl19938ieqn2.gif"/; but are different for &IMG ALIGN="MIDDLE" ALT="$2<\gamma<3$ " SRC="epl19938ieqn3.gif"/;. 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.
EPL, 2019, [Reviewed]

Disassortativity of percolating clusters in random networks
Shogo Mizutaka; Takehisa Hasegawa, Lead, 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≤0 for p≥pc. (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.
Physical Review E, 18 Dec. 2018, [Reviewed]

Fractality and degree correlations in scale-free networks
Yuka Fujiki; Shogo Mizutaka; Kousuke Yakubo, Fractal scale-free networks are empirically known to exhibit disassortative degree mixing. It is, however, not obvious whether a negative degree correlation between nearest neighbor nodes makes a scale-free network fractal. Here we examine the possibility that disassortativity in complex networks is the origin of fractality. To this end, maximally disassortative (MD) networks are prepared by rewiring edges while keeping the degree sequence of an initial uncorrelated scale-free network. We show that there are many MD networks with different topologies if the degree sequence is the same with that of the (u,v)-flower but most of them are not fractal. These results demonstrate that disassortativity does not cause the fractal property of networks. In addition, we suggest that fractality of scale-free networks requires a long-range repulsive correlation, in the sense of the shortest path distance, in similar degrees., SPRINGER
European Physical Journal B, 01 Jul. 2017, [Reviewed]

Structural instability of large-scale functional networks
Shogo Mizutaka; Kousuke Yakubo, Lead, We study how large functional networks can grow stably under possible cascading overload failures and evaluated the maximum stable network size above which even a small-scale failure would cause a fatal breakdown of the network. Employing a model of cascading failures induced by temporally fluctuating loads, the maximum stable size n has been calculated as a function of the load reduction parameter r that characterizes how quickly the total load is reduced during the cascade. If we reduce the total load sufficiently fast (r ≥ r ), the network can grow infinitely. Otherwise, n is finite and increases with r. For a fixed r(< r ), n for a scale-free network is larger than that for an exponential network with the same average degree. We also discuss how one detects and avoids the crisis of a fatal breakdown of the network from the relation between the sizes of the initial network and the largest component after an ordinarily occurring cascading failure. max c max c max, PUBLIC LIBRARY SCIENCE
PLoS ONE, Jul. 2017, [Reviewed]

Robustness analysis of bimodal networks in the whole range of degree correlation
Shogo Mizutaka; Toshihiro Tanizawa, Lead, We present an exact analysis of the physical properties of bimodal networks specified by the two peak degree distribution fully incorporating the degree-degree correlation between node connections. The structure of the correlated bimodal network is uniquely determined by the Pearson coefficient of the degree correlation, keeping its degree distribution fixed. The percolation threshold and the giant component fraction of the correlated bimodal network are analytically calculated in the whole range of the Pearson coefficient from -1 to 1 against two major types of node removal, which are the random failure and the degree-based targeted attack. The Pearson coefficient for next-nearest-neighbor pairs is also calculated, which always takes a positive value even when the correlation between nearest-neighbor pairs is negative. From the results, it is confirmed that the percolation threshold is a monotonically decreasing function of the Pearson coefficient for the degrees of nearest-neighbor pairs increasing from -1 and 1 regardless of the types of node removal. In contrast, the node fraction of the giant component for bimodal networks with positive degree correlation rapidly decreases in the early stage of random failure, while that for bimodal networks with negative degree correlation remains relatively large until the removed node fraction reaches the threshold. In this sense, bimodal networks with negative degree correlation are more robust against random failure than those with positive degree correlation., AMER PHYSICAL SOC
Physical Review E, 17 Aug. 2016, [Reviewed]

Fractal and small-world networks formed by self-organized critical dynamics
Akitomo Watanabe; Shogo Mizutaka; Kousuke Yakubo, We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution., PHYSICAL SOC JAPAN
Journal of the Physical Society of Japan, 15 Nov. 2015, [Reviewed]

Robustness of scale-free networks to cascading failures induced by fluctuating loads
Shogo Mizutaka; Kousuke Yakubo, Lead, Taking into account the fact that overload failures in real-world functional networks are usually caused by extreme values of temporally fluctuating loads that exceed the allowable range, we study the robustness of scale-free networks against cascading overload failures induced by fluctuating loads. In our model, loads are described by random walkers moving on a network and a node fails when the number of walkers on the node is beyond the node capacity. Our results obtained by using the generating function method show that scale-free networks are more robust against cascading overload failures than Erdos-Rényi random graphs with homogeneous degree distributions. This conclusion is contrary to that predicted by previous works, which neglect the effect of fluctuations of loads., AMER PHYSICAL SOC
Physical Review E, 20 Jul. 2015, [Reviewed]

Network fragility to overload failures: Influence of the scale-free property
Shogo Mizutaka; Kousuke Yakubo, Lead, Recently, it has been found that scale-free networks are fragile against overload failures. More precisely, the critical node-removal fraction f or the critical total load W at the percolation transition by removing overloaded nodes is an increasing function of the scale-free exponent γ describing the asymptotic degree distribution function as P(k) ∝ k . In the previous work, however, the decrease of the exponent γ reduces the total tolerance for loads. Thus, it is not clear whether scale-free networks are actually fragile against overload failures if the total tolerance dose not change. In this paper, we investigate network robustness against overload failures by calculating analytically the percolation transition point of scale-free networks for various values of γ with keeping the total tolerance constant. Our results show that scale-free networks are surely fragile to overload failures even after excluding the effect of the total tolerance. c c -γ, Oxford University Press
Journal of Complex Networks, 01 Dec. 2014, [Reviewed]

Structural robustness of scale-free networks against overload failures
Shogo Mizutaka; Kousuke Yakubo, Lead, We study the structural robustness of scale-free networks against overload failures induced by loads exceeding the node capacity, based on analytical and numerical approaches to the percolation problem in which a fixed number of nodes are removed according to the overload probability. Modeling fluctuating loads by random walkers in a network, we find that the degree dependence of the overload probability drastically changes with respect to the total load. We also elucidate that there exist two types of structural robustness of networks against overload failures. One is measured by the critical total load W and the other is by the critical node removal fraction f . Enhancing the scale-free property, networks become fragile in both senses of W and f . By contrast, increasing the node tolerance, scale-free networks become robust in the sense of the critical total load, while they come to be fragile in the sense of the critical node removal fraction. Furthermore, we show that these trends are not affected by degree-degree correlations, although assortative mixing makes networks robust in both senses of W and f . © 2013 American Physical Society. c c c c c c
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 08 Jul. 2013, [Reviewed]

Overload network failures: An approach from the random-walk model
Shogo Mizutaka; Kousuke Yakubo, Lead, We investigate analytically the robustness of scale-free networks against overload failures in percolation and cascading processes. Modeling fluctuating loads in a network by random walkers, it is found that there exist two kinds of robustness of networks against overload failures in the percolation process. One is measured by the critical total load W-rmc and the other is by the critical node removal fraction f-rmc. Enhancing the scale-free property, networks become fragile in both senses. In contrast, the increase of the node tolerance parameter makes scale-free networks robust in the sense of W-rmc but fragile in the sense of f-rmc. In addition, we treat the cascading overload process by considering it as a chain of percolation processes. We show that scale-free networks are robust against cascading overload failures modeled by random walkers, which is opposed to most of the previous results. © 2013 IEEE.
Proceedings - 2013 International Conference on Signal-Image Technology and Internet-Based Systems, SITIS 2013, 2013, [Reviewed]

Testing the order parameter of the anderson transition
Kousuke Yakubo; Shogo Mizutaka, The typical value of the local density of states φ , known as a candidate of the order parameter of the Anderson transition, of two-dimensional disordered electron systems with spin-orbit interactions is studied for assessing the validity of φ as the order parameter. We show that φ behaves critically as φ α (Ec - E)β with β = 0:466 ± 0:095 near the transition point. It is also found that the exponent β satisfies the scaling relation with the Lipschitz- Hölder exponent α0. Furthermore, we demonstrate that fluctuations in φ at the Anderson transition obey the generalized Gumbel distribution, which is expected to be the universal distribution function of critical fluctuations of order parameters. These results suggest that φ is a suitable quantity of the order parameter of the Anderson transition. © 2012 The Physical Society of Japan. typ typ typ typ typ typ, PHYSICAL SOC JAPAN
Journal of the Physical Society of Japan, Oct. 2012, [Reviewed]

ランダムネットワークの長距離次数相関　　　　　　　　　　　　　　　
水高将吾
電子情報通信学会 企画シンポジウム ネットワーク科学の最前線 〜数理で読み解くネットワーク〜, 27 Mar. 2025, [Invited]

情報社会とネットワーク科学：創造社会に向けて　　　　　　　　　　　　　　　
水高将吾
茨城県情報サービス産業協会 先進技術セミナー, 28 Nov. 2024, [Invited]

Analysis of distortions in voter dynamics by adversarial attacks　　　　　　　　　　　　　　　
Shogo Mizutaka
NetSci2024
20240616, 20240621

Correlated structure in projection of bipartite networks　　　　　　　　　　　　　　　
Shogo Mizutaka
Roles of Heterogeneity in Nonequilibrium Collective Dynamics 2022, [Invited]
20220916, 20220917

Fertility and survivorship in Jomon and Yayoi populations　　　　　　　　　　　　　　　
Yuriko Igarashi; Kunio Shimizu; Shogo Mizutaka
American Association of Physical Anthropologists 89th Annual Meeting
20200415, 20200418

複雑ネットワークの構造的性質〜次数相関、フラクタル性とその連関〜　　　　　　　　　　　　　　　
水高将吾
量子・古典における複雑系の物理と普遍性, [Invited]
20200217

Long-range degree correlations of fractal clusters in random networks　　　　　　　　　　　　　　　
Shogo Mizutaka; Takehisa Hasegawa
The 8th International Conference on Complex Networks and their Application
20191210, 20191212

Intrinsic long-range degree correlation of random nwtwroks near criticality　　　　　　　　　　　　　　　
Shogo Mizutaka
Roles of Heterogeneity in Non-equilibrium collective dynamics, [Invited]
20190715, 20190717

Degree correlations of percolating clusters in random networks　　　　　　　　　　　　　　　
Shogo Mizutaka; Takehisa Hasegawa
StatPhys27
20190706, 20190710

Negative degree correlations of percolating clusters in random networks　　　　　　　　　　　　　　　
Shogo Mizutaka; Takehisa Hasegawa
Critical and collective effects in graphs and networks 2019
20190506, 20190510

Degree correlations of percolating clusters in uncorrelated random networks　　　　　　　　　　　　　　　
Shogo Mizutaka
The 4th Workshop on Self-Organization and Robustness of Evolving Many-Body Systems, [Invited]
20190319

Fractal network formation based on self-organized critical dynamics　　　　　　　　　　　　　　　
Shogo Mizuaka
Workshop on dynamical processes on networks, [Invited]
20180718

Fractal networks formed by self-organized critical dynamics and its universality class　　　　　　　　　　　　　　　
Shogo Mizutaka
NetSci2018
20180613, 20180615

Degree correlations in scale-free fractal networks　　　　　　　　　　　　　　　
Yuka Fujiki; Shogo Mizutaka; Kosuke Yakubo
The 2nd Workshop on Self-Organization and Robustness of Evolving Many-Body Systems
20170908, 20170909

Fractality of Complex Networks Emerging from Self-Organaized Critical Dynamics　　　　　　　　　　　　　　　
Kosuke Yakubo; Akitomo Watanabe; Shogo Mizutaka
International Conference on statistical Physics (SigmaPhi2017)
20170714, 20170717

Disassortative degree mixing and fractality of scale-free networks　　　　　　　　　　　　　　　
Yuka Fujiki; Shogo Mizutaka; Kosuke Yakubo
International Conference on statistical Physics (SigmaPhi2017)
20170710, 20170714

Cascading Failures by Fluctuating Loads in Scale-Free Networks　　　　　　　　　　　　　　　
Kosuke Yakubo; Shogo Mizutaka
NetSci2016
20160530, 20160603

Exact Calculation of Robustness Properties of Correlated Bimodal Networks　　　　　　　　　　　　　　　
Toshihiro Tanizawa; Shogo Mizutaka
NetSci2016
20160530, 20160603

負荷揺らぎに起因したカスケード故障の定式化とネットワークの頑強性　　　　　　　　　　　　　　　
水高将吾
複雑ネットワーク・サマースクール2014, [Invited]
20140818, 20140821

Overload Network Failures: An Approach from the Random-Walk Model　　　　　　　　　　　　　　　
Shogo Mizutaka; Kosuke Yakubo
SITIS 2013
20131202, 20131205

Network robustness to overload failures　　　　　　　　　　　　　　　
Shogo Mizutaka; Kosuke Yakubo
International Workshop on Phase Transition, Critical Phenomena and Related Topics in Complex Networks
20130909, 20130911

Percolation on Scale-free Networks by Overload Failures　　　　　　　　　　　　　　　
Shogo Mizutaka; Kosuke Yakubo
StatPhys 25
20130722, 20130726

Algorithms and Data structures　　　　　　　　　　　　　　　
Apr. 2025 - Present
Ibaraki University

2024 - Present
茨城大学工学部

2024 - Present
茨城大学工学部

2024 - Present
茨城大学工学部

2024 - Present
茨城大学工学部

2023 - Present
茨城大学工学部

2023 - Present
茨城大学工学部

2023 - Present
茨城大学工学部

2023 - Present
茨城大学大学院理工学研究科

2023 - Present
茨城大学大学院理工学研究科

2023 - Present
茨城大学大学院理工学研究科

2023
茨城大学工学部

2020 - 2022
北陸先端科学技術大学院大学

2018 - 2019
茨城大学理学部

2018 - 2019
茨城大学理学部

日本物理学会

ゲルを説明可能な開放系複雑ネットワーク科学の創出　　　　　　　　　　　　　　　
Oct. 2024 - Mar. 2030

スケーリング理論によるフラクタル性を有する複雑ネットワークの理解
基盤研究(C)
茨城大学
01 Apr. 2024 - 31 Mar. 2027

Relation between fractality and degree correlation in complex networks
Grant-in-Aid for Early-Career Scientists
Japan Advanced Institute of Science and Technology
01 Apr. 2021 - 31 Mar. 2024

ネットワーク科学研究会
2023

自己組織化臨界ネットワークの数理構造解明とその応用　　　　　　　　　　　　　　　
Grant-in-Aid for JSPS Fellows
Ibaraki University
25 Apr. 2018 - 31 Mar. 2021

Evaluating and enhancing the robustness of artificial systems based on complex network theory
Grant-in-Aid for Scientific Research (C)
Ibaraki University
Jul. 2018 - Mar. 2021

次数相関を有するネットワークの頑健性とその最適構造の解明　　　　　　　　　　　　　　　
Grant-in-Aid for Early-Career Scientists
Ibaraki University
01 Apr. 2018 - 31 Mar. 2020

ネットワーク科学セミナー
2018

ネットワーク科学セミナー
2017

過負荷故障カスケードに対する複雑ネットワークの頑強性　　　　　　　　　　　　　　　
Grant-in-Aid for JSPS Fellows
Hokkaido University
25 Apr. 2014 - 31 Mar. 2016