We demonstrate that the susceptible-infected-susceptible (SIS) model on complex networks can have an inactive Griffiths phase characterized by a slow relaxation dynamics. It contrasts with the mean-field theoretical prediction that the SIS model on complex networks is active at any nonzero infection rate. The dynamic fluctuation of infected nodes, ignored in the mean field approach, is responsible for the inactive phase. It is proposed that the question whether the epidemic threshold of the SIS model on complex networks is zero or not can be resolved by the percolation threshold in a model where nodes are occupied in degree-descending order. Our arguments are supported by the numerical studies on scale-free network models.

A new cellular automaton traffic model is presented. The focus is on mechanical restrictions of vehicles realized by limited acceleration and deceleration capabilities. These features are incorporated into the model in order to construct the condition of collision-free movement. The strict collision-free criterion imposed by the mechanical restrictions is softened in certain traffic situations, reflecting human overreaction. It is shown that the present model reliably reproduces most empirical findings including synchronized flow, the so-called pinch effect, and the time-headway distribution of free flow. The findings suggest that many free flow phenomena can be attributed to the platoon formation of vehicles (platoon effect).

The explosive percolation problem on the complete graph is investigated via extensive numerical simulations. We obtain the cluster-size distribution at the moment when the cluster size heterogeneity becomes maximum. The distribution is found to be well described by the power-law form with the decay exponent τ = 2.06(2), followed by a hump. We then use the finite-size scaling method to make all the distributions at various system sizes up to N = 2 37 collapse perfectly onto a scaling curve characterized solely by the single exponent τ . We also observe that the instant of that collapse converges to a well-defined percolation threshold from below as N → ∞. Based on these observations, we show that the explosive percolation transition in the model should be continuous, contrary to the widely-spread belief of its discontinuity.PACS numbers: 64.60.ah, 64.60.aq, 36.40.Ei The term explosive percolation was proposed in Ref.[1] to describe a sudden appearance of a macroscopic cluster in a network growth model with the so-called product rule considered on the complete graph. This growth rule, named as the Achlioptas process (AP), is then studied on the two-dimensional lattice [2,3] and on the scalefree networks [4][5][6] as well, yielding similar results. That suddenness has been widely believed to indicate a discontinuity at the percolation transition in the thermodynamic limit [7,8], and the similar explosiveness has been observed with the other growth rules proposed later [9][10][11][12][13]. These observations of the explosiveness have drawn much interest due to the striking difference from the wellknown continuous transition in the standard percolation models [14]. However, in our point of view, the explosiveness has not been carefully investigated as yet enough to draw a decisive conclusion on the discontinuity, and possibly just represents an extremely steep but still continuous transition.Friedman and Landsberg [9] have suggested the argument of the powder keg as a circumstantial description to explain the apparent discontinuity of the explosive percolation transition. Meanwhile, da Costa et al. [15] have reported that the explosive percolation is actually continuous for a modified version of the AP by analytically deriving the critical scaling relations based on numerical observations of power-law critical distribution of cluster size [16]. In this Letter, we try to unmask the (dis)continuity in a systematic and direct way by performing a careful finite-size-scaling analysis at newly introduced pseudo-transition points for finite systems and show that the explosive percolation transition on the complete graph is indeed continuous in the thermodynamic limit.The model we study is the AP with the product rule on the complete graph [1]. Start with N nodes with all links unoccupied. At each step, choose two possible unoccupied links randomly between nodes. Then, select the link merging two clusters with a smaller product of the two cluster sizes. Here, a cluster is defined as a set of nodes connected each other via...

We show that the total entropy production in stochastic processes with odd-parity variables (under time reversal) is separated into three parts, only two of which satisfy the integral fluctuation theorems in general. One is the usual excess entropy production, which can appear only transiently and is called nonadiabatic. Another one is attributed solely to the breakage of detailed balance. The last part not satisfying the fluctuation theorem comes from the steady-state distribution asymmetry for odd-parity variables, which is activated in a non-transient manner. The latter two contributions combine together as the house-keeping (adiabatic) entropy production, whose positivity is not guaranteed except when the excess entropy production completely vanishes. where P r is the probability of a sequence r. As a corollary, the Jensen's inequality guarantees R ≥ 0. Consider r as a path or trajectory in state space, generated during a time interval by a stochastic dynamics. In case when its functional R [7] represents the total entropy production during the process, the FT has been derived for various nonequilibrium(NEQ) processes, and the thermodynamic 2nd law ∆S tot ≥ 0 automatically follows [3,4,8].More recently, Hatano and Sasa found that a part of the total entropy production (excess entropy), ∆S ex , also satisfies the FT, which represents the entropy production associated with transitions between steady states [9,10]. Later, Speck and Seifert showed that the remaining part (house-keeping entropy), ∆S hk , also satisfies the FT, which is required to maintain the NEQ steady state (NESS) [11,12]. In case of (quasi-static) reversible processes, the system stays at equilibrium almost always during the process, then the house-keeping entropy production vanishes, ∆S eq hk = 0. Most recently, Esposito et. al.[6] interpreted the house-keeping entropy as an adiabatic part and the excess entropy as a nonadiabatic part of the total entropy production, through a time-scale argument.Most of findings about the FTs so far hold only when all state variables have even parity under time reversal, such as position variables. A typical example is the driven Brownian motion in the over-damped limit. Including odd-parity variables, such as momentum, the mathematical description becomes more complicated in particular for NEQ processes. Recently, Spinney and Ford suggested a separation of the total entropy production into three terms for the stochastic system with odd-parity variables [13]. The excess entropy production can be cleanly separated out (in fact, exactly the same as in the case with even-parity variables only) and it satisfies the FT. However, the house-keeping part composes of two different terms and only one term satisfies the FT. Especially, the other term not satisfying the FT turns out to be transient, which seems inconsistent with the usual adiabatic feature of the house-keeping entropy. Thus, it was concluded that the physical interpretation of separated entropies is not as clear as in the even-variable only case (adiab...