Many sociological networks, as well as biological and technological ones, can be represented in terms of complex networks with a heterogeneous connectivity pattern. Dynamical processes taking place on top of them can be very much influenced by this topological fact. In this paper we consider a paradigmatic model of non-equilibrium dynamics, namely the forest fire model, whose relevance lies in its capacity to represent several epidemic processes in a general parametrization. We study the behavior of this model in complex networks by developing the corresponding heterogeneous mean-field theory and solving it in its steady state. We provide exact and approximate expressions for homogeneous networks and several instances of heterogeneous networks. A comparison of our analytical results with extensive numerical simulations allows to draw the region of the parameter space in which heterogeneous mean-field theory provides an accurate description of the dynamics, and enlights the limits of validity of the mean-field theory in situations where dynamical correlations become important.
一维Bak-Chen-Tang森林火灾模型中的火以孤立子的形式传播，类似于Burgers湍流中的激波。孤立子的分支，创造新的火灾，是平衡的成对湮灭相反移动的孤立子。在树木生长速率$p$为零的极限处，出现了两个不同的、发散的长度尺度。孤子的宽度$w$以幂律$1/p$发散，而孤子之间的平均距离以$ d \sim \exp(π^2/12p)$发散得更快。
研究了一个以空场密度为控制参数的森林火灾模型。该模型表现出三个阶段，由一个一阶相变和一个“混合”相变，其中显示的临界行为，只有一侧和滞后分离。临界行为是自组织临界林火模型的临界行为[B. Drossel和F. Schwabl，Phys. Rev. Lett. 69，1629（1992）]，而在相邻的阶段，人们发现的螺旋波的Bak等森林火灾模型[P. Bak，K. Chen和C. Tang，Phys. Lett. A 147，297（1990）]。在第三阶段，人们观察到树木的集群，火在集群的边缘燃烧。解释了螺旋态密度分布与逾渗阈值之间的关系，并讨论了任意可激发系统中螺旋波定态的意义。此外，我们评论的可能性映射自组织的关键系统到“普通”的关键系统。
Vermont Complex Systems Institute and Department of Computer Science, University of Vermont（佛蒙特大学复杂系统研究所与计算机科学系）；Complexity Science Hub（复杂性科学中心）；Santa Fe Institute（圣塔菲研究所）
