The study of the response of complex dynamical social, biological, or technological networks to external perturbations has numerous applications. Random Boolean Networks (RBNs) are commonly used a simple generic model for certain dynamics of complex systems. Traditionally, RBNs are interconnected randomly and without considering any spatial extension and arrangement of the links and nodes. However, most real-world networks are spatially extended and arranged with regular, power-law, small-world, or other non-random connections. Here we explore the RBN network topology between extreme local connections, random small-world, and pure random networks, and study the damage spreading with small perturbations. We find that spatially local connections change the scaling of the relevant component at very low connectivities ($\bar{K} \ll 1$) and that the critical connectivity of stability $K_s$ changes compared to random networks. At higher $\bar{K}$, this scaling remains unchanged. We also show that the relevant component of spatially local networks scales with a power-law as the system size N increases, but with a different exponent for local and small-world networks. The scaling behaviors are obtained by finite-size scaling. We further investigate the wiring cost of the networks. From an engineering perspective, our new findings provide the key design trade-offs between damage spreading (robustness), the network's wiring cost, and the network's communication characteristics.
Deep Dive into Damage Spreading in Spatial and Small-world Random Boolean Networks.
The study of the response of complex dynamical social, biological, or technological networks to external perturbations has numerous applications. Random Boolean Networks (RBNs) are commonly used a simple generic model for certain dynamics of complex systems. Traditionally, RBNs are interconnected randomly and without considering any spatial extension and arrangement of the links and nodes. However, most real-world networks are spatially extended and arranged with regular, power-law, small-world, or other non-random connections. Here we explore the RBN network topology between extreme local connections, random small-world, and pure random networks, and study the damage spreading with small perturbations. We find that spatially local connections change the scaling of the relevant component at very low connectivities ($\bar{K} \ll 1$) and that the critical connectivity of stability $K_s$ changes compared to random networks. At higher $\bar{K}$, this scaling remains unchanged. We also show
The robustness against failures, the wiring cost, and the communication characteristics are key measures of most complex, finite-size real-world networks. For example, the electrical power grid needs to be robust against a variety of failures, minimize the wiring cost, and minimize the transmission losses. Similarly, the neural circuitry in the human brain requires efficient signal transmission and robustness against damage while being constrained in volume.
In this letter, we use random Boolean networks (RBNs) as a simple model to study the (1) robustness, i.e., the damage spreading, (2) the wiring cost, and (3) the communication characteristics as a function of different network topologies (local, small-world, random), different connectivities K, and different network sizes N . More generally speaking, this allows us to answer the question of how much and what type of interconnectivity a complex network-in our case RBNs-needs in order to satisfy given restrictions on the robustness against certain types of failure, the (wiring) cost, and the (communication) efficiency. The work presented here extends previous work by Rohlf et al. [17] to new network topologies, which are more biologically plausible, such as for example smallworld topologies.
RBNs were originally introduced by Kauffman as simplified models of gene regulation networks [6,7]. In its simplest form, an RBN is discrete dynamical system, also called N K network (or model), composed of N automata (or nodes), each of which receives inputs from K (either exact or average) randomly chosen other automata. Each automaton is a Boolean variable with two possible states: {0, 1}, and the dynamics is such that
where F = (f 1 , …, f i , …, f N ), and each f i is represented by a look-up table of K i inputs randomly chosen from the set of N automata. Initially, K i neighbors and a look-table are assigned to each automaton at random.
An automaton state σ t i ∈ {0, 1} is updated using its corresponding Boolean function:
We randomly initialize the states of the automata (initial condition of the RBN). The automata are updated synchronously using their corresponding Boolean functions.
In the thermodynamic limit, RBNs exhibit a dynamical order-disorder transition at a sparse critical connectivity K c [4]. For a finite system size N , the dynamics of RBNs converge to periodic attractors after a finite number of updates. At K c , the phase space structure in terms of attractor periods [1], the number of different attractors [20] and the distribution of basins of attraction [2] is complex, showing many properties reminiscent of biological networks [7].
The study of the response of complex dynamical networks to external perturbations, also referred to as damage, has numerous applications, e.g., the spreading of disease through a population [11,14], the spreading of a computer virus on the internet [3], failure propagation in power grids [18], the perturbation of gene expression patterns in a cell due to mutations [16], or the intermittent stationary state in economic decision networks triggered by the mutation of strategy from a few individual agents [13]. Mean-field approaches, e.g., the annealed approximation (AA) introduced by Derrida and Pomeau [4], allow for an analytical treatment of damage spreading and exact determination of the critical connectivity K c under various constraints [9,21]. However, these approximations rely on the assumption that N → ∞, which, for an application to real-world problems, is often an irrelevant limit. A number of studies [8,10] has recently focused on the finite-size scaling of (un-)frozen and/or relevant nodes in RBN with respect to N with the goal to go beyond the annealed approximation. Only a few studies, however, consider finite-size scaling of damage spreading in RBNs [16,17,19]. Of particular interest is the “sparse percolation (SP) limit” [19], where the initial perturbation size d(0) does not scale up with the network size N , i.e., the relative size of perturbations tends to zero for large N . In [17] Rohlf et al. have identified a new characteristic connectivity K s for RBNs, at which the average number of damaged nodes d, after a large number of dynamical updates, is independent of N . This limit is particularly relevant to information and damage propagation in many technological and natural networks. The work in this letter extends these new findings and systematically studies damage spreading in RBNs as a function of new network topologies, namely local and small-world, different connectivities K, and different network sizes N .
For our purpose, we measure the expected damage d as the Hamming distance between two different initial system configurations after a large number of T system updates. The randomly chosen initial conditions differ by one bit, i.e., the damage size is 1. As introduced in [17], let N be a randomly sampled set (ensemble) of z N networks with average degree K, I n a set of z I random
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