Directed network modules

A search technique locating network modules, i.e., internally densely connected groups of nodes in directed networks is introduced by extending the Clique Percolation Method originally proposed for undirected networks. After giving a suitable definition for directed modules we investigate their percolation transition in the Erdos-Renyi graph both analytically and numerically. We also analyse four real-world directed networks, including Google's own webpages, an email network, a word association graph and the transcriptional regulatory network of the yeast Saccharomyces cerevisiae. The obtained directed modules are validated by additional information available for the nodes. We find that directed modules of real-world graphs inherently overlap and the investigated networks can be classified into two major groups in terms of the overlaps between the modules. Accordingly, in the word-association network and among Google's webpages the overlaps are likely to contain in-hubs, whereas the modules in the email and transcriptional regulatory networks tend to overlap via out-hubs.Comment: 21 pages, 10 figures, version 2: added two paragaph

Kauffman Boolean model in undirected scale free networks

We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including scale-free topology of connections $P(k)\sim k^{-\gamma}$. We show that in infinite scale-free networks the transition between frozen and chaotic phase occurs for $3<\gamma < 3.5$. The observation is interesting for two reasons. First, since most of critical phenomena in scale-free networks reveal their non-trivial character for $\gamma<3$, the position of the critical line in Kauffman model seems to be an important exception from the rule. Second, since gene regulatory networks are characterized by scale-free topology with $\gamma<3$, the observation that in finite-size networks the mentioned transition moves towards smaller $\gamma$ is an argument for Kauffman model as a good starting point to model real systems. We also explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena

Computational core and fixed-point organisation in Boolean networks

In this paper, we analyse large random Boolean networks in terms of a constraint satisfaction problem. We first develop an algorithmic scheme which allows to prune simple logical cascades and under-determined variables, returning thereby the computational core of the network. Second we apply the cavity method to analyse number and organisation of fixed points. We find in particular a phase transition between an easy and a complex regulatory phase, the latter one being characterised by the existence of an exponential number of macroscopically separated fixed-point clusters. The different techniques developed are reinterpreted as algorithms for the analysis of single Boolean networks, and they are applied to analysis and in silico experiments on the gene-regulatory networks of baker's yeast (saccaromices cerevisiae) and the segment-polarity genes of the fruit-fly drosophila melanogaster.Comment: 29 pages, 18 figures, version accepted for publication in JSTA

Boolean Dynamics with Random Couplings

This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical Sciences Serie

Dynamical Instability in Boolean Networks as a Percolation Problem

Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially. We show and numerically verify that this phase transition in the dynamics can be mapped onto a static percolation problem which predicts the long-time average Hamming distance between perturbed and unperturbed orbits

Random model for RNA interference yields scale free network

We introduce a random bit-string model of post-transcriptional genetic regulation based on sequence matching. The model spontaneously yields a scale free network with power law scaling with $\gamma=-1$ and also exhibits log-periodic behaviour. The in-degree distribution is much narrower, and exhibits a pronounced peak followed by a Gaussian distribution. The network is of the smallest world type, with the average minimum path length independent of the size of the network, as long as the network consists of one giant cluster. The percolation threshold depends on the system size.Comment: 9 pages, 13 figures, submitted to Midterm Conference COSIN on ``Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems'', Rome, 1-5 September 200

Damage Spreading and Criticality in Finite Random Dynamical Networks

We systematically study and compare damage spreading at the sparse percolation (SP) limit for random boolean and threshold networks with perturbations that are independent of the network size $N$. This limit is relevant to information and damage propagation in many technological and natural networks. Using finite size scaling, we identify a new characteristic connectivity $K_s$, at which the average number of damaged nodes $\bar d$, after a large number of dynamical updates, is independent of $N$. Based on marginal damage spreading, we determine the critical connectivity $K_c^{sparse}(N)$ for finite $N$ at the SP limit and show that it systematically deviates from $K_c$, established by the annealed approximation, even for large system sizes. Our findings can potentially explain the results recently obtained for gene regulatory networks and have important implications for the evolution of dynamical networks that solve specific computational or functional tasks.Comment: 4 pages, 4 eps figure

When correlations matter - response of dynamical networks to small perturbations

We systematically study and compare damage spreading for random Boolean and threshold networks under small external perturbations (damage), a problem which is relevant to many biological networks. We identify a new characteristic connectivity $K_s$, at which the average number of damaged nodes after a large number of dynamical updates is independent of the total number of nodes $N$. We estimate the critical connectivity for finite $N$ and show that it systematically deviates from the annealed approximation. Extending the approach followed in a previous study, we present new results indicating that internal dynamical correlations tend to increase not only the probability for small, but also for very large damage events, leading to a broad, fat-tailed distribution of damage sizes. These findings indicate that the descriptive and predictive value of averaged order parameters for finite size networks - even for biologically highly relevant sizes up to several thousand nodes - is limited.Comment: 4 pages, 4 figures. Accepted for the "Workshop on Computational Systems Biology", Leipzig 200