The phase diagram of random threshold networks

Threshold networks are used as models for neural or gene regulatory networks. They show a rich dynamical behaviour with a transition between a frozen and a chaotic phase. We investigate the phase diagram of randomly connected threshold networks with real-valued thresholds h and a fixed number of inputs per node. The nodes are updated according to the same rules as in a model of the cell-cycle network of Saccharomyces cereviseae [PNAS 101, 4781 (2004)]. Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the "on" and "off" state, and for the sensitivity $\lambda$. The results are compared with simulations of quenched networks. We find that for integer values of h the simulations show marked deviations from the annealed approximation even for large networks. This can be attributed to the particular choice of the updating rule.Comment: 8 pages, 6 figure

Inter-arrival times of message propagation on directed networks

One of the challenges in fighting cybercrime is to understand the dynamics of message propagation on botnets, networks of infected computers used to send viruses, unsolicited commercial emails (SPAM) or denial of service attacks. We map this problem to the propagation of multiple random walkers on directed networks and we evaluate the inter-arrival time distribution between successive walkers arriving at a target. We show that the temporal organization of this process, which models information propagation on unstructured peer to peer networks, has the same features as SPAM arriving to a single user. We study the behavior of the message inter-arrival time distribution on three different network topologies using two different rules for sending messages. In all networks the propagation is not a pure Poisson process. It shows universal features on Poissonian networks and a more complex behavior on scale free networks. Results open the possibility to indirectly learn about the process of sending messages on networks with unknown topologies, by studying inter-arrival times at any node of the network.Comment: 9 pages, 12 figure

Evolution of a population of random Boolean networks

87.23.Kg Dynamics of evolution, 89.75.Hc Networks and genealogical trees, 87.15.A- Theory, modeling, and computer simulation,

Robust Ordering of Anaphase Events by Adaptive Thresholds and Competing Degradation Pathways

The splitting of chromosomes in anaphase and their delivery into the daughter cells needs to be accurately executed to maintain genome stability. Chromosome splitting requires the degradation of securin, whereas the distribution of the chromosomes into the daughter cells requires the degradation of cyclin B. We show that cells encounter and tolerate variations in the abundance of securin or cyclin B. This makes the concurrent onset of securin and cyclin B degradation insufficient to guarantee that early anaphase events occur in the correct order. We uncover that the timing of chromosome splitting is not determined by reaching a fixed securin level, but that this level adapts to the securin degradation kinetics. In conjunction with securin and cyclin B competing for degradation during anaphase, this provides robustness to the temporal order of anaphase events. Our work reveals how parallel cell-cycle pathways can be temporally coordinated despite variability in protein concentrations

Comparing the evolution of canalyzing and threshold networks

We study the influence of the type of update functions on the evolution of Boolean networks under selection for dynamical robustness. The chosen types of functions are canalyzing functions and threshold functions. Starting from a random initial network, we evolve the network by an adaptive walk. During the first time period, where the networks evolve to the plateau of 100 percent fitness, we find that both type of update functions give the same behavior, albeit for different network sizes and connectedness. However, on the long run, as the networks continue to evolve on the fitness plateau, the different types of update functions give rise to different network structure, due to their different mutational robustness. When both types of update functions occur together, none of them is preferred under long-term evolution. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011

Evolving critical boolean networks

Random Boolean networks are a widely acknowledged model for cell dynamics. Previous studies have shown the possibility of achieving Boolean networks with given characteristics by means of evolutionary techniques. In this work we make a further step towards more biologically plausible models by aiming at evolving networks with a given fraction of active nodes along the attractors, while constraining the evolutionary process to move across critical networks. Results show that this path along criticality does not impede to climb the mount of improbable, yet biologically realistic requirements