stability of the kauffman model random boolean networks the kauffman model are revisited by means of a novel decimation algorithm which removes variables that cannot be relevant to the asymptotic dynamics of the system the major part of the removed variables have the same fixed state in all limit cycles these variables are denoted as the stable core of the network and their number grows approximately linearly with n the number of variables in the original network the sensitivity of the attractors to perturbations is investigated we find that reduced networks lack the well known insensitivity observed in full kauffman networks we conclude that somewhat counterintuitive this remarkable property of full kauffman networks is generated by the dynamics of their stable core the decimation method is also used to simulate large critical kauffman networks for networks up to n 32 we perform full enumeration studies strong evidence is provided that the number of limit cycles grows linearly with n this result is in sharp contrast to the often cited square root of n behavior

stability of boolean networks the joint effects of topology and update rules we study the stability of orbits in large boolean networks we treat the case in which the network has a given complex topology and we do not assume a specific form for the update rules which may be correlated with local topological properties of the network while recent past work has addressed the separate effects of complex network topology and certain classes of update rules on stability only crude results exist about how these effects interact we present a widely applicable solution to this problem numerical simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems

the effect of network topology on the stability of discrete state models of genetic control boolean networks have been proposed as potentially useful models for genetic control an important aspect of these networks is the stability of their dynamics in response to small perturbations previous approaches to stability have assumed uncorrelated random network structure real gene networks typically have nontrivial topology significantly different from the random network paradigm to address such situations we present a general method for determining the stability of large boolean networks of any specified network topology and predicting their steady state behavior in response to small perturbations additionally we generalize to the case where individual genes have a distribution of expression biases and we consider a nonsynchronous update as well as extension of our method to non boolean models in which there are more than two possible gene states we find that stability is governed by the maximum eigenvalue of a modified adjacency matrix and we test this result by comparison with numerical simulations we also discuss the possible application of our work to experimentally inferred gene networks

number and length of attractors in a critical kauffman model with connectivity one the kauffman model describes a system of randomly connected nodes with dynamics based on boolean update functions though it is a simple model it exhibits very complex behavior for critical parameter values at the boundary between a frozen and a disordered phase and is therefore used for studies of real network problems we prove here that the mean number and mean length of attractors in critical random boolean networks with connectivity one both increase faster than any power law with network size we derive these results by generating the networks through a growth process and by calculating lower bounds

a formulation for the 4 node quadrilateral element a formulation for the plane 4 node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body incompatible modes are added to the standard displacement field then expressions for gradient operators are obtained from an expansion of the basis functions into a second order taylor series in the physical co ordinates the internal degrees of freedom of the incompatible modes are eliminated on the element level a modified change of variables is used to integrate the element matrices for a linear elastic material the element stiffness matrix can be separated into two parts these are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix the formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials further the approach is suitable for non linear analysis there an application is given for the calculation of inelastic problems in physically non linear elasticity the element is efficient to implement and it is frame invariant locking effects and zero energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion a high accuracy is obtained for numerical solutions in displacements and stresses

exhaustive percolation on random networks we consider propagation models that describe the spreading of an attribute called damage through the nodes of a random network in some systems the average fraction of nodes that remain undamaged vanishes in the large system limit a phenomenon we refer to as exhaustive percolation we derive scaling law exponents and exact results for the distribution of the number of undamaged nodes valid for a broad class of random networks at the exhaustive percolation transition and in the exhaustive percolation regime this class includes processes that determine the set of frozen nodes in random boolean networks with indegree distributions that decay sufficiently rapidly with the number of inputs connections between our calculational methods and previous studies of percolation beginning from a single initial node are also pointed out central to our approach is the observation that key aspects of damage spreading on a random network are fully characterized by a single function specifying the probability that a given node will be damaged as a function of the fraction of damaged nodes in addition to our analytical investigations of random networks we present a numerical example of exhaustive percolation on a directed lattice less

relevant elements magnetization and dynamical properties in kauffman networks a numerical study abstract this is the first of two papers about the structure of kauffman networks in this paper we define the relevant elements of random networks of automata following previous work by flyvbjerg j phys a 21 1988 l955 l960 and flyvbjerg and kjaer j phys a 21 1988 1695 1718 and we study numerically their probability distributions in the chaotic phase and on the critical line of the model a simple approximate argument predicts that their number scales as n on the critical line while it is linear with n in the chaotic phase and independent on system size in the frozen phase this argument is confirmed by numerical results the study of the relevant elements gives useful information about the properties of the attractors in critical networks where the pictures coming from either approximate computation methods or from simulations are not very clear

genetic networks with canalyzing boolean rules are always stable we determine stability and attractor properties of random boolean genetic network models with canalyzing rules for a variety of architectures for all power law exponential and flat in degree distributions we find that the networks are dynamically stable furthermore for architectures with few inputs per node the dynamics of the networks is close to critical in addition the fraction of genes that are active decreases with the number of inputs per node these results are based upon investigating ensembles of networks using analytical methods also for different in degree distributions the numbers of fixed points and cycles are calculated with results intuitively consistent with stability analysis fewer inputs per node implies more cycles and vice versa there are hints that genetic networks acquire broader degree distributions with evolution and hence our results indicate that for single cells the dynamics should become more stable with evolution however such an effect is very likely compensated for by multicellular dynamics because one expects less stability when interactions among cells are included we verify this by simulations of a simple model for interactions among cells

phase transition for the threshold contact process an approximation of heterogeneous random boolean networks we consider a model for heterogeneous gene regulatory networks that is a generalization of the model proposed by chatterjee and durrett random struct algorithms 39 2 228 246 2011 as an annealed approximation of kauffmann s j theor biol 22 3 437 467 1969 random boolean networks in this model genes are represented by the vertices of a random directed graph mathscr g n on n vertices having specified in degree distribution mathbf p text in resp joint distribution mathbf p text in out of in degree and out degree and the expression bias the expected fraction of 1 s in the boolean functions p is same for all vertices following chatterjee and durrett random struct algorithms 39 2 228 246 2011 and a standard practice in the physics literature we use a discrete time threshold contact process with parameter q 2p 1 p in which vertices are either occupied or vacant a vertex with at least one resp no occupied input at time t will be occupied at time t 1 with probability q resp 0 on mathscr g n to approximate the dynamics of the boolean network we find a parameter r which has an explicit expression in terms of certain moments of mathbf p text in resp mathbf p text in out such that with probability tending to 1 as n goes to infinity if r cdot 2p 1 p 1 then starting from all occupied sites the threshold contact process on mathscr g n maintains a positive quasi stationary density pi mathbf p text in resp pi mathbf p text in out of occupied sites till time which is exponential in n whereas if r cdot 2p 1 p 1 then the persistence time of the threshold contact process on mathscr g n is at most logarithmic in n these two phases correspond to the chaotic and ordered behavior of the gene networks respectively

broad edge of chaos in strongly heterogeneous boolean networks the dynamic stability of the boolean networks representing a model for the gene transcriptional regulation kauffman model is studied by calculating analytically and numerically the hamming distance between two evolving configurations this turns out to behave in a universal way close to the phase boundary only for in degree distributions with a finite second moment in degree distributions of the form pd k k with 2 3 thus having a diverging second moment lead to a slower increase of the hamming distance when moving towards the unstable phase and to a broadening of the phase boundary for finite n with decreasing we conclude that the heterogeneous regulatory network connectivity facilitates the balancing between robustness and evolvability in living organisms

a stabilization procedure for the quadrilateral plate element with one point quadrature a stabilization procedure is developed for controlling the kinematic modes of the four node bilinear quadrilateral element when single point quadrature is used these kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness in this stabilization procedure additional generalized strains are defined which are activated by the kinematic modes these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral by using a scaling law developed in an earlier paper the stabilization parameters are defined so they do not adversely affect the element s performance several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

on the incompressible constraint of the 4 node quadrilateral element the standard plane 4 node element is written as the summation of a constant gradient matrix usually obtained from underintegration and a stabilization matrix the split is based on a taylor series expansion of element basis functions in the incompressible limit the locking effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher order term in the taylor series the incompressibility condition is formulated in a weak sense so that the element displacement field is divergence free when integrated over the element volume the resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first order terms of the taylor series the corresponding eigenvalue enforces incompressibility implicitly by means of a penalty constraint analytical expressions for that constant dilatation eigenpair are derived for arbitrary element geometries it is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner for several classical and recent elements the eigensystems are analysed numerically it is shown that most of the formulations reflect the incompressible constraint identically in the incompressible limit the numerical accuracies of the elements are compared

finite size scaling in complex networks a finite size scaling fss theory is proposed for various models in complex networks in particular we focus on the fss exponent which plays a crucial role in analyzing numerical data for finite size systems based on the droplet excitation hyperscaling argument we conjecture the values of the fss exponents for the ising model the susceptible infected susceptible model and the contact process all of which are confirmed reasonably well in numerical simulations

mean field and non mean field behaviors in scale free networks with random boolean dynamics we study two types of simplified boolean dynamics in scale free networks both with a synchronous update assigning only the boolean functions and and xor to the nodes with probabilities 1 p and p respectively we are able to analyze the density of 1 s and the hamming distance on the network by numerical simulations and by a mean field approximation annealed approximation we show that the behavior is quite different if the node always enters in the dynamics as its own input self regulation or not the same conclusion holds for the kauffman nk model moreover the simulation results and the mean field ones i agree well when there is no self regulation and ii disagree for small p when self regulation is present in the model

efficient linear and nonlinear heat conduction with a quadrilateral element a method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one point quadrature the stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter it is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values in fact the values 3 2 and 1 for the normalized stabilization parameter lead to the 5 point finite difference 9 point finite difference and fully integrated finite element operators respectively for rectangular meshes numerical experiments reported here show that the three have identical rates of convergence in the l2 norm eigenvalues of the element matrices which are needed for stability limits are also given numerical applications are used to show that the method yields accurate solutions with large increases in efficiency particularly in nonlinear problems

stable and unstable attractors in boolean networks boolean networks at the critical point have been a matter of debate for many years as e g the scaling of numbers of attractors with system size recently it was found that this number scales superpolynomially with system size contrary to a common earlier expectation of sublinear scaling we point out here that these results are obtained using deterministic parallel update where a large fraction of attractors are an artifact of the updating scheme this limits the significance of these results for biological systems where noise is omnipresent here we take a fresh look at attractors in boolean networks with the original motivation of simplified models for biological systems in mind we test the stability of attractors with respect to infinitesimal deviations from synchronous update and find that most attractors are artifacts arising from synchronous clocking the remaining fraction of attractors are stable against fluctuating delays the average number of these stable attractors grows sublinearly with system size in the numerically tractable range

multiple quadrature underintegrated finite elements new multiple quadrature point underintegrated finite elements with hourglass control are developed the elements are selectively underintegrated to avoid volumetric and shear locking and save computational time an approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co ordinates so that the elements require no stabilization parameter to improve accuracy over the traditional one point quadrature elements several quadrature points are used to integrate the internal forces especially for tracing the plastic fronts in the mesh during loading and unloading in elastic plastic analysis two and four point quadrature elements are proposed for use in the two and three dimensional elements respectively other multiple quadrature points can also be employed several numerical examples such as thin beam plate and shell problems are presented to demonstrate the applicability of the proposed elements

scaling in a general class of critical random boolean networks we derive analytically the scaling behavior in the thermodynamic limit of the number of nonfrozen and relevant nodes in the most general class of critical kauffman networks for any number of inputs per node and for any choice of the probability distribution for the boolean functions by defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes in any critical network with more than one input per node scales with the network size n as n2 3 with only n1 3 nonfrozen nodes having two nonfrozen inputs and the number of nonfrozen nodes with more than two inputs being finite in the thermodynamic limit using these results we can conclude that the mean number of relevant nodes increases for large n as n1 3 with only a finite number of relevant nodes having two relevant inputs and a vanishing fraction of nodes having more than three of them it follows that all relevant components apart from a finite number are simple loops and that the mean number and length of attractors increases faster than any power law with network size

scaling laws in critical random boolean networks with general in and out degree distributions we evaluate analytically and numerically the size of the frozen core and various scaling laws for critical boolean networks that have a power law in and or out degree distribution to this purpose we generalize an efficient method that has previously been used for conventional random boolean networks and for networks with power law in degree distributions with this generalization we can also deal with power law out degree distributions when the power law exponent is between 2 and 3 the second moment of the distribution diverges with network size and the scaling exponent of the nonfrozen nodes depends on the degree distribution exponent furthermore the exponent depends also on the dependence of the cutoff of the degree distribution on the system size altogether we obtain an impressive number of different scaling laws depending on the type of cutoff as well as on the exponents of the in and out degree distributions we confirm our scaling arguments and analytical considerations by numerical investigations

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

kirillov models and integral structures in p adic smooth representations of gl2 f abstract let f be a local field of residual characteristic p and a smooth irreducible representation of gl 2 f realized over the algebraic closure of q p studying its kirillov model we exhibit a necessary and sufficient criterion for the existence of an integral structure in we apply our criterion to tamely ramified principal series and get a new proof of a theorem of m f vigneras

percolation transition in networks with degree degree correlation we introduce an exponential random graph model for networks with a fixed degree distribution and a tunable degree degree correlation we then investigate the nature of the percolation transition in a correlated network with a poisson degree distribution it is found that negative correlation is irrelevant in that the percolation transition in the disassortative network belongs to the same universality class as in the uncorrelated network positive correlation turns out to be relevant the percolation transition in the assortative network is characterized by the nondiverging mean size of finite clusters and power law scalings of the density of the largest cluster and the cluster size distribution in the nonpercolating phase as well as at the critical point our results suggest that the unusual type of percolation transition in the growing network models reported recently may be inherited from the assortative degree degree correlation

the modular structure of kauffman networks this is the second paper of a series of two about the structural properties that influence the asymptotic dynamics of random boolean networks here we study the functionally independent clusters in which the relevant elements introduced and studied in our first paper 3 are subdivided we show that the phase transition in random boolean networks can also be described as a percolation transition the statistical properties of the clusters of relevant elements that we call modules give an insight on the scaling behavior of the attractors of the critical networks that according to kauffman have a biological analogy as a model of genetic regulatory systems

the stability of boolean network with transmission sensitivity abstract boolean network has been widely used in modeling biological systems and one of the key problems is its stability in response to small perturbations based on the hypothesis that the states of all nodes are homogenously updated great progress has been made in previous works in real biological networks however the updates of genes typically show much heterogeneity to address such conditions we introduce transmission sensitivity into boolean network model by the method of semi annealed approximation we illustrate that in a homogenous network the critical condition of stability has no connection with its transmission sensitivity as for heterogeneous networks it reveals that correlations between network topology and transmission sensitivity can have profound effects on the its stability this result shows a new mechanism that affects the stability of boolean network which could be used to control the dynamics in real biological systems

self organized critical neural networks a mechanism for self organization of the degree of connectivity in model neural networks is studied network connectivity is regulated locally on the basis of an order parameter of the global dynamics which is estimated from an observable at the single synapse level this principle is studied in a two dimensional neural network with randomly wired asymmetric weights in this class of networks network connectivity is closely related to a phase transition between ordered and disordered dynamics a slow topology change is imposed on the network through a local rewiring rule motivated by activity dependent synaptic development neighbor neurons whose activity is correlated on average develop a new connection while uncorrelated neighbors tend to disconnect as a result robust self organization of the network towards the order disorder transition occurs convergence is independent of initial conditions robust against thermal noise and does not require fine tuning of parameters

emergent criticality from coevolution in random boolean networks the coevolution of network topology and dynamics is studied in an evolutionary boolean network model that is a simple model of a gene regulatory network we find that a critical state emerges spontaneously resulting from the interplay between topology and dynamics during the evolution the final evolved state is shown to be independent of initial conditions the network appears to be driven to a random boolean network with uniform in degree of 2 in the large network limit however for biologically realized network sizes significant finite size effects are observed including a broad in degree distribution and an average in degree connections between 2 and 3 these results may be important for explaining the properties of gene regulatory networks

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

synchronization transition in scale free networks clusters of synchrony we study the synchronization transition in scale free networks that display power law asymptotic behaviors in their degree distributions the critical coupling strength and the order parameter critical exponent derived by the mean field approach depend on the degree exponent ensuremath lambda which implies a close connection between structural organization and the emergence of dynamical order in complex systems we also derive the finite size scaling behavior of the order parameter finding that the giant cluster of synchronized nodes is formed in different ways between scale free networks with 2l ensuremath lambda l3 and those with ensuremath lambda g3

first and second order phase transitions in scale free networks we study first and second order phase transitions of ferromagnetic lattice models on scale free networks with a degree exponent gamma using the example of the q state potts model we derive a general self consistency relation within the frame of the weiss molecular field approximation which presumably leads to exact critical singularities depending on the value of gamma we have found three different regimes of the phase diagram as a general trend first order transitions soften with decreasing gamma and the critical singularities at the second order transitions are gamma dependent

effects of alternative connectivity on behavior of randomly constructed boolean networks random boolean networks rbns were first introduced and used to model regulatory gene networks by kauffman in 1969 and since then have been frequently used to model networks at various levels of biological organization the standard model has a characteristic topology in which the network has n genes with an average of k input connections selected at random we have used rbns to examine regulatory gene networks with four different topologies which are characterized by different rank distributions of output connections that vary from uniform to highly skewed among these is the topology of the standard model which is included for purposes of comparison and a topology with a power law rank distribution which is based on recent data for the regulatory gene network of the bacterium escherichia coli we also examine effects of bias in the distribution of boolean functions for the network the dynamical properties and mutual information of these networks depend not only on their size but also on their topology and boolean functions networks with the more uniform rank distributions exhibit longer lengths of attractors larger numbers of attractors and less mutual information networks with the more skewed rank distributions have complementary properties when viewed as biological decision making networks those with either the most uniform or the most skewed rank distributions have disadvantageous properties the intermediate rank distribution exhibited by the regulatory gene network of e coli avoids these disadvantages

scaling in critical random boolean networks we derive mostly analytically the scaling behavior of the number of nonfrozen and relevant nodes in critical kauffman networks with two inputs per node in the thermodynamic limit by defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes scales with the network size n as n 2 3 with only n 1 3 nonfrozen nodes having two nonfrozen inputs we also show the probability distributions for the numbers of these nodes using a different stochastic process we determine the scaling behavior of the number of relevant nodes their mean number increases for large n as n 1 3 and only a finite number of relevant nodes have two relevant inputs it follows that all relevant components apart from a finite number are simple loops and that the mean number and length of attractors increases faster than any power law with network size