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The Effect of Negative Feedback Loops on the Dynamics of Boolean Networks
, 2008
"... Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distancetopositivefeedback which, in essence, captures the number of independent negative feedback loops in the network, a ..."
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Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distancetopositivefeedback which, in essence, captures the number of independent negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks, it is shown that distancetopositivefeedback has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same number of nodes and same connectivity.
On Parallel vs. Sequential Threshold Cellular Automata
 IN PROCEEDINGS OF THE FIRST EUROPEAN CONFERENCE ON COMPLEX SYSTEMS ECCS’05, EUROPEAN COMPLEX SYSTEMS SOCIETY
, 2005
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On Counting Fixed Point Configurations in Star Networks
 Advances in Parallel & Distributed Computational Models Workshop (APDCM’05), within The 19th IEEE Int’l Parallel & Distributed Processing Symp
, 2005
"... We study herewith some aspects related to predictability of the longterm global behavior of the star topology based infrastructures when all the nodes, including the central node, are assumed to function reliably, faultlessly and synchronously. In particular, we use the nonlinear complex systems co ..."
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We study herewith some aspects related to predictability of the longterm global behavior of the star topology based infrastructures when all the nodes, including the central node, are assumed to function reliably, faultlessly and synchronously. In particular, we use the nonlinear complex systems concepts and methodology, coupled with those of computational complexity, to show that, simple as the star topology is, determining and predicting the longterm global behavior of the starbased infrastructures are computationally challenging tasks. More formally, determining various configuration space properties of the appropriate star network abstractions is shown to be hard in general. We particularly focus herein on the computational (in)tractability of counting the “fixed point ” configurations of a class of formal discrete dynamical systems defined over the star interconnection topology. 1
Large attractors in cooperative biquadratic Boolean networks
, 2007
"... Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960’s as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular impor ..."
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Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960’s as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper we consider cooperative systems, i.e. systems with positive feedback interconnections among all variables, which in a continuous setting guarantees a very stable dynamics. In Part I of this paper we presented a construction that shows that for an arbitrary constant 0 < c < 2 and sufficiently large n there exist ndimensional Boolean cooperative networks in which both the indegree and outdegree of each for each variable is bounded by two (biquadratic networks) and which nevertheless contain periodic orbits of length at least c n. In this part, we prove an inverse result showing that for sufficiently large n and for 0 < c < 2 sufficiently close to 2, any ndimensional cooperative, biquadratic Boolean network with a cycle of length at least c n must have a large proportion of variables with indegree 1. Such systems therefore share a structural similarity to the systems constructed in Part I.
Computational Complexity of Some Enumeration Problems About Uniformly Sparse Boolean Network Automata
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 159 (2006)
, 2006
"... We study the computational complexity of counting the fixed point configurations (FPs), the predecessor configurations and the ancestor configurations in certain classes of graph or network automata viewed as discrete dynamical systems. Some early results of this investigation are presented in two r ..."
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We study the computational complexity of counting the fixed point configurations (FPs), the predecessor configurations and the ancestor configurations in certain classes of graph or network automata viewed as discrete dynamical systems. Some early results of this investigation are presented in two recent ECCC reports [39, 40]. In particular, it is proven in [40] that both exact and approximate counting of FPs in the two closely related classes of Boolean network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems when each node is required to update according to a monotone Boolean function. In the present paper, we further strengthen those results by showing that the intractability of exact enumeration of FPs of a monotone Boolean SDS or SyDS still holds even when (i) the monotone update rules are restricted to linear threshold functions, and (ii) the underlying graph is uniformly sparse. By uniform sparseness we mean that every node in the graph has its degree bounded by for a small value of constant. In particular, we prove that exactly enumerating FPs in such SDSs and SyDSs remains #Pcomplete even when no node degree exceeds. Among other consequences, we show that this result also implies intractability of determining the exact memory capacity of discrete Hopfield networks with uniformly sparse and nonnegative integer weight matrices.
On the Computational Complexity of Predicting Dynamical Evolution of Large Agent Ensembles
"... We study global behavior of large ensembles of simple reactive agents. We do so by applying computational complexity tools to the analysis of formal complex systems and their dynamics. Since we are interested in the global dynamics and emerging behavior of large agent ensembles, rather than in an in ..."
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We study global behavior of large ensembles of simple reactive agents. We do so by applying computational complexity tools to the analysis of formal complex systems and their dynamics. Since we are interested in the global dynamics and emerging behavior of large agent ensembles, rather than in an individual agent’s deliberation, learning or other cognitive abilities, the discrete complex systems we study are based on the communicating finite state machine (CFSM) abstraction. In particular, we show that counting the number of possible evolutions of a particular class of CFSMs is computationally intractable, even when those CFSMs are very severely restricted both in terms of an individual agent’s behavior (that is, the local update rules), and the interagent interaction pattern (that is, the underlying communication network topology). We use this abstract framework to formally prove the wellknown intuition about multiagent systems (MAS) that a complex and, in general, unpredictable global behavior may arise from coupling of rather simple local behaviors and interactions.
On Modeling and Analyzing Sparsely Networked LargeScale Multiagent Systems with Cellular and Graph Automata
 ICCS
, 2006
"... Modeling, designing and analyzing large scale multiagent systems (MAS) with anywhere from tens of thousands to millions of autonomous agents will require mathematical and computational theories and models substantially different from those underlying the study of small to mediumscale MAS made of ..."
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Modeling, designing and analyzing large scale multiagent systems (MAS) with anywhere from tens of thousands to millions of autonomous agents will require mathematical and computational theories and models substantially different from those underlying the study of small to mediumscale MAS made of only dozens, or perhaps hundreds, of agents. In this paper, we study certain aspects of the global behavior of large ensembles of simple reactive agents. We do so by analyzing the collective dynamics of several related models of discrete complex systems based on cellular automata. We survey our recent results on dynamical properties of the complex systems of interest, and discuss some useful ways forward in modeling and analysis of largescale MAS via appropriately modified versions of the classical cellular automata.
On complexity of counting fixed point configurations in certain classes of graph automata
 Electronic Colloquium on Computational Complexity
"... We study computational complexity of counting the fixed point configurations (FPs) in certain discrete dynamical systems. We prove that counting FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) is computationally intractable, even when each node is required to updat ..."
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We study computational complexity of counting the fixed point configurations (FPs) in certain discrete dynamical systems. We prove that counting FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) is computationally intractable, even when each node is required to update according to a symmetric Boolean function. We also show that counting the garden of Eden configurations (GEs), as well as all transient configurations, is just as hard in this setting. Moreover, the hardness of enumerating FPs holds even in some severely restricted cases, such as when the nodes of an SDS or SyDS use only two different symmetric Boolean update rules, and when each node has a neighborhood size bounded by a small constant.
On the complexity of counting fixed points and gardens of Eden in sequential dynamical systems on planar bipartite graphs
 International Journal of Foundations of Computer Science
"... We study counting various types of configurations in certain classes of graph automata viewed as discrete dynamical systems. The graph automata models of our interest are Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively). These models have been proposed as a mathematical fo ..."
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We study counting various types of configurations in certain classes of graph automata viewed as discrete dynamical systems. The graph automata models of our interest are Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively). These models have been proposed as a mathematical foundation for a theory of largescale simulations of complex multiagent systems. Our emphasis in this paper is on the computational complexity of counting the fixed point and the garden of Eden configurations in Boolean SDSs and SyDSs. We have shown in [47] that counting fixed points is, in general, computationally intractable. We show in the present report that this intractability still holds when both the underlying graphs and the node update rules of these SDSs and SyDSs are severely restricted. In particular, we prove that the problems of exactly counting fixed points, gardens of Eden and two other types of S(y)DS configurations are all #Pcomplete, even if the SDSs and SyDSs are defined over planar bipartite graphs, and each of their nodes updates its state according to a monotone update rule given as a Boolean formula. We thus add these formal discrete dynamical systems to the list of those problem domains for which counting the combinatorial structures of interest is intractable even when the related decision problems are known to be efficiently solvable.
On computational complexity of counting fixed points in symmetric boolean graph automata
 IN PROCEEDINGS OF THE 4TH INTERNATIONAL CONFERENCE ON UNCONVENTIONAL COMPUTATION (UC’05), VOLUME 3699 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We study computational complexity of counting the fixed point configurations (FPs) in certain classes of graph automata viewed as discrete dynamical systems. We prove that both exact and approximate counting of FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) are c ..."
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We study computational complexity of counting the fixed point configurations (FPs) in certain classes of graph automata viewed as discrete dynamical systems. We prove that both exact and approximate counting of FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) are computationally intractable, even when each node is required to update according to a symmetric Boolean function. We also show that the problems of counting exactly the garden of Eden configurations (GEs), as well as all transient configurations, are in general intractable, as well. Moreover, exactly enumerating FPs or GEs remains hard even in some severely restricted cases, such as when the nodes of an SDS or SyDS use only two different symmetric Boolean update rules, and every node has a neighborhood size bounded by a small constant.