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21
A Geometrical Hierarchy of Graphs via Cellular Automata
, 1998
"... Historically, cellular automata were defined on the lattices Z n , but the definition can be extended to bounded degree graphs. Given a notion of simulation between cellular automata defined on different structures (namely graphs of automata), we can deduce an order on graphs. In this paper, w ..."
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Historically, cellular automata were defined on the lattices Z n , but the definition can be extended to bounded degree graphs. Given a notion of simulation between cellular automata defined on different structures (namely graphs of automata), we can deduce an order on graphs. In this paper, we link this order to graph properties and explicit the order for most of the common graphs.
Predecessor and permutation existence problems for sequential dynamical systems
 In Proceedings of the Conference on Discrete Models for Complex Systems (DMCS’03), volume AB of Discrete Mathematics and Theoretical Computer Science Proceedings
, 2003
"... 3 Part of the work was done while the authors were visiting the Basic and Applied Simulation Sciences Group (NISAC) ..."
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3 Part of the work was done while the authors were visiting the Basic and Applied Simulation Sciences Group (NISAC)
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 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 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 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.
Computational aspects of analyzing social network dynamics
 in Proc. International Joint Conference on Artificial Intellligence (IJCAI
"... Motivated by applications such as the spread of epidemics and the propagation of influence in social networks, we propose a formal model for analyzing the dynamics of such networks. Our model is a stochastic version of discrete dynamical systems. Using this model, we formulate and study the computat ..."
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Motivated by applications such as the spread of epidemics and the propagation of influence in social networks, we propose a formal model for analyzing the dynamics of such networks. Our model is a stochastic version of discrete dynamical systems. Using this model, we formulate and study the computational complexity of two fundamental problems (called reachability and predecessor existence problems) which arise in the context of social networks. We also point out the implications of our results on other computational models such as Hopfield networks, communicating finite state machines and systolic arrays. 1
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.