Rhythmic activity in complex systems is generated and sustained through interactions among the constituent units. In this paper we study the interplay between topology and dynamics of excitable nodes on random networks. The nodal dynamics are discrete, each node being in three possible states: active, refractory, or silent. Loading rules are defined whereby a subset of active nodes may be able to convert a silent node into an active one at the next time step. In the case of simple loading (SL) a silent node becomes active if it receives input from any neighbor. In the majority rules (MR) loading, a silent node fires when the majority of its neighbors are active. We address the question of whether a particular network design pattern confers dynamical advantage for the generation and sustainment of rhythmic activity. We find that the intrinsic properties of a node and the rules for interaction between them determine which structural features of the graph permit sustained activity. With SL the level of activity in the graph increases monotonically with the probability of connections between nodes, while for MR, the level of activity may be either monotonic or nonmonotonic, depending on parameters.

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

We examine driven nonlinear dynamical systems that are known to be in a state of generalized synchronization with an external drive. The chaotic time series of the response system are subject to empirical mode decomposition analysis. The instantaneous intrinsic mode frequencies (and their variance) present in these signals provide suitable order parameters for detecting the transition between the regimes of strong and weak generalized synchrony. Application is made to a variety of chaotically driven flows as well as maps.

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We study the interplay of topology and dynamics of excitable nodes on random networks. Comparison is made between systems grown by purely random (Erdős-Rényi) rules and those grown by the Achlioptas process. For a given size, the growth mechanism affects both the thresholds for the emergence of different structural features as well as the level of dynamical activity supported on the network.

We study the collective behavior of interacting agents in a simple model of market economics originally introduced by Nørrelykke and Bak. A general theoretical framework for interacting traders on an arbitrary network is presented, with the interaction consisting of buying (namely, consumption) and selling (namely, production) of commodities. Extremal dynamics is introduced by having the agent with least profit in the market readjust prices, causing the market to self-organize. We study this model market on regular lattices in two-dimension as well as on random complex networks; in the critical state fluctuations in an activity signal exhibit properties that are characteristic of avalanches observed in models of self-organized criticality, and these can be described by power-law distributions.

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