Largenet2: an object-oriented programming library for simulating large adaptive networks

The largenet2 C++ library provides an infrastructure for the simulation of large dynamic and adaptive networks with discrete node and link states. The library is released as free software. It is available at http://rincedd.github.com/largenet2. Largenet2 is licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License.Comment: 2 pages, 1 figur

Early warning signs for saddle-escape transitions in complex networks

Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.Comment: revised versio

Adaptive-network models of swarm dynamics

We propose a simple adaptive-network model describing recent swarming experiments. Exploiting an analogy with human decision making, we capture the dynamics of the model by a low-dimensional system of equations permitting analytical investigation. We find that the model reproduces several characteristic features of swarms, including spontaneous symmetry breaking, noise- and density-driven order-disorder transitions that can be of first or second order, and intermittency. Reproducing these experimental observations using a non-spatial model suggests that spatial geometry may have a lesser impact on collective motion than previously thought.Comment: 8 pages, 3 figure

Adaptive-network models of collective dynamics

Complex systems can often be modelled as networks, in which their basic units are represented by abstract nodes and the interactions among them by abstract links. This network of interactions is the key to understanding emergent collective phenomena in such systems. In most cases, it is an adaptive network, which is defined by a feedback loop between the local dynamics of the individual units and the dynamical changes of the network structure itself. This feedback loop gives rise to many novel phenomena. Adaptive networks are a promising concept for the investigation of collective phenomena in different systems. However, they also present a challenge to existing modelling approaches and analytical descriptions due to the tight coupling between local and topological degrees of freedom. In this thesis, I present a simple rule-based framework for the investigation of adaptive networks, using which a wide range of collective phenomena can be modelled and analysed from a common perspective. In this framework, a microscopic model is defined by the local interaction rules of small network motifs, which can be implemented in stochastic simulations straightforwardly. Moreover, an approximate emergent-level description in terms of macroscopic variables can be derived from the microscopic rules, which we use to analyse the system\'s collective and long-term behaviour by applying tools from dynamical systems theory. We discuss three adaptive-network models for different collective phenomena within our common framework. First, we propose a novel approach to collective motion in insect swarms, in which we consider the insects\' adaptive interaction network instead of explicitly tracking their positions and velocities. We capture the experimentally observed onset of collective motion qualitatively in terms of a bifurcation in this non-spatial model. We find that three-body interactions are an essential ingredient for collective motion to emerge. Moreover, we show what minimal microscopic interaction rules determine whether the transition to collective motion is continuous or discontinuous. Second, we consider a model of opinion formation in groups of individuals, where we focus on the effect of directed links in adaptive networks. Extending the adaptive voter model to directed networks, we find a novel fragmentation mechanism, by which the network breaks into distinct components of opposing agents. This fragmentation is mediated by the formation of self-stabilizing structures in the network, which do not occur in the undirected case. We find that they are related to degree correlations stemming from the interplay of link directionality and adaptive topological change. Third, we discuss a model for the evolution of cooperation among self-interested agents, in which the adaptive nature of their interaction network gives rise to a novel dynamical mechanism promoting cooperation. We show that even full cooperation can be achieved asymptotically if the networks\' adaptive response to the agents\' dynamics is sufficiently fast

Early fragmentation in the adaptive voter model on directed networks

We consider voter dynamics on a directed adaptive network with fixed out-degree distribution. A transition between an active phase and a fragmented phase is observed. This transition is similar to the undirected case if the networks are sufficiently dense and have a narrow out-degree distribution. However, if a significant number of nodes with low out degree is present, then fragmentation can occur even far below the estimated critical point due to the formation of self-stabilizing structures that nucleate fragmentation. This process may be relevant for fragmentation in current political opinion formation processes.Comment: 9 pages, 8 figures as published in Phys. Rev.

Adaptive-network models of collective dynamics

Complex systems can often be modelled as networks, in which their basic units are represented by abstract nodes and the interactions among them by abstract links. This network of interactions is the key to understanding emergent collective phenomena in such systems. In most cases, it is an adaptive network, which is defined by a feedback loop between the local dynamics of the individual units and the dynamical changes of the network structure itself. This feedback loop gives rise to many novel phenomena. Adaptive networks are a promising concept for the investigation of collective phenomena in different systems. However, they also present a challenge to existing modelling approaches and analytical descriptions due to the tight coupling between local and topological degrees of freedom. In this thesis, I present a simple rule-based framework for the investigation of adaptive networks, using which a wide range of collective phenomena can be modelled and analysed from a common perspective. In this framework, a microscopic model is defined by the local interaction rules of small network motifs, which can be implemented in stochastic simulations straightforwardly. Moreover, an approximate emergent-level description in terms of macroscopic variables can be derived from the microscopic rules, which we use to analyse the system\'s collective and long-term behaviour by applying tools from dynamical systems theory. We discuss three adaptive-network models for different collective phenomena within our common framework. First, we propose a novel approach to collective motion in insect swarms, in which we consider the insects\' adaptive interaction network instead of explicitly tracking their positions and velocities. We capture the experimentally observed onset of collective motion qualitatively in terms of a bifurcation in this non-spatial model. We find that three-body interactions are an essential ingredient for collective motion to emerge. Moreover, we show what minimal microscopic interaction rules determine whether the transition to collective motion is continuous or discontinuous. Second, we consider a model of opinion formation in groups of individuals, where we focus on the effect of directed links in adaptive networks. Extending the adaptive voter model to directed networks, we find a novel fragmentation mechanism, by which the network breaks into distinct components of opposing agents. This fragmentation is mediated by the formation of self-stabilizing structures in the network, which do not occur in the undirected case. We find that they are related to degree correlations stemming from the interplay of link directionality and adaptive topological change. Third, we discuss a model for the evolution of cooperation among self-interested agents, in which the adaptive nature of their interaction network gives rise to a novel dynamical mechanism promoting cooperation. We show that even full cooperation can be achieved asymptotically if the networks\' adaptive response to the agents\' dynamics is sufficiently fast

Adaptive-network models of collective dynamics

Complex systems can often be modelled as networks, in which their basic units are represented by abstract nodes and the interactions among them by abstract links. This network of interactions is the key to understanding emergent collective phenomena in such systems. In most cases, it is an adaptive network, which is defined by a feedback loop between the local dynamics of the individual units and the dynamical changes of the network structure itself. This feedback loop gives rise to many novel phenomena. Adaptive networks are a promising concept for the investigation of collective phenomena in different systems. However, they also present a challenge to existing modelling approaches and analytical descriptions due to the tight coupling between local and topological degrees of freedom. In this thesis, I present a simple rule-based framework for the investigation of adaptive networks, using which a wide range of collective phenomena can be modelled and analysed from a common perspective. In this framework, a microscopic model is defined by the local interaction rules of small network motifs, which can be implemented in stochastic simulations straightforwardly. Moreover, an approximate emergent-level description in terms of macroscopic variables can be derived from the microscopic rules, which we use to analyse the system\'s collective and long-term behaviour by applying tools from dynamical systems theory. We discuss three adaptive-network models for different collective phenomena within our common framework. First, we propose a novel approach to collective motion in insect swarms, in which we consider the insects\' adaptive interaction network instead of explicitly tracking their positions and velocities. We capture the experimentally observed onset of collective motion qualitatively in terms of a bifurcation in this non-spatial model. We find that three-body interactions are an essential ingredient for collective motion to emerge. Moreover, we show what minimal microscopic interaction rules determine whether the transition to collective motion is continuous or discontinuous. Second, we consider a model of opinion formation in groups of individuals, where we focus on the effect of directed links in adaptive networks. Extending the adaptive voter model to directed networks, we find a novel fragmentation mechanism, by which the network breaks into distinct components of opposing agents. This fragmentation is mediated by the formation of self-stabilizing structures in the network, which do not occur in the undirected case. We find that they are related to degree correlations stemming from the interplay of link directionality and adaptive topological change. Third, we discuss a model for the evolution of cooperation among self-interested agents, in which the adaptive nature of their interaction network gives rise to a novel dynamical mechanism promoting cooperation. We show that even full cooperation can be achieved asymptotically if the networks\' adaptive response to the agents\' dynamics is sufficiently fast

A homoclinic route to full cooperation in adaptive networks and its failure

We consider the evolutionary dynamics of a cooperative game on an adaptive network, where the strategies of agents (cooperation or defection) feed back on their local interaction topology. While mutual cooperation is the social optimum, unilateral defection yields a higher payoff and undermines the evolution of cooperation. Although no a priori advantage is given to cooperators, an intrinsic dynamical mechanism can lead asymptotically to a state of full cooperation. In finite systems, this state is characterized by long periods of strong cooperation interrupted by sudden episodes of predominant defection, suggesting a possible mechanism for the systemic failure of cooperation in real-world systems.Comment: 11 pages, 5 figure