Stefan Bornholdt

Handbook of Graphs and Networks: From the Genome to the Internet

From the Publisher: nThis book defines the field of complex interacting networks in its infancy and presents the dynamics of networks and their structure as a key concept across disciplines. The contributions present common underlying principles of network dynamics and their theoretical description and are of interest to specialists as well as to the non-specialized reader looking for an introduction to this new exciting field. nTheoretical concepts include modeling networks as dynamical systems with numerical methods and new graph theoretical approaches, but also focus on networks that change their topology as in morphogenesis and self-organization. The authors offer concepts to model network structures and dynamics, focussing on approaches applicable across disciplines.

Emergence of a small world from local interactions: modeling acquaintance networks.

How do we make acquaintances? A simple observation from everyday experience is that often one of our acquaintances introduces us to one of his or her acquaintances. Such a simple triangle interaction may be viewed as the basis of the evolution of many social networks. Here, it is demonstrated that this assumption is sufficient to reproduce major nontrivial features of social networks: short path length, high clustering, and scale-free or exponential link distributions.

Coevolutionary games on networks.

We study agents on a network playing an iterated Prisoners dilemma against their neighbors. The resulting spatially extended coevolutionary game exhibits stationary states which are Nash equilibria. After perturbation of these equilibria, avalanches of mutations reestablish a stationary state. Scale-free avalanche distributions are observed that are in accordance with calculations from the Nash equilibria and a confined branching process. The transition from subcritical to critical avalanche dynamics can be traced to a change in the degeneracy of the cooperative macrostate and is observed for many variants of this game.

Topology of biological networks and reliability of information processing.

Survival of living cells and organisms is largely based on highly reliable function of their regulatory networks. However, the elements of biological networks, e.g., regulatory genes in genetic networks or neurons in the nervous system, are far from being reliable dynamical elements. How can networks of unreliable elements perform reliably? We here address this question in networks of autonomous noisy elements with fluctuating timing and study the conditions for an overall system behavior being reproducible in the presence of such noise. We find a clear distinction between reliable and unreliable dynamical attractors. In the reliable case, synchrony is sustained in the network, whereas in the unreliable scenario, fluctuating timing of single elements can gradually desynchronize the system, leading to nonreproducible behavior. The likelihood of reliable dynamical attractors strongly depends on the underlying topology of a network. Comparing with the observed architectures of gene regulation networks, we find that those 3-node subgraphs that allow for reliable dynamics are also those that are more abundant in nature, suggesting that specific topologies of regulatory networks may provide a selective advantage in evolution through their resistance against noise.

Transient dynamics increasing network vulnerability to cascading failures.

We study cascading failures in networks using a dynamical flow model based on simple conservation and distribution laws. It is found that considering the flow dynamics may imply reduced network robustness compared to previous static overload failure models. This is due to the transient oscillations or overshooting in the loads, when the flow dynamics adjusts to the new (remaining) network structure. The robustness of networks showing cascading failures is generally given by a complex interplay between the network topology and flow dynamics.

Robustness as an evolutionary principle

We suggest simulating evolution of complex organisms using a model constrained solely by the requirement of robustness in its expression patterns. This scenario is illustrated by evolving discrete logical networks with epigenetic properties. Evidence for dynamical features in the evolved networks is found that can be related to biological observables.

Dynamics of price and trading volume in a spin model of stock markets with heterogeneous agents

The dynamics of a stock market with heterogeneous agents is discussed in the framework of a recently proposed spin model for the emergence of bubbles and crashes. We relate the log-returns of stock prices to magnetization in the model and find that it is closely related to trading volume as observed in real markets. The cumulative distribution of log-returns exhibits scaling with exponents steeper than 2 and scaling is observed in the distribution of transition times between bull and bear markets.

World Wide Web scaling exponent from Simon's 1955 model.

The statistical properties of the World Wide Web have attracted considerable attention recently since self-similar regimes were first observed in the scaling of its link structure. One characteristic quantity is the number of (in-)links k that point to a particular web page. Its probability distribution P(k) shows a pronounced power-law scaling P(k) approximately k(-gamma) that is not readily explained by standard random graph theory. Here, we recall a simple and elegant model for scaling phenomena in general copy- and growth-processes as proposed by Simon in 1955. When combined with an experimental measurement of network growth in the World Wide Web, this classical model is able to model the in-link dynamics and predicts the scaling exponent gamma=2.1 in accordance with observation.

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.

Neutral mutations and punctuated equilibrium in evolving genetic networks

Neutral mutations and punctuated equilibrium in evolving genetic networks searching for evolutionary origin of differentiation this was not what I wanted but might lateron be interesting. Strongly relates to Kauffman RBN