Oscillations of complex networks

Binary dynamics on star networks under external perturbations

We study a binary dynamical process that is a representation of the voter model with two candidates and opinion makers. The voters are represented by nodes of a network of social contacts with internal states labeled 0 or 1 and nodes that are connected can influence each other. The network is also perturbed by opinion makers, a set of external nodes whose states are frozen in 0 or 1 and that can influence all nodes of the network. The quantity of interest is the probability of finding m nodes in state 1 at time t. Here we study this process on star networks, which are simple representations of hubs found in complex systems, and compare the results with those obtained for networks that are fully connected. In both cases a transition from disordered to ordered equilibrium states is observed as the number of external nodes becomes small. For fully connected networks the probability distribution becomes uniform at the critical point. For star networks, on the other hand, we show that the equilibrium distribution splits in two peaks, reflecting the two possible states of the central node. We obtain approximate analytical solutions for the equilibrium distribution that clarify the role of the central node in the process. We show that the network topology also affects the time scale of oscillations in single realizations of the dynamics, which are much faster for the star network. Finally, extending the analysis to two stars we compare our results with simulations in simple scale-free networks.

Graph analysis of epileptogenic networks in human partial epilepsy

PURPOSE

The current gold standard for the localization of the cortical regions responsible for the initiation and propagation of the ictal activity is through the use of invasive electrocorticography (ECoG). This method is utilized to guide surgical intervention in cases of medically intractable epilepsy by identifying the location and extent of the epileptogenic focus. Recent studies have proposed mechanisms in which the activity of epileptogenic cortical networks, rather than discrete focal sources, contributes to the generation of the ictal state. If true, selective modulation of key network components could be employed for the prevention and termination of the ictal state.

METHODS

Here, we have applied graph theory methods as a means to identify critical network nodes in cortical networks during both ictal and interictal states. ECoG recordings were obtained from a cohort of 25 patients undergoing presurgical monitoring for the treatment of intractable epilepsy at the Mayo Clinic (Rochester, MN, U.S.A.).

KEY FINDINGS

One graph measure, the betweenness centrality, was found to correlate with the location of the resected cortical regions in patients who were seizure-free following surgical intervention. Furthermore, these network interactions were also observed during random nonictal periods as well as during interictal spike activity. These network characteristics were found to be frequency dependent, with high frequency gamma band activity most closely correlated with improved postsurgical outcome as has been reported in previous literature.

SIGNIFICANCE

These findings could lead to improved understanding of epileptogenesis. In addition, this theoretically allows for more targeted therapeutic interventions through the selected modulation or disruption of these epileptogenic networks.

An alternative approach to characterize the topology of complex networks and its application in epidemic spreading

Based on the mean-field approach, epidemic spreading has been well studied. However, the mean-field approach cannot show the detailed contagion process, which is important in the control of epidemic. To fill this gap, we present a novel approach to study how the topological structure of complex network influences the concrete process of epidemic spreading. After transforming the network structure into hierarchical layers, we introduce a set of new parameters, i.e., the average fractions of degree for outgoing, ingoing, and remaining in the same layer, to describe the infection process. We find that this set of parameters are closely related to the degree distribution and the clustering coefficient but are more convenient than them in describing the process of epidemic spreading. Moreover, we find that the networks with exponential distribution have slower spreading speed than the networks with power-law degree distribution. Numerical simulations have confirmed the theoretical predictions.