A New Measure of Centrality for Brain Networks

Modular organization and degree-degree correlations are ubiquitous in the connectivity structure of biological, technological, and social interacting systems. So far most studies have concentrated on unveiling both features in real world networks, but a model that succeeds in generating them simultaneously is needed. We consider a network of interacting phase oscillators, and an adaptation mechanism for the coupling that promotes the connection strengths between those elements that are dynamically correlated. We show that, under these circumstances, the dynamical organization of the oscillators shapes the topology of the graph in such a way that modularity and assortativity features emerge spontaneously and simultaneously. In turn, we prove that such an emergent structure is associated with an asymptotic arrangement of the collective dynamical state of the network into cluster synchronization. Natural networking systems [1] are vastly characterized by a modular organization of their connectivity structure [2], and by nontrivial correlation features in the way units with a given number of connections (degree) tend to link with members of the same degree (assortativity), or with units with different degrees (disassortativity). Modularity is clearly the result of the need for social, biological, and technological systems to optimize their parallel, yet integrated, functioning [3,4] by means of an organization into mesoscale structures, such as communities, i.e., groups of highly interconnected nodes that are sparsely connected to the rest of the graph [5]. Degreedegree correlation reflects the observed tendency of natural networks to organize the main topology on top of a backbone of nodes that may be starlike (disassortativity) or of highly connected hubs (assortativity).Whereas the emphasis typically has been on devising suitable tools and algorithms to unveil, identify, and quantify both modular and correlation features in real world networks [1,2], a model that has proven to be capable of generating these features simultaneously is needed. Modularity and assortativity are usually imposed separately, in a second step, on top of already generated networks by means of ad hoc algorithms and methods [6,7].In this Rapid Communication, we show that all these features may spontaneously emerge in an adaptive network of interacting oscillators as the result of a delicate interplay between synchronization processes and coevolution of the connectivity structure. When the connectivity dynamics is such that links coupling the nodes with synchronous (nonsynchronous) dynamics are promoted (weakened), we prove that an initially unstructured clique configuration evolves in time toward an emerging structured network displaying both modularity and assortativity.Let us then start by considering an initial ensemble of N all-to-all coupled Kuramoto oscillators [8,9]. Each unit of the ensemble n = 1,..., N is characterized by its phase 9", whose dynamics is ruled byw nm sin(0 m -6"), (1) where £2 n is the natural frequency of the wth ...

The brain can be regarded as a network: a connected system where nodes, or units, represent different specialized regions and links, or connections, represent communication pathways. From a functional perspective, communication is coded by temporal dependence between the activities of different brain areas. In the last decade, the abstract representation of the brain as a graph has allowed to visualize functional brain networks and describe their non-trivial topological properties in a compact and objective way. Nowadays, the use of graph analysis in translational neuroscience has become essential to quantify brain dysfunc- tions in terms of aberrant reconfiguration of functional brain networks. Despite its evident impact, graph analysis of functional brain networks is not a simple toolbox that can be blindly applied to brain signals. On the one hand, it requires the know-how of all the methodological steps of the pipeline that manipulate the input brain signals and extract the functional network prop- erties. On the other hand, knowledge of the neural phenomenon under study is required to perform physiologically relevant analysis. The aim of this review is to provide practical indications to make sense of brain network analysis and contrast counterproductive attitudes

Although graph theory has been around since the 18th century, the field of network science is more recent and continues to gain popularity, particularly in the field of neuroimaging. The field was propelled forward when Watts and Strogatz introduced their small-world network model, which described a network that provided regional specialization with efficient global information transfer. This model is appealing to the study of brain connectivity, as the brain can be viewed as a system with various interacting regions that produce complex behaviors. In practice, graph metrics such as clustering coefficient, path length, and efficiency measures are often used to characterize system properties. Centrality metrics such as degree, betweenness, closeness, and eigenvector centrality determine critical areas within the network. Community structure is also essential for understanding network organization and topology. Network science has led to a paradigm shift in the neuroscientific community, but it should be viewed as more than a simple ''tool du jour.'' To fully appreciate the utility of network science, a greater understanding of how network models apply to the brain is needed. An integrated appraisal of multiple network analyses should be performed to better understand network structure rather than focusing on univariate comparisons to find significant group differences; indeed, such comparisons, popular with traditional functional magnetic resonance imaging analyses, are arguably no longer relevant with graph-theory based approaches. These methods necessitate a philosophical shift toward complexity science. In this context, when correctly applied and interpreted, network scientific methods have a chance to revolutionize the understanding of brain function.