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In this paper, we study the limit of compactness which is a graph index originally introduced for measuring structural characteristics of hypermedia. Applying compactness to large scale small-world graphs (Mehler, 2008) observed its limit behaviour to be equal 1. The striking question concerning this finding was whether this limit behaviour resulted from the specifics of small-world graphs or was simply an artefact. In this paper, we determine the necessary and sufficient conditions for any sequence of connected graphs resulting in a limit value of CB = 1 which can be generalized with some consideration for the case of disconnected graph classes (Theorem 3). This result can be applied to many well-known classes of connected graphs. Here, we illustrate it by considering four examples. In fact, our proof-theoretical approach allows for quickly obtaining the limit value of compactness for many graph classes sparing computational costs.
Attention-deficit/hyperactivity disorder (ADHD) is one of the most common neurodevelopmental disorders with significant and often lifelong effects on social, emotional, and cognitive functioning. Influential neurocognitive models of ADHD link behavioral symptoms to altered connections between and within functional brain networks. Here, we investigate whether network-based theories of ADHD can be generalized to understanding variations in ADHD-related behaviors within the normal (i.e., clinically unaffected) adult population. In a large and representative sample, self-rated presence of ADHD symptoms varied widely; only eight out of 291 participants scored in the clinical range. Subject-specific brain-network graphs were modeled from functional MRI resting-state data and revealed significant associations between (non-clinical) ADHD symptoms and region-specific profiles of between-module and within-module connectivity. Effects were located in brain regions associated with multiple neuronal systems including the default-mode network, the salience network, and the central executive system. Our results are consistent with network perspectives of ADHD and provide further evidence for the relevance of an appropriate information transfer between task-negative (default-mode) and task-positive brain regions. More generally, our findings support a dimensional conceptualization of ADHD and contribute to a growing understanding of cognition as an emerging property of functional brain networks.
Two-person neuroscience (2 PN) is a recently introduced conceptual and methodological framework used to investigate the neural basis of human social interaction from simultaneous neuroimaging of two or more subjects (hyperscanning). In this study, we adopted a 2 PN approach and a multiple-brain connectivity model to investigate the neural basis of a form of cooperation called joint action. We hypothesized different intra-brain and inter-brain connectivity patterns when comparing the interpersonal properties of joint action with non-interpersonal conditions, with a focus on co-representation, a core ability at the basis of cooperation. 32 subjects were enrolled in dual-EEG recordings during a computerized joint action task including three conditions: one in which the dyad jointly acted to pursue a common goal (joint), one in which each subject interacted with the PC (PC), and one in which each subject performed the task individually (Solo).
A combination of multiple-brain connectivity estimation and specific indices derived from graph theory allowed to compare interpersonal with non-interpersonal conditions in four different frequency bands. Our results indicate that all the indices were modulated by the interaction, and returned a significantly stronger integration of multiple-subject networks in the joint vs. PC and Solo conditions. A subsequent classification analysis showed that features based on multiple-brain indices led to a better discrimination between social and non-social conditions with respect to single-subject indices. Taken together, our results suggest that multiple-brain connectivity can provide a deeper insight into the understanding of the neural basis of cooperation in humans.