Temporal stability of network partitions

We present a method to find the best temporal partition at any time-scale and rank the relevance of partitions found at different time-scales. This method is based on random walkers coevolving with the network and as such constitutes a generalization of partition stability to the case of temporal networks. We show that, when applied to a toy model and real datasets, temporal stability uncovers structures that are persistent over meaningful time-scales as well as important isolated events, making it an effective tool to study both abrupt changes and gradual evolution of a network mesoscopic structures.Comment: 15 pages, 12 figure

Directed Abelian sandpile with multiple downward neighbors

We study the directed Abelian sandpile model on a square lattice, with $K$ downward neighbors per site, $K > 2$. The $K=3$ case is solved exactly, which extends the earlier known solution for the $K=2$ case. For $K>2$, the avalanche clusters can have holes and side-branches and are thus qualitatively different from the $K=2$ case where avalance clusters are compact. However, we find that the critical exponents for $K>2$ are identical with those for the $K=2$ case, and the large scale structure of the avalanches for $K>2$ tends to the $K=2$ case.Comment: Accepted for publication in PR

An Odyssey with complexity and network science: From the brain to social organisation

Complexity science is the study of systems that give rise to a priori unexpected macroscopic patterns, at different scales, emerging from the simple microscopic rules governing the evolution of the system. Method to explore complex systems are based on tools from a wide range of sciences, including statistical mechanics. Recently, the study of the emerging properties of complex systems has been enriched by a new toolbox derived by an extension of graph theory, namely, complex network science. We use both approaches to investigate two complex systems, the human brain and the communications patterns of a social network. The brain could be considered as an archetypical complex system; its fundamental constituting units, the neurones, communicate via simple inhibitory and excitatory interactions. These give rise to an extraordinarily rich hierarchical complex system, the human mind, that enables one to apprehend and interact with the universe. Understanding how the brain functions is essential, both for the general knowledge of humanity, but also for medical purposes; a better understanding of brain functions could lead to finding cures. We investigate the dynamics of brain activity at the smallest scale accessible by functional Magnetic Resonance Imaging, the voxel, while subjects are at the resting-state. We apply real-space renormalisation from the statistical mechanics toolbox, and our findings confirm that brain dynamics displays characteristic signatures of a critical system. At a coarser level, we study the structural differences in the functional networks of a healthy cohort and one made of people at-risk of developing a mental disorder during a verbal fluency task. We find that a key brain region plays a different role in the network organisation of the two populations, which is in agreement with previous findings on the disease schizophrenia. Finally, we investigate community structure in complex systems. Social interactions in humans are also a prime example of a system with emerging structure, the nature of which is dependent on the types of interactions between individuals. We use and develop new methods for community detection to uncover structures due to spatial and linguistic interactions in a mobile phone network

Connecting Hodge and Sakaguchi-Kuramoto through a mathematical framework for coupled oscillators on simplicial complexes

Phase synchronizations in models of coupled oscillators such as the Kuramoto model have been widely studied with pairwise couplings on arbitrary topologies, showing many unexpected dynamical behaviors. Here, based on a recent formulation the Kuramoto model on weighted simplicial complexes with phases supported on simplices of any order k, we introduce linear and non-linear frustration terms independent of the orientation of the k + 1 simplices, as a natural generalization of the Sakaguchi-Kuramoto model to simplicial complexes. With increasingly complex simplicial complexes, we study the the dynamics of the edge simplicial Sakaguchi-Kuramoto model with nonlinear frustration to highlight the complexity of emerging dynamical behaviors. We discover various dynamical phenomena, such as the partial loss of synchronization in subspaces aligned with the Hodge subspaces and the emergence of simplicial phase re-locking in regimes of high frustration

Altered dynamical integration/segregation balance during anesthesia-induced loss of consciousness

In recent years, brain imaging studies have begun to shed light on the neural correlates of physiologically-reversible altered states of consciousness such as deep sleep, anesthesia, and psychedelic experiences. The emerging consensus is that normal waking consciousness requires the exploration of a dynamical repertoire enabling both global integration i.e., long-distance interactions between brain regions, and segregation, i.e., local processing in functionally specialized clusters. Altered states of consciousness have notably been characterized by a tipping of the integration/segregation balance away from this equilibrium. Historically, functional MRI (fMRI) has been the modality of choice for such investigations. However, fMRI does not enable characterization of the integration/segregation balance at sub-second temporal resolution. Here, we investigated global brain spatiotemporal patterns in electrocorticography (ECoG) data of a monkey (Macaca fuscata) under either ketamine or propofol general anesthesia. We first studied the effects of these anesthetics from the perspective of band-specific synchronization across the entire ECoG array, treating individual channels as oscillators. We further aimed to determine whether synchrony within spatially localized clusters of oscillators was differently affected by the drugs in comparison to synchronization over spatially distributed subsets of ECoG channels, thereby quantifying changes in integration/segregation balance on physiologically-relevant time scales. The findings reflect global brain dynamics characterized by a loss of long-range integration in multiple frequency bands under both ketamine and propofol anesthesia, most pronounced in the beta (13–30 Hz) and low-gamma bands (30–80 Hz), and with strongly preserved local synchrony in all bands

Self-similar correlation function in brain resting-state fMRI

Adaptive behavior, cognition and emotion are the result of a bewildering variety of brain spatiotemporal activity patterns. An important problem in neuroscience is to understand the mechanism by which the human brain's 100 billion neurons and 100 trillion synapses manage to produce this large repertoire of cortical configurations in a flexible manner. In addition, it is recognized that temporal correlations across such configurations cannot be arbitrary, but they need to meet two conflicting demands: while diverse cortical areas should remain functionally segregated from each other, they must still perform as a collective, i.e., they are functionally integrated. Here, we investigate these large-scale dynamical properties by inspecting the character of the spatiotemporal correlations of brain resting-state activity. In physical systems, these correlations in space and time are captured by measuring the correlation coefficient between a signal recorded at two different points in space at two different times. We show that this two-point correlation function extracted from resting-state fMRI data exhibits self-similarity in space and time. In space, self-similarity is revealed by considering three successive spatial coarse-graining steps while in time it is revealed by the 1/f frequency behavior of the power spectrum. The uncovered dynamical self-similarity implies that the brain is spontaneously at a continuously changing (in space and time) intermediate state between two extremes, one of excessive cortical integration and the other of complete segregation. This dynamical property may be seen as an important marker of brain well-being both in health and disease.Comment: 14 pages 13 figures; published online before print September 2

A unified framework for Simplicial Kuramoto models

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.Comment: 36 pages, 11 figure

Functional brain networks before the onset of psychosis: A prospective fMRI study with graph theoretical analysis☆☆☆

Individuals with an at-risk mental state (ARMS) have a risk of developing a psychotic disorder significantly greater than the general population. However, it is not currently possible to predict which ARMS individuals will develop psychosis from clinical assessment alone. Comparison of ARMS subjects who do, and do not, develop psychosis can reveal which factors are critical for the onset of illness. In the present study, 37 patients with an ARMS were followed clinically at least 24 months subsequent to initial referral. Functional MRI data were collected at the beginning of the follow-up period during performance of an executive task known to recruit frontal lobe networks and to be impaired in psychosis. Graph theoretical analysis was used to compare the organization of a functional brain network in ARMS patients who developed a psychotic disorder following the scan (ARMS-T) to those who did not become ill during the same follow-up period (ARMS-NT) and aged-matched controls. The global properties of each group's representative network were studied (density, efficiency, global average path length) as well as regionally-specific contributions of network nodes to the organization of the system (degree, farness-centrality, betweenness-centrality). We focused our analysis on the dorsal anterior cingulate cortex (ACC), a region known to support executive function that is structurally and functionally impaired in ARMS patients. In the absence of between-group differences in global network organization, we report a significant reduction in the topological centrality of the ACC in the ARMS-T group relative to both ARMS-NT and controls. These results provide evidence that abnormalities in the functional organization of the brain predate the onset of psychosis, and suggest that loss of ACC topological centrality is a potential biomarker for transition to psychosis

Altered dynamical integration/segregation balance during anesthesia-induced loss of consciousness

In recent years, brain imaging studies have begun to shed light on the neural correlates of physiologically-reversible altered states of consciousness such as deep sleep, anesthesia, and psychedelic experiences. The emerging consensus is that normal waking consciousness requires the exploration of a dynamical repertoire enabling both global integration i.e., long-distance interactions between brain regions, and segregation, i.e., local processing in functionally specialized clusters. Altered states of consciousness have notably been characterized by a tipping of the integration/segregation balance away from this equilibrium. Historically, functional MRI (fMRI) has been the modality of choice for such investigations. However, fMRI does not enable characterization of the integration/segregation balance at sub-second temporal resolution. Here, we investigated global brain spatiotemporal patterns in electrocorticography (ECoG) data of a monkey (Macaca fuscata) under either ketamine or propofol general anesthesia. We first studied the effects of these anesthetics from the perspective of band-specific synchronization across the entire ECoG array, treating individual channels as oscillators. We further aimed to determine whether synchrony within spatially localized clusters of oscillators was differently affected by the drugs in comparison to synchronization over spatially distributed subsets of ECoG channels, thereby quantifying changes in integration/segregation balance on physiologically-relevant time scales. The findings reflect global brain dynamics characterized by a loss of long-range integration in multiple frequency bands under both ketamine and propofol anesthesia, most pronounced in the beta (13–30 Hz) and low-gamma bands (30–80 Hz), and with strongly preserved local synchrony in all bands