Sizing the length of complex networks

Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short

Reciprocity of Networks with Degree Correlations and Arbitrary Degree Sequences

Although most of the real networks contain a mixture of directed and bidirectional (reciprocal) connections, the reciprocity $r$ has received little attention as a subject of theoretical understanding. We study the expected reciprocity of networks with an arbitrary degree sequence and a broad class of degree correlations by means of statistical ensemble approach. We demonstrate that degree correlations are crucial to understand the reciprocity in real networks and a hierarchy of correlation contributions to $r$ is revealed. Numerical experiments using novel network randomization methods show very good agreement to our analytical estimations.Comment: 8 pages, 3 figures, added a new table and a new figure, accepted for publication in Phys.Rev.

Cortical Hubs Form a Module for Multisensory Integration on Top of the Hierarchy of Cortical Networks

Sensory stimuli entering the nervous system follow particular paths of processing, typically separated (segregated) from the paths of other modal information. However, sensory perception, awareness and cognition emerge from the combination of information (integration). The corticocortical networks of cats and macaque monkeys display three prominent characteristics: (i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and potential function of the cortical hubs by graph analysis and information theoretical methods. We find that the cortical hubs form a spatially delocalised, but topologically central module with the capacity to integrate multisensory information in a collaborative manner. With this, we resolve the underlying anatomical substrate that supports the simultaneous capacity of the cortex to segregate and to integrate multisensory information

From modular to centralized organization of synchronization in functional areas of the cat cerebral cortex

Recent studies have pointed out the importance of transient synchronization between widely distributed neural assemblies to understand conscious perception. These neural assemblies form intricate networks of neurons and synapses whose detailed map for mammals is still unknown and far from our experimental capabilities. Only in a few cases, for example the C. elegans, we know the complete mapping of the neuronal tissue or its mesoscopic level of description provided by cortical areas. Here we study the process of transient and global synchronization using a simple model of phase-coupled oscillators assigned to cortical areas in the cerebral cat cortex. Our results highlight the impact of the topological connectivity in the developing of synchronization, revealing a transition in the synchronization organization that goes from a modular decentralized coherence to a centralized synchronized regime controlled by a few cortical areas forming a Rich-Club connectivity pattern.Comment: 24 pages, 8 figures. Final version published in PLoS On

Identification and neuromodulation of brain states to promote recovery of consciousness

Experimental and clinical studies of consciousness identify brain states (i.e., transient, relevant features of the brain associated with the state of consciousness) in a non-systematic manner and largely independent from the research into the induction of state changes. In this narrative review with a focus on patients with a disorder of consciousness (DoC), we synthesize advances on the identification of brain states associated with consciousness in animal models and physiological (sleep), pharmacological (anesthesia) and pathological (DoC) states of altered consciousness in human. We show that in reduced consciousness the frequencies in which the brain operates are slowed down and that the pattern of functional communication in the brain is sparser, less efficient, and less complex. The results also highlight damaged resting state networks, in particular the default mode network, decreased connectivity in long-range connections and in the thalamocortical loops. Next, we show that therapeutic approaches to treat DoC, through pharmacology (e.g., amantadine, zolpidem), and (non-)invasive brain stimulation (e.g., transcranial current stimulation, deep brain stimulation) have shown some effectiveness to promote consciousness recovery. It seems that these deteriorated features of conscious brain states may improve in response to these neuromodulation approaches, yet, targeting often remains non-specific and does not always lead to (behavioral) improvements. Furthermore, in silico model-based approaches allow the development of personalized assessment of the effect of treatment on brain-wide dynamics. Although still in infancy, the fields of brain state identification and neuromodulation of brain states in relation to consciousness are showing fascinating developments that, when united, might propel the development of new and better targeted techniques for DoC. For example, brain states could be identified in a predictive setting, and the theoretical and empirical testing (i.e., in animals, under anesthesia and patients with a DoC) of neuromodulation techniques to promote consciousness could be investigated. This review further helps to identify where challenges and opportunities lay for the maturation of brain state research in the context of states of consciousness. Finally, it aids in recognizing possibilities and obstacles for the clinical translation of these diagnostic techniques and neuromodulation treatment options across both the multi-modal and multi-species approaches outlined throughout the review. This paper presents interactive figures, supported by the Live Paper initiative of the Human Brain Project, enabling the interaction with data and figures illustrating the concepts in the paper through EBRAINS (go to https://wiki.ebrains.eu/bin/view/Collabs/live-paper-states-altered-consciousness and get started with an EBRAINS account).NA is research fellow, OG is Research Associate, and SL is research director at FRS-FNRS. JA is postdoctoral fellow at the FWO. The study was further supported by the University and University Hospital of Liège, the BIAL Foundation, the Belgian National Funds for Scientific Research (FRS-FNRS), the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3), the FNRS PDR project (T.0134.21), the ERA-Net FLAG-ERA JTC2021 project ModelDXConsciousness (Human Brain Project Partnering Project), the fund Generet, the King Baudouin Foundation, the Télévie Foundation, the European Space Agency (ESA) and the Belgian Federal Science Policy Office (BELSPO) in the framework of the PRODEX Programme, the Public Utility Foundation 'Université Européenne du Travail', "Fondazione Europea di Ricerca Biomedica", the BIAL Foundation, the Mind Science Foundation, the European Commission, the Fondation Leon Fredericq, the Mind-Care foundation, the DOCMA project (EU-H2020-MSCA–RISE–778234), the National Natural Science Foundation of China (Joint Research Project 81471100) and the European Foundation of Biomedical Research FERB Onlus

Cortical hubs form a module for multisensory integration on top of the hierarchy of cortical networks

Sensory stimuli entering the nervous system follow particular paths of processing, typically separated (segregated) from the paths of other modal information. However, sensory perception, awareness and cognition emerge from the combination of information (integration). The corticocortical networks of cats and macaque monkeys display three prominent characteristics: (i) modular organisation (facilitating the segregation), (ii) abundant alternative processing paths and (iii) the presence of highly connected hubs. Here, we study in detail the organisation and potential function of the cortical hubs by graph analysis and information theoretical methods. We find that the cortical hubs form a spatially delocalised, but topologically central module with the capacity to integrate multisensory information in a collaborative manner. With this, we resolve the underlying anatomical substrate that supports the simultaneous capacity of the cortex to segregate and to integrate multisensory information

Framework based on communicability and flow to analyze complex network dynamics

Graph theory constitutes a widely used and established field providing powerful tools for the characterization of complex networks. The intricate topology of networks can also be investigated by means of the collective dynamics observed in the interactions of self-sustained oscillations (synchronization patterns) or propagationlike processes such as random walks. However, networks are often inferred from real-data-forming dynamic systems, which are different from those employed to reveal their topological characteristics. This stresses the necessity for a theoretical framework dedicated to the mutual relationship between the structure and dynamics in complex networks, as the two sides of the same coin. Here we propose a rigorous framework based on the network response over time (i.e., Green function) to study interactions between nodes across time. For this purpose we define the flow that describes the interplay between the network connectivity and external inputs. This multivariate measure relates to the concepts of graph communicability and the map equation. We illustrate our theory using the multivariate Ornstein-Uhlenbeck process, which describes stable and non-conservative dynamics, but the formalism can be adapted to other local dynamics for which the Green function is known. We provide applications to classical network examples, such as small-world ring and hierarchical networks. Our theory defines a comprehensive framework that is canonically related to directed and weighted networks, thus paving a way to revise the standards for network analysis, from the pairwise interactions between nodes to the global properties of networks including community detection.M.G. acknowledges funding from the Marie SkłodowskaCurie Action (Grant No. H2020-MSCA-656547) of the European Commission. G.Z.L., N.E.K., and G.D. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 720270 (HBP SGA1). G.D. also acknowledges funding from the European Research Council Advanced Grant DYSTRUCTURE (No. 295129) and the Spanish Research Project (No. PSI2013-42091-P). N.E.K. acknowledges support by the “MOVE-IN Louvain” fellowship cofunded by the Marie Skłodowska-Curie Action of the European Commission