Approximating the largest eigenvalue of network adjacency matrices

Eigenvalue spectra of finely structured random matrices

Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are applicable in a myriad of different contexts. However, it is often assumed that all components of the complex system in question are statistically equivalent, which is unrealistic in many applications. Here we introduce the concept of a finely structured random matrix. These are random matrices with element-specific statistics, which can be used to model systems in which the individual components are statistically distinct. By supposing that the degree of "fine structure" in the matrix is small, we arrive at a succinct "modified" elliptical law. We demonstrate the direct applicability of our results to the niche and cascade models in theoretical ecology, as well as a model of a neural network, and a directed network with arbitrary degree distribution. The simple closed form of our central results allow us to draw broad qualitative conclusions about the effect of fine structure on stability.

Strength and resilience of developing brain circuits predict adolescent emotional and stress responses during the COVID-19 pandemic

The COVID-19 pandemic has had profound but incompletely understood adverse effects on youth. To elucidate the role of brain circuits in how adolescents responded to the pandemic's stressors, we investigated their prepandemic organization as a predictor of mental/emotional health in the first ~15 months of the pandemic. We analyzed resting-state networks from n = 2,641 adolescents [median age (interquartile range) = 144.0 (13.0) months, 47.7% females] in the Adolescent Brain Cognitive Development study, and longitudinal assessments of mental health, stress, sadness, and positive affect, collected every 2 to 3 months from May 2020 to May 2021. Topological resilience and/or network strength predicted overall mental health, stress and sadness (but not positive affect), at multiple time points, but primarily in December 2020 and May 2021. Higher resilience of the salience network predicted better mental health in December 2020 (β = 0.19, 95% CI = [0.06, 0.31], P = 0.01). Lower connectivity of left salience, reward, limbic, and prefrontal cortex and its thalamic, striatal, amygdala connections, predicted higher stress (β = -0.46 to -0.20, CI = [-0.72, -0.07], P < 0.03). Lower bilateral robustness (higher fragility) and/or connectivity of these networks predicted higher sadness in December 2020 and May 2021 (β = -0.514 to -0.19, CI = [-0.81, -0.05], P < 0.04). These findings suggest that the organization of brain circuits may have played a critical role in adolescent stress and mental/emotional health during the pandemic.

Modulatory effects of fMRI acquisition time of day, week and year on adolescent functional connectomes across spatial scales: Implications for inference

Metabolic, hormonal, autonomic and physiological rhythms may have a significant impact on cerebral hemodynamics and intrinsic brain synchronization measured with fMRI (the resting-state connectome). The impact of their characteristic time scales (hourly, circadian, seasonal), and consequently scan timing effects, on brain topology in inherently heterogeneous developing connectomes remains elusive. In a cohort of 4102 early adolescents with resting-state fMRI (median age = 120.0 months; 53.1 % females) from the Adolescent Brain Cognitive Development Study, this study investigated associations between scan time-of-day, time-of-week (school day vs weekend) and time-of-year (school year vs summer vacation) and topological properties of resting-state connectomes at multiple spatial scales. On average, participants were scanned around 2 pm, primarily during school days (60.9 %), and during the school year (74.6 %). Scan time-of-day was negatively correlated with multiple whole-brain, network-specific and regional topological properties (with the exception of a positive correlation with modularity), primarily of visual, dorsal attention, salience, frontoparietal control networks, and the basal ganglia. Being scanned during the weekend (vs a school day) was correlated with topological differences in the hippocampus and temporoparietal networks. Being scanned during the summer vacation (vs the school year) was consistently positively associated with multiple topological properties of bilateral visual, and to a lesser extent somatomotor, dorsal attention and temporoparietal networks. Time parameter interactions suggested that being scanned during the weekend and summer vacation enhanced the positive effects of being scanned in the morning. Time-of-day effects were overall small but spatially extensive, and time-of-week and time-of-year effects varied from small to large (Cohen's f ≤ 0.1, Cohen's d<0.82, p < 0.05). Together, these parameters were also positively correlated with temporal fMRI signal variability but only in the left hemisphere. Finally, confounding effects of scan time parameters on relationships between connectome properties and cognitive task performance were assessed using the ABCD neurocognitive battery. Although most relationships were unaffected by scan time parameters, their combined inclusion eliminated associations between properties of visual and somatomotor networks and performance in the Matrix Reasoning and Pattern Comparison Processing Speed tasks. Thus, scan time of day, week and year may impact measurements of adolescent brain's functional circuits, and should be accounted for in studies on their associations with cognitive performance, in order to reduce the probability of incorrect inference.

Excess BMI in early adolescence adversely impacts maturating functional circuits supporting high-level cognition and their structural correlates

BACKGROUND/OBJECTIVES

Adverse effects of excess BMI (affecting 1 in 5 children in the US) on brain circuits during neurodevelopmentally vulnerable periods are incompletely understood. This study investigated BMI-related alterations in maturating functional networks and their underlying brain structures, and high-level cognition in early adolescence.

SUBJECTS/METHODS

Cross-sectional resting-state fMRI, structural sMRI, neurocognitive task scores, and BMI from 4922 youth [median (IQR) age = 120.0 (13.0) months, 2572 females (52.25%)] from the Adolescent Brain Cognitive Development (ABCD) cohort were analyzed. Comprehensive topological and morphometric network properties were estimated from fMRI and sMRI, respectively. Cross-validated linear regression models assessed correlations with BMI. Results were reproduced across multiple fMRI datasets.

RESULTS

Almost 30% of youth had excess BMI, including 736 (15.0%) with overweight and 672 (13.7%) with obesity, and statistically more Black and Hispanic compared to white, Asian and non-Hispanic youth (p < 0.01). Those with obesity or overweight were less physically active, slept less than recommended, snored more frequently, and spent more time using an electronic device (p < 0.01). They also had lower topological efficiency, resilience, connectivity, connectedness and clustering in Default-Mode, dorsal attention, salience, control, limbic, and reward networks (p ≤ 0.04, Cohen's d: 0.07-0.39). Lower cortico-thalamic efficiency and connectivity were estimated only in youth with obesity (p < 0.01, Cohen's d: 0.09-0.19). Both groups had lower cortical thickness, volume and white matter intensity in these networks' constituent structures, particularly anterior cingulate, entorhinal, prefrontal, and lateral occipital cortices (p < 0.01, Cohen's d: 0.12-0.30), which also mediated inverse relationships between BMI and regional functional topologies. Youth with obesity or overweight had lower scores in a task measuring fluid reasoning - a core aspect of cognitive function, which were partially correlated with topological changes (p ≤ 0.04).

CONCLUSIONS

Excess BMI in early adolescence may be associated with profound aberrant topological alterations in maturating functional circuits and underdeveloped brain structures that adversely impact core aspects of cognitive function.

Emergence of protective behaviour under different risk perceptions to disease spreading

The behaviour of individuals is a main actor in the control of the spread of a communicable disease and, in turn, the spread of an infectious disease can trigger behavioural changes in a population. Here, we study the emergence of individuals' protective behaviours in response to the spread of a disease by considering two different social attitudes within the same population: concerned and risky. Generally speaking, concerned individuals have a larger risk aversion than risky individuals. To study the emergence of protective behaviours, we couple, to the epidemic evolution of a susceptible-infected-susceptible model, a decision game based on the perceived risk of infection. Using this framework, we find the effect of the protection strategy on the epidemic threshold for each of the two subpopulations (concerned and risky), and study under which conditions risky individuals are persuaded to protect themselves or, on the contrary, can take advantage of a herd immunity by remaining healthy without protecting themselves, thanks to the shield provided by concerned individuals. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Hypergraph assortativity: A dynamical systems perspective

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.

Percolation in networks with local homeostatic plasticity

Percolation is a process that impairs network connectedness by deactivating links or nodes. This process features a phase transition that resembles paradigmatic critical transitions in epidemic spreading, biological networks, traffic and transportation systems. Some biological systems, such as networks of neural cells, actively respond to percolation-like damage, which enables these structures to maintain their function after degradation and aging. Here we study percolation in networks that actively respond to link damage by adopting a mechanism resembling synaptic scaling in neurons. We explain critical transitions in such active networks and show that these structures are more resilient to damage as they are able to maintain a stronger connectedness and ability to spread information. Moreover, we uncover the role of local rescaling strategies in biological networks and indicate a possibility of designing smart infrastructures with improved robustness to perturbations.

Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks

We discuss here the application of the simultaneous block diagonalization (SBD) of matrices to the study of the stability of both complete and cluster synchronization in random (generic) networks. For both problems, we define indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. We then see that in the case of random networks the extent of the dimensionality reduction achievable is the same as that produced by application of a trivial transformation.

Phase fMRI defines brain resting-state functional hubs within central and posterior regions

From a brain functional connectivity (FC) matrix, we can identify the hub nodes by a new method of eigencentrality mapping, which not only counts for one node's centrality but also all other nodes' centrality values through correlation connections in an eigenvector of the FC matrix. For the resting-state functional MRI (fMRI) data (complex-valued EPI images in nature), both magnitude and phase images are useful for brain FC analysis. We herein report on brain functional hubness analysis by constructing the FC matrix from phase fMRI data and identifying the hub nodes by eigencentrality mapping. In our study, we collected a cohort of 160 complex-valued fMRI dataset (consisting of magnitude and phase in pairs), and performed independent component analysis (ICA), FC matrix calculation (in size of 50 × 50) and FC matrix eigen decomposition; thereby obtained the 50-node eigencentrality values in the eigenvector associated with the largest eigenvalue. We also compared the hub structures inferred from FC matrices under different thresholding. Alternatively, we obtained the geometric hubs among p value the 50 nodes involved in the FC matrix through the use of harmonic centrality metric. Our results showed that phase fMRI data analysis defines the resting-state brain functional hubs primarily in the central region (subcortex) and the posterior region (parieto-occipital lobes and cerebella). The brain central hubness was supported by the geometric central hubness, which, however, is distinct from the magnitude-inferred hubness in brain superior region (frontal and parietal lobes). Our findings pose a new understanding of (or a debate over) brain functional connectivity architecture.

Persistent individual bias in a voter model with quenched disorder

Many theoretical studies of the voter model (or variations thereupon) involve order parameters that are population-averaged. While enlightening, such quantities may obscure important statistical features that are only apparent on the level of the individual. In this work, we ask which factors contribute to a single voter maintaining a long-term statistical bias for one opinion over the other in the face of social influence. To this end, a modified version of the network voter model is proposed, which also incorporates quenched disorder in the interaction strengths between individuals and the possibility of antagonistic relationships. We find that a sparse interaction network and heterogeneity in interaction strengths give rise to the possibility of arbitrarily long-lived individual biases, even when there is no population-averaged bias for one opinion over the other. This is demonstrated by calculating the eigenvalue spectrum of the weighted network Laplacian using the theory of sparse random matrices.

Role of anaxonic local neurons in the crossover to continuously varying exponents for avalanche activity

Local anaxonic neurons with graded potential release are important ingredients of nervous systems, present in the olfactory bulb system of mammalians and in the human visual system, as well as in arthropods and nematodes. We develop a neuronal network model including both axonic and anaxonic neurons and monitor the activity tuned by the following parameters: the decay length of the graded potential in local neurons, the fraction of local neurons, the largest eigenvalue of the adjacency matrix, and the range of connections of the local neurons. Tuning the fraction of local neurons, we derive the phase diagram including two transition lines: a critical line separating subcritical and supercritical regions, characterized by power-law distributions of avalanche sizes and durations, and a bifurcation line. We find that the overall behavior of the system is controlled by a parameter tuning the relevance of local neuron transmission with respect to the axonal one. The statistical properties of spontaneous activity are affected by local neurons at large fractions and on the condition that the graded potential transmission dominates the axonal one. In this case the scaling properties of spontaneous activity exhibit continuously varying exponents, rather than the mean-field branching model universality class.

Widespread Positive Direct and Indirect Effects of Regular Physical Activity on the Developing Functional Connectome in Early Adolescence

Adolescence is a period of profound but incompletely understood changes in the brain's neural circuitry (the connectome), which is vulnerable to risk factors such as unhealthy weight, but may be protected by positive factors such as regular physical activity. In 5955 children (median age = 120 months; 50.86% females) from the Adolescent Brain Cognitive Development (ABCD) cohort, we investigated direct and indirect (through impact on body mass index [BMI]) effects of physical activity on resting-state networks, the backbone of the functional connectome that ubiquitously affects cognitive function. We estimated significant positive effects of regular physical activity on network connectivity, efficiency, robustness and stability (P ≤ 0.01), and on local topologies of attention, somatomotor, frontoparietal, limbic, and default-mode networks (P < 0.05), which support extensive processes, from memory and executive control to emotional processing. In contrast, we estimated widespread negative BMI effects in the same network properties and brain regions (P < 0.05). Additional mediation analyses suggested that physical activity could also modulate network topologies leading to better control of food intake, appetite and satiety, and ultimately lower BMI. Thus, regular physical activity may have extensive positive effects on the development of the functional connectome, and may be critical for improving the detrimental effects of unhealthy weight on cognitive health.

Network science characteristics of brain-derived neuronal cultures deciphered from quantitative phase imaging data

Understanding the mechanisms by which neurons create or suppress connections to enable communication in brain-derived neuronal cultures can inform how learning, cognition and creative behavior emerge. While prior studies have shown that neuronal cultures possess self-organizing criticality properties, we further demonstrate that in vitro brain-derived neuronal cultures exhibit a self-optimization phenomenon. More precisely, we analyze the multiscale neural growth data obtained from label-free quantitative microscopic imaging experiments and reconstruct the in vitro neuronal culture networks (microscale) and neuronal culture cluster networks (mesoscale). We investigate the structure and evolution of neuronal culture networks and neuronal culture cluster networks by estimating the importance of each network node and their information flow. By analyzing the degree-, closeness-, and betweenness-centrality, the node-to-node degree distribution (informing on neuronal interconnection phenomena), the clustering coefficient/transitivity (assessing the “small-world” properties), and the multifractal spectrum, we demonstrate that murine neurons exhibit self-optimizing behavior over time with topological characteristics distinct from existing complex network models. The time-evolving interconnection among murine neurons optimizes the network information flow, network robustness, and self-organization degree. These findings have complex implications for modeling neuronal cultures and potentially on how to design biological inspired artificial intelligence.

Dismantling complex networks based on the principal eigenvalue of the adjacency matrix

The connectivity of complex networks is usually determined by a small fraction of key nodes. Earlier works successfully identify an influential single node, yet have some problems for the case of multiple ones. In this paper, based on the matrix spectral theory, we propose the collective influence of multiple nodes. An interesting finding is that some traditionally influential nodes have strong internal coupling interactions that reduce their collective influence. We then propose a greedy algorithm to dismantle complex networks by optimizing the collective influence of multiple nodes. Experimental results show that our proposed method outperforms the state of the art methods in terms of the principal eigenvalue and the giant component of the remaining networks.

Uncertainty propagation in complex networks: From noisy links to critical properties

Many complex networks are built up from empirical data prone to experimental error. Thus, the determination of the specific weights of the links is a noisy measure. Noise propagates to those macroscopic variables researchers are interested in, such as the critical threshold for synchronization of coupled oscillators or for the spreading of a disease. Here, we apply error propagation to estimate the macroscopic uncertainty in the critical threshold for some dynamical processes in networks with noisy links. We obtain closed form expressions for the mean and standard deviation of the critical threshold depending on the properties of the noise and the moments of the degree distribution of the network. The analysis provides confidence intervals for critical predictions when dealing with uncertain measurements or intrinsic fluctuations in empirical networked systems. Furthermore, our results unveil a nonmonotonous behavior of the uncertainty of the critical threshold that depends on the specific network structure.

Network structure and risk-based surveillance algorithms for live shrimp movements in Thailand

Live shrimp movements pose a potential route for site-to-site transmission of acute hepatopancreatic necrosis disease (AHPND) and other shrimp diseases. We present the first application of network theory to study shrimp epizootiology, providing quantitative information about the live shrimp movement network of Thailand (LSMN), and supporting practical and policy implementations of disease surveillance and control measures. We examined the LSMN over a 13-month period from March 2013 to March 2014, with data obtained from the Thailand Department of Fisheries. The LSMN had a mixture of characteristics both limiting and facilitating disease spread. Importantly, the LSMN exhibited power-law distributions of in and out degrees with exponents of 2.87 and 2.17, respectively. This characteristic indicates that the LSMN behaves like a scale-free network and suggests that an effective strategy to control disease spread in the Thai shrimp farming sector can be achieved by removing a small number of targeted inter-site connections (arcs between nodes). Specifically, a disease-control algorithm based on betweenness centrality (defined as the number of shortest paths between node pairs that traverse a given arc) is proposed here to prioritize targets for disease surveillance and control measures.

Spectral properties of complex networks

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

Fisher information and criticality in the Kuramoto model of nonidentical oscillators

We use the Fisher information to provide a lens on the transition to synchronization of the Kuramoto model of nonidentical frequencies on a variety of undirected graphs. We numerically solve the equations of motion for a N=400 complete graph and N=1000 small-world, scale-free, uniform random, and random regular graphs. For large but finite graphs of small average diameter the Fisher information F as a function of coupling shows a peak closely coinciding with the critical point as determined by Kuramoto's order parameter or synchronization measure r. However, for graphs of larger average diameter the position of the peak in F differs from the critical point determined by estimates of r. On the one hand, this is a finite-size effect even at N=1000; however, we show across a range of topologies that the Fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase-synchronized clusters, often directly from metastable to stable frequency synchronization. Solving explicitly for a two-cluster ansatz subject to Gaussian noise shows that the Fisher infomation peaks at such a transition. We discuss the implications for Fisher information as an indicator for edge-of-chaos phenomena in finite-coupled oscillator systems.

Age-Related Changes in Topological Degradation of White Matter Networks and Gene Expression in Chronic Schizophrenia

Current hypotheses stipulate core symptoms of schizophrenia (SZ) result from the brain's incapacity to integrate neural processes. Converging diffusion magnetic resonance imaging and graph theory studies provide evidence of macrostructural alterations in SZ. However, age-related topological changes within and between white matter (WM) networks and its relationship to gene expression with disease progression remain incompletely understood. This cross-sectional study uses network modeling to investigate changes in WM network organization with disease progression in chronic SZ as well its relationship with gene expression in healthy brains. First, we replicate prior findings demonstrating altered global WM network topology in SZ. Novel results show significantly altered age-related network degradation patterns in patients compared with controls. Specifically, controls show stereotyped, linear global network decline with age. In contrast, patients show nonlinear network decline with age. Further analysis reveals lack of significant topological decline in younger adult patients, which is subsequently followed by stereotyped linear decline in older adult patients. Node-specific analyses show significant topological differences in frontal and limbic regions of younger adult patients compared with age-matched controls, which become less pronounced with age in older adult patients compared with age-matched controls. Lastly, we show several gene expression profiles, including DISC1, are associated with age-related changes in WM disconnectivity. Together, these findings provide novel WM topological and genetic evidence supporting neurodevelopmental models of SZ, suggesting that network remodeling continues throughout the third decade of life before stabilizing.

A dynamic vaccination strategy to suppress the recurrent epidemic outbreaks

Efficient vaccination strategy is crucial for controlling recurrent epidemic spreading on networks. In this paper, based on the analysis of real epidemic data and simulations, it's found that the risk indicator of recurrent epidemic outbreaks could be determined by the ratio of the epidemic infection rate of the year to the average infected density of the former year. According to the risk indicator, the dynamic vaccination probability of each year can be designed to suppress the epidemic outbreaks. Our simulation results show that the dynamic vaccination strategy could effectively decrease the maximal and average infected density, and meanwhile increase the time intervals of epidemic outbreaks and individuals attacked by epidemic. In addition, our results indicate that to depress the influenza outbreaks, it is not necessary to keep the vaccination probability high every year; and adjusting the vaccination probability at right time could decrease the outbreak risks with lower costs. Our findings may present a theoretical guidance for the government and the public to control the recurrent epidemic outbreaks.

Refractory period in network models of excitable nodes: self-sustaining stable dynamics, extended scaling region and oscillatory behavior

Networks of excitable nodes have recently attracted much attention particularly in regards to neuronal dynamics, where criticality has been argued to be a fundamental property. Refractory behavior, which limits the excitability of neurons is thought to be an important dynamical property. We therefore consider a simple model of excitable nodes which is known to exhibit a transition to instability at a critical point (λ = 1), and introduce refractory period into its dynamics. We use mean-field analytical calculations as well as numerical simulations to calculate the activity dependent branching ratio that is useful to characterize the behavior of critical systems. We also define avalanches and calculate probability distribution of their size and duration. We find that in the presence of refractory period the dynamics stabilizes while various parameter regimes become accessible. A sub-critical regime with λ < 1.0, a standard critical behavior with exponents close to critical branching process for λ = 1, a regime with 1 < λ < 2 that exhibits an interesting scaling behavior, and an oscillating regime with λ > 2.0. We have therefore shown that refractory behavior leads to a wide range of scaling as well as periodic behavior which are relevant to real neuronal dynamics.

Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment

This study highlights the relevance of network-guided controllability analysis as a precision oncology tool. Target controllability through networks is potentially relevant to cancer research for the identification of therapeutic targets. With reference to a recent study on multiple phenotypes from 22 osteosarcoma (OS) cell lines characterized both in vitro and in vivo, we found that a variety of critical proteins in OS regulation circuits were in part phenotype specific and in part shared. To generalize our inference approach and match cancer phenotypic heterogeneity, we employed multitype networks and identified targets in correspondence with protein sub-complexes. Therefore, we established the relevance for diagnostic and therapeutic purposes of inspecting interactive targets, namely those enriched by significant connectivity patterns in protein sub-complexes. Emerging targets appeared with reference to the OS microenvironment, and relatively to small leucine-rich proteoglycan members and D-type cyclins, among other collagen, laminin, and keratin proteins. These described were evidences shared across all phenotypes; instead, specific evidences were provided by critical proteins including IGFBP7 and PDGFRA in the invasive phenotype, and FGFR3 and THBS1 in the colony forming phenotype.

Correlations induced by depressing synapses in critically self-organized networks with quenched dynamics

In a recent work, mean-field analysis and computer simulations were employed to analyze critical self-organization in networks of excitable cellular automata where randomly chosen synapses in the network were depressed after each spike (the so-called annealed dynamics). Calculations agree with simulations of the annealed version, showing that the nominal branching ratio σ converges to unity in the thermodynamic limit, as expected of a self-organized critical system. However, the question remains whether the same results apply to the biological case where only the synapses of firing neurons are depressed (the so-called quenched dynamics). We show that simulations of the quenched model yield significant deviations from σ=1 due to spatial correlations. However, the model is shown to be critical, as the largest eigenvalue of the synaptic matrix approaches unity in the thermodynamic limit, that is, λ_{c}=1. We also study the finite size effects near the critical state as a function of the parameters of the synaptic dynamics.

Mean-field equations for neuronal networks with arbitrary degree distributions

The emergent dynamics in networks of recurrently coupled spiking neurons depends on the interplay between single-cell dynamics and network topology. Most theoretical studies on network dynamics have assumed simple topologies, such as connections that are made randomly and independently with a fixed probability (Erdös-Rényi network) (ER) or all-to-all connected networks. However, recent findings from slice experiments suggest that the actual patterns of connectivity between cortical neurons are more structured than in the ER random network. Here we explore how introducing additional higher-order statistical structure into the connectivity can affect the dynamics in neuronal networks. Specifically, we consider networks in which the number of presynaptic and postsynaptic contacts for each neuron, the degrees, are drawn from a joint degree distribution. We derive mean-field equations for a single population of homogeneous neurons and for a network of excitatory and inhibitory neurons, where the neurons can have arbitrary degree distributions. Through analysis of the mean-field equations and simulation of networks of integrate-and-fire neurons, we show that such networks have potentially much richer dynamics than an equivalent ER network. Finally, we relate the degree distributions to so-called cortical motifs.

EIGENVECTOR-BASED CENTRALITY MEASURES FOR TEMPORAL NETWORKS. MULTISCALE MODELING & SIMULATION : A SIAM INTERDISCIPLINARY JOURNAL 2017;15:537-574. [PMID: 29046619 PMCID: PMC5643020 DOI: 10.1137/16m1066142]

Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a principled generalization of network centrality measures that is valid for any eigenvector-based centrality. We consider a temporal network with N nodes as a sequence of T layers that describe the network during different time windows, and we couple centrality matrices for the layers into a supra-centrality matrix of size NT × NT whose dominant eigenvector gives the centrality of each node i at each time t. We refer to this eigenvector and its components as a joint centrality, as it reflects the importances of both the node i and the time layer t. We also introduce the concepts of marginal and conditional centralities, which facilitate the study of centrality trajectories over time. We find that the strength of coupling between layers is important for determining multiscale properties of centrality, such as localization phenomena and the time scale of centrality changes. In the strong-coupling regime, we derive expressions for time-averaged centralities, which are given by the zeroth-order terms of a singular perturbation expansion. We also study first-order terms to obtain first-order-mover scores, which concisely describe the magnitude of nodes' centrality changes over time. As examples, we apply our method to three empirical temporal networks: the United States Ph.D. exchange in mathematics, costarring relationships among top-billed actors during the Golden Age of Hollywood, and citations of decisions from the United States Supreme Court.

Multiplex visibility graphs to investigate recurrent neural network dynamics

A recurrent neural network (RNN) is a universal approximator of dynamical systems, whose performance often depends on sensitive hyperparameters. Tuning them properly may be difficult and, typically, based on a trial-and-error approach. In this work, we adopt a graph-based framework to interpret and characterize internal dynamics of a class of RNNs called echo state networks (ESNs). We design principled unsupervised methods to derive hyperparameters configurations yielding maximal ESN performance, expressed in terms of prediction error and memory capacity. In particular, we propose to model time series generated by each neuron activations with a horizontal visibility graph, whose topological properties have been shown to be related to the underlying system dynamics. Successively, horizontal visibility graphs associated with all neurons become layers of a larger structure called a multiplex. We show that topological properties of such a multiplex reflect important features of ESN dynamics that can be used to guide the tuning of its hyperparamers. Results obtained on several benchmarks and a real-world dataset of telephone call data records show the effectiveness of the proposed methods.

Modeling the network dynamics of pulse-coupled neurons

We derive a mean-field approximation for the macroscopic dynamics of large networks of pulse-coupled theta neurons in order to study the effects of different network degree distributions and degree correlations (assortativity). Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)], we obtain a reduced system of ordinary differential equations describing the mean-field dynamics, with significantly lower dimensionality compared with the complete set of dynamical equations for the system. We find that, for sufficiently large networks and degrees, the dynamical behavior of the reduced system agrees well with that of the full network. This dimensional reduction allows for an efficient characterization of system phase transitions and attractors. For networks with tightly peaked degree distributions, the macroscopic behavior closely resembles that of fully connected networks previously studied by others. In contrast, networks with highly skewed degree distributions exhibit different macroscopic dynamics due to the emergence of degree dependent behavior of different oscillators. For nonassortative networks (i.e., networks without degree correlations), we observe the presence of a synchronously firing phase that can be suppressed by the presence of either assortativity or disassortativity in the network. We show that the results derived here can be used to analyze the effects of network topology on macroscopic behavior in neuronal networks in a computationally efficient fashion.

Spectral properties of the hierarchical product of graphs

The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large coupling, as well as the leading-order relaxation to these values in terms of the eigenvalues and eigenvectors of the two smaller graphs. Given its central roll in the structural and dynamical properties of networks, we study in detail the Perron-Frobenius, or largest, eigenvalue. Finally, as an example application we use our theory to predict the epidemic threshold of the susceptible-infected-susceptible model on a hierarchical product of two graphs.

Reducing the Spectral Radius of a Torus Network by Link Removal

The optimal link removal (OLR) problem aims at removing a given number of links of a network so that the spectral radius of the residue network obtained by removing the links from the network attains the minimum. Torus networks are a class of regular networks that have witnessed widespread applications. This paper addresses three subproblems of the OLR problem for torus networks, where two or three or four edges are removed. For either of the three subproblems, a link-removing scheme is described. Exhaustive searches show that, for small-sized tori, each of the proposed schemes produces an optimal solution to the corresponding subproblem. Monte-Carlo simulations demonstrate that, for medium-sized tori, each of the three schemes produces a solution to the corresponding subproblem, which is optimal when compared to a large set of randomly produced link-removing schemes. Consequently, it is speculated that each of the three schemes produces an optimal solution to the corresponding subproblem for all torus networks. The set of links produced by each of our schemes is evenly distributed over a network, which may be a common feature of an optimal solution to the OLR problem for regular networks.

Neural Network Spectral Robustness under Perturbations of the Underlying Graph

Recent studies have been using graph-theoretical approaches to model complex networks (such as social, infrastructural, or biological networks) and how their hardwired circuitry relates to their dynamic evolution in time. Understanding how configuration reflects on the coupled behavior in a system of dynamic nodes can be of great importance, for example, in the context of how the brain connectome is affecting brain function. However, the effect of connectivity patterns on network dynamics is far from being fully understood. We study the connections between edge configuration and dynamics in a simple oriented network composed of two interconnected cliques (representative of brain feedback regulatory circuitry). In this article our main goal is to study the spectra of the graph adjacency and Laplacian matrices, with a focus on three aspects in particular: (1) the sensitivity and robustness of the spectrum in response to varying the intra- and intermodular edge density, (2) the effects on the spectrum of perturbing the edge configuration while keeping the densities fixed, and (3) the effects of increasing the network size. We study some tractable aspects analytically, then simulate more general results numerically, thus aiming to motivate and explain our further work on the effect of these patterns on the network temporal dynamics and phase transitions. We discuss the implications of such results to modeling brain connectomics. We suggest potential applications to understanding synaptic restructuring in learning networks and the effects of network configuration on function of regulatory neural circuits.

Low-rank network decomposition reveals structural characteristics of small-world networks

Small-world networks occur naturally throughout biological, technological, and social systems. With their prevalence, it is particularly important to prudently identify small-world networks and further characterize their unique connection structure with respect to network function. In this work we develop a formalism for classifying networks and identifying small-world structure using a decomposition of network connectivity matrices into low-rank and sparse components, corresponding to connections within clusters of highly connected nodes and sparse interconnections between clusters, respectively. We show that the network decomposition is independent of node indexing and define associated bounded measures of connectivity structure, which provide insight into the clustering and regularity of network connections. While many existing network characterizations rely on constructing benchmark networks for comparison or fail to describe the structural properties of relatively densely connected networks, our classification relies only on the intrinsic network structure and is quite robust with respect to changes in connection density, producing stable results across network realizations. Using this framework, we analyze several real-world networks and reveal new structural properties, which are often indiscernible by previously established characterizations of network connectivity.

Hamiltonian mean field model: Effect of network structure on synchronization dynamics

The Hamiltonian mean field model of coupled inertial Hamiltonian rotors is a prototype for conservative dynamics in systems with long-range interactions. We consider the case where the interactions between the rotors are governed by a network described by a weighted adjacency matrix. By studying the linear stability of the incoherent state, we find that the transition to synchrony begins when the coupling constant K is inversely proportional to the largest eigenvalue of the adjacency matrix. We derive a closed system of equations for a set of local order parameters to study the effect of network heterogeneity on the synchronization of the rotors. When K is just beyond the transition to synchronization, we find that the degree of synchronization is highly dependent on the network's heterogeneity, but that for large K the degree of synchronization is robust to changes in the degree distribution. Our results are illustrated with numerical simulations on Erdös-Renyi networks and networks with power-law degree distributions.

Effects of random rewiring on the degree correlation of scale-free networks

Random rewiring is used to generate null networks for the purpose of analyzing the topological properties of scale-free networks, yet the effects of random rewiring on the degree correlation are subject to contradicting interpretations in the literature. We comprehensively analyze the degree correlation of randomly rewired scale-free networks and show that random rewiring increases disassortativity by reducing the average degree of the nearest neighbors of high-degree nodes. The effect can be captured by the measures of the degree correlation that consider all links in the network, but not by analogous measures that consider only links between degree peers, hence the potential for contradicting interpretations. We furthermore find that random and directional rewiring affect the topology of a scale-free network differently, even if the degree correlation of the rewired networks is the same. Consequently, the network dynamics is changed, which is proven here by means of the biased random walk.

The effects of connectivity on metapopulation persistence: network symmetry and degree correlations

A spatial metapopulation is a mosaic of interconnected patch populations. The complex routes of colonization between the patches are governed by the metapopulation's dispersal network. Over the past two decades, there has been considerable interest in uncovering the effects of dispersal network topology and its symmetry on metapopulation persistence. While most studies find that the level of symmetry in dispersal pattern enhances persistence, some have reached the conclusion that symmetry has at most a minor effect. In this work, we present a new perspective on the debate. We study properties of the in- and out-degree distribution of patches in the metapopulation which define the number of dispersal routes into and out of a particular patch, respectively. By analysing the spectral radius of the dispersal matrices, we confirm that a higher level of symmetry has only a marginal impact on persistence. We continue to analyse different properties of the in-out degree distribution, namely the 'in-out degree correlation' (IODC) and degree heterogeneity, and find their relationship to metapopulation persistence. Our analysis shows that, in contrast to symmetry, the in-out degree distribution and particularly, the IODC are dominant factors controlling persistence.

Stability of Boolean networks: the joint effects of topology and update rules

We study the stability of orbits in large Boolean networks. We treat the case in which the network has a given complex topology, and we do not assume a specific form for the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of complex network topology and certain classes of update rules on stability, only crude results exist about how these effects interact. We present a widely applicable solution to this problem. Numerical simulations confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.

Inhibition causes ceaseless dynamics in networks of excitable nodes

The collective dynamics of a network of excitable nodes changes dramatically when inhibitory nodes are introduced. We consider inhibitory nodes which may be activated just like excitatory nodes but, upon activating, decrease the probability of activation of network neighbors. We show that, although the direct effect of inhibitory nodes is to decrease activity, the collective dynamics becomes self-sustaining. We explain this counterintuitive result by defining and analyzing a "branching function" which may be thought of as an activity-dependent branching ratio. The shape of the branching function implies that, for a range of global coupling parameters, dynamics are self-sustaining. Within the self-sustaining region of parameter space lies a critical line along which dynamics take the form of avalanches with universal scaling of size and duration, embedded in a ceaseless time series of activity. Our analyses, confirmed by numerical simulation, suggest that inhibition may play a counterintuitive role in excitable networks.

The robustness analysis of wireless sensor networks under uncertain interference

Based on the complex network theory, robustness analysis of condition monitoring wireless sensor network under uncertain interference is present. In the evolution of the topology of sensor networks, the density weighted algebraic connectivity is taken into account, and the phenomenon of removing and repairing the link and node in the network is discussed. Numerical simulation is conducted to explore algebraic connectivity characteristics and network robustness performance. It is found that nodes density has the effect on algebraic connectivity distribution in the random graph model; high density nodes carry more connections, use more throughputs, and may be more unreliable. Moreover, the results show that, when network should be more error tolerant or robust by repairing nodes or adding new nodes, the network should be better clustered in median and high scale wireless sensor networks and be meshing topology in small scale networks.

Structure-dynamics relationships in bursting neuronal networks revealed using a prediction framework

The question of how the structure of a neuronal network affects its functionality has gained a lot of attention in neuroscience. However, the vast majority of the studies on structure-dynamics relationships consider few types of network structures and assess limited numbers of structural measures. In this in silico study, we employ a wide diversity of network topologies and search among many possibilities the aspects of structure that have the greatest effect on the network excitability. The network activity is simulated using two point-neuron models, where the neurons are activated by noisy fluctuation of the membrane potential and their connections are described by chemical synapse models, and statistics on the number and quality of the emergent network bursts are collected for each network type. We apply a prediction framework to the obtained data in order to find out the most relevant aspects of network structure. In this framework, predictors that use different sets of graph-theoretic measures are trained to estimate the activity properties, such as burst count or burst length, of the networks. The performances of these predictors are compared with each other. We show that the best performance in prediction of activity properties for networks with sharp in-degree distribution is obtained when the prediction is based on clustering coefficient. By contrast, for networks with broad in-degree distribution, the maximum eigenvalue of the connectivity graph gives the most accurate prediction. The results shown for small () networks hold with few exceptions when different neuron models, different choices of neuron population and different average degrees are applied. We confirm our conclusions using larger () networks as well. Our findings reveal the relevance of different aspects of network structure from the viewpoint of network excitability, and our integrative method could serve as a general framework for structure-dynamics studies in biosciences.

Scaling behavior in probabilistic neuronal cellular automata

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely, spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint, and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

Social climber attachment in forming networks produces a phase transition in a measure of connectivity

The formation and fragmentation of networks are typically studied using percolation theory, but most previous research has been restricted to studying a phase transition in cluster size, examining the emergence of a giant component. This approach does not study the effects of evolving network structure on dynamics that occur at the nodes, such as the synchronization of oscillators and the spread of information, epidemics, and neuronal excitations. We introduce and analyze an alternative link-formation rule, called social climber (SC) attachment, that may be combined with arbitrary percolation models to produce a phase transition using the largest eigenvalue of the network adjacency matrix as the order parameter. This eigenvalue is significant in the analyses of many network-coupled dynamical systems in which it measures the quality of global coupling and is hence a natural measure of connectivity. We highlight the important self-organized properties of SC attachment and discuss implications for controlling dynamics on networks.

A network function-based definition of communities in complex networks

We consider an alternate definition of community structure that is functionally motivated. We define network community structure based on the function the network system is intended to perform. In particular, as a specific example of this approach, we consider communities whose function is enhanced by the ability to synchronize and/or by resilience to node failures. Previous work has shown that, in many cases, the largest eigenvalue of the network's adjacency matrix controls the onset of both synchronization and percolation processes. Thus, for networks whose functional performance is dependent on these processes, we propose a method that divides a given network into communities based on maximizing a function of the largest eigenvalues of the adjacency matrices of the resulting communities. We also explore the differences between the partitions obtained by our method and the modularity approach (which is based solely on consideration of network structure). We do this for several different classes of networks. We find that, in many cases, modularity-based partitions do almost as well as our function-based method in finding functional communities, even though modularity does not specifically incorporate consideration of function.

Statistical properties of avalanches in networks

We characterize the distributions of size and duration of avalanches propagating in complex networks. By an avalanche we mean the sequence of events initiated by the externally stimulated excitation of a network node, which may, with some probability, then stimulate subsequent excitations of the nodes to which it is connected, resulting in a cascade of excitations. This type of process is relevant to a wide variety of situations, including neuroscience, cascading failures on electrical power grids, and epidemiology. We find that the statistics of avalanches can be characterized in terms of the largest eigenvalue and corresponding eigenvector of an appropriate adjacency matrix that encodes the structure of the network. By using mean-field analyses, previous studies of avalanches in networks have not considered the effect of network structure on the distribution of size and duration of avalanches. Our results apply to individual networks (rather than network ensembles) and provide expressions for the distributions of size and duration of avalanches starting at particular nodes in the network. These findings might find application in the analysis of branching processes in networks, such as cascading power grid failures and critical brain dynamics. In particular, our results show that some experimental signatures of critical brain dynamics (i.e., power-law distributions of size and duration of neuronal avalanches) are robust to complex underlying network topologies.

Stability of Boolean networks with generalized canalizing rules

Boolean networks are discrete dynamical systems in which the state (0 or 1) of each node is updated at each time t to a state determined by the states at time t-1 of those nodes that have links to it. When these systems are used to model genetic control, the case of canalizing update rules is of particular interest. A canalizing rule is one for which a node state at time t is determined by the state at time t-1 of a single one of its inputs when that inputting node is in its canalizing state. Previous work on the order-disorder transition in Boolean networks considered complex, nonrandom network topology. In the current paper we extend this previous work to account for canalizing behavior.

Network extreme eigenvalue: from mutimodal to scale-free networks

The extreme eigenvalues of adjacency matrices are important indicators on the influence of topological structures to the collective dynamical behavior of complex networks. Recent findings on the ensemble averageability of the extreme eigenvalue have further authenticated its applicability to the study of network dynamics. However, the ensemble average of extreme eigenvalue has only been solved analytically up to the second order correction. Here, we determine the ensemble average of the extreme eigenvalue and characterize its deviation across the ensemble through the discrete form of random scale-free network. Remarkably, the analytical approximation derived from the discrete form shows significant improvement over previous results, which implies a more accurate prediction of the epidemic threshold. In addition, we show that bimodal networks, which are more robust against both random and targeted removal of nodes, are more vulnerable to the spreading of diseases.