Model of brain activation predicts the neural collective influence map of the brain

Noise cleaning the precision matrix of short time series

We present a comparison between various algorithms of inference of covariance and precision matrices in small data sets of real vectors of the typical length and dimension of human brain activity time series retrieved by functional magnetic resonance imaging (fMRI). Assuming a Gaussian model underlying the neural activity, the problem consists of denoising the empirically observed matrices to obtain a better estimator of the (unknown) true precision and covariance matrices. We consider several standard noise-cleaning algorithms and compare them on two types of data sets. The first type consists of synthetic time series sampled from a generative Gaussian model of which we can vary the fraction of dimensions per sample q and the strength of off-diagonal correlations. The second type consists of time series of fMRI brain activity of human subjects at rest. The reliability of each algorithm is assessed in terms of test-set likelihood and, in the case of synthetic data, of the distance from the true precision matrix. We observe that the so-called optimal rotationally invariant estimator, based on random matrix theory, leads to a significantly lower distance from the true precision matrix in synthetic data and higher test likelihood in natural fMRI data. We propose a variant of the optimal rotationally invariant estimator in which one of its parameters is optimzed by cross-validation. In the severe undersampling regime (large q) typical of fMRI series, it outperforms all the other estimators. We furthermore propose a simple algorithm based on an iterative likelihood gradient ascent, leading to very accurate estimations in weakly correlated synthetic data sets.

Identifying properties of pattern completion neurons in a computational model of the visual cortex

Neural ensembles are found throughout the brain and are believed to underlie diverse cognitive functions including memory and perception. Methods to activate ensembles precisely, reliably, and quickly are needed to further study the ensembles' role in cognitive processes. Previous work has found that ensembles in layer 2/3 of the visual cortex (V1) exhibited pattern completion properties: ensembles containing tens of neurons were activated by stimulation of just two neurons. However, methods that identify pattern completion neurons are underdeveloped. In this study, we optimized the selection of pattern completion neurons in simulated ensembles. We developed a computational model that replicated the connectivity patterns and electrophysiological properties of layer 2/3 of mouse V1. We identified ensembles of excitatory model neurons using K-means clustering. We then stimulated pairs of neurons in identified ensembles while tracking the activity of the entire ensemble. Our analysis of ensemble activity quantified a neuron pair's power to activate an ensemble using a novel metric called pattern completion capability (PCC) based on the mean pre-stimulation voltage across the ensemble. We found that PCC was directly correlated with multiple graph theory parameters, such as degree and closeness centrality. To improve selection of pattern completion neurons in vivo, we computed a novel latency metric that was correlated with PCC and could potentially be estimated from modern physiological recordings. Lastly, we found that stimulation of five neurons could reliably activate ensembles. These findings can help researchers identify pattern completion neurons to stimulate in vivo during behavioral studies to control ensemble activation.

Unusual Mathematical Approaches Untangle Nervous Dynamics

The massive amount of available neurodata suggests the existence of a mathematical backbone underlying neuronal oscillatory activities. For example, geometric constraints are powerful enough to define cellular distribution and drive the embryonal development of the central nervous system. We aim to elucidate whether underrated notions from geometry, topology, group theory and category theory can assess neuronal issues and provide experimentally testable hypotheses. The Monge’s theorem might contribute to our visual ability of depth perception and the brain connectome can be tackled in terms of tunnelling nanotubes. The multisynaptic ascending fibers connecting the peripheral receptors to the neocortical areas can be assessed in terms of knot theory/braid groups. Presheaves from category theory permit the tackling of nervous phase spaces in terms of the theory of infinity categories, highlighting an approach based on equivalence rather than equality. Further, the physical concepts of soft-matter polymers and nematic colloids might shed new light on neurulation in mammalian embryos. Hidden, unexpected multidisciplinary relationships can be found when mathematics copes with neural phenomena, leading to novel answers for everlasting neuroscientific questions. For instance, our framework leads to the conjecture that the development of the nervous system might be correlated with the occurrence of local thermal changes in embryo–fetal tissues.

Modeling multi-scale data via a network of networks

MOTIVATION

Prediction of node and graph labels are prominent network science tasks. Data analyzed in these tasks are sometimes related: entities represented by nodes in a higher-level (higher scale) network can themselves be modeled as networks at a lower level. We argue that systems involving such entities should be integrated with a 'network of networks' (NoNs) representation. Then, we ask whether entity label prediction using multi-level NoN data via our proposed approaches is more accurate than using each of single-level node and graph data alone, i.e. than traditional node label prediction on the higher-level network and graph label prediction on the lower-level networks. To obtain data, we develop the first synthetic NoN generator and construct a real biological NoN. We evaluate accuracy of considered approaches when predicting artificial labels from the synthetic NoNs and proteins' functions from the biological NoN.

RESULTS

For the synthetic NoNs, our NoN approaches outperform or are as good as node- and network-level ones depending on the NoN properties. For the biological NoN, our NoN approaches outperform the single-level approaches for just under half of the protein functions, and for 30% of the functions, only our NoN approaches make meaningful predictions, while node- and network-level ones achieve random accuracy. So, NoN-based data integration is important.

AVAILABILITY AND IMPLEMENTATION

The software and data are available at https://nd.edu/~cone/NoNs.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Characteristic functional cores revealed by hyperbolic disc embedding and k-core percolation on resting-state fMRI

Hyperbolic disc embedding and k-core percolation reveal the hierarchical structure of functional connectivity on resting-state fMRI (rsfMRI). Using 180 normal adults' rsfMRI data from the human connectome project database, we visualized inter-voxel relations by embedding voxels on the hyperbolic space using the [Formula: see text] model. We also conducted k-core percolation on 30 participants to investigate core voxels for each individual. It recursively peels the layer off, and this procedure leaves voxels embedded in the center of the hyperbolic disc. We used independent components to classify core voxels, and it revealed stereotypes of individuals such as visual network dominant, default mode network dominant, and distributed patterns. Characteristic core structures of resting-state brain connectivity of normal subjects disclosed the distributed or asymmetric contribution of voxels to the kmax-core, which suggests the hierarchical dominance of certain IC subnetworks characteristic of subgroups of individuals at rest.

Link-usage asymmetry and collective patterns emerging from rich-club organization of complex networks

In models of excitable dynamics on graphs, excitations can travel in both directions of an undirected link. However, as a striking interplay of dynamics and network topology, excitations often establish a directional preference. Some of these cases of "link-usage asymmetry" are local in nature and can be mechanistically understood, for instance, from the degree gradient of a link (i.e., the difference in node degrees at both ends of the link). Other contributions to the link-usage asymmetry are instead, as we show, self-organized in nature, and strictly nonlocal. This is the case for excitation waves, where the preferential propagation of excitations along a link depends on its orientation with respect to a hub acting as a source, even if the link in question is several steps away from the hub itself. Here, we identify and quantify the contribution of such self-organized patterns to link-usage asymmetry and show that they extend to ranges significantly longer than those ascribed to local patterns. We introduce a topological characterization, the hub-set-orientation prevalence of a link, which indicates its average orientation with respect to the hubs of a graph. Our numerical results show that the hub-set-orientation prevalence of a link strongly correlates with the preferential usage of the link in the direction of propagation away from the hub core of the graph. Our methodology is embedding-agnostic and allows for the measurement of wave signals and the sizes of the cores from which they originate.

Critical behaviors of high-degree adaptive and collective-influence percolation

How the giant component of a network disappears under attacking nodes or links addresses a key aspect of network robustness, which can be framed into percolation problems. Various strategies to select the node to be deactivated have been studied in the literature, for instance, a simple random failure or high-degree adaptive (HDA) percolation. Recently, a new attack strategy based on a quantity called collective-influence (CI) has been proposed from the perspective of optimal percolation. By successively deactivating the node having the largest CI-centrality value, it was shown to be able to dismantle a network more quickly and abruptly than many of the existing methods. In this paper, we focus on the critical behaviors of the percolation processes following degree-based attack and CI-based attack on random networks. Through extensive Monte Carlo simulations assisted by numerical solutions, we estimate various critical exponents of the HDA percolation and those of the CI percolations. Our results show that these attack-type percolation processes, despite displaying apparently more abrupt collapse, nevertheless exhibit standard mean-field critical behaviors at the percolation transition point. We further discover an extensive degeneracy in top-centrality nodes in both processes, which may provide a hint for understanding the observed results.

Current methods and new directions in resting state fMRI

Resting state functional connectivity magnetic resonance imaging (rsfcMRI) has become a key component of investigations of neurocognitive and psychiatric behaviors. Over the past two decades, several methods and paradigms have been adopted to utilize and interpret data from resting-state fluctuations in the brain. These findings have increased our understanding of changes in many disease states. As the amount of resting state data available for research increases with big datasets and data-sharing projects, it is important to review the established traditional analysis methods and recognize areas where research methodology can be adapted to better accommodate the scale and complexity of rsfcMRI analysis. In this paper, we review established methods of analysis as well as areas that have been receiving increasing attention such as dynamic rsfcMRI, independent vector analysis, multiband rsfcMRI and network of networks.

Influencer identification in dynamical complex systems

The integrity and functionality of many real-world complex systems hinge on a small set of pivotal nodes, or influencers. In different contexts, these influencers are defined as either structurally important nodes that maintain the connectivity of networks, or dynamically crucial units that can disproportionately impact certain dynamical processes. In practice, identification of the optimal set of influencers in a given system has profound implications in a variety of disciplines. In this review, we survey recent advances in the study of influencer identification developed from different perspectives, and present state-of-the-art solutions designed for different objectives. In particular, we first discuss the problem of finding the minimal number of nodes whose removal would breakdown the network (i.e. the optimal percolation or network dismantle problem), and then survey methods to locate the essential nodes that are capable of shaping global dynamics with either continuous (e.g. independent cascading models) or discontinuous phase transitions (e.g. threshold models). We conclude the review with a summary and an outlook.

Interconnections between networks acting like an external field in a first-order percolation transition

Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we show how the effects of such interconnections can be described as an external field for interdependent networks experiencing a first-order percolation transition. We find that the critical exponents γ and δ, related to the external field, can also be defined for first-order transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents (for first and second order) can even be found within a single model of interdependent networks, depending on the dependency coupling strength. Nevertheless, in both cases both sets satisfy Widom's identity, δ-1=γ/β, which further supports the validity of their definitions. Furthermore, we find that both Erdős-Rényi and scale-free networks have the same values of the exponents in the first-order regime, implying that these models are in the same universality class. In addition, we find that in k-core percolation the values of the critical exponents related to the field are the same as for interdependent networks, suggesting that these systems also belong to the same universality class.

K-core robustness in ecological and financial networks

In many real-world networks, the ability to withstand targeted or global attacks; extinctions; or shocks is vital to the survival of the network itself, and of dependent structures such as economies (for financial networks) or even the planet (for ecosystems). Previous attempts to characterise robustness include nestedness of mutualistic networks or exploration of degree distribution. In this work we present a new approach for characterising the stability and robustness of networks with all-positive interactions by studying the distribution of the k-shell of the underlying network. We find that high occupancy of nodes in the inner and outer k-shells and low occupancy in the middle shells of financial and ecological networks (yielding a "U-shape" in a histogram of k-shell occupancy) provide resilience against both local targeted and global attacks. Investigation of this highly-populated core gives insights into the nature of a network (such as sharp transitions in the core composition of the stock market from a mix of industries to domination by one or two in the mid-1990s) and allow predictions of future network stability, e.g., by monitoring populations of "core" species in an ecosystem or noting when stocks in the core-dominant sector begin to move in lock-step, presaging a dramatic move in the market. Moreover, this "U-shape" recalls core-periphery structure, seen in a wide range of networks including opinion and internet networks, suggesting that the "U-shaped" occupancy histogram and its implications for network health may indeed be universal.

Functionability in complex networks: Leading nodes for the transition from structural to functional networks through remote asynchronization

Complex networks are essentially heterogeneous not only in the basic properties of the constituent nodes, such as their degree, but also in the effects that these have on the global dynamical properties of the network. Networks of coupled identical phase oscillators are good examples for analyzing these effects, since an overall synchronized state can be considered a reference state. A small variation of intrinsic node parameters may cause the system to move away from synchronization, and a new phase-locked stationary state can be achieved. We propose a measure of phase dispersion that quantifies the functional response of the system to a given local perturbation. As a particular implementation, we propose a variation of the standard Kuramoto model in which the nodes of a complex network interact with their neighboring nodes, by including a node-dependent frustration parameter. The final stationary phase-locked state now depends on the particular frustration parameter at each node and also on the network topology. We exploit this scenario by introducing individual frustration parameters and measuring what their effect on the whole network is, measured in terms of the phase dispersion, which depends only on the topology of the network and on the choice of the particular node that is perturbed. This enables us to define a characteristic of the node, its functionability, that can be computed analytically in terms of the network topology. Finally, we provide a thorough comparison with other centrality measures.

TSSCM: A synergism-based three-step cascade model for influence maximization on large-scale social networks

Identification of the most influential spreaders that maximize information propagation in social networks is a classic optimization problem, called the influence maximization (IM) problem. A reasonable diffusion model that can accurately simulate information propagation in social networks is the key step to efficiently solving the IM problem. Synergism of neighbor nodes plays an important role in information propagation dynamics. Some known diffusion models have considered the reinforcement mechanism in defining the activation threshold. Most of these models focus on the synergetic effects of nodes on their common neighbors, but the accumulation of synergism has been neglected in previous studies. Inspired by these facts, we first discuss the catalytic role of synergism in the spreading dynamics of social networks and then propose a novel diffusion model called the synergism-based three-step cascade model (TSSCM) based on the above analysis and the three-degree influence theory. Finally, we devise an algorithm for solving the IM problem based on the TSSCM. Experiments on five real large-scale social networks demonstrate the efficacy of our method, which achieves competitive results in terms of influence spreading compared to the four other algorithms tested.

Topological phase transitions in functional brain networks

Functional brain networks are often constructed by quantifying correlations between time series of activity of brain regions. Their topological structure includes nodes, edges, triangles, and even higher-dimensional objects. Topological data analysis (TDA) is the emerging framework to process data sets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. The geometric nature of the transitions can be interpreted, under certain hypotheses, as an extension of percolation to high-dimensional objects. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives toward establishing reliable topological and geometrical markers for group and possibly individual differences in functional brain network organization.

Functional Translocation of Broca's Area in a Low-Grade Left Frontal Glioma: Graph Theory Reveals the Novel, Adaptive Network Connectivity

We describe frontal language reorganization in a 50–60 year-old right-handed patient with a low-grade left frontotemporal insular glioma. Pre-operative fMRI revealed robust activation in the left superior temporal gyrus (Wernicke Area, WA) and in the right inferior frontal gyrus (right anatomical homolog of Broca Area, BA). Intra-operative cortical stimulation of the left inferior frontal gyrus and adjacent cortices elicited no speech deficits, and gross total resection including the expected location of BA resulted in no speech impairment. We employed statistical inference methods to reconstruct the functional brain network and determined how different brain areas connect with one another. We found that the right homolog of the BA in this patient functionally connected to the same areas as the left BA in a typical healthy control. As opposed to the functional connection of the left BA in a healthy brain, the right BA did not connect directly with the left WA, but connected indirectly, mediated by the pre-Supplementary Motor Area and the Middle Frontal Gyrus. This case illustrates that pre-surgical fMRI may be used to identify atypical hemispheric language reorganization in the presence of brain tumor and that network theory opens the possibility for future insight into the neural mechanism underlying the language reorganization.

Critical transitions in heterogeneous networks: Loss of low-degree nodes as an early warning signal

A large number of real networks show abrupt phase transition phenomena in response to environmental changes. In this case, cascading phenomena can induce drastic and discontinuous changes in the system state and lead to collapse. Although complex network theory has been used to investigate these drastic events, we are still unable to predict them effectively. We here analyze collapse phenomena by proposing a minimal two-state dynamic on a complex network and introducing the effect of local connectivities on the evolution of network nodes. We find that a heterogeneous system of interconnected components presents a mixed response to stress and can serve as a control indicator. In particular, before the critical transition point is reached a severe loss of low-degree nodes is observed, masked by the minimal failure of higher-degree nodes. Accordingly, we suggest that a significant reduction in less connected nodes can indicate impending global failure.

The multidimensional brain

Brain activity takes place in three spatial-plus time dimensions. This rather obvious claim has been recently questioned by papers that, taking into account the big data outburst and novel available computational tools, are starting to unveil a more intricate state of affairs. Indeed, various brain activities and their correlated mental functions can be assessed in terms of trajectories embedded in phase spaces of dimensions higher than the canonical ones. In this review, I show how further dimensions may not just represent a convenient methodological tool that allows a better mathematical treatment of otherwise elusive cortical activities, but may also reflect genuine functional or anatomical relationships among real nervous functions. I then describe how to extract hidden multidimensional information from real or artificial neurodata series, and make clear how our mind dilutes, rather than concentrates as currently believed, inputs coming from the environment. Finally, I argue that the principle "the higher the dimension, the greater the information" may explain the occurrence of mental activities and elucidate the mechanisms of human diseases associated with dimensionality reduction.

On the stability of multilayer Boolean networks under targeted immunization

In this paper, we study targeted immunization in a multilayer Boolean network model for genetic regulatory networks. Given a specific set of nodes immune to perturbations, we find that the stability of a multilayer Boolean network is determined by the largest eigenvalue of the weighted non-backtracking matrix of corresponding aggregated network. Aimed to minimize this largest eigenvalue, we developed the metric of multilayer collective influence (MCI) to quantify the impact of immunizing individual nodes on the stability of the system. Compared with other competing heuristics, immunizing nodes with high MCI scores can stabilize an unstable multilayer network with higher efficiency on both synthetic and real-world networks. Moreover, despite that coupling nodes can exert direct influence across multiple layers, they are found to exhibit less importance as measured by the MCI score. Our work reveals the mechanism of maintaining the stability of multilayer Boolean networks and provides an efficient targeted immunization strategy, which can be potentially applied to the location of pathogenesis of diseases and the development of targeted therapy.

Resilience of networks with community structure behaves as if under an external field

Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic- paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.

Finding influential nodes for integration in brain networks using optimal percolation theory

Global integration of information in the brain results from complex interactions of segregated brain networks. Identifying the most influential neuronal populations that efficiently bind these networks is a fundamental problem of systems neuroscience. Here, we apply optimal percolation theory and pharmacogenetic interventions in vivo to predict and subsequently target nodes that are essential for global integration of a memory network in rodents. The theory predicts that integration in the memory network is mediated by a set of low-degree nodes located in the nucleus accumbens. This result is confirmed with pharmacogenetic inactivation of the nucleus accumbens, which eliminates the formation of the memory network, while inactivations of other brain areas leave the network intact. Thus, optimal percolation theory predicts essential nodes in brain networks. This could be used to identify targets of interventions to modulate brain function.

Correlated network of networks enhances robustness against catastrophic failures

Networks in nature rarely function in isolation but instead interact with one another with a form of a network of networks (NoN). A network of networks with interdependency between distinct networks contains instability of abrupt collapse related to the global rule of activation. As a remedy of the collapse instability, here we investigate a model of correlated NoN. We find that the collapse instability can be removed when hubs provide the majority of interconnections and interconnections are convergent between hubs. Thus, our study identifies a stable structure of correlated NoN against catastrophic failures. Our result further suggests a plausible way to enhance network robustness by manipulating connection patterns, along with other methods such as controlling the state of node based on a local rule.

Connectivity and complex systems: learning from a multi-disciplinary perspective

In recent years, parallel developments in disparate disciplines have focused on what has come to be termed connectivity; a concept used in understanding and describing complex systems. Conceptualisations and operationalisations of connectivity have evolved largely within their disciplinary boundaries, yet similarities in this concept and its application among disciplines are evident. However, any implementation of the concept of connectivity carries with it both ontological and epistemological constraints, which leads us to ask if there is one type or set of approach(es) to connectivity that might be applied to all disciplines. In this review we explore four ontological and epistemological challenges in using connectivity to understand complex systems from the standpoint of widely different disciplines. These are: (i) defining the fundamental unit for the study of connectivity; (ii) separating structural connectivity from functional connectivity; (iii) understanding emergent behaviour; and (iv) measuring connectivity. We draw upon discipline-specific insights from Computational Neuroscience, Ecology, Geomorphology, Neuroscience, Social Network Science and Systems Biology to explore the use of connectivity among these disciplines. We evaluate how a connectivity-based approach has generated new understanding of structural-functional relationships that characterise complex systems and propose a 'common toolbox' underpinned by network-based approaches that can advance connectivity studies by overcoming existing constraints.

Biological conservation law as an emerging functionality in dynamical neuronal networks

Scientists strive to understand how functionalities, such as conservation laws, emerge in complex systems. Living complex systems in particular create high-ordered functionalities by pairing up low-ordered complementary processes, e.g., one process to build and the other to correct. We propose a network mechanism that demonstrates how collective statistical laws can emerge at a macro (i.e., whole-network) level even when they do not exist at a unit (i.e., network-node) level. Drawing inspiration from neuroscience, we model a highly stylized dynamical neuronal network in which neurons fire either randomly or in response to the firing of neighboring neurons. A synapse connecting two neighboring neurons strengthens when both of these neurons are excited and weakens otherwise. We demonstrate that during this interplay between the synaptic and neuronal dynamics, when the network is near a critical point, both recurrent spontaneous and stimulated phase transitions enable the phase-dependent processes to replace each other and spontaneously generate a statistical conservation law-the conservation of synaptic strength. This conservation law is an emerging functionality selected by evolution and is thus a form of biological self-organized criticality in which the key dynamical modes are collective.

The interdependent network of gene regulation and metabolism is robust where it needs to be

Despite being highly interdependent, the major biochemical networks of the living cell-the networks of interacting genes and of metabolic reactions, respectively-have been approached mostly as separate systems so far. Recently, a framework for interdependent networks has emerged in the context of statistical physics. In a first quantitative application of this framework to systems biology, here we study the interdependent network of gene regulation and metabolism for the model organism Escherichia coli in terms of a biologically motivated percolation model. Particularly, we approach the system's conflicting tasks of reacting rapidly to (internal and external) perturbations, while being robust to minor environmental fluctuations. Considering its response to perturbations that are localized with respect to functional criteria, we find the interdependent system to be sensitive to gene regulatory and protein-level perturbations, yet robust against metabolic changes. We expect this approach to be applicable to a range of other interdependent networks.Although networks of interacting genes and metabolic reactions are interdependent, they have largely been treated as separate systems. Here the authors apply a statistical framework for interdependent networks to E. coli, and show that it is sensitive to gene and protein perturbations but robust against metabolic changes.

Emergence of robustness in networks of networks

A model of interdependent networks of networks (NONs) was introduced recently [Proc. Natl. Acad. Sci. (USA) 114, 3849 (2017)PNASA60027-842410.1073/pnas.1620808114] in the context of brain activation to identify the neural collective influencers in the brain NON. Here we investigate the emergence of robustness in such a model, and we develop an approach to derive an exact expression for the random percolation transition in Erdös-Rényi NONs of this kind. Analytical calculations are in agreement with numerical simulations, and highlight the robustness of the NON against random node failures, which thus presents a new robust universality class of NONs. The key aspect of this robust NON model is that a node can be activated even if it does not belong to the giant mutually connected component, thus allowing the NON to be built from below the percolation threshold, which is not possible in previous models of interdependent networks. Interestingly, the phase diagram of the model unveils particular patterns of interconnectivity for which the NON is most vulnerable, thereby marking the boundary above which the robustness of the system improves with increasing dependency connections.

Collective Influence of Multiple Spreaders Evaluated by Tracing Real Information Flow in Large-Scale Social Networks

Identifying the most influential spreaders that maximize information flow is a central question in network theory. Recently, a scalable method called "Collective Influence (CI)" has been put forward through collective influence maximization. In contrast to heuristic methods evaluating nodes' significance separately, CI method inspects the collective influence of multiple spreaders. Despite that CI applies to the influence maximization problem in percolation model, it is still important to examine its efficacy in realistic information spreading. Here, we examine real-world information flow in various social and scientific platforms including American Physical Society, Facebook, Twitter and LiveJournal. Since empirical data cannot be directly mapped to ideal multi-source spreading, we leverage the behavioral patterns of users extracted from data to construct "virtual" information spreading processes. Our results demonstrate that the set of spreaders selected by CI can induce larger scale of information propagation. Moreover, local measures as the number of connections or citations are not necessarily the deterministic factors of nodes' importance in realistic information spreading. This result has significance for rankings scientists in scientific networks like the APS, where the commonly used number of citations can be a poor indicator of the collective influence of authors in the community.

Collective Influence Algorithm to find influencers via optimal percolation in massively large social media

We elaborate on a linear-time implementation of Collective-Influence (CI) algorithm introduced by Morone, Makse, Nature 524, 65 (2015) to find the minimal set of influencers in networks via optimal percolation. The computational complexity of CI is O(N log N) when removing nodes one-by-one, made possible through an appropriate data structure to process CI. We introduce two Belief-Propagation (BP) variants of CI that consider global optimization via message-passing: CI propagation (CIP) and Collective-Immunization-Belief-Propagation algorithm (CIBP) based on optimal immunization. Both identify a slightly smaller fraction of influencers than CI and, remarkably, reproduce the exact analytical optimal percolation threshold obtained in Random Struct. Alg. 21, 397 (2002) for cubic random regular graphs, leaving little room for improvement for random graphs. However, the small augmented performance comes at the expense of increasing running time to O(N(2)), rendering BP prohibitive for modern-day big-data. For instance, for big-data social networks of 200 million users (e.g., Twitter users sending 500 million tweets/day), CI finds influencers in 2.5 hours on a single CPU, while all BP algorithms (CIP, CIBP and BDP) would take more than 3,000 years to accomplish the same task.