Flaviano Morone

Influence maximization in complex networks through optimal percolation

The whole frame of interconnections in complex networks hinges on a specific set of structural nodes, much smaller than the total size, which, if activated, would cause the spread of information to the whole network, or, if immunized, would prevent the diffusion of a large scale epidemic. Localizing this optimal, that is, minimal, set of structural nodes, called influencers, is one of the most important problems in network science. Despite the vast use of heuristic strategies to identify influential spreaders, the problem remains unsolved. Here we map the problem onto optimal percolation in random networks to identify the minimal set of influencers, which arises by minimizing the energy of a many-body system, where the form of the interactions is fixed by the non-backtracking matrix of the network. Big data analyses reveal that the set of optimal influencers is much smaller than the one predicted by previous heuristic centralities. Remarkably, a large number of previously neglected weakly connected nodes emerges among the optimal influencers. These are topologically tagged as low-degree nodes surrounded by hierarchical coronas of hubs, and are uncovered only through the optimal collective interplay of all the influencers in the network. The present theoretical framework may hold a larger degree of universality, being applicable to other hard optimization problems exhibiting a continuous transition from a known phase.

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(N2), 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.

Efficient collective influence maximization in cascading processes with first-order transitions

In many social and biological networks, the collective dynamics of the entire system can be shaped by a small set of influential units through a global cascading process, manifested by an abrupt first-order transition in dynamical behaviors. Despite its importance in applications, efficient identification of multiple influential spreaders in cascading processes still remains a challenging task for large-scale networks. Here we address this issue by exploring the collective influence in general threshold models of cascading process. Our analysis reveals that the importance of spreaders is fixed by the subcritical paths along which cascades propagate: the number of subcritical paths attached to each spreader determines its contribution to global cascades. The concept of subcritical path allows us to introduce a scalable algorithm for massively large-scale networks. Results in both synthetic random graphs and real networks show that the proposed method can achieve larger collective influence given the same number of seeds compared with other scalable heuristic approaches.In spreading dynamics in social networks, there exists an optimal set of influencers whose activation can induce a global-scale cascade of information. To find the optimal, or minimal, set of spreaders, a method based on collective influence theory has been proposed for spreading dynamics with a continuous phase transition that can be mapped to optimal percolation. However, when it comes to diffusion processes exhibiting a first-order, or discontinuous transition, identifying the set of optimal spreaders with a linear algorithm for large-scale networks still remains a challenging task. Here we address this issue by exploring the collective influence in general threshold models of opinion cascading. Our analysis reveals that the importance of spreaders is fixed by the subcritical paths along which cascades propagate: the number of subcritical paths attached to each spreader determines its contribution to global cascades. The concept of subcritical path allows us to introduce a linearly scalable algorithm for massively large-scale networks. Results in both synthetic random graphs and real networks show that the set of spreaders predicted by our method is smaller than those identified by other linearly scalable heuristic approaches.

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.

Corrigendum: Influence maximization in complex networks through optimal percolation

This corrects the article DOI: 10.1038/nature14604

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

Significance Evidence suggests that the brain is arranged in functionally specialized modules to form a network of networks (NoN). Understanding how functionality emerges from the integration of such modular architectures is one of the greatest scientific challenges. We introduce a model of brain NoN, which is robust against random node failures, captures the integration of functionally specialized modules in the brain, and provides falsifiable predictions about the locations of the most influential nodes, called neural collective influencers, in the brain network—predictions that are impossible in existing but fragile models of interdependent NoN. If confirmed by experiment, our results may pave the way for applications of clinical interest. Efficient complex systems have a modular structure, but modularity does not guarantee robustness, because efficiency also requires an ingenious interplay of the interacting modular components. The human brain is the elemental paradigm of an efficient robust modular system interconnected as a network of networks (NoN). Understanding the emergence of robustness in such modular architectures from the interconnections of its parts is a longstanding challenge that has concerned many scientists. Current models of dependencies in NoN inspired by the power grid express interactions among modules with fragile couplings that amplify even small shocks, thus preventing functionality. Therefore, we introduce a model of NoN to shape the pattern of brain activations to form a modular environment that is robust. The model predicts the map of neural collective influencers (NCIs) in the brain, through the optimization of the influence of the minimal set of essential nodes responsible for broadcasting information to the whole-brain NoN. Our results suggest intervention protocols to control brain activity by targeting influential neural nodes predicted by network theory.

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.Complex networks can be used to model brain networks. Here the authors identify the essential nodes in a model of a brain network and then validate these predictions by means of in vivo pharmacogenetic interventions. They find that the nucleus accumbens is a central region for brain integration.

Theories for Influencer Identification in Complex Networks

In social and biological systems, the structural heterogeneity of interaction networks gives rise to the emergence of a small set of influential nodes, or influencers, in a series of dynamical processes. Although much smaller than the entire network, these influencers were observed to be able to shape the collective dynamics of large populations in different contexts. As such, the successful identification of influencers should have profound implications in various real-world spreading dynamics such as viral marketing, epidemic outbreaks, and cascading failure. In this chapter, we first summarize the centrality-based approach in finding single influencers in complex networks, and then discuss the more complicated problem of locating multiple influencers from a collective point of view. Progress rooted in collective influence theory, belief-propagation, and computer science will be presented. Finally, we present some applications of influencer identification in diverse real-world systems, including online social platforms, scientific publication, brain networks, and socioeconomic systems.

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 Behaviour in Video Viewing: A Thermodynamic Analysis of Gaze Position

Videos and commercials produced for large audiences can elicit mixed opinions. We wondered whether this diversity is also reflected in the way individuals watch the videos. To answer this question, we presented 65 commercials with high production value to 25 individuals while recording their eye movements, and asked them to provide preference ratings for each video. We find that gaze positions for the most popular videos are highly correlated. To explain the correlations of eye movements, we model them as “interactions” between individuals. A thermodynamic analysis of these interactions shows that they approach a “critical” point such that any stronger interaction would put all viewers into lock-step and any weaker interaction would fully randomise patterns. At this critical point, groups with similar collective behaviour in viewing patterns emerge while maintaining diversity between groups. Our results suggest that popularity of videos is already evident in the way we look at them, and that we maintain diversity in viewing behaviour even as distinct patterns of groups emerge. Our results can be used to predict popularity of videos and commercials at the population level from the collective behaviour of the eye movements of a few viewers.