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Register a new userInfluence maximization in complex networks through optimal percolation
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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 [1]; or, if immunized, would prevent the diffusion of a large scale epidemic [2,3]. Localizing this optimal, i.e. minimal, set of structural nodes, called influencers, is one of the most important problems in network science [4,5]. Despite the vast use of heuristic strategies to identify influential spreaders [6-14], 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 [15] 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. Eventually, 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 [16].
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Controlling complex networks is of paramount importance in science and engineering. Despite the recent development of structural-controllability theory, we continue to lack a framework to control undirected complex networks, especially given link weights. Here we introduce an exact-controllability paradigm based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The framework reproduces the structural controllability of directed networks characterized by structural matrices. We explore the controllability of a large number of real and model networks, finding that dense networks with identical weights are difficult to be controlled. An efficient and accurate tool is offered to assess the controllability of large sparse and dense networks. The exact-controllability framework enables a comprehensive understanding of the impact of network properties on controllability, a fundamental problem towards our ultimate control of complex systems.
We elaborate on a linear time implementation of the Collective Influence (CI) algorithm introduced by Morone, Makse, Nature 524, 65 (2015) to find the minimal set of influencers in a network via optimal percolation. We show that the computational complexity of CI is O(N log N) when removing nodes one-by-one, with N the number of nodes. This is made possible by using an appropriate data structure to process the CI values, and by the finite radius l of the CI sphere. Furthermore, we introduce a simple extension of CI when l is infinite, the CI propagation (CI_P) algorithm, that considers the global optimization of influence via message passing in the whole network and identifies a slightly smaller fraction of influencers than CI. Remarkably, CI_P is able to reproduce the exact analytical optimal percolation threshold obtained by Bau, Wormald, Random Struct. Alg. 21, 397 (2002) for cubic random regular graphs, leaving little improvement left for random graphs. We also introduce the Collective Immunization Belief Propagation algorithm (CI_BP), a belief-propagation (BP) variant of CI based on optimal immunization, which has the same performance as CI_P. However, this small augmented performance of the order of 1-2 % in the low influencers tail comes at the expense of increasing the computational complexity from O(N log N) to O(N^2 log N), rendering both, CI_P and CI_BP, prohibitive for finding influencers in modern-day big-data. The same nonlinear running time drawback pertains to a recently introduced BP-decimation (BPD) algorithm by Mugisha, Zhou, arXiv:1603.05781. For instance, we show that for big-data social networks of typically 200 million users (eg, active Twitter users sending 500 million tweets per day), CI finds the influencers in less than 3 hours running on a single CPU, while the BP algorithms (CI_P, CI_BP and BDP) would take more than 3,000 years to accomplish the same task.
We present a model that takes into account the coupling between evolutionary game dynamics and social influence. Importantly, social influence and game dynamics take place in different domains, which we model as different layers of a multiplex network. We show that the coupling between these dynamical processes can lead to cooperation in scenarios where the pure game dynamics predicts defection. In addition, we show that the structure of the network layers and the relation between them can further increase cooperation. Remarkably, if the layers are related in a certain way, the system can reach a polarized metastable state.These findings could explain the prevalence of polarization observed in many social dilemmas.
The largest eigenvalue of a networks adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: The hub with its immediate neighbors and the densely connected set of nodes with maximum $K$-core index. We validate this formula showing that it predicts with good accuracy the largest eigenvalue of a large set of synthetic and real-world topologies. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a byproduct, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.
Higher order interactions are increasingly recognised as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraph as well as simplicial complexes capture the higher-order interactions of complex systems and allow to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks and K-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.
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