Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

Spreading of information, ideas or diseases can be conveniently modelled in the context of complex networks. An analysis now reveals that the most efficient spreaders are not always necessarily the most connected agents in a network. Instead, the position of an agent relative to the hierarchical topological organization of the network might be as important as its connectivity.

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Identification of influential spreaders in complex networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

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Critical phenomena in complex networks

1. Overview 2. Erdos-Renyi random graphs 3. Fixed degree distributions 4. Power laws 5. Small worlds 6. Random walks 7. CHKNS model.

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Random graph dynamics

How will a virus propagate in a real network? Does an epidemic threshold exist for a finite graph? How long does it take to disinfect a network given particular values of infection rate and virus death rate? We answer the first question by providing equations that accurately model virus propagation in any network including real and synthesized network graphs. We propose a general epidemic threshold condition that applies to arbitrary graphs: we prove that, under reasonable approximations, the epidemic threshold for a network is closely related to the largest eigenvalue of its adjacency matrix. Finally, for the last question, we show that infections tend to zero exponentially below the epidemic threshold. We show that our epidemic threshold model subsumes many known thresholds for special-case graphs (e.g., Erdos-Renyi, BA power-law, homogeneous); we show that the threshold tends to zero for infinite power-law graphs. We show that our threshold condition holds for arbitrary graphs.

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Epidemic spreading in real networks: an eigenvalue viewpoint

The vanishing epidemic threshold for viruses spreading on scale-free networks indicate that traditional methods, aiming to decrease a virus' spreading rate cannot succeed in eradicating an epidemic. We demonstrate that policies that discriminate between the nodes, curing mostly the highly connected nodes, can restore a finite epidemic threshold and potentially eradicate a virus. We find that the more biased a policy is towards the hubs, the more chance it has to bring the epidemic threshold above the virus' spreading rate. Furthermore, such biased policies are more cost effective, requiring less cures to eradicate the virus.

Halting viruses in scale-free networks.

Networks with complex topology describe systems as diverse as the cell or the World Wide Web. The emergence of these networks is driven by self-organizing processes that are governed by simple but generic laws. In the last three years it became clear that many complex networks, such as the Internet, the cell, or the world wide web, share the same large-scale topology. Here we review recent advances in the characterization of complex networks, focusing the emergence of the scale-free and the hierarchical architecture. We also present empirical results to demonstrate that the scale-free and the hierarchical property are shared by a wide range of complex networks. Finally, we discuss the impact of the network topology on our ability to stop the spread of viruses in complex networks.

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Zoltán Dezső

Papers

Halting viruses in scale-free networks.

Scale‐Free and Hierarchical Structures in Complex Networks