Victor M. Preciado

Optimal Resource Allocation for Network Protection Against Spreading Processes

We study the problem of containing spreading processes in arbitrary directed networks by distributing protection resources throughout the nodes of the network. We consider that two types of protection resources are available: 1) preventive resources able to defend nodes against the spreading (such as vaccines in a viral infection process) and 2) corrective resources able to neutralize the spreading after it has reached a node (such as antidotes). We assume that both preventive and corrective resources have an associated cost and study the problem of finding the cost-optimal distribution of resources throughout the nodes of the network. We analyze these questions in the context of viral spreading processes in directed networks. We study the following two problems: 1) given a fixed budget, find the optimal allocation of preventive and corrective resources in the network to achieve the highest level of containment and 2) when a budget is not specified, find the minimum budget required to control the spreading process. We show that both the resource allocation problems can be solved in polynomial time using geometric programming (GP) for arbitrary directed graphs of nonidentical nodes and a wide class of cost functions. We illustrate our approach by designing optimal protection strategies to contain an epidemic outbreak that propagates through an air transportation network.

Optimal vaccine allocation to control epidemic outbreaks in arbitrary networks

We consider the problem of controlling the propagation of an epidemic outbreak in an arbitrary contact network by distributing vaccination resources throughout the network. We analyze a networked version of the Susceptible-Infected-Susceptible (SIS) epidemic model when individuals in the network present different levels of susceptibility to the epidemic. In this context, controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. We study the problem of finding the optimal distribution of vaccines throughout the network to control the spread of an epidemic outbreak. We propose a convex framework to find cost-optimal distribution of vaccination resources when different levels of vaccination are allowed.We illustrate our approaches with numerical simulations in a real social network.

Stability of Spreading Processes over Time-Varying Large-Scale Networks

In this paper, we analyze the dynamics of spreading processes taking place over time-varying networks. A common approach to model time-varying networks is via Markovian random graph processes. This modeling approach presents the following limitation: Markovian random graphs can only replicate switching patterns with exponential inter-switching times, while in real applications these times are usually far from exponential. In this paper, we introduce a flexible and tractable extended family of processes able to replicate, with arbitrary accuracy, any distribution of inter-switching times. We then study the stability of spreading processes in this extended family. We first show that a direct analysis based on Itôs formula provides stability conditions in terms of the eigenvalues of a matrix whose size grows exponentially with the number of edges. To overcome this limitation, we derive alternative stability conditions involving the eigenvalues of a matrix whose size grows linearly with the number of nodes. Based on our results, we also show that heuristics based on aggregated static networks approximate the epidemic threshold more accurately as the number of nodes grows, or the temporal volatility of the random graph process is reduced. Finally, we illustrate our findings via numerical simulations.

Topology Identification of Directed Dynamical Networks via Power Spectral Analysis

We address the problem of identifying the topology of an unknown weighted, directed network of LTI systems stimulated by wide-sense stationary noises of unknown power spectral densities. We propose several reconstruction algorithms by measuring the cross-power spectral densities of the network response to the input noises. The measurements are based on a series of node-knockout experiments where at each round the knocked out node broadcasts zero state without being eliminated from the network. Our first algorithm reconstructs the Boolean structure (i.e., existence and directions of links) of a directed network from a series of dynamical responses. Moreover, we propose a second algorithm to recover the exact structure of the network (including edge weights), as well as the power spectral density of the input noises, when an eigenvalue-eigenvector pair of the connectivity matrix is known (for example, Laplacian connectivity matrices). Finally, for the particular cases of nonreciprocal networks (i.e., networks with no directed edges pointing in opposite directions) and undirected networks, we propose specialized algorithms that result in a lower computational cost.

A convex framework for optimal investment on disease awareness in social networks

We consider the problem of controlling the propagation of an epidemic outbreak in an arbitrary network of contacts by investing on disease awareness throughout the network. We model the effect of agent awareness on the dynamics of an epidemic using the SAIS epidemic model, an extension of the SIS epidemic model that includes a state of “awareness”. This model allows to derive a condition to control the spread of an epidemic outbreak in terms of the eigenvalues of a matrix that depends on the network structure and the parameters of the model. We study the problem of finding the cost-optimal investment on disease awareness throughout the network when the cost function presents some realistic properties. We propose a convex framework to find cost-optimal allocation of resources. We validate our results with numerical simulations in a real online social network.

Spectral analysis of virus spreading in random geometric networks

In this paper, we study the dynamics of a viral spreading process in random geometric graphs (RGG). The spreading of the viral process we consider in this paper is closely related with the eigenvalues of the adjacency matrix of the graph. We deduce new explicit expressions for all the moments of the eigenvalue distribution of the adjacency matrix as a function of the spatial density of nodes and the radius of connection. We apply these expressions to study the behavior of the viral infection in an RGG. Based on our results, we deduce an analytical condition that can be used to design RGGs in order to tame an initial viral infection. Numerical simulations are in accordance with our analytical predictions.

Moment-based spectral analysis of large-scale networks using local structural information

The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the importance of nodes in the network. A common approach to study the relationship between the structure of a network and its eigenvalues is to use synthetic random networks in which structural properties of interest, such as degree distributions, are prescribed. Although very common, synthetic models present two major flaws: 1) These models are only suitable to study a very limited range of structural properties; and 2) they implicitly induce structural properties that are not directly controlled and can deceivingly influence the network eigenvalue spectrum. In this paper, we propose an alternative approach to overcome these limitations. Our approach is not based on synthetic models. Instead, we use algebraic graph theory and convex optimization to study how structural properties influence the spectrum of eigenvalues of the network. Using our approach, we can compute, with low computational overhead, global spectral properties of a network from its local structural properties. We illustrate our approach by studying how structural properties of online social networks influence their eigenvalue spectra.

Traffic optimization to control epidemic outbreaks in metapopulation models

We propose a novel framework to study viral spreading processes in metapopulation models. Large subpopulations (i.e., cities) are connected via metalinks (i.e., roads) according to a metagraph structure (i.e., the traffic infrastructure). The problem of containing the propagation of an epidemic outbreak in a metapopulation model by controlling the traffic between subpopulations is considered. Controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. Based on this spectral condition, we propose a convex optimization framework to find cost-optimal approaches to traffic control in epidemic outbreaks.

Stability analysis of generalized epidemic models over directed networks

In this paper we propose a generalized version of the Susceptible-Exposed-Infected-Vigilant (SEIV) disease spreading model over arbitrary directed graphs. In the standard SEIV model there is only one infectious state. Our model instead allows for the exposed state to also be infectious to healthy individuals. This model captures the fact that infected individuals may act differently when they are aware of their infection. For instance, when the individual is aware of the infection, different actions may be taken, such as staying home from work, causing less chance for spreading the infection. This model generalizes the standard SEIV model which is already known to generalize many other infection spreading models available. We use tools from nonlinear stability analysis to suggest a coordinate transformation that allows us to study the stability of the origin of a relevant linear system. We provide a necessary and sufficient condition for when the disease-free equilibrium is globally exponentially stable. We then extend the results to the case where the infection parameters are not homogeneous among the nodes of the network. Simulations illustrate our results.

Optimal information dissemination strategy to promote preventive behaviors in multilayer epidemic networks.

Launching a prevention campaign to contain the spread of infection requires substantial financial investments; therefore, a trade-off exists between suppressing the epidemic and containing costs. Information exchange among individuals can occur as physical contacts (e.g., word of mouth, gatherings), which provide inherent possibilities of disease transmission, and non-physical contacts (e.g., email, social networks), through which information can be transmitted but the infection cannot be transmitted. Contact network (CN) incorporates physical contacts, and the information dissemination network (IDN) represents non-physical contacts, thereby generating a multilayer network structure. Inherent differences between these two layers cause alerting through CN to be more effective but more expensive than IDN. The constraint for an epidemic to die out derived from a nonlinear Perron-Frobenius problem that was transformed into a semi-definite matrix inequality and served as a constraint for a convex optimization problem. This method guarantees a dying-out epidemic by choosing the best nodes for adopting preventive behaviors with minimum monetary resources. Various numerical simulations with network models and a real-world social network validate our method.