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In this work, we use the spectral properties of graphons to study stability and sensitivity to noise of deterministic SIS epidemics over large networks. We consider the presence of additive noise in a linearized SIS model and we derive a noise index to quantify the deviation from the disease-free state due to noise. For finite networks, we show that the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we find an approximation of the index that is tight when the network size goes to infinity. A numerical example is included to illustrate the results.
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This paper considers the susceptible-infected-susceptible (SIS) epidemic model with an underlying network structure among subpopulations and focuses on the effect of social distancing to regulate the epidemic level. We demonstrate that if each subpopulation is informed of its infection rate and reduces interactions accordingly, the fraction of the subpopulation infected can remain below half for all time instants. To this end, we first modify the basic SIS model by introducing a state dependent parameter representing the frequency of interactions between subpopulations. Thereafter, we show that for this modified SIS model, the spectral radius of a suitably-defined matrix being not greater than one causes all the agents, regardless of their initial sickness levels, to converge to the healthy state; assuming non-trivial disease spread, the spectral radius being greater than one leads to the existence of a unique endemic equilibrium, which is also asymptotically stable. Finally, by leveraging the aforementioned results, we show that the fraction of (sub)populations infected never exceeds half.
To improve the accuracy of network-based SIS models we introduce and study a multilayer representation of a time-dependent network. In particular, we assume that individuals have their long-term (permanent) contacts that are always present, identifying in this way the first network layer. A second network layer also exists, where the same set of nodes can be connected by occasional links, created with a given probability. While links of the first layer are permanent, a link of the second layer is only activated with some probability and under the condition that the two nodes, connected by this link, are simultaneously participating to the temporary link. We develop a model for the SIS epidemic on this time-dependent network, analyze equilibrium and stability of the corresponding mean-field equations, and shed some light on the role of the temporal layer on the spreading process.
In this paper, we first consider a pinning node selection and control gain co-design problem for complex networks. A necessary and sufficient condition for the synchronization of the pinning controlled networks at a homogeneous state is provided. A quantitative model is built to describe the pinning costs and to formulate the pinning node selection and control gain design problem for different scenarios into the corresponding optimization problems. Algorithms to solve these problems efficiently are presented. Based on the developed results, we take the existence of a malicious attacker into consideration and a resource allocation model for the defender and the malicious attacker is described. We set up a leader-follower Stackelberg game framework to study the behaviour of both sides and the equilibrium of this security game is investigated. Numerical examples and simulations are presented to demonstrate the main results.
Observability and estimation are closely tied to the system structure, which can be visualized as a system graph--a graph that captures the inter-dependencies within the state variables. For example, in social system graphs such inter-dependencies represent the social interactions of different individuals. It was recently shown that contractions, a key concept from graph theory, in the system graph are critical to system observability, as (at least) one state measurement in every contraction is necessary for observability. Thus, the size and number of contractions are critical in recovering for loss of observability. In this paper, the correlation between the average-size/number of contractions and the global clustering coefficient (GCC) of the system graph is studied. Our empirical results show that estimating systems with high GCC requires fewer measurements, and in case of measurement failure, there are fewer possible options to find substitute measurement that recovers the systems observability. This is significant as by tuning the GCC, we can improve the observability properties of large-scale engineered networks, such as social networks and smart grid.
Interpersonal influence estimation from empirical data is a central challenge in the study of social structures and dynamics. Opinion dynamics theory is a young interdisciplinary science that studies opinion formation in social networks and has a huge potential in applications, such as marketing, advertisement and recommendations. The term social influence refers to the behavioral change of individuals due to the interactions with others in a social system, e.g. organization, community, or society in general. The advent of the Internet has made a huge volume of data easily available that can be used to measure social influence over large populations. Here, we aim at qualitatively and quantitatively infer social influence from data using a systems and control viewpoint. First, we introduce some definitions and models of opinions dynamics and review some structural constraints of online social networks, based on the notion of sparsity. Then, we review the main approaches to infer the networks structure from a set of observed data. Finally, we present some algorithms that exploit the introduced models and structural constraints, focusing on the sample complexity and computational requirements.
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