Explosive phase transition in susceptible-infected-susceptible epidemics with arbitrary small but nonzero self-infection rate

The -susceptible-infected-susceptible (SIS) epidemic model on a graph adds an independent, Poisson self-infection process with rate to the "classical" Markovian SIS process. The steady state in the classical SIS process (with =0) on any finite graph is the absorbing or overall-healthy state, in which the virus is eradicated from the network. We report that there always exists a phase transition around τc=O-1N-1 in the -SIS process on the complete graph KN with N nodes, above which the effective infection rate τ>τc causes the average steady-state fraction of infected nodes to approach that of the mean-field approximation, no matter how small, but not zero, the self-infection rate is. For τ<τc and small, the network is almost overall healthy. The observation was found by mathematical analysis on the complete graph KN, but we claim that the phase transition of explosive type may also occur in any other finite graph. We thus conclude that the overall-healthy state of the classical Markovian SIS model is unstable in the -SIS process and, hence, unlikely to exist in reality, where "background" infection >0 is imminent.Network Architectures and Service

Origin of the fractional derivative and fractional non-Markovian continuous-time processes

A complex fractional derivative can be derived by formally extending the integer k in the kth derivative of a function, computed via Cauchy's integral, to complex α. This straightforward approach reveals fundamental problems due to inherent nonanalyticity. A consequence is that the complex fractional derivative is not uniquely defined. We explain in detail the anomalies (not closed paths, branch cut jumps) and try to interpret their meaning physically in terms of entropy, friction and deviations from ideal vector fields. Next, we present a class of non-Markovian continuous-time processes by replacing the standard derivative by a Caputo fractional derivative in the classical Chapman-Kolmogorov governing equation of a continuous-time Markov process. The fractional derivative leads to a replacement of the set of exponential base functions by a set of Mittag-Leffler functions, but also creates a complicated dependence structure between states. This fractional non-Markovian process may be applied to generalize the Markovian SIS epidemic process on a contact graph to a more realistic setting. Network Architectures and Service

Time Evolution of SIS epidemics in the Complete Graph

Network Architectures and Service

Moment closure approximations of susceptible-infected-susceptible epidemics on adaptive networks

The influence of people's individual responses to the spread of contagious phenomena, like the COVID-19 pandemic, is still not well understood. We investigate the Markovian Generalized Adaptive Susceptible-Infected-Susceptible (G-ASIS) epidemic model. The G-ASIS model comprises many contagious phenomena on networks, ranging from epidemics and information diffusion to innovation spread and human brain interactions. The connections between nodes in the G-ASIS model change adaptively over time, because nodes make decisions to create or break links based on the health state of their neighbors. Our contribution is fourfold. First, we rigorously derive the first-order and second-order mean-field approximations from the continuous-time Markov chain. Second, we illustrate that the first-order mean-field approximation fails to approximate the epidemic threshold of the Markovian G-ASIS model accurately. Third, we show that the second-order mean-field approximation is a qualitative good approximation of the Markovian G-ASIS model. Finally, we discuss the Adaptive Information Diffusion (AID) model in detail, which is contained in the G-ASIS model. We show that, similar to most other instances of the G-ASIS model, the AID model possesses a unique steady state, but that in the AID model, the convergence time toward the steady state is very large. Our theoretical results are supported by numerical simulations.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Network Architectures and Service

Predicting network dynamics without requiring the knowledge of the interaction graph

A network consists of two interdependent parts: the network topology or graph, consisting of the links between nodes and the network dynamics, specified by some governing equations. A crucial challenge is the prediction of dynamics on networks, such as forecasting the spread of an infectious disease on a human contact network. Unfortunately, an accurate prediction of the dynamics seems hardly feasible, because the network is often complicated and unknown. In this work, given past observations of the dynamics on a fixed graph, we show the contrary: Even without knowing the network topology, we can predict the dynamics. Specifically, for a general class of deterministic governing equations, we propose a two-step prediction algorithm. First, we obtain a surrogate network by fitting past observations of every nodal state to the dynamical model. Second, we iterate the governing equations on the surrogate network to predict the dynamics. Surprisingly, even though there is no similarity between the surrogate topology and the true topology, the predictions are accurate, for a considerable prediction time horizon, for a broad range of observation times, and in the presence of a reasonable noise level. The true topology is not needed for predicting dynamics on networks, since the dynamics evolve in a subspace of astonishingly low dimension compared to the size and heterogeneity of the graph. Our results constitute a fresh perspective on the broad field of nonlinear dynamics on complex networks.Network Architectures and Service

Co-eigenvector Graphs

Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product Ξ = X ◦ X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N ≥ 6. Coeigenvector graphs possess the same orthogonal eigenvector matrix X, but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N = 6 and N = 7 nodes. Finally, we list many open problems.Network Architectures and Service

The fastest spreader in SIS epidemics on networks

Identifying the fastest spreaders in epidemics on a network helps to ensure an efficient spreading. By ranking the average spreading time for different spreaders, we show that the fastest spreader may change with the effective infection rate of a SIS epidemic process, which means that the time-dependent influence of a node is usually strongly coupled to the dynamic process and the underlying network. With increasing effective infection rate, we illustrate that the fastest spreader changes from the node with the largest degree to the node with the shortest flooding time. (The flooding time is the minimum time needed to reach all other nodes if the process is reduced to a flooding process.) Furthermore, by taking the local topology around the spreader and the average flooding time into account, we propose the spreading efficiency as a metric to quantify the efficiency of a spreader and identify the fastest spreader, which is adaptive to different infection rates in general networks.Network Architectures and Service

Conditions That Impact the Complexity of QoS Routing

Finding a path in a network based on multiple constraints (the MCP problem) is often considered an integral part of quality of service (QoS) routing. QoS routing with constraints on multiple additive measures has been proven to be NP-complete. This proof has dramatically influenced the research community, resulting into the common belief that exact QoS routing is intractable in practice. However, to our knowledge, no one has ever examined which ¿worst cases¿ lead to intractability. In fact, the MCP problem is not strong NP-complete, suggesting that in practice an exact QoS routing algorithm may work in polynomial time. The goal of this paper is to argue that in practice QoS routing may be tractable. We will provide properties, an approximate analysis, and simulation results to indicate that NP-completeness hinges on four conditions, namely: 1) the topology; 2) the granularity of link weights; 3) the correlation between link weights; and 4) the constraints. We expect that, in practice, these conditions are manageable and therefore believe that exact QoS routing is tractable in practice.Old - EWI Sect. Telecomm. and Traffic-control SystemsNetwork Architectures and Service

Exact Network Reconstruction from Complete SIS Nodal State Infection Information Seems Infeasible

The SIS dynamics of the spread of a virus crucially depend on both the network topology and the spreading parameters. Since neither the topology nor the spreading parameters are known for the majority of applications, they have to be inferred from observations of the viral spread. We propose an inference method for both topology and spreading parameters based on a maximum-a-posteriori estimation approach for the sampled-time Markov chain of an SIS process. The resulting estimation problem, given by a mixed-integer optimisation problem, results in exponential computational time if a brute-force approach is employed. By introducing an efficient and accurate, polynomial-time heuristic, the topology of the network can almost always be exactly reconstructed. Notwithstanding, reconstructing the network with a reasonably high accuracy requires a subexponentially increasing number of observations and an exponentially increasing computation time with respect to the number of nodes N. Such long observation periods are hardly realistic, which justifies the claim in the title.Network Architectures and Service

Tighter spectral bounds for the cut size, based on Laplacian eigenvectors

The cut-set ∂V in a graph is defined as the set of all links between a set of nodes V and all other nodes in that graph. Finding bounds for the size of a cut-set |∂V| is an important problem, and is related to mixing times, connectedness and spreading processes on networks. A standard way to bound the number of links in a cut-set |∂V| relies on Laplacian eigenvalues, which approximate the largest and smallest possible cut-sets for a given size of the set V. In this article, we extend the standard spectral approximations by including information about the Laplacian eigenvectors. This additional information leads to provably tighter bounds compared to the standard spectral bounds. We apply our new method to find improved spectral bounds for the well-known Cheeger constant, the Max Cut problem and the expander mixing lemma. We also apply our bounds to study cut sizes in the hypercube graph, and describe an application related to the spreading of epidemics on networks. We further illustrate the performance of our new bounds using simulations, revealing that a significant improvement over the standard bounds is possible.Accepted author manuscriptNetwork Architectures and Service