Evolving nature of human contact networks with its impact on epidemic processes

Human contact networks are constituted by a multitude of individuals and pairwise contacts among them. However, the dynamic nature, which generates the evolution of human contact networks, of contact patterns is not known yet. Here, we analyse three empirical datasets and identify two crucial mechanisms of the evolution of temporal human contact networks, i.e. the activity state transition laws for an individual to be socially active, and the contact establishment mechanism that active individuals adopt. We consider both of the two mechanisms to propose a temporal network model, named the memory driven (MD) model, of human contact networks. Then we study the susceptible-infected (SI) spreading processes on empirical human contact networks and four corresponding temporal network models, and compare the full prevalence time of SI processes with various infection rates on the networks. The full prevalence time of SI processes in the MD model is the same as that in real-world human contact networks. Moreover, we find that the individual activity state transition promotes the spreading process, while, the contact establishment of active individuals suppress the prevalence. Apart from this, we observe that even a small percentage of individuals to explore new social ties is able to induce an explosive spreading on networks. The proposed temporal network framework could help the further study of dynamic processes in temporal human contact networks, and offer new insights to predict and control the diffusion processes on networks

Sync in Complex Dynamical Networks: Stability, Evolution, Control, and Application

In the past few years, the discoveries of small-world and scale-free properties of many natural and artificial complex networks have stimulated significant advances in better understanding the relationship between the topology and the collective dynamics of complex networks. This paper reports recent progresses in the literature of synchronization of complex dynamical networks including stability criteria, network synchronizability and uniform synchronous criticality in different topologies, and the connection between control and synchronization of complex networks as well. The economic-cycle synchronous phenomenon in the World Trade Web, a scale-free type of social economic networks, is used to illustrate an application of the network synchronization mechanism.Comment: 23 pages, 13 figure

WW-entropy, super Perelman Ricci flows and (K,m)(K, m)-Ricci solitons

In this paper, we prove the characterization of the $(K, \infty)$-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dimension free Harnack inequality on manifolds with $(K, \infty)$-super Perelman Ricci flows. Based on a new second order differential inequality on the Boltzmann-Shannon entropy for the heat equation of the Witten Laplacian, we introduce a new $W$-entropy quantity and prove its monotonicity for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(K, \infty)$-condition and on compact manifolds with $(K, \infty)$-super Perelman Ricci flows. Our results characterize the $(K, \infty)$-Ricci solitons and the $(K, \infty)$-Perelman Ricci flows. We also prove a second order differential entropy inequality on $(K, m)$-super Ricci flows, which can be used to characterize the $(K, m)$-Ricci solitons and the $(K, m)$-Ricci flows. Finally, we give a probabilistic interpretation of the $W$-entropy for the heat equation of the Witten Laplacian on manifolds with the $CD(K, m)$-condition.Comment: We remove Section 5 from the previous version and add two new results in Section 5. arXiv admin note: text overlap with arXiv:1412.703

WW-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds

We introduce Perelman's $W$-entropy and prove the $W$-entropy formula along geodesic flow on the Wasserstein space $P^\infty_2(M, \mu)$ over compact Riemannian manifolds equipped with Otto's infinite dimensional Riemannian metric. As a corollary, we recapture Lott and Villani's result on the displacement convexity of $t{\rm Ent}+mt\log t$ on $P^\infty_2(M, \mu)$ over Riemannian manifolds with Bakry-Emery's curvature-dimension $CD(0, m)$-condition. To better understand the similarity between the $W$-entropy formula for the geodesic flow on the Wasserstein space and the $W$-entropy formula for the heat equation of the Witten Laplacian on the underlying manifolds, we introduce the Langevin deformation of flows on the Wasserstein space, which interpolates the geodesic flow and the gradient flow of the Boltzmann-Shannon entropy on the Wasserstein space over Riemannian manifolds, and can be regarded as the potential flow of the compressible Euler equation with damping on manifolds. We prove the local and global existence, uniqueness and regularity of the potential flow on compact Riemannian manifolds, and prove an analogue of the Perelman type $W$-entropy formula along the Langevin deformation of flows on the Wasserstein space on compact Riemannian manifolds. We also prove a rigidity theorem for the $W$-entropy for the geodesic flow and provide the rigidity models for the $W$-entropy for the Langevin deformation of flows on the Wasserstein space over complete Riemannian manifolds with the $CD(0, m)$-condition. Finally, we prove the $W$-entropy inequalities along the geodesic flow, gradient flow and the Langevin deformation of flows on the Wasserstein space over compact Riemannian manifolds with Erbar-Kuwada-Sturm's entropic curvature-dimension $CD_{\rm Ent}(K, N)$-condition

Hamilton differential Harnack inequality and WW-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition, where $m\in [n, \infty)$ and $K\geq 0$ are two constants. Moreover, we introduce the $W$-entropy and prove the $W$-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(-K, m)$-condition and on compact manifolds equipped with $(-K, m)$-super Ricci flows.Comment: To appear in Journal of Functional Analysis. This paper is an improved version of a part of our previous preprint [14] (arxiv:1412.7034, version1 (22 December 2014) and version 2 (7 February 2016)

Dynamical transitions in a modulated Landau-Zener model with finite driving fields

We investigate a special time-dependent quantum model which assumes the Landau-Zener driving form but with an overall modulation of the intensity of the pulsing field. We demonstrate that the dynamics of the system, including the two-level case as well as its multi-level extension, is exactly solvable analytically. Differing from the original Landau-Zener model, the nonadiabatic effect of the evolution in the present driving process does not destroy the desired population transfer. As the sweep protocol employs only the finite driving fields which tend to zero asymptotically, the cutoff error due to the truncation of the driving pulse to the finite time interval turns out to be negligibly small. Furthermore, we investigate the noise effect on the driving protocol due to the dissipation of the surrounding environment. The losses of the fidelity in the protocol caused by both the phase damping process and the random spin flip noise are estimated by solving numerically the corresponding master equations within the Markovian regime.Comment: 6 pages, 4 figure

Harnack inequalities and WW-entropy formula for Witten Laplacian on Riemannian manifolds with KK-super Perelman Ricci flow

In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a rigidity theorem on complete Riemannian manifolds satisfying the $CD(K, m)$ condition, and extend the $W$-entropy formula to time dependent Witten Laplacian on compact Riemannian manifolds with $(K, m)$-super Perelman Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$ are two constants. Finally, we prove the Li-Yau and the Li-Yau-Hamilton Harnack inequalities for positive solutions to the heat equation $\partial_t u=Lu$ associated to the time dependent Witten Laplacian on compact or complete manifolds equipped with variants of the $(K, m)$-super Ricci flow.Comment: Theorem 1.1 in the first version has been improved. In the case of time dependent metrics and potentials, an error in the proof of Theorem 1.5 (i.e., Theorem 2.2), Theorem 2.3 and Theorem 2.4 in the first version has been corrected. See Section 4 and Section

The Impact of Information Dissemination on Vaccination in Multiplex Networks

The impact of information dissemination on epidemic control is essentially subject to individual behaviors. Unlike information-driven behaviors, vaccination is determined by many cost-related factors, whose correlation with the information dissemination should be better understood. To this end, we propose an evolutionary vaccination game model in multiplex networks by integrating an information-epidemic spreading process into the vaccination dynamics, and explore how information dissemination influences vaccination. The spreading process is described by a two-layer coupled susceptible-alert-infected-susceptible (SAIS) model, where the strength coefficient between two layers is defined to characterize the tendency and intensity of information dissemination. We find that information dissemination can increase the epidemic threshold, however, more information transmission cannot promote vaccination. Specifically, increasing information dissemination even leads to a decline of the vaccination equilibrium and raises the final infection density. Moreover, we study the impact of strength coefficient and individual sensitivity on social cost, and unveil the role of information dissemination in controlling the epidemic with numerical simulations

WW-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the $W$-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the $W$-entropy formula for the heat equation associated with the Witten Laplacian on $n$-dimensional complete Riemannian manifolds with the $CD(K, m)$-condition, and the $W$-entropy formula for the heat equation associated with the time dependent Witten Laplacian on $n$-dimensional compact manifolds equipped with a $(K, m)$-super Ricci flow, where $K\in \mathbb{R}$ and $m\in [n, \infty]$. Furthermore, we prove an analogue of the $W$-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two $W$-entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the $W$-entropy formula for the Langevin deformation. Finally, we make a discussion on the $W$-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.Comment: Survey paper. Submitted to Science China Mathematics. arXiv admin note: text overlap with arXiv:1604.02596, arXiv:1706.0704

Theoretical Analysis of Compressive Sensing via Random Filter

In this paper, the theoretical analysis of compressive sensing via random filter, firstly outlined by J. Romberg [compressive sensing by random convolution, submitted to SIAM Journal on Imaging Science on July 9, 2008], has been refined or generalized to the design of general random filter used for compressive sensing. This universal CS measurement consists of two parts: one is from the convolution of unknown signal with a random waveform followed by random time-domain subsampling; the other is from the directly time-domain subsampling of the unknown signal. It has been shown that the proposed approach is a universally efficient data acquisition strategy, which means that the n-dimensional signal which is S sparse in any sparse representation can be exactly recovered from Slogn measurements with overwhelming probability