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The topic of finding effective strategy to halt virus in complex network is of current interest. We propose an immunization strategy for seasonal epidemics that occur periodically. Based on the local information of the infection status from the previous epidemic season, the selection of vaccinated nodes is optimized gradually. The evolution of vaccinated nodes during iterations demonstrates that the immunization tends to locate in both global hubs and local hubs. We analyze the epidemic prevalence by a heterogeneous mean-field method and present numerical simulations of our model. This immunization performs superiorly to some other previously known strategies. Our work points out a new direction in immunization of seasonal epidemics., Comment: 8 pages, 9 figures

A number of predictors have been suggested to detect the most influential spreaders of information in online social media across various domains such as Twitter or Facebook. In particular, degree, PageRank, k-core and other centralities have been adopted to rank the spreading capability of users in information dissemination media. So far, validation of the proposed predictors has been done by simulating the spreading dynamics rather than following real information flow in social networks. Consequently, only model-dependent contradictory results have been achieved so far for the best predictor. Here, we address this issue directly. We search for influential spreaders by following the real spreading dynamics in a wide range of networks. We find that the widely-used degree and PageRank fail in ranking users' influence. We find that the best spreaders are consistently located in the k-core across dissimilar social platforms such as Twitter, Facebook, Livejournal and scientific publishing in the American Physical Society. Furthermore, when the complete global network structure is unavailable, we find that the sum of the nearest neighbors' degree is a reliable local proxy for user's influence. Our analysis provides practical instructions for optimal design of strategies for "viral" information dissemination in relevant applications., Comment: 12 pages, 7 figures

Counting the solution number of combinational optimization problems is an important topic in the study of computational complexity, especially on the #P-complete complexity class. In this paper, we first investigate some organizations of Vertex-Cover unfrozen subgraphs by the underlying connectivity and connected components of unfrozen vertices. Then, a Vertex-Cover Solution Number Counting Algorithm is proposed and its complexity analysis is provided, the results of which fit very well with the simulations and have better performance than those by 1-RSB in a neighborhood of c = e for random graphs. Base on the algorithm, variation and fluctuation on the solution number statistics are studied to reveal the evolution mechanism of the solution numbers. Besides, marginal probability distributions on the solution space are investigated on both random graph and scale-free graph to illustrate different evolution characteristics of their solution spaces. Thus, doing solution number counting based on graph expression of solution space should be an alternative and meaningful way to study the hardness of NP-complete and #P-complete problems, and appropriate algorithm design can help to achieve better approximations of solving combinational optimization problems and the corresponding counting problems., Comment: 17 pages, 6 figures

We investigate the collective dynamics of excitatory-inhibitory excitable networks in response to external stimuli. How to enhance dynamic range, which represents the ability of networks to encode external stimuli, is crucial to many applications. We regard the system as a two-layer network (E-Layer and I-Layer) and explore the criticality and dynamic range on diverse networks. Interestingly, we find that phase transition occurs when the dominant eigenvalue of E-layer's weighted adjacency matrix is exactly one, which is only determined by the topology of E-Layer. Meanwhile, it is shown that dynamic range is maximized at critical state. Based on theoretical analysis, we propose an inhibitory factor for each excitatory node. We suggest that if nodes with high inhibitory factors are cut out from I-Layer, dynamic range could be further enhanced. However, because of the sparseness of networks and passive function of inhibitory nodes, the improvement is relatively small compared tooriginal dynamic range. Even so, this provides a strategy to enhance dynamic range., Comment: 7 pages, 9 figures

We have investigated the Josephson coupling between two s-wave superconductors separated by the ferromagnetic trilayers with noncollinear magnetization. We find that the long-range triplet critical current will oscillate with the strength of the exchange field and the thickness of the interface layer, when the interface magnetizations are orthogonal to the central magnetization. This feature is induced by the spatial oscillations of the spin-triplet state |\uparrow\downarrow>+|\downarrow\uparrow> in the interface layer. Moreover, the critical current can exhibit a characteristic nonmonotonic behavior, when the misalignment angle between interface magnetization and central ferromagnet increases from 0 to \pi/2. This peculiar behavior will take place under the condition that the original state of the junction with the parallel magnetizations is the \pi state.

The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a `Devil's staircase' of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erdos-Renyi percolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erdos-Renyi and other processes exhibiting global continuity, including models exhibiting an `explosive' transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions, which is a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.

Recently, the modified BFW model on random graph [Phys. Rev. Lett., 106, 115701 (2011)], which shows a strongly discontinuous percolation transition with multiple giant components, has attracted much attention from physicists, statisticians and materials scientists. In this paper, by establishing the theoretical expression of evolution equations on the modified BFW model, the steady-state and evolution process are analyzed and a close correspondence is built between the values of parameter \alpha and the number of giant components in steady-states, which fits very well with the numerical simulations. In fact, with the value of \alpha decreasing to 0.25, the error between theoretical and numerical results is smaller than 4% and trends to 0 rapidly. Furthermore, the sizes of giant components for different evolution strategies can also be obtained by solving some constraints derived from the evolution equations. The analysis of the steady-state and evolution process is of great help to explain why the percolation of modified BFW model is explosive and how explosive it is., Comment: 12 pages, 5 figures

To solve the combinatorial optimization problems especially the minimal Vertex-cover problem with high efficiency, is a significant task in theoretical computer science and many other subjects. Aiming at detecting the solution space of Vertex-cover, a new structure named interaction between nodes is defined and discovered for random graph, which results in the emergence of the frustration and long-range correlation phenomenon. Based on the backbones and interactions with a node adding process, we propose an Interaction and Backbone Evolution Algorithm to achieve the reduced solution graph, which has a direct correspondence to the solution space of Vertex-cover. By this algorithm, the whole solution space can be obtained strictly when there is no leaf-removal core on the graph and the odd cycles of unfrozen nodes bring great obstacles to its efficiency. Besides, this algorithm possesses favorable exactness and has good performance on random instances even with high average degrees. The interaction with the algorithm provides a new viewpoint to solve Vertex-cover, which will have a wide range of applications to different types of graphs, better usage of which can lower the computational complexity for solving Vertex-cover.

Understanding what types of phenomena lead to discontinuous phase transitions in the connectivity of random networks is an outstanding challenge. Here we show that a simple stochastic model of graph evolution leads to a discontinuous percolation transition and we derive the underlying mechanism responsible: growth by overtaking. Starting from a collection of $n$ isolated nodes, potential edges chosen uniformly at random from the complete graph are examined one at a time while a cap, $k$, on the maximum allowed component size is enforced. Edges whose addition would exceed $k$ can be simply rejected provided the accepted fraction of edges never becomes smaller than a function which decreases with $k$ as $g(k) = 1/2 + (2k)^{-\beta}$. We show that if $\beta < 1$ it is always possible to reject a sampled edge and the growth in the largest component is dominated by an overtaking mechanism leading to a discontinuous transition. If $\beta > 1$, once $k \ge n^{1/\beta}$, there are situations when a sampled edge must be accepted leading to direct growth dominated by stochastic fluctuations and a "weakly" discontinuous transition. We also show that the distribution of component sizes and the evolution of component sizes are distinct from those previously observed and show no finite size effects for the range of $\beta$ studied., Comment: 6 pages. Final version appearing in EPL (2012)

Sequences with good correlation properties are widely used in engineering applications, especially in the area of communications. Among the known sequences, cyclotomic families have the optimal autocorrelation property. In this paper, we decide the cross-correlation function of the known cyclotomic sequences completely. Moreover, to get our results, the relations between the multiplier group and the decimations of the characteristic sequence are also established for an arbitrary difference set.

We introduce a non-interacting boson model to investigate topological structure of complex networks in the present paper. By exactly solving this model, we show that it provides a powerful analytical tool in uncovering the important properties of real-world networks. We find that the ground state degeneracy of this model is equal to the number of connected components in the network and the square of coefficients in the expansion of ground state gives the averaged time for a random walker spending at each node in the infinite time limit. Furthermore, the first excited state appears always on its largest connected component. To show usefulness of this approach in practice, we carry on also numerical simulations on some concrete complex networks. Our results are completely consistent with the previous conclusions derived by graph theory methods., Comment: 4 pages, 3 figures

This paper analyzes the scaling window of a random CSP model (i.e. model RB) for which we can identify the threshold points exactly, denoted by $r_{cr}$ or $p_{cr}$. For this model, we establish the scaling window $W(n,\delta)=(r_{-}(n,\delta), r_{+}(n,\delta))$ such that the probability of a random instance being satisfiable is greater than $1-\delta$ for $rr_{+}(n,\delta)$. Specifically, we obtain the following result $$W(n,\delta)=(r_{cr}-\Theta(\frac{1}{n^{1-\epsilon}\ln n}), \ r_{cr}+\Theta(\frac{1}{n\ln n})),$$ where $0\leq\epsilon<1$ is a constant. A similar result with respect to the other parameter $p$ is also obtained. Since the instances generated by model RB have been shown to be hard at the threshold, this is the first attempt, as far as we know, to analyze the scaling window of such a model with hard instances.