Diffusion in scale-free networks with annealed disorder

Formation and Evolution of Ideal Interfirm Collaborative Innovation Networks Based on Decision-Making Rules for Partner Selection

On the basis of an external and static perspective on the topological structure of collaborative innovation networks, it is extremely difficult to answer the two most important concerns, namely, which structure is ideal and how to develop it in practice. By contrast, this study transfers to internal and dynamic perspectives, and then proposes that the essence of developing the ideal network lies in choosing the best partners. Therefore, we firstly propose the basic decision-making rules for selecting partners. In order of priority: knowledge distance, knowledge complementarity and barter exchange. Secondly, a model is constructed to describe this process of selecting partners and exchanging knowledge. Thirdly, the simulation results show that a small-world network is ideal in the initial stage of collaborative innovation. However, a random network is ideal in the mature periods. This result shows that the ideal network structure is not fixed, but affected by the life cycle of collaborative innovation alliance. Furthermore, this supports the notion that a small world is spontaneously generated in the real world, and also confirms that the formation of a small-world network will be driven intrinsically by a firm’s demand for external knowledge, and not necessarily by the external driving force of social capital. Finally, these findings solve the above two most important questions.

Epidemic Threshold in Continuous-Time Evolving Networks

Current understanding of the critical outbreak condition on temporal networks relies on approximations (time scale separation, discretization) that may bias the results. We propose a theoretical framework to compute the epidemic threshold in continuous time through the infection propagator approach. We introduce the weak commutation condition allowing the interpretation of annealed networks, activity-driven networks, and time scale separation into one formalism. Our work provides a coherent connection between discrete and continuous time representations applicable to realistic scenarios.

A knowledge generation model via the hypernetwork

The influence of the statistical properties of the network on the knowledge diffusion has been extensively studied. However, the structure evolution and the knowledge generation processes are always integrated simultaneously. By introducing the Cobb-Douglas production function and treating the knowledge growth as a cooperative production of knowledge, in this paper, we present two knowledge-generation dynamic evolving models based on different evolving mechanisms. The first model, named “HDPH model,” adopts the hyperedge growth and the hyperdegree preferential attachment mechanisms. The second model, named “KSPH model,” adopts the hyperedge growth and the knowledge stock preferential attachment mechanisms. We investigate the effect of the parameters on the total knowledge stock of the two models. The hyperdegree distribution of the HDPH model can be theoretically analyzed by the mean-field theory. The analytic result indicates that the hyperdegree distribution of the HDPH model obeys the power-law distribution and the exponent is . Furthermore, we present the distributions of the knowledge stock for different parameters . The findings indicate that our proposed models could be helpful for deeply understanding the scientific research cooperation.

Phase transitions with infinitely many absorbing states in complex networks

We investigate the properties of the threshold contact process (TCP), a process showing an absorbing-state phase transition with infinitely many absorbing states, on random complex networks. The finite-size scaling exponents characterizing the transition are obtained in a heterogeneous mean-field (HMF) approximation and compared with extensive simulations, particularly in the case of heterogeneous scale-free networks. We observe that the TCP exhibits the same critical properties as the contact process, which undergoes an absorbing-state phase transition to a single absorbing state. The accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a HMF theory is a universal feature of absorbing-state phase transitions in complex networks, depending only on the locality of the interactions and independent of the number of absorbing states. The conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible-infected-susceptible epidemic spreading model, which in fact belongs to a different universality class in complex networks.

Epidemic thresholds of the susceptible-infected-susceptible model on networks: a comparison of numerical and theoretical results

Recent work has shown that different theoretical approaches to the dynamics of the susceptible-infected-susceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size N. We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show that the latter is in general more accurate, scaling with N with the correct exponent, but often failing to capture the correct prefactor.

Random walks on temporal networks

Many natural and artificial networks evolve in time. Nodes and connections appear and disappear at various time scales, and their dynamics has profound consequences for any processes in which they are involved. The first empirical analysis of the temporal patterns characterizing dynamic networks are still recent, so that many questions remain open. Here, we study how random walks, as a paradigm of dynamical processes, unfold on temporally evolving networks. To this aim, we use empirical dynamical networks of contacts between individuals, and characterize the fundamental quantities that impact any general process taking place upon them. Furthermore, we introduce different randomizing strategies that allow us to single out the role of the different properties of the empirical networks. We show that the random walk exploration is slower on temporal networks than it is on the aggregate projected network, even when the time is properly rescaled. In particular, we point out that a fundamental role is played by the temporal correlations between consecutive contacts present in the data. Finally, we address the consequences of the intrinsically limited duration of many real world dynamical networks. Considering the fundamental prototypical role of the random walk process, we believe that these results could help to shed light on the behavior of more complex dynamics on temporally evolving networks.

Influence of zero range process interaction on diffusion

We study the aspects of diffusion for the case of zero range process interaction on scale-free networks, through statistical quantities such as the mean first passage time, coverage, mean square displacement etc., and pay attention to how the interaction, especially the resulted condensation, influences the diffusion. By mean-field theory we show that the statistical quantities of diffusion can be significantly reduced by the condensation and can be figured out by the waiting time of a particle staying at a node. Numerical simulations have confirmed the theoretical predictions.

Mean-field diffusive dynamics on weighted networks

Diffusion is a key element of a large set of phenomena occurring on natural and social systems modeled in terms of complex weighted networks. Here, we introduce a general formalism that allows to easily write down mean-field equations for any diffusive dynamics on weighted networks. We also propose the concept of annealed weighted networks, in which such equations become exact. We show the validity of our approach addressing the problem of the random walk process, pointing out a strong departure of the behavior observed in quenched real scale-free networks from the mean-field predictions. Additionally, we show how to employ our formalism for more complex dynamics. Our work sheds light on mean-field theory on weighted networks and on its range of validity, and warns about the reliability of mean-field results for complex dynamics.

Langevin approach for the dynamics of the contact process on annealed scale-free networks

We study the dynamics of the contact process, one of the simplest nonequilibrium stochastic processes, taking place on a scale-free network. We consider the network topology as annealed, i.e., all links are rewired at each microscopic time step, so that no dynamical correlation can build up. This is a practical implementation of the absence of correlations assumed by mean-field approaches. We present a detailed analysis of the contact process in terms of a Langevin equation, including explicitly the effects of stochastic fluctuations in the number of particles in finite networks. This allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs. The fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has a hard upper bound. When the upper bound is soft, the presence of outliers with huge connectivity perturbs the picture even more, inducing an apparent shift of the critical point. In light of these findings, recent theoretical and numerical results in the literature are critically reviewed.

Random walks on complex trees

We study the properties of random walks on complex trees. We observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts. First, both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case. Second, the mean first passage time (MFPT) displays a logarithmic degree dependence, in contrast to the inverse degree shape exhibited in looped networks. This deviation can be ascribed to the dominance of source-target topological distance in trees. To show this, we study the distance dependence of a symmetrized MFPT and derive its logarithmic profile, obtaining good agreement with simulation results. These unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees.

Generation of arbitrarily two-point-correlated random networks

Random networks are intensively used as null models to investigate properties of complex networks. We describe an efficient and accurate algorithm to generate arbitrarily two-point degree-degree correlated undirected random networks without self-edges or multiple edges among vertices. With the goal to systematically investigate the influence of two-point correlations, we furthermore develop a formalism to construct a joint degree distribution P(j,k) , which allows one to fix an arbitrary degree distribution P(k) and an arbitrary average nearest neighbor function k_{nn}(k) simultaneously. Using the presented algorithm, this formalism is demonstrated with scale-free networks [P(k) proportional, variantk;{-gamma}] and empirical complex networks [ P(k) taken from network] as examples. Finally, we generalize our algorithm to annealed networks which allows networks to be represented in a mean-field-like manner.

Exploring complex networks through random walks

Most real complex networks--such as protein interactions, social contacts, and the Internet--are only partially known and available to us. While the process of exploring such networks in many cases resembles a random walk, it becomes a key issue to investigate and characterize how effectively the nodes and edges of such networks can be covered by different strategies. At the same time, it is critically important to infer how well can topological measurements such as the average node degree and average clustering coefficient be estimated during such network explorations. The present article addresses these problems by considering random, Barabási-Albert (BA), and geographical network models with varying connectivity explored by three types of random walks: traditional, preferential to untracked edges, and preferential to unvisited nodes. A series of relevant results are obtained, including the fact that networks of the three studied models with the same size and average node degree allow similar node and edge coverage efficiency, the identification of linear scaling with the size of the network of the random walk step at which a given percentage of the nodes/edges is covered, and the critical result that the estimation of the averaged node degree and clustering coefficient by random walks on BA networks often leads to heavily biased results. Many are the theoretical and practical implications of such results.