Temporal networks

A great variety of systems in nature, society and technology–from the web of sexual contacts to the Internet, from the nervous system to power grids–can be modeled as graphs of vertices coupled by edges. The network structure, describing how the graph is wired, helps us understand, predict and optimize the behavior of dynamical systems. In many cases, however, the edges are not continuously active. As an example, in networks of communication via e-mail, text messages, or phone calls, edges represent sequences of instantaneous or practically instantaneous contacts. In some cases, edges are active for non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can be represented by a graph where an edge between two individuals is on throughout the time they are at the same ward. Like network topology, the temporal structure of edge activations can affect dynamics of systems interacting through the network, from disease contagion on the network of patients to information diffusion over an e-mail network. In this review, we present the emergent field of temporal networks, and discuss methods for analyzing topological and temporal structure and models for elucidating their relation to the behavior of dynamical systems. In the light of traditional network theory, one can see this framework as moving the information of when things happen from the dynamical system on the network, to the network itself. Since fundamental properties, such as the transitivity of edges, do not necessarily hold in temporal networks, many of these methods need to be quite different from those for static networks. The study of temporal networks is very interdisciplinary in nature. Reflecting this, even the object of study has many names—temporal graphs, evolving graphs, time-varying graphs, time-aggregated graphs, time-stamped graphs, dynamic networks, dynamic graphs, dynamical graphs, and so on. This review covers different fields where temporal graphs are considered, but does not attempt to unify related terminology—rather, we want to make papers readable across disciplines.     

Renormalizing Sznajd model on complex networks taking into account the effects of growth mechanisms

We present a renormalization approach to solve the Sznajd opinion formation model on complex networks. For the case of two opinions, we present an expression of the probability of reaching consensus for a given opinion as a function of the initial fraction of agents with that opinion. The calculations reproduce the sharp transition of the model on a fixed network, as well as the recently observed smooth function for the model when simulated on a growing complex networks.     

Superpolynomial growth in the number of attractors in Kauffman networks

The Kauffman model describes a particularly simple class of random Boolean networks. Despite the simplicity of the model, it exhibits complex behavior and has been suggested as a model for real world network problems. We introduce a novel approach to analyzing attractors in random Boolean networks, and applying it to Kauffman networks we prove that the average number of attractors grows faster than any power law with system size.     

Scale-free networks without growth

In this paper, we proposed an ungrowing scale-free network model, indicating the growth may not be a necessary condition of the self-organization of a network in a scale-free structure. The analysis shows that the degree distributions of the present model can varying from the Poisson form to the power-law form with the decrease of a free parameter α. This model provides a possible mechanism for the evolution of some scale-free networks with fixed size, such as the friendship networks of school children and the functional networks of the human brain.     

Damage growth in random fuse networks

The correlations among elements that break in random fuse network fracture are studied, with disorder strong enough to allow for volume damage before final failure. The growth of microfractures is found to be uncorrelated above a lengthscale, that increases as the final breakdown approaches. Since the fuse network strength decreases with sample size, asymptotically the process resembles more and more mean-field-like (“democratic fiber bundle”) fracture. This is found from the microscopic dynamics of avalanches or microfractures, from a study of damage localization via entropy, and from the final damage profile. In particular, the last one is statistically constant, except exactly at the final crack zone, in spite of the fact that the fracture surfaces are self-affine. This also implies that the correlations in damage are not extensive.     

Elastic effects in disordered nematic networks

Elastic effects in a model of disordered nematic elastomers are numerically investigated in two dimensions. Networks crosslinked in the isotropic phase exhibit an unusual soft mechanical response against stretching. It arises from a gradual alignment of orientationally correlated regions that are elongated along the director. A sharp crossover to a macroscopically aligned state is obtained on further stretching. The effect of random internal stress is also discussed.     

Collective effects in networks of negatron elements

Results are presented from numerical simulations and experimental studies of a network system consisting of elements with negative differential resistance (negatrons). It is shown that systems of this type have valuable functional characteristics owing to a cooperative interaction of the elements and can be used to create information systems for pattern analysis and recognition. A variational principle is introduced which uses the macroscopic positions to determine the direction of evolution of the state of the network under the influence of applied signals. The possibility of integrating negatron networks into engineering systems may eventually lead to the creation of new materials of the synergetic type for use in electronic technology. Zh. Tekh. Fiz. 68, 1–8 (May 1998)     

Temporal and dimensional effects in evolutionary graph theory

The spread in time of a mutation through a population is studied analytically and computationally in fully connected networks and on spatial lattices. The time t* for a favorable mutation to dominate scales with the population size N as N(D+1)/D in D-dimensional hypercubic lattices and as NlnN in fully-connected graphs. It is shown that the surface of the interface between mutants and nonmutants is crucial in predicting the dynamics of the system. Network topology has a significant effect on the equilibrium fitness of a simple population model incorporating multiple mutations and sexual reproduction.     

Mean-field effects on the aging transitions in the damaged networks

We investigate the dynamical robustness property of the damaged network of active and inactive oscillators under the influence of the mean-field diffusion. The tolerance of dynamical activity of the entire coupled network has realized through the aging transition in the coupled dynamical network. We analytically derived the critical threshold of mean-field density and coupling values for the appearance of the aging transition in the damaged network. By using the critical values as a quantifiable measure of dynamical robustness of the damaged network, we showed that higher mean-field value is favorable to increase the dynamical robustness of the entire network. We also perform the numerical experiment on the network of Stuart-Landau oscillators and the obtained numerical results have an excellent agreement with the analytical findings. Finally, we extend our investigation into the coupled time-delayed network and discussed the affirmative influence of the mean-field parameter on the dynamical robustness of the network.     

Temporal decorrelation of collective oscillations in neural networks with local inhibition and long-range excitation

We consider two neuronal networks coupled by long-range excitatory interactions. Oscillations in the gamma frequency band are generated within each network by local inhibition. When long-range excitation is weak, these oscillations phase lock with a phase shift dependent on the strength of local inhibition. Increasing the strength of long-range excitation induces a transition to chaos via period doubling or quasiperiodic scenarios. In the chaotic regime, oscillatory activity undergoes fast temporal decorrelation. The generality of these dynamical properties is assessed in firing-rate models as well as in large networks of conductance-based neurons.     

Temporal effects in millimetre wave aerosol spectra and the influence of infrared radiation

Time variations of millimetre wave maser emission from water aerosols on the time scale of an hour appear to be related to the incident infrared flux. They explain spectral window variability which often affects their spectra.     

Migration-driven aggregate growth on scale-free networks

We study the kinetics of migration-driven aggregate growth on completely connected scale-free networks. A reversible migration system is considered with the size-dependent rate kernel K(k; l/i;j) approximately k(u)i(v)(lj)(v), at which an i-mer aggregate located on the node with j links gains one monomer from a k-mer aggregate on the node with l links. The results show that the evolution behavior of the aggregate size distribution is drastically different from that for the corresponding same system in normal space. This model can be used to mimic some phenomena such as the distribution of city populations. Moreover, we verify our analytic results in good agreement with the data of the population distributions of all U.S. counties.     

Cut-offs and finite size effects in scale-free networks

We analyze the degree distributions cut-off in finite size scale-free networks. We show that the cut-off behavior with the number of vertices N is ruled by the topological constraints induced by the connectivity structure of the network. Even in the simple case of uncorrelated networks, we obtain an expression of the structural cut-off that is smaller than the natural cut-off obtained by means of extremal theory arguments. The obtained results are explicitly applied in the case of the configuration model to recover the size scaling of tadpoles and multiple edges.Received: 18 November 2003, Published online: 24 February 2004PACS: 89.75.-k Complex systems - 87.23.Ge Dynamics of social systems - 05.70.Ln Nonequilibrium and irreversible thermodynamics     

Modeling the effects of social impact on epidemic spreading in complex networks

We investigate by mean-field analysis and extensive simulations the effects of social impact on epidemic spreading in various typical networks with two types of nodes: active nodes and passive nodes, of which the behavior patterns are modeled according to the social impact theory. In this study, nodes are not only the media to spread the virus, but also disseminate their opinions on the virus—whether there is a need for certain self-protection measures to be taken to reduce the risk of being infected. Our results indicate that the interaction between epidemic spreading and opinion dynamics can have significant influences on the spreading of infectious diseases and related applications, such as the implementation of prevention and control measures against the infectious diseases.     

Communication activity in social networks: growth and correlations

We investigate the timing of messages sent in two online communities with respect to growth fluctuations and long-term correlations. We find that the timing of sending and receiving messages comprises pronounced long-term persistence. Considering the activity of the community members as growing entities, i.e. the cumulative number of messages sent (or received) by the individuals, we identify non-trivial scaling in the growth fluctuations which we relate to the long-term correlations. We find a connection between the scaling exponents of the growth and the long-term correlations which is supported by numerical simulations based on peaks over threshold. In addition, we find that the activity on directed links between pairs of members exhibits long-term correlations, indicating that communication activity with the most liked partners may be responsible for the long-term persistence in the timing of messages. Finally, we show that the number of messages, M, and the number of communication partners, K, of the individual members are correlated following a power-law, K ~ M ^λ , with exponent λ ≈ 3 / 4.     

Effects of accelerating growth on the evolution of weighted complex networks

Many real systems possess accelerating statistics where the total number of edges grows faster than the network size. In this paper, we propose a simple weighted network model with accelerating growth. We derive analytical expressions for the evolutions and distributions for strength, degree, and weight, which are relevant to accelerating growth. We also find that accelerating growth determines the clustering coefficient of the networks. Interestingly, the distributions for strength, degree, and weight display a transition from scale-free to exponential form when the parameter with respect to accelerating growth increases from a small to large value. All the theoretical predictions are successfully contrasted with numerical simulations.     

Nonlinear growth in weighted networks with neighborhood preferential attachment

We propose a nonlinear growing model for weighted networks with two significant characteristics: (i) the new weights triggered by new edges at each time step grow nonlinearly with time; and (ii) a neighborhood local-world exists for local preferential attachment, which is defined as one selected node and its neighbors. Global strength-driven and local weight-driven preferential attachment mechanisms are involved in our model. We study the evolution process through both mathematical analysis and numerical simulation, and find that the model exhibits a wide-range power-law distribution for node degree, strength, and weight. In particular, a nonlinear degree–strength relationship is obtained. This nonlinearity implies that accelerating growth of new weights plays a nontrivial role compared with accelerating growth of edges. Because of the specific local-world model, a small-world property emerges, and a significant hierarchical organization, independent of the parameters, is observed.