Heterogeneous Mean-field Approximation Research Articles

Generation of Scale-Free Assortative Networks via Newman Rewiring for Simulation of Diffusion Phenomena

By collecting and expanding several numerical recipes developed in previous work, we implement an object-oriented Python code, based on the networkX library, for the realization of the configuration model and Newman rewiring. The software can be applied to any kind of network and “target” correlations, but it is tested with focus on scale-free networks and assortative correlations. In order to generate the degree sequence we use the method of “random hubs”, which gives networks with minimal fluctuations. For the assortative rewiring we use the simple Vazquez-Weigt matrix as a test in the case of random networks; since it does not appear to be effective in the case of scale-free networks, we subsequently turn to another recipe which generates matrices with decreasing off-diagonal elements. The rewiring procedure is also important at the theoretical level, in order to test which types of statistically acceptable correlations can actually be realized in concrete networks. From the point of view of applications, its main use is in the construction of correlated networks for the solution of dynamical or diffusion processes through an analysis of the evolution of single nodes, i.e., beyond the Heterogeneous Mean Field approximation. As an example, we report on an application to the Bass diffusion model, with calculations of the time tmax of the diffusion peak. The same networks can additionally be exported in environments for agent-based simulations like NetLogo.

Emergence of chaotic cluster synchronization in heterogeneous networks.

Many real-world complex systems rely on cluster synchronization to function properly. A cluster of nodes exhibits synchronous behavior, while others behave erratically. Predicting the emergence of these clusters and understanding the mechanism behind their structure and variation in response to a parameter change is a daunting task in networks that lack symmetry. We unravel the mechanism for the emergence of cluster synchronization in heterogeneous random networks. We develop heterogeneous mean-field approximation together with a self-consistent theory to determine the onset and stability of the cluster. Our analysis shows that cluster synchronization occurs in a wide variety of heterogeneous networks, node dynamics, and coupling functions. The results could lead to a new understanding of the dynamical behavior of networks ranging from neural to social.

Investigation on the influence of heterogeneous synergy in contagion processes on complex networks.

Synergistic contagion in a networked system occurs in various forms in nature and human society. While the influence of network's structural heterogeneity on synergistic contagion has been well studied, the impact of individual-based heterogeneity on synergistic contagion remains unclear. In this work, we introduce individual-based heterogeneity with a power-law form into the synergistic susceptible-infected-susceptible model by assuming the synergistic strength as a function of individuals' degree and investigate this synergistic contagion process on complex networks. By employing the heterogeneous mean-field (HMF) approximation, we analytically show that the heterogeneous synergy significantly changes the critical threshold of synergistic strength σc that is required for the occurrence of discontinuous phase transitions of contagion processes. Comparing to the synergy without individual-based heterogeneity, the value of σc decreases with degree-enhanced synergy and increases with degree-suppressed synergy, which agrees well with Monte Carlo prediction. Next, we compare our heterogeneous synergistic contagion model with the simplicial contagion model [Iacopini et al., Nat. Commun. 10, 2485 (2019)], in which high-order interactions are introduced to describe complex contagion. Similarity of these two models are shown both analytically and numerically, confirming the ability of our model to statistically describe the simplest high-order interaction within HMF approximation.

Contagion dynamics on adaptive multiplex networks with awareness-dependent rewiring* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11601294 and 61873154), Shanxi Scholarship Council of China (Grant No. 2016-011), the Shanxi Province Science Foundation for Youths (Grant Nos. 201601D021012, 201801D221011, 201901D211159, 201801D221007 and 201801D221003), and the 1331 Engineering Project of Shanxi Province, China.

Over the last few years, the interplay between contagion dynamics of social influences (e.g., human awareness, risk perception, and information dissemination) and biological infections has been extensively investigated within the framework of multiplex networks. The vast majority of existing multiplex network spreading models typically resort to heterogeneous mean-field approximation and microscopic Markov chain approaches. Such approaches usually manifest richer dynamical properties on multiplex networks than those on simplex networks; however, they fall short of a subtle analysis of the variations in connections between nodes of the network and fail to account for the adaptive behavioral changes among individuals in response to epidemic outbreaks. To transcend these limitations, in this paper we develop a highly integrated effective degree approach to modeling epidemic and awareness spreading processes on multiplex networks coupled with awareness-dependent adaptive rewiring. This approach keeps track of the number of nearest neighbors in each state of an individual; consequently, it allows for the integration of changes in local contacts into the multiplex network model. We derive a formula for the threshold condition of contagion outbreak. Also, we provide a lower bound for the threshold parameter to indicate the effect of adaptive rewiring. The threshold analysis is confirmed by extensive simulations. Our results show that awareness-dependent link rewiring plays an important role in enhancing the transmission threshold as well as lowering the epidemic prevalence. Moreover, it is revealed that intensified awareness diffusion in conjunction with enhanced link rewiring makes a greater contribution to disease prevention and control. In addition, the critical phenomenon is observed in the dependence of the epidemic threshold on the awareness diffusion rate, supporting the metacritical point previously reported in literature. This work may shed light on understanding of the interplay between epidemic dynamics and social contagion on adaptive networks.

Scalar model of flocking dynamics on complex social networks.

We investigate the effects of long-range social interactions in flocking dynamics by studying the dynamics of a scalar model of collective motion embedded in a complex network representing a pattern of social interactions, as observed in several social species. In this scalar model we find a phenomenology analogous to that observed in the classic Vicsek model: In networks with low heterogeneity, a phase transition separates an ordered from a disordered phase. At high levels of heterogeneity, instead, the transition is suppressed, and the system is always ordered. This observation is backed up analytically by the solution of a modified scalar model within an heterogeneous mean-field approximation. Our work extends the understanding of the effects of social interactions in flocking dynamics and opens the path to the analytical study of more complex topologies of social ties.

Synergistic effects in threshold models on networks.

Network structure can have a significant impact on the propagation of diseases, memes, and information on social networks. Different types of spreading processes (and other dynamical processes) are affected by network architecture in different ways, and it is important to develop tractable models of spreading processes on networks to explore such issues. In this paper, we incorporate the idea of synergy into a two-state ("active" or "passive") threshold model of social influence on networks. Our model's update rule is deterministic, and the influence of each meme-carrying (i.e., active) neighbor can-depending on a parameter-either be enhanced or inhibited by an amount that depends on the number of active neighbors of a node. Such a synergistic system models social behavior in which the willingness to adopt either accelerates or saturates in a way that depends on the number of neighbors who have adopted that behavior. We illustrate that our model's synergy parameter has a crucial effect on system dynamics, as it determines whether degree-k nodes are possible or impossible to activate. We simulate synergistic meme spreading on both random-graph models and networks constructed from empirical data. Using a heterogeneous mean-field approximation, which we derive under the assumption that a network is locally tree-like, we are able to determine which synergy-parameter values allow degree-k nodes to be activated for many networks and for a broad family of synergistic models.

Synchronization and long-time memory in neural networks with inhibitory hubs and synaptic plasticity.

We investigate the dynamical role of inhibitory and highly connected nodes (hub) in synchronization and input processing of leaky-integrate-and-fire neural networks with short term synaptic plasticity. We take advantage of a heterogeneous mean-field approximation to encode the role of network structure and we tune the fraction of inhibitory neurons f_{I} and their connectivity level to investigate the cooperation between hub features and inhibition. We show that, depending on f_{I}, highly connected inhibitory nodes strongly drive the synchronization properties of the overall network through dynamical transitions from synchronous to asynchronous regimes. Furthermore, a metastable regime with long memory of external inputs emerges for a specific fraction of hub inhibitory neurons, underlining the role of inhibition and connectivity also for input processing in neural networks.

Tristable and multiple bistable activity in complex random binary networks of two-state units

We study complex networks of stochastic two-state units. Our aim is to model discrete stochastic excitable dynamics with a rest and an excited state. Both states are assumed to possess different waiting time distributions. The rest state is treated as an activation process with an exponentially distributed life time, whereas the latter in the excited state shall have a constant mean which may originate from any distribution. The activation rate of any single unit is determined by its neighbors according to a random complex network structure. In order to treat this problem in an analytical way, we use a heterogeneous mean-field approximation yielding a set of equations generally valid for uncorrelated random networks. Based on this derivation we focus on random binary networks where the network is solely comprised of nodes with either of two degrees. The ratio between the two degrees is shown to be a crucial parameter. Dependent on the composition of the network the steady states show the usual transition from disorder to homogeneously ordered bistability as well as new scenarios that include inhomogeneous ordered and disordered bistability as well as tristability. The various steady states differ in their spiking activity expressed by a state dependent spiking rate. Numerical simulations agree with analytic results of the heterogeneous mean-field approximation.

Neural networks with excitatory and inhibitory components: Direct and inverse problems by a mean-field approach.

We study the dynamics of networks with inhibitory and excitatory leak-integrate-and-fire neurons with short-term synaptic plasticity in the presence of depressive and facilitating mechanisms. The dynamics is analyzed by a heterogeneous mean-field approximation, which allows us to keep track of the effects of structural disorder in the network. We describe the complex behavior of different classes of excitatory and inhibitory components, which give rise to a rich dynamical phase diagram as a function of the fraction of inhibitory neurons. Using the same mean-field approach, we study and solve a global inverse problem: reconstructing the degree probability distributions of the inhibitory and excitatory components and the fraction of inhibitory neurons from the knowledge of the average synaptic activity field. This approach unveils new perspectives on the numerical study of neural network dynamics and the possibility of using these models as a test bed for the analysis of experimental data.

Nature of synchronization transitions in random networks of coupled oscillators

We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform, unimodal, and bimodal distribution. We employ a heterogeneous mean-field approximation based on the annealed networks and also perform numerical simulations on the quenched Erdös-Rényi networks. We find that the connectivity disorder drastically changes the nature of the synchronization transitions. In particular, the quenched randomness completely wipes away the diversity of the transition nature, and only a continuous transition appears with the same mean-field exponent for all types of frequency distributions. The physical origin of this unexpected result is discussed.

Susceptible-infected-susceptible model: A comparison ofN-intertwined and heterogeneous mean-field approximations

We introduce the ε-susceptible-infected-susceptible (SIS) spreading model, which is taken as a benchmark for the comparison between the N-intertwined approximation and the Pastor-Satorras and Vespignani heterogeneous mean-field (HMF) approximation of the SIS model. The N-intertwined approximation, the HMF approximation, and the ε-SIS spreading model are compared for different graph types. We focus on the epidemic threshold and the steady-state fraction of infected nodes in networks with different degree distributions. Overall, the N-intertwined approximation is superior to the HMF approximation. The N-intertwined approximation is exactly the same as the HMF approximation in regular graphs. However, for some special graph types, such as the square lattice graph and the path graph, the two mean-field approximations are both very different from the ε-SIS spreading model.

Heterogenous mean-field analysis of a generalized voter-like model on networks

We propose a generalized framework for the study of voter models in complex networks at the the heterogeneous mean-field (HMF) level that (i) yields a unified picture for existing copy/invasion processes and (ii) allows for the introduction of further heterogeneity through degree-selectivity rules. In the context of the HMF approximation, our model is capable of providing straightforward estimates for central quantities such as the exit probability and the consensus/fixation time, based on the statistical properties of the complex network alone. The HMF approach has the advantage of being readily applicable also in those cases in which exact solutions are difficult to work out. Finally, the unified formalism allows one to understand previously proposed voter-like processes as simple limits of the generalized model.