Heterogeneous Mean Field Research Articles

An application of complex networks on predicting the behavior of infectious disease on campus

Networks are widely used in understanding the spread of infectious disease. Universities and colleges are characterized by frequent social gatherings and extensive interpersonal communication, making them ideal applications of complex networks. In this paper, a compound model consisting of a homogeneous (small world) network and a heterogeneous (scale free) network is established, the spreading characteristics of epidemics are analyzed by using the mean‐field (MF) and the heterogeneous mean‐field (HMF) approaches, and the effects of various factors such as the total node number (N), the randomly reconnection probability (p), the initial proportion of infected node (p0), the average degree (k), and the shared nodes (s) are simulated by numerical calculations. The simulation results are in consistent with theoretical predictions, indicating that the randomness effect decreases with increasing k and N and is small when k (k = 8) and N (N = 200) are relatively large; the randomness effect is more significant in the scale free (SF) network than in the small world (SW) network, and the disease spread faster in the SF network; shared nodes can weakly increase the disease spread, but when the shared nodes are involved in an infected and an uninfected network, it can induce the disease outbreak in the uninfected network.

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.

Innovation of Mental Health Education Path in Colleges and Universities under the Perspective of “Internet +”

Abstract The traditional mental health education model has made it difficult to cope with the increasingly serious psychological problems of college students. In this paper, under the perspective of “Internet+”, based on the social network of college students, we constructed a propagation dynamics model of depression in colleges and universities and utilized the heterogeneous mean field to detect the propagation status of students’ depression. After calculating the comprehensive centrality degree of students with depression, the community mining of peer psychological adjustment is completed. The C-P similarity of students under the mental health perspective was defined by the ordering of the comprehensive centrality degree and the friend recommendation results were obtained by the distance measure of similarity. In the network of students without depressed mood, after a period of oscillation initially, the end state can be basically reached within 1000 steps, and the infected nodes reach 1000. When the depressed mood factor is small, such as S=0.02 and S=0.04, it has been able to prevent the transformation of depressed mood to the state of all 0, and the proportion occupied by positive and negative moods in the network is basically stable. Internet technology effectively improves the problem of identifying students with psychological abnormalities and implementing targeted mental health education.

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.

Heterogeneous pair-approximation analysis for susceptible-infectious-susceptible epidemics on networks.

The pair heterogeneous mean-field (PHMF) model has been used extensively in previous studies to investigate the dynamics of susceptible-infectious-susceptible epidemics on complex networks. However, the approximate treatment of the classical or reduced PHMF models lacks a rigorous theoretical analysis. By means of the standard and full PHMF models, we first derived the equivalent conditions for the approximate model treatment. Furthermore, we analytically derived a novel epidemic threshold for the PHMF model, and we demonstrated via numerical simulations that this threshold condition differs from all those reported in earlier studies. Our findings indicate that both the reduced and full PHMF models agree well with continuous-time stochastic simulations, especially when infection is spreading at considerably higher rates.

Heterogeneous mean-field theory for two-species symbiotic processes on networks.

A simple model to study cooperation is the two-species symbiotic contact process (2SCP), in which two different species spread on a graph and interact by a reduced death rate if both occupy the same vertex, representing a symbiotic interaction. The 2SCP is known to exhibit a complex behavior with a rich phase diagram, including continuous and discontinuous transitions between the active phase and extinction. In this work, we advance the understanding of the phase transition of the 2SCP on uncorrelated networks by developing a heterogeneous mean-field (HMF) theory, in which the heterogeneity of contacts is explicitly reckoned. The HMF theory for networks with power-law degree distribution shows that the region of bistability (active and inactive phases) in the phase diagram shrinks as the heterogeneity level is increased by reducing the degree exponent. Finite-size analysis reveals a complex behavior where a pseudodiscontinuous transition at a finite size can be converted into a continuous one in the thermodynamic limit, depending on degree exponent and symbiotic coupling. The theoretical results are supported by extensive numerical simulations.

Binary-state dynamics on complex networks: Stochastic pair approximation and beyond

Theoretical approaches to binary-state models on complex networks are generally restricted to infinite size systems, where a set of non-linear deterministic equations is assumed to characterize its dynamics and stationary properties. We develop in this work the stochastic formalism of the different compartmental approaches, these are: approximate master equation (AME), pair approximation (PA) and heterogeneous mean field (HMF), in descending order of accuracy. Using different system-size expansions of a general master equation, we are able to obtain approximate solutions of the fluctuations and finite-size corrections of the global state. On the one hand, far from criticality, the deviations from the deterministic solution are well captured by a Gaussian distribution whose properties we derive, including its correlation matrix and corrections to the average values. On the other hand, close to a critical point there are non-Gaussian statistical features that can be described by the finite-size scaling functions of the models. We show how to obtain the scaling functions departing only from the theory of the different approximations. We apply the techniques for a wide variety of binary-state models in different contexts, such as epidemic, opinion and ferromagnetic models.

Constrained evolutionary algorithms for epidemic spreading curing policy

The design and developments of policies aiming to control and contain spreading processes when resources are limited is an important problem in many application domains dealing with resource allocation, such as public health and network security. This problem, referred as Optimal Curing Policy (OCP) problem, can be formalized as a constrained minimization problem by relying on the approximated heterogeneous N-Intertwined Mean-Field Approximation (NIMFA) model of the SIS spreading process. In this paper, an approach which combines Differential Evolution and Genetic Algorithms is proposed to solve the OCP problem. The hybridization leverages the best characteristics of the two methods to produce high quality solutions in an efficient and effective way. An extensive experimentation on both real-world and synthetic networks shows that the approach is able to outperform a standard solver for semidefinite programming.

Concurrency and reachability in treelike temporal networks

Network properties govern the rate and extent of various spreading processes, from simple contagions to complex cascades. Recently, the analysis of spreading processes has been extended from static networks to temporal networks, where nodes and links appear and disappear. We focus on the effects of accessibility, whether there is a temporally consistent path from one node to another, and reachability, the density of the corresponding accessibility graph representation of the temporal network. The level of reachability thus inherently limits the possible extent of any spreading process on the temporal network. We study reachability in terms of the overall levels of temporal concurrency between edges and the structural cohesion of the network agglomerating over all edges. We use simulation results and develop heterogeneous mean-field model predictions for random networks to better quantify how the properties of the underlying temporal network regulate reachability.

An Improved Heterogeneous Mean-Field Theory for the Ising Model on Complex Networks**Supported by the National Natural Science Foundation of China under Grant Nos. 11875069, 11405001, and the Key Scientific Research Fund of Anhui Provincial Education Department under Grant No. KJ2019A0781

Heterogeneous mean-field theory is commonly used methodology to study dynamical processes on complex networks, such as epidemic spreading and phase transitions in spin models. In this paper, we propose an improved heterogeneous mean-field theory for studying the Ising model on complex networks. Our method shows a more accurate prediction in the critical temperature of the Ising model than the previous heterogeneous mean-field theory. The theoretical results are validated by extensive Monte Carlo simulations in various types of networks.

Searching efficiency of multiple walkers on the weighted networks

The weighted networks are realistic forms of networks, where a weight is attached to each link. In order to better study searching efficiency of multiple walkers we extend the study on the binary networks to the weighted networks. In this paper, the main aim is to measure the searching efficiency of multiple walkers on the weighted networks. Firstly, we review the theoretical foundations related to our study. Secondly, we introduce the heterogeneous mean-field (HMF) theory and the annealed network approach on the weighted networks. Then, we deduce the analysis formula of mean first parallel passage time (MFPPT). Finally, we study the global mean first parallel passage time (GMFPPT) and compare it to the searching efficiency of a single walker previously studied. The obtained result shows that the GMFPPT follows a uniform power law with the number of walkers. The key of this paper is to apply the heterogeneous mean-field (HMF) theory and the annealed weighted network approach to replace the weighted uncorrelated networks with the weighted fully connected networks and then construct a general probability transfer matrix on the weighted fully connected networks.

Onset of synchronization of Kuramoto oscillators in scale-free networks.

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent 2<γ≤3, in line with what has been observed for other dynamical processes in such networks. For γ>3, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.

Precisely identifying the epidemic thresholds in real networks via asynchronous updating

Two numerical simulation methods, asynchronous updating and synchronous updating, are applied to mimic epidemic spreading and identify epidemic threshold. As a continuous time Markov process, asynchronous updating makes only one node be selected to change its state in each time step, and thus reflects the fact that nodes are updated independently, which is more reasonable to describe the real dynamic process of disease spreading. Unlike previous studies based on prevalent synchronous updating, in this paper, we mainly apply asynchronous updating to precisely identify epidemic thresholds of SIR spreading dynamics in real networks. Meanwhile, we also use four benchmark theoretical methods, i.e., the heterogeneous mean-field (HMF), the quenched mean-field (QMF), the dynamical message passing (DMP) and the connectivity matrix (CM), to verify the identification accuracy based on asynchronous updating. The extensive numerical experiments in 41 real networks show that the identification accuracy approaches more closely to the theoretical results obtained from the CM because the CM incorporates network topology with dynamic correlations. In addition, because asynchronous updating is high time complexity comparing with synchronous updating, we further investigate the approximation of synchronous updating to asynchronous updating by modulating very small time step. When the time step of synchronous updating is set with 0.2, it can approach closely to the identification accuracy based asynchronous updating, and guarantee a lower time complexity.

Reactive explorers to unravel network topology

A procedure is developed and tested to recover the distribution of connectivity of an a priori unknown network, by sampling the dynamics of an ensemble made of reactive walkers. The relative weight between reaction and relocation is gauged by a scalar control parameter, which can be adjusted at will. Different equilibria are attained by the system, following the externally imposed modulation, and reflecting the interplay between reaction and diffusion terms. The information gathered on the observation node is used to predict the stationary density as displayed by the system, via a direct implementation of the celebrated Heterogeneous Mean Field (HMF) approximation. This knowledge translates into a linear problem which can be solved to return the entries of the sought distribution. A variant of the model is then considered which consists in assuming a localized source where the reactive constituents are injected, at a rate that can be adjusted as a stepwise function of time. The linear problem obtained when operating in this setting allows one to recover a fair estimate of the underlying system size. Numerical experiments are carried so as to challenge the predictive ability of the theory.

Efficient message passing for cascade size distributions

How big is the risk that a few initial failures of networked nodes amplify to large cascades that endanger the functioning of the system? Common answers refer to the average final cascade size. Two analytic approaches allow its computation: (a) (heterogeneous) mean field approximation and (b) belief propagation. The former applies to (infinitely) large locally tree-like networks, while the latter is exact on finite trees. Yet, cascade sizes can have broad and multi-modal distributions that are not well represented by their average. Full distribution information is essential to identify likely events and to estimate the tail risk, i.e. the probability of extreme events. We therefore present an efficient message passing algorithm that calculates the cascade size distribution in finite networks. It is exact on finite trees and for a large class of cascade processes. An approximate version applies to any network structure and performs well on locally tree-like networks, as we show with several examples.

Effective degree theory for awareness and epidemic spreading on multiplex networks

Epidemic spreading processes on multiplex networks have richer dynamical properties than those on single layered networks. To describe the intertwined processes on such networks, heterogeneous mean field (HMF) approach for continuous-time processes and microscopic Markov chain approach for discrete-time processes have been proposed. However, it has been shown that the time evolution of infected individuals and the final epidemic size obtained from these approaches have noticeable discrepancy comparing to those from Monte Carlo simulations. In this paper, we extend the approach of effective degree theory (EDT) on multiplex networks. We will show that predictions obtained from the EDT have excellent agreement with Monte Carlo simulations. Moreover, since the dynamics on multiplex networks involve more dynamical variables, which may invoke more computations, to reduce the computational burden, we further develop an approach based on partial effective degree theory (PEDT) for analyzing the dynamics on multiplex networks, where one layer adopts EDT and the other layer adopts the HMF. Our results show that PEDT has a good performance in predicting the target dynamical process.

Effective Degree Theory on Multiplex Networks for Concurrent Three-State Spreading Dynamics

We propose a model of concurrent three-state spreading dynamics on multiplex networks to study the co-evolution process of epidemic spreading and information diffusion. The spreading dynamics and information diffusion process are both described by three-state models which can be used to address a wide range of common spreading behaviors. As accurate prediction is important in understanding the behaviors of complicated spreading dynamics, based on our proposed model, we develop a continuous-time effective degree theory (EDT) to delicately analyze the concurrent dynamics. We show that compared to the Monte Carlo simulations, this developed theory could predict the behavior of the dynamics in high accuracy, outperforming the dominantly adopted heterogeneous mean field theory applied on relevant dynamics on multiplex networks.

Risk-Sensitive Mean Field Games via the Stochastic Maximum Principle

In this paper, we consider risk-sensitive mean field games via the risk-sensitive maximum principle. The problem is analyzed through two sequential steps: (i) risk-sensitive optimal control for a fixed probability measure, and (ii) the associated fixed-point problem. For step (i), we use the risk-sensitive maximum principle to obtain the optimal solution, which is characterized in terms of the associated forward–backward stochastic differential equation (FBSDE). In step (ii), we solve for the probability law induced by the state process with the optimal control in step (i). In particular, we show the existence of the fixed point of the probability law of the state process determined by step (i) via Schauder’s fixed-point theorem. After analyzing steps (i) and (ii), we prove that the set of N optimal distributed controls obtained from steps (i) and (ii) constitutes an approximate Nash equilibrium or $$\epsilon $$-Nash equilibrium for the N player risk-sensitive game, where $$\epsilon \rightarrow 0$$ as $$N \rightarrow \infty $$ at the rate of $$O(\frac{1}{N^{1/(n+4)}})$$. Finally, we discuss extensions to heterogeneous (non-symmetric) risk-sensitive mean field games.

Correlations between thresholds and degrees: An analytic approach to model attacks and failure cascades.

Two node variables determine the evolution of cascades in random networks: a node's degree and threshold. Correlations between both fundamentally change the robustness of a network, yet they are disregarded in standard analytic methods as local tree or heterogeneous mean field approximations, since order statistics are difficult to capture analytically because of their combinatorial nature. We show how they become tractable in the thermodynamic limit of infinite network size. This enables the analytic description of node attacks that are characterized by threshold allocations based on node degree. Using two examples, we discuss possible implications of irregular phase transitions and different speeds of cascade evolution for the control of cascades.

Social hotspot propagation dynamics model based on heterogeneous mean field and evolutionary games

In the field of social network analysis, information diffusion is a focus of current research. Taking into account the real topological relations among the participants and the psychological characteristics of the users, in this paper, a hotspot propagation model based on heterogeneous mean field and evolutionary games is proposed. First, in real social networks, the changes of hotspot’s trend could lead to the dynamic changes of users’ willingness to participate in the hot topic. This effect is reflected in the dynamic behaviors among the users. In this work, based on the evolutionary games, a dynamic evolution mechanism for users’ willingness to participate in hotspot is constructed and dynamically adjusts the infection rate of information dissemination model. Second, in view of the heterogeneity of the real network structure and the complexity of the heterogeneous mean field, graphical evolutionary game is introduced to improve the heterogeneous mean field. Thus, a new dynamics model of information dissemination is constructed based on graphical evolutionary game. Finally, considering the dynamic behavior among the nodes and the heterogeneity of real social networks, we obtain a hotspot propagation model based on dynamic evolution mechanism and improved heterogeneous mean field. To verify the proposed model, we perform simulations over synthetic networks and real network. Experiments show that the model effectively reveal the impact of different driving factors on information dissemination, and predict the trend of information dissemination in social networks.