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Private data, being larger and quality-higher than public data, can greatly improve large language models (LLM). However, due to privacy concerns, this data is often dispersed in multiple silos, making its secure utilization for LLM training a challenge. Federated learning (FL) is an ideal solution for training models with distributed private data, but traditional frameworks like FedAvg are unsuitable for LLM due to their high computational demands on clients. An alternative, split learning, offloads most training parameters to the server while training embedding and output layers locally, making it more suitable for LLM. Nonetheless, it faces significant challenges in security and efficiency. Firstly, the gradients of embeddings are prone to attacks, leading to potential reverse engineering of private data. Furthermore, the server's limitation of handle only one client's training request at a time hinders parallel training, severely impacting training efficiency. In this paper, we propose a Federated Learning framework for LLM, named FL-GLM, which prevents data leakage caused by both server-side and peer-client attacks while improving training efficiency. Specifically, we first place the input block and output block on local client to prevent embedding gradient attacks from server. Secondly, we employ key-encryption during client-server communication to prevent reverse engineering attacks from peer-clients. Lastly, we employ optimization methods like client-batching or server-hierarchical, adopting different acceleration methods based on the actual computational capabilities of the server. Experimental results on NLU and generation tasks demonstrate that FL-GLM achieves comparable metrics to centralized chatGLM model, validating the effectiveness of our federated learning framework.

The sparsity of reward feedback remains a challenging problem in online deep reinforcement learning (DRL). Previous approaches have utilized offline demonstrations to achieve impressive results in multiple hard tasks. However, these approaches place high demands on demonstration quality, and obtaining expert-like actions is often costly and unrealistic. To tackle these problems, we propose a simple and efficient algorithm called Policy Optimization with Smooth Guidance (POSG), which leverages a small set of state-only demonstrations (where only state information is included in demonstrations) to indirectly make approximate and feasible long-term credit assignments and facilitate exploration. Specifically, we first design a trajectory-importance evaluation mechanism to determine the quality of the current trajectory against demonstrations. Then, we introduce a guidance reward computation technology based on trajectory importance to measure the impact of each state-action pair. We theoretically analyze the performance improvement caused by smooth guidance rewards and derive a new worst-case lower bound on the performance improvement. Extensive results demonstrate POSG's significant advantages in control performance and convergence speed in four sparse-reward environments, including the grid-world maze, Hopper-v4, HalfCheetah-v4, and Ant maze. Notably, the specific metrics and quantifiable results are investigated to demonstrate the superiority of POSG., Comment: 31 pages, 23 figures

Deep reinforcement learning (DRL) faces significant challenges in addressing the hard-exploration problems in tasks with sparse or deceptive rewards and large state spaces. These challenges severely limit the practical application of DRL. Most previous exploration methods relied on complex architectures to estimate state novelty or introduced sensitive hyperparameters, resulting in instability. To mitigate these issues, we propose an efficient adaptive trajectory-constrained exploration strategy for DRL. The proposed method guides the policy of the agent away from suboptimal solutions by leveraging incomplete offline demonstrations as references. This approach gradually expands the exploration scope of the agent and strives for optimality in a constrained optimization manner. Additionally, we introduce a novel policy-gradient-based optimization algorithm that utilizes adaptively clipped trajectory-distance rewards for both single- and multi-agent reinforcement learning. We provide a theoretical analysis of our method, including a deduction of the worst-case approximation error bounds, highlighting the validity of our approach for enhancing exploration. To evaluate the effectiveness of the proposed method, we conducted experiments on two large 2D grid world mazes and several MuJoCo tasks. The extensive experimental results demonstrate the significant advantages of our method in achieving temporally extended exploration and avoiding myopic and suboptimal behaviors in both single- and multi-agent settings. Notably, the specific metrics and quantifiable results further support these findings. The code used in the study is available at \url{https://github.com/buaawgj/TACE}., Comment: 35 pages, 36 figures; accepted by Knowledge-Based Systems, not published

Indirect reciprocity unveils how social cooperation is founded upon moral systems. Under the frame of dyadic games based on individual reputations, the "leading-eight" social norms distinguish themselves in promoting and sustaining cooperation. However, in real societies, people constantly interact within larger groups. They are assessed not only by their own behaviors, but also the reputations of the groups they come from and interact with. Here, we study indirect reciprocity in public goods games with group reputations. We show that varying criteria of group reputation assessment destabilize the consensus within leading-eight populations. We find the presence of an optimal, moderately set social assessment criterion that is most conductive to promoting cooperation. As the payoff of group cooperation increases, the assessment criterion evolves to relaxation following two dynamical patterns, indicating that social prosperity encourages people to be tolerate. Our work identifies the role of group reputation assessment to boost cooperation in social populations.

Efficient link prediction in continuous-time dynamic networks is a challenging problem that has attracted much research attention in recent years. A widely used approach to dynamic network link prediction is to extract the local structure of the target link through temporal random walk on the network and learn node features using a coding model. However, this approach often assumes that candidate temporal neighbors follow some certain types of distributions, which may be inappropriate for real-world networks, thereby incurring information loss. To address this limitation, we propose a framework in continuous-time dynamic networks based on Optimal Transmission (OT) and Metropolis Hastings (MH) sampling (COM). Specifically, we use optimal transmission theory to calculate the Wasserstein distance between the current node and the time-valid candidate neighbors to minimize information loss in node information propagation. Additionally, we employ the MH algorithm to obtain higher-order structural relationships in the vicinity of the target link, as it is a Markov Chain Monte Carlo method and can flexibly simulate target distributions with complex patterns. We demonstrate the effectiveness of our proposed method through experiments on eight datasets from different fields., Comment: 11 pages, 7 figures

Autonomous individuals establish a structural complex system through pairwise connections and interactions. Notably, the evolution reflects the dynamic nature of each complex system since it recodes a series of temporal changes from the past, the present into the future. Different systems follow distinct evolutionary trajectories, which can serve as distinguishing traits for system classification. However, modeling a complex system's evolution is challenging for the graph model because the graph is typically a snapshot of the static status of a system, and thereby hard to manifest the long-term evolutionary traits of a system entirely. To address this challenge, we suggest utilizing a heat-driven method to generate temporal graph augmentation. This approach incorporates the physics-based heat kernel and DropNode technique to transform each static graph into a sequence of temporal ones. This approach effectively describes the evolutional behaviours of the system, including the retention or disappearance of elements at each time point based on the distributed heat on each node. Additionally, we propose a dynamic time-wrapping distance GDTW to quantitatively measure the distance between pairwise evolutionary systems through optimal matching. The resulting approach, called the Evolution Kernel method, has been successfully applied to classification problems in real-world structural graph datasets. The results yield significant improvements in supervised classification accuracy over a series of baseline methods.

Molecular fingerprints are significant cheminformatics tools to map molecules into vectorial space according to their characteristics in diverse functional groups, atom sequences, and other topological structures. In this paper, we set out to investigate a novel molecular fingerprint \emph{Anonymous-FP} that possesses abundant perception about the underlying interactions shaped in small, medium, and large molecular scale links. In detail, the possible inherent atom chains are sampled from each molecule and are extended in a certain anonymous pattern. After that, the molecular fingerprint \emph{Anonymous-FP} is encoded in virtue of the Natural Language Processing technique \emph{PV-DBOW}. \emph{Anonymous-FP} is studied on molecular property identification and has shown valuable advantages such as rich information content, high experimental performance, and full structural significance. During the experimental verification, the scale of the atom chain or its anonymous manner matters significantly to the overall representation ability of \emph{Anonymous-FP}. Generally, the typical scale $r = 8$ enhances the performance on a series of real-world molecules, and specifically, the accuracy could level up to above $93\%$ on all NCI datasets., Comment: 11 pages

Network modeling characterizes the underlying principles of structural properties and is of vital significance for simulating dynamical processes in real world. However, bridging structure and dynamics is always challenging due to the multiple complexities in real systems. Here, through introducing the individual's activity rate and the possibility of group interaction, we propose a probabilistic activity driven (PAD) model that could generate temporal higher-order networks with both power-law and high-clustering characteristics, which successfully links the two most critical structural features and a basic dynamical pattern in extensive complex systems. Surprisingly, the power-law exponents and the clustering coefficients of the aggregated PAD network could be tuned in a wide range by altering a set of model parameters. We further provide an approximation algorithm to select the proper parameters that can generate networks with given structural properties, the effectiveness of which is verified by fitting various real-world networks. Lastly, we explore the co-evolution of PAD model and higher-order contagion dynamics, and analytically derive the critical conditions for phase transition and bistable phenomenon. Our model provides a basic tool to reproduce complex structural properties and to study the widespread higher-order dynamics, which has great potential for applications across fields.

Environmental changes play a critical role in determining the evolution of social dilemmas in many natural or social systems. Generally, the environmental changes include two prominent aspects: the global time-dependent fluctuations and the local strategy-dependent feedbacks. However, the impacts of these two types of environmental changes have only been studied separately, a complete picture of the environmental effects exerted by the combination of these two aspects remains unclear. Here we develop a theoretical framework that integrates group strategic behaviors with their general dynamic environments, where the global environmental fluctuations are associated with a nonlinear factor in public goods game and the local environmental feedbacks are described by the `eco-evolutionary game'. We show how the coupled dynamics of local game-environment evolution differs in static and dynamic global environments. In particular, we find the emergence of cyclic evolutions of group cooperation and local environment, which forms an interior irregular loop in the phase plane, depending on the relative changing speed of both global and local environments compared to the strategic change. Our results provide important insights toward how diverse evolutionary outcomes could emerge from the nonlinear interactions between strategies and the changing environments.

The brain's structural connectome supports signal propagation between neuronal elements, shaping diverse coactivation patterns that can be captured as functional connectivity. While the link between structure and function remains an ongoing challenge, the prevailing hypothesis is that the structure-function relationship may itself be gradually decoupled along a macroscale functional gradient spanning unimodal to transmodal regions. However, this hypothesis is strongly constrained by the underlying models which may neglect requisite signaling mechanisms. Here, we transform the structural connectome into a set of orthogonal eigenmodes governing frequency-specific diffusion patterns and show that regional structure-function relationships vary markedly under different signaling mechanisms. Specifically, low-frequency eigenmodes, which are considered sufficient to capture the essence of the functional network, contribute little to functional connectivity reconstruction in transmodal regions, resulting in structure-function decoupling along the unimodal-transmodal gradient. In contrast, high-frequency eigenmodes, which are usually on the periphery of attention due to their association with noisy and random dynamical patterns, contribute significantly to functional connectivity prediction in transmodal regions, inducing gradually convergent structure-function relationships from unimodal to transmodal regions. Although the information in high-frequency eigenmodes is weak and scattered, it effectively enhances the structure-function correspondence by 35% in unimodal regions and 56% in transmodal regions. Altogether, our findings suggest that the structure-function divergence in transmodal areas may not be an intrinsic property of brain organization, but can be narrowed through multiplexed and regionally specialized signaling mechanisms.

Deep learning utilizing deep neural networks (DNNs) has achieved a lot of success recently in many important areas such as computer vision, natural language processing, and recommendation systems. The lack of convexity for DNNs has been seen as a major disadvantage of many optimization methods, such as stochastic gradient descent, which greatly reduces the genelization of neural network applications. We realize that the convexity make sense in the neural network and propose the exponential multilayer neural network (EMLP), a class of parameter convex neural network (PCNN) which is convex with regard to the parameters of the neural network under some conditions that can be realized. Besides, we propose the convexity metric for the two-layer EGCN and test the accuracy when the convexity metric changes. For late experiments, we use the same architecture to make the exponential graph convolutional network (EGCN) and do the experiment on the graph classificaion dataset in which our model EGCN performs better than the graph convolutional network (GCN) and the graph attention network (GAT)., Comment: 10 pages

Binary decision-making process is ubiquitous in social life and is of vital significance in many real-world issues, ranging from public health to political campaigns. While continuous opinion evolution independent of discrete choice behavior has been extensively studied, few works unveil how the group binary decision-making result is determined by the coupled dynamics of these two processes. To this end, we propose an agent-based model to study the collective behaviors of individual binary decision-making process through competitive opinion dynamics on social networks. Three key factors are considered: bounded confidence that describes the cognitive scope of the population, stubbornness level that characterizes the opinion updating speed, and the opinion strength that represents the asymmetry power or attractiveness of the two choices. We find that bounded confidence plays an important role in determining competing evolution results. As bounded confidence grows, population opinions experience polarization to consensus, leading to the emergence of phase transition from co-existence to winner-takes-all state under binary decisions. Of particular interest, we show how the combined effects of bounded confidence and asymmetry opinion strength may reverse the initial supportive advantage in competitive dynamics. Notably, our model qualitatively reproduces the important dynamical pattern during a brutal competition, namely, cascading collapse, as observed by real data. Finally and intriguingly, we find that individual cognitive heterogeneity can bring about randomness and unpredictability in binary decision-making process, leading to the emergence of indeterministic oscillation. Our results reveal how the diverse behavioral patterns of binary decision-making can be interpreted by the complicated interactions of the proposed elements, which provides important insights toward competitive dynamics

In high speed CNC (Compute Numerical Control) machining, the feed rate scheduling has played an important role to ensure machining quality and machining efficiency. In this paper, a novel feed rate scheduling method is proposed for generating smooth feed rate profile conveniently with the consideration of both geometric error and kinematic error. First, a relationship between feed rate value and chord error is applied to determine the feed rate curve. Then, breaking points, which can split whole curve into several blocks, can be found out using proposed two step screening method. For every block, a feed rate profile based on the Sigmoid function is generated. With the consideration of kinematic limitation and machining efficiency, a time-optimal feed rate adjustment algorithm is proposed to further adjust feed rate value at breaking points. After planning feed rate profile for each block, all blocks feed rate profile will be connected smoothly. The resulting feed rate profile is more concise compared with the polynomial profile and more efficient compared with the trigonometric profile. Finally, simulations with two free-form NURBS curves are conducted and comparison with the sine-curve method are carried out to verify the feasibility and applicability of the proposed method.

First-order methods like stochastic gradient descent(SGD) are recently the popular optimization method to train deep neural networks (DNNs), but second-order methods are scarcely used because of the overpriced computing cost in getting the high-order information. In this paper, we propose the Damped Newton Stochastic Gradient Descent(DN-SGD) method and Stochastic Gradient Descent Damped Newton(SGD-DN) method to train DNNs for regression problems with Mean Square Error(MSE) and classification problems with Cross-Entropy Loss(CEL), which is inspired by a proved fact that the hessian matrix of last layer of DNNs is always semi-definite. Different from other second-order methods to estimate the hessian matrix of all parameters, our methods just accurately compute a small part of the parameters, which greatly reduces the computational cost and makes convergence of the learning process much faster and more accurate than SGD. Several numerical experiments on real datesets are performed to verify the effectiveness of our methods for regression and classification problems., Comment: 10 pages

Feedrate scheduling is a key step in computer numerical control (CNC) machining, as it has a close relationship with machining time and surface quality, and has now become a hot issue in industry and academia. To reduce high chord errors and round-off errors, and generate continuous velocity, acceleration, and jerk profile of parametric interpolation, a novel and complete S-shape based feedrate scheduling algorithm is presented in this paper. The algorithm consists of three modules: bidirectional scanning module, velocity scheduling module and round-off error elimination module. The bidirectional scanning module with the limitations of chord error, normal acceleration/jerk and command feedrate aims to guarantee the continuity of the feed rate at the junctions between successive NURBS sub-curves. After the NURBS sub-curves have been classified into two cases, the velocity scheduling module firstly calculates the actual maximum federate, and then generates the feed rate profiles of all NURBS sub-curves according to our velocity scheduling function. Later, the round-off error elimination module is proposed to make the total interpolating time become an integer multiple of the interpolation period, which leads to the elimination of round-off errors. Finally, benchmarks are conducted to verify the applicability of the proposed method compared with some other methods.

B-splines are widely used in the fields of reverse engineering and computer-aided design, due to their superior properties. Traditional B-spline surface interpolation algorithms usually assume regularity of the data distribution. In this paper, we introduce a novel B-spline surface interpolation algorithm: KPI, which can interpolate sparsely and non-uniformly distributed data points. As a two-stage algorithm, our method generates the dataset out of the sparse data using Kriging, and uses the proposed KPI (Key-Point Interpolation) method to generate the control points. Our algorithm can be extended to higher dimensional data interpolation, such as reconstructing dynamic surfaces. We apply the method to interpolating the temperature of Shanxi Province. The generated dynamic surface accurately interpolates the temperature data provided by the weather stations, and the preserved dynamic characteristics can be useful for meteorology studies.

In recent years, Win-Stay-Lose-Learn rule has attracted wide attention as an effective strategy updating rule, and voluntary participation is proposed by introducing a third strategy in Prisoner's dilemma game. Some researches show that combining Win-Stay-Lose-Learn rule with voluntary participation could promote cooperation more significantly under moderate temptation values, however, cooperators' survival under high aspiration levels and high temptation values is still a challenging problem. In this paper, inspired by Achievement Motivation Theory, a Dynamic-Win-Stay-Lose-Learn rule with voluntary participation is investigated, where a dynamic aspiration process is introduced to describe the co-evolution of individuals' strategies and aspirations. It is found that cooperation is extremely promoted and defection is almost extinct in our model, even when the initial aspiration levels and temptation values are high. The combination of dynamic aspiration and voluntary participation plays an active role since loners could survive under high initial aspiration levels and they will expand stably because of their fixed payoffs. The robustness of our model is also discussed and some adverse structures are found which should be alerted in the evolutionary process. Our work provides a more realistic model and shows that cooperators may prevail defectors in an unfavorable initial environment., Comment: 13 pages,9 figures

Graphs are often used to organize data because of their simple topological structure, and therefore play a key role in machine learning. And it turns out that the low-dimensional embedded representation obtained by graph representation learning are extremely useful in various typical tasks, such as node classification, content recommendation and link prediction. However, the existing methods mostly start from the microstructure (i.e., the edges) in the graph, ignoring the mesoscopic structure (high-order local structure). Here, we propose wGCN -- a novel framework that utilizes random walk to obtain the node-specific mesoscopic structures of the graph, and utilizes these mesoscopic structures to reconstruct the graph And organize the characteristic information of the nodes. Our method can effectively generate node embeddings for previously unseen data, which has been proven in a series of experiments conducted on citation networks and social networks (our method has advantages over baseline methods). We believe that combining high-order local structural information can more efficiently explore the potential of the network, which will greatly improve the learning efficiency of graph neural network and promote the establishment of new learning models., Comment: 8 pages, 3 figures. arXiv admin note: text overlap with arXiv:2007.15838

Confirmation bias and peer pressure both have substantial impacts on the formation of collective decisions. Nevertheless, few attempts have been made to study how the interplay between these two mechanisms affects public opinion evolution. Here we propose an agent-based model of opinion dynamics which incorporates the conjugate effect of confirmation bias (characterized by the population identity scope and individual stubbornness) and peer pressure (described by a susceptibility threshold). We show that the number of opinion fragments first increases and then decreases to one as population identity scope becomes larger in a homogeneous population. Further, in heterogeneous situations, we find that even a small fraction of impressionable individuals who are sensitive to peer pressure could help eliminate public polarization when population identity scope is relatively large. Intriguingly, we highlight the emergence of 'impressionable moderates' who are easily influenced, hold wavering opinions, and are of vital significance in competitive campaigns.

Community detection is a fundamental and important problem in network science, as community structures often reveal both topological and functional relationships between different components of the complex system. In this paper, we first propose a gradient descent framework of modularity optimization called vector-label propagation algorithm (VLPA), where a node is associated with a vector of continuous community labels instead of one label. Retaining weak structural information in vector-label, VLPA outperforms some well-known community detection methods, and particularly improves the performance in networks with weak community structures. Further, we incorporate stochastic gradient strategies into VLPA to avoid stuck in the local optima, leading to the stochastic vector-label propagation algorithm (sVLPA). We show that sVLPA performs better than Louvain Method, a widely used community detection algorithm, on both artificial benchmarks and real-world networks. Our theoretical scheme based on vector-label propagation can be directly applied to high-dimensional networks where each node has multiple features, and can also be used for optimizing other partition measures such as modularity with resolution parameters.

Prisoner's dilemma game is the most commonly used model of spatial evolutionary game which is considered as a paradigm to portray competition among selfish individuals. In recent years, Win-Stay-Lose-Learn, a strategy updating rule base on aspiration, has been proved to be an effective model to promote cooperation in spatial prisoner's dilemma game, which leads aspiration to receive lots of attention. But in many research the assumption that individual's aspiration is fixed is inconsistent with recent results from psychology. In this paper, according to Expected Value Theory and Achievement Motivation Theory, we propose a dynamic aspiration model based on Win-Stay-Lose-Learn rule in which individual's aspiration is inspired by its payoff. It is found that dynamic aspiration has a significant impact on the evolution process, and different initial aspirations lead to different results, which are called Stable Coexistence under Low Aspiration, Dependent Coexistence under Moderate aspiration and Defection Explosion under High Aspiration respectively. Furthermore, a deep analysis is performed on the local structures which cause cooperator's existence or defector's expansion, and the evolution process for different parameters including strategy and aspiration. As a result, the intrinsic structures leading to defectors' expansion and cooperators' survival are achieved for different evolution process, which provides a penetrating understanding of the evolution. Compared to fixed aspiration model, dynamic aspiration introduces a more satisfactory explanation on population evolution laws and can promote deeper comprehension for the principle of prisoner's dilemma., Comment: 17 pages, 13 figures

The graph structure is a commonly used data storage mode, and it turns out that the low-dimensional embedded representation of nodes in the graph is extremely useful in various typical tasks, such as node classification, link prediction , etc. However, most of the existing approaches start from the binary relationship (i.e., edges) in the graph and have not leveraged the higher order local structure (i.e., motifs) of the graph. Here, we propose mGCMN -- a novel framework which utilizes node feature information and the higher order local structure of the graph to effectively generate node embeddings for previously unseen data. Through research we have found that different types of networks have different key motifs. And the advantages of our method over the baseline methods have been demonstrated in a large number of experiments on citation network and social network datasets. At the same time, a positive correlation between increase of the classification accuracy and the clustering coefficient is revealed. It is believed that using high order structural information can truly manifest the potential of the network, which will greatly improve the learning efficiency of the graph neural network and promote a brand-new learning mode establishment., Comment: 7 pages, 4 figures

Community detection is of fundamental significance for understanding the topology characters and the spreading dynamics on complex networks. While random walk is widely used and is proven effective in many community detection algorithms, there still exists two major defects: (i) the maximal length of random walk is too large to distinguish the clustering information if using the average step of all possible random walks; (ii) the useful community information at all other step lengths are missed if using a pre-assigned maximal length. In this paper, we propose a novel community detection method based on the first passage probabilities (FPPM), equipped with a new similarity measure that incorporates the complete structural information within the maximal step length. Here the diameter of the network is chosen as an appropriate boundary of random walks which is adaptive to different networks. Then we use the hierarchical clustering to group the vertices into communities and further select the best division through the corresponding modularity values. Finally, a post-processing strategy is designed to integrate the unreasonable small communities, which significantly improves the accuracy of community division. Surprisingly, the numerical simulations show that FPPM performs best compared to several classic algorithms on both synthetic benchmarks and real-world networks, which reveals the universality and effectiveness of our method.

Social reinforcement mechanism, which characterizes the promoting effects when exposing to multiple sources in social contagion process, is ubiquitous in information-technology ecosystem and has aroused great attention in recent years. While the impacts of social reinforcement on single-layer networks are well-documented, extension to multilayer networks is needed to study how reinforcement from different social circles influences the spreading dynamics. To this end, we incorporate multilayer social reinforcement into ignorant-spreader-ignorant (SIS) model on multiplex networks. Our theoretical analysis combines pairwise method and mean-field theory and agrees well with large-scale simulations. Surprisingly, we find this complex social contagion mechanism triggers the emergence of bistability phenomena, where extinction and outbreak states coexist. In particular, the hysteresis loop of stationary prevalence occurs in this bistable region, explaining why the fight towards the spread of rumors is protracted and difficult in modern society. Further, we show that the final state of bistable regions depends on the initial density of adopters, the critical value of which decreases as the contagion transmissibility or the multilayer reinforcement increases. In particular, we highlight two possible conditions for the outbreak of social contagion: to possess large contagion transmissibility, or to possess large initial density of adopters with strong multilayer reinforcement. Our results unveil the non-negligible power of social reinforcement on multiplex networks, which sheds lights on designing efficient strategies in spreading behaviors such as marketing and promoting innovations.

In recent years, social media has increasingly become an important platform for political campaigns, especially elections. It remains elusive how exactly public discourse is driven by the intricate interplay between individual socio-cognitive biases, dueling campaign efforts, and social media platforms. We examine this complex socio-political process by integrating observed retweet networks from the 2016 political networks with an agent-based model of political opinion formation and network structure. Here we show that the range of political viewpoints individuals are willing to consider is a key determinant in the formation of polarized networks and the emergence of echo chambers. We also find that winning majority support in public discourse is determined by both the effort exerted by campaigns and the relative ideological positioning of opposing campaigns. Our results demonstrate how public discourse and political polarization can be modeled as an interactive process of shifting individual opinions, evolving social networks, and political campaigns.

In this paper, solution space organization of minimum vertex-cover problem is deeply investigated using the K\"{o}nig-Eg\'{e}rvary (KE) graph and theorem, in which a hierarchical decomposition mechanism named KE-layer structure of general graphs is proposed to reveal the complexity of vertex-cover. An algorithm to verify the KE graph is given by the solution space expression of vertex-cover, and the relation between multi-layer KE graphs and maximal matching is illustrated and proved. Furthermore, a framework to calculate the KE-layer number and approximate the minimal vertex-cover is provided, with different strategies of switching nodes and counting energy. The phase transition phenomenon between different KE-layers are studied with the transition points located, and searching of vertex-cover got by this strategy presents comparable advantage against several other methods. The graphical representation and hierarchical decomposition provide a new perspective to illustrate the intrinsic complexity for large-scale graphs/systems recognition., Comment: 20 pages, 6 figures

Competitive information diffusion on large-scale social networks reveals fundamental characteristics of rumor contagions and has profound influence on public opinion formation. There has been growing interest in exploring dynamical mechanisms of the competing evolutions recently. Nevertheless, the impacts of population homophily, which determines powerful collective human behaviors, remains unclear. In this paper, we incorporate homophily effects into a modified competitive ignorant-spreader-ignorant (SIS) rumor diffusion model with generalized population preference. Using microscopic Markov chain approach, we first derive the phase diagram of competing diffusion results and examine how competitive information spreads and evolves on social networks. We then explore the detailed effects of homophily, which is modeled by a rewiring mechanism. Results show that homophily promotes the formation of divided "echo chambers" and protects the disadvantaged information from extinction, which further changes or even reverses the evolutionary advantage, i.e., the difference of final proportions of the competitive information. We highlight the conclusion that the reversals may happen only when the initially disadvantaged information has stronger transmission ability, owning diffusion advantage over the other one. Our framework provides profound insight into competing dynamics with population homophily, which may pave ways for further controlling misinformation and guiding public belief systems. Moreover, the reversing condition sheds light on designing effective competing strategies in many real scenarios.