探讨无标度网络(图)的平衡集

无标度网络的无标度性导致其各顶点之间的连接状况(度数)具有严重的不均匀分布性,无法给出无标度网络的具体结构,不能直接观察信息传播的具体路径。基于利用生成树来研究无标度网络(图)的拓扑结构思想,尝试寻找与时间和次要节点无关的无标度网络(图)的普适性结构,研究与生成树密切相关的平衡集,给出一个寻找具有较多叶子生成树的算法。     

Unified spatial scaling of species and their trophic interactions

Two largely independent bodies of scaling theory address the quantitative relationships between habitat area, species diversity and trophic interactions. Spatial theory within macroecology addresses how species richness scales with area in landscapes, while typically ignoring interspecific interactions. Complexity theory within community ecology addresses how trophic links scale with species richness in food webs, while typically ignoring spatial considerations. Recent studies suggest unifying these theories by demonstrating how spatial patterns influence food-web structure and vice versa. Here, we follow this suggestion by developing and empirically testing a more unified scaling theory. On the basis of power-law species-area relationships, we develop link-area and non-power-law link-species models that accurately predict how trophic links scale with area and species richness of microcosms, lakes and streams from community to metacommunity levels. In contrast to previous models that assume that species richness alone determines the number of trophic links, these models include the species' spatial distribution, and hence extend the domain of complexity theory to metacommunity scales. This generality and predictive success shows how complexity theory and spatial theory can be unified into a much more general theory addressing new domains of ecology.     

Simple rules yield complex food webs

Several of the most ambitious theories in ecology describe food webs that document the structure of strong and weak trophic links that is responsible for ecological dynamics among diverse assemblages of species. Early mechanism-based theory asserted that food webs have little omnivory and several properties that are independent of species richness. This theory was overturned by empirical studies that found food webs to be much more complex, but these studies did not provide mechanistic explanations for the complexity. Here we show that a remarkably simple model fills this scientific void by successfully predicting key structural properties of the most complex and comprehensive food webs in the primary literature. These properties include the fractions of species at top, intermediate and basal trophic levels, the means and variabilities of generality, vulnerability and food-chain length, and the degrees of cannibalism, omnivory, looping and trophic similarity. Using only two empirical parameters, species number and connectance, our 'niche model' extends the existing 'cascade model and improves its fit ten-fold by constraining species to consume a contiguous sequence of prey in a one-dimensional trophic niche.     

Compartments revealed in food-web structure

Compartments in food webs are subgroups of taxa in which many strong interactions occur within the subgroups and few weak interactions occur between the subgroups. Theoretically, compartments increase the stability in networks, such as food webs. Compartments have been difficult to detect in empirical food webs because of incompatible approaches or insufficient methodological rigour. Here we show that a method for detecting compartments from the social networking science identified significant compartments in three of five complex, empirical food webs. Detection of compartments was influenced by food web resolution, such as interactions with weights. Because the method identifies compartmental boundaries in which interactions are concentrated, it is compatible with the definition of compartments. The method is rigorous because it maximizes an explicit function, identifies the number of non-overlapping compartments, assigns membership to compartments, and tests the statistical significance of the results. A graphical presentation reveals systemic relationships and taxa-specific positions as structured by compartments. From this graphic, we explore two scenarios of disturbance to develop a hypothesis for testing how compartmentalized interactions increase stability in food webs.     

Uncovering the overlapping community structure of complex networks in nature and society

Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins, industrial sectors and groups of people) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties.     

Functional cartography of complex metabolic networks

High-throughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enables us to extract and display information contained in complex networks. Specifically, we demonstrate that we can find functional modules in complex networks, and classify nodes into universal roles according to their pattern of intra- and inter-module connections. The method thus yields a 'cartographic representation' of complex networks. Metabolic networks are among the most challenging biological networks and, arguably, the ones with most potential for immediate applicability. We use our method to analyse the metabolic networks of twelve organisms from three different superkingdoms. We find that, typically, 80% of the nodes are only connected to other nodes within their respective modules, and that nodes with different roles are affected by different evolutionary constraints and pressures. Remarkably, we find that metabolites that participate in only a few reactions but that connect different modules are more conserved than hubs whose links are mostly within a single module.     

Allometric degree distributions facilitate food-web stability

In natural ecosystems, species are linked by feeding interactions that determine energy fluxes and create complex food webs. The stability of these food webs enables many species to coexist and to form diverse ecosystems. Recent theory finds predator-prey body-mass ratios to be critically important for food-web stability. However, the mechanisms responsible for this stability are unclear. Here we use a bioenergetic consumer-resource model to explore how and why only particular predator-prey body-mass ratios promote stability in tri-trophic (three-species) food chains. We find that this 'persistence domain' of ratios is constrained by bottom-up energy availability when predators are much smaller than their prey and by enrichment-driven dynamics when predators are much larger. We also find that 97% of the tri-trophic food chains across five natural food webs exhibit body-mass ratios within the predicted persistence domain. Further analyses of randomly rewired food webs show that body mass and allometric degree distributions in natural food webs mediate this consistency. The allometric degree distributions hold that the diversity of species' predators and prey decreases and increases, respectively, with increasing species' body masses. Our results demonstrate how simple relationships between species' body masses and feeding interactions may promote the stability of complex food webs.     

Spatial scaling laws yield a synthetic theory of biodiversity.

Ecologists still search for common principles that predict well-known responses of biological diversity to different factors. Such factors include the number of available niches in space, productivity, area, species' body size and habitat fragmentation. Here we show that all these patterns can arise from simple constraints on how organisms acquire resources in space. We use spatial scaling laws to describe how species of different sizes find food in patches of varying size and resource concentration. We then derive a mathematical rule for the minimum similarity in size of species that share these resources. This packing rule yields a theory of species diversity that predicts relations between diversity and productivity more effectively than previous models. Size and diversity patterns for locally coexisting East African grazing mammals and North American savanna plants strongly support these predictions. The theory also predicts relations between diversity and area and between diversity and habitat fragmentation. Thus, spatial scaling laws provide potentially unifying first principles that may explain many important patterns of species diversity.     

Fuzzy网络中生成树的优化问题

生成树的优化问题在网络理论中，是一个具有广泛应用的重要课题，本文将Ｆｕｚｚｙ网络中有关生成树的优化问题推广到Ｆｕｚｚｙ网络中，建立了三个Ｆｕｚｚｙ网络生成树的优化问题模型；α－ＭＣ模型、ＭＦＣ模型和ＭＦＥＣ模型，并给出了相应问题模型的救解算法以及算法复杂性分析。     

Link Prediction in Brain Networks Based on a Hierarchical Random Graph Model

Link prediction attempts to estimate the likelihood of the existence of links between nodes based on available brain network information, such as node attributes and observed links. In response to the problem of the poor efficiency of general link prediction methods applied to brain networks, this paper proposes a hierarchical random graph model based on maximum likelihood estimation. This algorithm uses brain network data to create a hierarchical random graph model. Then, it samples the space of all possible dendrograms using a Markov-chain Monte Carlo algorithm. Finally, it calculates the average connection probability. It also employs an evaluation index.Comparing link prediction in a brain network with link prediction in three different networks(Treponemapallidum metabolic network, terrorist networks, and grassland species food webs) using the hierarchical random graph model, experimental results show that the algorithm applied to the brain network has the highest prediction accuracy in terms of AUC scores. With the increase of network scale, AUC scores of the brain network reach 0.8 before gradually leveling off. In addition, the results show AUC scores of various algorithms computed in networks of eight different scales in 28 normal people. They show that the HRG algorithm is far better than random prediction and the ACT global index, and slightly inferior to local indexes CN and LP. Although the HRG algorithm does not produce the best results, its forecast effect is obvious, and shows good time complexity.     

应用TFLS算法寻找SBEGN模型具最多叶子生成树

用动态网络模型来逼近和模拟现实网络是研究网络的重要手段之一,从构造单边界增长网络模型入手,验证该模型具有增长性、层次性．设计了时间优先层次搜索算法,利用该算法来寻找单边界增长网络模型具有最多叶子的生成树,并证明了所找到的具有最多叶子生成树具有无标度性这一重要的网络拓扑结构．     

利用对偶图求平面图的生成树数目

图的生成树数目是图的一个重要参数，求连通图生成树数目的方法有很多．本文利用平面图的对偶图的Kirchhoff矩阵来求一些平面图的生成树数目，求这类平面图的生成树数目比直接利用收缩边和去边得到递推公式的方法要简单，该方法对于平面图可以进一步推广.     

Size and form in efficient transportation networks.

Many biological processes, from cellular metabolism to population dynamics, are characterized by allometric scaling (power-law) relationships between size and rate. An outstanding question is whether typical allometric scaling relationships--the power-law dependence of a biological rate on body mass--can be understood by considering the general features of branching networks serving a particular volume. Distributed networks in nature stem from the need for effective connectivity, and occur both in biological systems such as cardiovascular and respiratory networks and plant vascular and root systems, and in inanimate systems such as the drainage network of river basins. Here we derive a general relationship between size and flow rates in arbitrary networks with local connectivity. Our theory accounts in a general way for the quarter-power allometric scaling of living organisms, recently derived under specific assumptions for particular network geometries. It also predicts scaling relations applicable to all efficient transportation networks, which we verify from observational data on the river drainage basins. Allometric scaling is therefore shown to originate from the general features of networks irrespective of dynamical or geometric assumptions.     

树扩图生成树数的界

连通图的生成树是指该图的极小连通生成子图．在Cayley公式的基础上，给出树扩图生成树数的上下界．     

Symbiotic fungal endophytes control insect host-parasite interaction webs

Symbiotic microorganisms that live intimately associated with terrestrial plants affect both the quantity and quality of resources, and thus the energy supply to consumer populations at higher levels in the food chain. Empirical evidence on resource limitation of food webs points to primary productivity as a major determinant of consumer abundance and trophic structure. Prey quality plays a critical role in community regulation. Plants infected by endophytic fungi are known to be chemically protected against herbivore consumption. However, the influence of this microbe-plant association on multi-trophic interactions remains largely unexplored. Here we present the effects of fungal endophytes on insect food webs that reflect limited energy transfer to consumers as a result of low plant quality, rather than low productivity. Herbivore-parasite webs on endophyte-free grasses show enhanced insect abundance at alternate trophic levels, higher rates of parasitism, and increased dominance by a few trophic links. These results mirror predicted effects of increased productivity on food-web dynamics. Thus 'hidden' microbial symbionts can have community-wide impacts on the pattern and strength of resource-consumer interactions.     

Reconciling complexity with stability in naturally assembling food webs

Understanding how complex food webs assemble through time is fundamental both for ecological theory and for the development of sustainable strategies of ecosystem conservation and restoration. The build-up of complexity in communities is theoretically difficult, because in random-pattern models complexity leads to instability. There is growing evidence, however, that nonrandom patterns in the strengths of the interactions between predators and prey strongly enhance system stability. Here we show how such patterns explain stability in naturally assembling communities. We present two series of below-ground food webs along natural productivity gradients in vegetation successions. The complexity of the food webs increased along the gradients. The stability of the food webs was captured by measuring the weight of feedback loops of three interacting 'species' locked in omnivory. Low predator-prey biomass ratios in these omnivorous loops were shown to have a crucial role in preserving stability as productivity and complexity increased during succession. Our results show the build-up of food-web complexity in natural productivity gradients and pin down the feedback loops that govern the stability of whole webs. They show that it is the heaviest three-link feedback loop in a network of predator-prey effects that limits its stability. Because the weight of these feedback loops is kept relatively low by the biomass build-up in the successional process, complexity does not lead to instability.     

广度优先搜索算法在交叉立方体中的应用

给出了互连网络上的广度优先搜索算法，将其应用到交叉立方体上可以得到交叉立方体的广度优先生成树。连通图的广度优先生成树的树高不会超过该图其他同根生成树的高度。利用这一性质，通过分析交叉立方体的广度优先生成树的特征，给出了n维交叉立方体CQ的直径为[(n 1)／2]的另外一种证明方法；该算法可以用来求解单源节点最短路径问题。并为讨论新的互连网络拓扑结构的直径和故障直径问题以及单源广播算法提供了一条新的思路。     

Structural asymmetry and the stability of diverse food webs

Untangling the influence of human activities on food-web stability and persistence is complex given the large numbers of species and overwhelming number of interactions within ecosystems. Although biodiversity has been associated with stability, the actual structures and processes that confer stability to diverse food webs remain largely unknown. Here we show that real food webs are structured such that top predators act as couplers of distinct energy channels that differ in both productivity and turnover rate. Our theoretical analysis shows that coupled fast and slow channels convey both local and non-local stability to food webs. Alarmingly, the same human actions that have been implicated in the loss of biodiversity also directly erode the very structures and processes that we show to confer stability on food webs.     

突变量表的数学定义及应用

突变现象普遍存在。为了描述突变现象,提出了包括突变点和突变特征函数等概念的突变量表的数学定义。通过以不同水平的偏序集（Aa,≤）、（Ab,≤）和（Ac,≤）表达广西各市石漠化程度的差异,阐述其应用。突变量表的提出,使传统的包括定名量表,顺序量表和其他两种量表的系统得到了理论上的发展和完善。     

Systemic risk in banking ecosystems

In the run-up to the recent financial crisis, an increasingly elaborate set of financial instruments emerged, intended to optimize returns to individual institutions with seemingly minimal risk. Essentially no attention was given to their possible effects on the stability of the system as a whole. Drawing analogies with the dynamics of ecological food webs and with networks within which infectious diseases spread, we explore the interplay between complexity and stability in deliberately simplified models of financial networks. We suggest some policy lessons that can be drawn from such models, with the explicit aim of minimizing systemic risk.