time delays are unavoidable in many practical applications . for example , in networked systems , the time each node takes to communicate with its neighbors is dependent on the physical distance and the technical specifications of the system .these communication delays affect the collective performance of the network .delay can be a result of measurements , which are often used as feedback .feedback delay may harm not only the performance but also the stability of a control system .another common delay type is input delays .these are caused , for example , by actual transport in chemical processes or by the internal dynamics of actuators .input delays are known to have strong destabilizing effects on control systems and pose challenging problems .early work sought compensation for input delay is based on smith predictor for single - input - single - output open - loop stable systems .finite spectrum assignment and artstein reduction extended smith predictor to multi - input - multi - output open - loop unstable systems .however , the efficacy of these methods requires high fidelity of the system model and the exact knowledge of the delay value .more recent techniques include robust and adaptive control schemes that are able to deal with uncertainties and unknown delays in the systems .for example , the research documented in proposes robust controllers to deal with input delay for uncertain systems .delay - dependent state feedback controllers is designed in for linear systems with actuator delay .model transformation is a common technique used in the adaptive control of systems with input delays .however , this requires exact information of the delay value in the control laws . in recent important developments on the topic , develop a control framework in terms of predictor - based formulation to transform systems with input delays into partial differential equations of transport type .work in concerns with the problem of guaranteed cost control for a class of fuzzy systems with input delay . in , a state feedback scheme that depends on the range of delay valuesis constructed for system with constraints in the control input .a finite - frequency approach is formulated in for active suspension systems with actuator delays .work in proposes an interval observation technique for the design of a linear feedback scheme to stabilize linear - time invariant system with a time - varying input delay .see for extensive reviews on control techniques for delay systems .we are interested in designing an input - delay compensation scheme that can adapt quickly to the system uncertainty .this problem is challenging because it is well known that fast adaptation often degrades the system s ability to tolerate time delays in the control loop , especially input delays .work in develops a control framework that facilitates fast adaptation with guaranteed robustness .the applications of the control framework in flight control systems is described in .the delay robustness of the framework is analyzed numerically in and rigorously in .  in this paper, we present the analysis for a modification to this control architecture to accommodate large actuator delays by purposely introducing on a delay in the state predictor .this modification was suggested to the authors by dr .naira hovakimyan .we establish a stability condition that is a function of the actuator delay as well as the delay - compensation term .the formulation suggests that there exists a range for the compensation delay around the input delay on which the condition is satisfied . within this range ,the closed - loop adaptive system is rigorously proven to admit a transient performance bound via the bounded - input - bounded - output stability of a non - adaptive reference system .the remainder of this paper is organized as follows .the problem statement is presented in section [ 8sec : plant ] . in section [ 8sec : refsys ] , we design a nonadaptive reference system , formulate a delay - dependent stability condition , and show the bounded - input - bounded - output stability of the reference system for certain values of the delays .sections [ 8sec : transbnd ] and [ sec : trans ] discuss the design of the adaptive controller and establish the transient performance bounds when the stability condition is satisfied .the effect of the input delay on the system response , as well as the efficacy of the delay compensation is illustrated in section [ 8sec : num ] .here , we also present the numerical analysis based on pad approximants and continuation techniques to construct an approximation for the region of values of the input and compensation delays where the stability condition is satisfied .section [ 8conc ] gives concluding remarks .consider the following system where , , , is a hurwitz matrix , , , is an unknown input delay , and are unknown time - varying parameters such that and .the control objective is for the system output to track a desired trajectory , which is the output of the following desired system with being a reference trajectory : similar control objectives were shown in to be achievable using the adaptive control framework with fast adaptation for less than some critical value . in this paper , we present and investigate a modification to the framework that appears to support stable operation with even larger value of the input delay .in this section , we analyze a nonadaptive reference system that represents the ideal behavior for the desired closed - loop control system .the control law of this reference system includes a delay - compensation term .the analysis establishes a delay - dependent stability condition for the reference system .it then shows that , for sufficiently large values of a design parameter , there exists a range for the input delay and a range for the compensation delay around the value of , where the stability condition is satisfied .consider a reference system with state and output , identical to , but with input given by the solution to the delay - differential equation ,\nonumber\end{aligned}\ ] ] here , the delay is introduced to alleviate the destabilizing effect of the input delay .it is trivial to see that when , the reference system recovers a special case of the one analyzed in .moreover , when , equation becomes a stable ordinary differential equation .   in the frequency domain , where and and are the laplace transforms of and , respectively .we can see from that since has only one pole , which is negative , is exponentially stable . furthermore , the rightmost roots of an lti delay differential equation depend continuously on the delay , pp . 11 - 12 . therefore ,for a given value of , there exist and such that is an exponentially stable transfer function for all  ] because on this interval is exponentially stable .hence , on this interval ,   similarly , it follows from that where , and using the same method that proves lemmas 3 and 4 in , we have is continuous in on  ] and  ] , since is exponentially stable on this delay interval and is hurwitz .[ 8thm1 ] _ suppose is selected such that is satisfied for  ] .there exists a value of the filter bandwidth such that the reference system is bounded - input bounded - output stable with respect to the desired trajectory and the initial condition . _taking the norm of yields since the stability condition in is satisfied by appropriately selecting and for  ] , it follows that this and imply that the reference system is bounded - input bounded - output stable with respect to and .  when , hence , for with less than two zero poles , the above limit is equal to zero .in addition , we have in such cases , it follows that , for large   since , it follows that when and has less than two zero poles .no such conclusion follows otherwise .in this section , we first present the adaptive controller from with a modification to mitigate the effect of the input delay for the system of interest . let the control input be the solution to the differential equation where  ] , and the bandwidth , the adaptive gain , and the bounds and are chosen appropriately , then the state and control input of the closed - loop control system governed by and - follow those of the reference system closely . in particular ,  [ 8thm2 ] _ suppose is chosen as in lemma [ lem : stbcnd ] and assume that \mbox { and } \hat{\tau}\in[\tau-\underline{\delta } , \tau+\bar { \delta}] ] .consequently ,    the claim then follows .[ 8lem : etatb ] _ let . when  ] , it follows that the norm is bounded .thus , yields and the claim follows .  we proceed to prove theorem [ 8thm2 ] .  since it follows by continuity that there exists a , such that and .theorem [ 8thm1 ] then leads to .it follows that for some , which is independent of .next , we write the control law in in a similar form as in the control law of the reference system : and , consequently , it follows from and that it further follows from the definitions of and that and , using lemma [ 8lem : etatb ] ,   from , we now obtain together with and lemma [ 8lem : etatb ] , this results in the bound where , which is finite for  ] , the stability condition in holds and implies that .thus , from and , we conclude that and the claim then follows by choosing and such that the right - hand side of is strictly less than 1 and .  in the next section, we will demonstrate the destabilizing effect of input delay as well as the efficacy of .in addition , we use pad approximants to visualize the stability condition in and to estimate ranges and , and for the time delay margin by monitoring the stability condition in when the delay is being gradually increased .consider the controller with the delay compensation in - applied to system with ( cf . ) ^\top,\nonumber\\     \sigma(t)&=\sin(0.5\pi t ) .   \end{aligned}\ ] ] since it is trivial to see that , we set .the control parameters are set to : , , , , , and .the states of the actual adaptive system and the desired system are initialized at ^\top ] , respectively .the system output is tasked to track the output of a desired system with .we first set the input delay and turn off the delay compensation , i.e. , .figure [ fig : delayp5 ] shows the result of the simulation .we can see that the output is able to track closely the output of the desired system . even without the delay compensation ,the controller is able to tolerate some small input delay in the limit of large adaptive gain .this is consistent with the fact that the controller is robust with fast adaptation as widely reported in the literature .the observed deviation between the two trajectories is traced back to the need to keep a finite filter bandwidth in order to maintain system robustness . in the absence of the input delay, the deviation gets smaller when is increased .this agrees with the result in .however , increasing , while improving tracking performance , deteriorates the controller s delay robustness .therefore , there is a trade - off between tracking performance and system robustness in the control framework .  .]  when the input delay is large enough , the tracking performance of the unmodified control system is degraded .for instance , with delay , the system output deviates from the desired trajectory and exhibits high - frequency oscillations as seen in figure [ fig : delayp52 ] .further increases in input delay eventually result in unbounded system output .  :high frequency oscillations with large tracking errors . ]we next illustrate the efficacy of the delay compensation modification represented by in .the input delay is kept at , while is set to . the resultant output in figure [ fig : wtauhat ]shows that the system recovers desired performance with small tracking error in the presence of the compensation delay .in fact , the control system remains stable for values of as high as provided that is tuned to a value near .   and : desired tracking is recovered ]   in this section , we discuss a method for numerically analyzing the sufficient stability condition in .this method will result in a stability chart based on techniques of parameter continuation .the result will also demonstrate the effect of the compensation delay as well as the dependence of the delay compensation performance on the low - pass filter bandwidth .it is practically impossible to determine the explicit form of the impulse response for the transfer function in the definition of in due to the delay terms and , which result in infinite number of poles of the transfer function . to overcome this challenge, we employ a pad approximant to arrive at the following approximation where   the convergence and the error bounds of pad approximants are discussed in .a pad approximant allows for the expansion of the delay terms as rational transfer functions .this enables the estimation of the norm in the definition of in , given values of and , via the algorithm proposed in with a tolerant of .the control parameters remain unchanged from the previous simulations . while increasing from zero , for each value of the input delay , we vary from until the stability condition is violated .we compute the stability chart for all and that satisfy the stability condition in for the example system of interest in this section .this is performed using the continuation toolbox called coco .the continuation tolerant is set to . for a given value of the filter bandwidth ,we compute the implicitly defined solution manifold of the equation   figure [ fig : k25 ] shows the result for a bandwidth .the solid line represents the pairs values that satisfy .the use of the continuation techniques allows for the automatic computation of the complicated folds on the curve , which may be highly challenging for traditional root - finding methods , for example , the bisection method .on one side of the curve , the stability condition holds and the stability of the modified controller is guaranteed .in contrast , on the other side of the manifold , the condition is violated and stability is no longer guaranteed .the dash - dot line indicates the identity line .the continuation result confirms the stability behavior suggested by the theoretical analysis in section [ sec : trans ] .in particular , along the identity line , , there is a such that when , the system stability is guaranteed . for the solution manifold in figure [ fig : k25 ] with , we obtain .in fact , the manifold suggests that stability is guaranteed for even higher values of off the identity line .hence , the largest possible that guarantees stability if predicted by the theoretical analysis may be conservative .in addition , the numerical results also validate this paper s claim that for a given , there are and such that the system stability is guaranteed if $ ] . for example , in figure [ fig : k25 ] with , at , we have and . in addition ,when , the value band for is significantly larger as compared to that when .when is around , the manifold exhibits a dramatic change and the value of varies greatly .another observation is that when , the value of , represented by the brown solid line in figure [ fig : k25 ] , equals zero since is non - negative .this brown solid line also indicates the lower bound for time - delay margin of the adaptive control system in the absence of the delay - compensation modification .    to study the dependence of the stability chart on the filter bandwidth , the same continuation processis performed for and .the results , given by figures [ fig : k50 ] and [ fig : k100 ] , respectively , illustrate several points .first , in the presence of the delay compensation , the time - delay margin of the unmodified controller , indicated by the brown solid lines in figures [ fig : k25 ] , [ fig : k50 ] , and [ fig : k100 ] , is decreased as the bandwidth increases .this is consistent with the result in .when the input delay is near zero , the maximum allowable is large and increasing with increasing bandwidth .this implies that over treatment when the delay is small is not a problem since stability is guaranteed even for a large deviation between the compensation delay and the actuator delay .furthermore , for input delay far away from zero , the value band of that guarantees stability gets narrower for higher bandwidth .for example , when , the allowable value band for is , , and with , , and , respectively .another point to remark is that increasing filter bandwidth surprisingly improves the time - delay margin of the adaptive system along the identity line in the plane .specifically , the time - delay margin along the identity line , and with , , and , respectively .this result implies an interesting property of the modified controller . in the unmodified control framework ,increasing the filter bandwidth deteriorates system robustness ( see for the detailed analysis ) . in contrast , with the delay - compensation modification , when it is possible to set , increasing bandwidth indeed improves system robustness , indicated by the ability to tolerate input delay .we have suggested and analyzed a modification to a controller with fast adaptation to compensate for an actuator delay in the plant .the idea is simple : we inject a delay in the control input of the predictor so that it emulates the structure of the actual plant equation .we first prove that a nonadaptive reference system is bounded - input - bounded - output stable provided that a delay - dependent stability condition satisfied . in that case , when the adaptive gain is chosen sufficiently large , the adaptive system output is shown to follow closely that of the nonadaptive reference system. the stability condition may be satisfied for a range of values of the compensation delay about the input delay , for sufficiently large filter bandwidth and sufficiently small values of the input delay .the set of values of the delays that satisfy the stability condition may be estimated numerically using pad approximants .the efficacy of the delay - compensation term has not been rigorously proved though it is demonstrated by the illustrative example and numerical analysis .the method of proof does not produce an explicit estimate for the upper bound on the range of input delay for which stable behavior may be achieved .in addition , a closed - form expression for the allowable range of around for the system stability has not been formulated .furthermore , the predictable dependence of the system s time - delay margin on or has not been completely investigated .the main challenge is how to obtain an analyzable form of the norm in terms of the delays in the stability condition .finding a way to overcome this challenge will be the focus of future work .this work is supported by agriculture and food research initiative competitive grant no .2014 - 67021 - 22109 from the usda national institute of food and agriculture .nguyen kd , dankowicz h , hovakimyan n. marginal stability in l1-adaptive control of manipulators . _ asme international conference on multibody systems , nonlinear dynamics , and control _ 2013 , detc2013 - 12744 , portland , or .

Abstract: a modification to the control framework for uncertain systems with actuator delay is presented . specifically , a time delay is introduced in the control input of the state predictor to compensate for the destabilizing effect of input delay in the plant . for this modified framework , the analysis shows that the output of the adaptive system closely follows the behavior of a suitably defined , nonadaptive , stable reference system provided that a delay - dependent stability condition is satisfied and the adaptive gain is chosen sufficiently large . the set of combinations of input delay and compensation delay for which the stability condition is satisfied contains an open set of pairs of positive values provided that a filter bandwidth , characteristic of adaptive control is chosen sufficiently large . the efficacy of the delay compensation is illustrated by a simple example . a numerical continuation is also performed to explore the stability region for a case where this can be approximated a priori .


an embedding is a map from input data to points in euclidean space .machine learning researchers and developers often need to explore the properties of a specific embedding to understand the behavior of their model . an engineer who creates an embedding of songs for a recommendation system might want to verify that the nearest neighbors of `` stairway to heaven '' include `` whole lotta love '' and not `` let it go '' from frozen .meanwhile , a researcher may be interested in global geometric properties , such as linear relationships between meaningful subsets of embedded points . for both sets of users , gainingan understanding of embedding geometry is a key step in interpreting a machine learning model .an appealing approach to analyzing embeddings is to visualize them .since embeddings often exist in a space of hundreds of dimensions , an essential step is `` dimensionality reduction '' which projects points to a more approachable two or three dimensions .many tools exist to perform various types of dimensionality reduction , but they are largely non - interactive ( e.g. , matplotlib [ 3 ] , or the code released in [ 5 ] ) .unfortunately , these static views are generally inadequate for exploring high - dimensional data although an important first step in the process , users then typically want to switch quickly between many views , zooming and filtering , and then closely inspecting details .indeed , there have been many toolkits created with the express purpose of exploring high - dimensional data ( e.g. , [ 2 ] , [ 8 ] ) , and a key lesson has been that rich interactivity coupled with multiple linked views is extremely helpful .  at the same time , the embeddings that arise in machine learning differ from the kind of data sets seen in conventional `` high - dimensional '' visualization .most notably , traditional systems often assume the underlying dimensions have a particular meaning ( the expression level of a given gene , or the age of a person ) . on the other hand ,the basis vectors in a typical embedding are typically not meaningful in fact , discovering semantically significant directions can be a goal in itself .  in this paper , we present the embedding projector , a system for interactive visualization and analysis of high - dimensional data . although designed to be a general - purpose tool , it is optimized for the use cases that arise in machine learning and includes special features to explore meaningful directions in a data set .we held informal interviews with engineers and researchers within our organization to learn how they worked ( or wanted to work ) with embeddings .we also looked at usage of existing internal tools .three themes arose consistently .[ [ task-1.-exploring - local - neighborhoods ] ] task 1 . exploring local neighborhoods + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  many users wanted to inspect the nearest neighbors of a given point .confirming that nearby points are semantically related represented an important step in establishing trust in an algorithm .[ [ task-2.-viewing - global - geometry - and - finding - clusters ] ] task 2 .viewing global geometry and finding clusters + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  several users were interested in finding large clusters of related points , as well as seeing the global geometry of the embeddings .  [ [ task-3.-finding - meaningful - directions ] ] task 3 . finding meaningful `` directions '' + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  researchers have found that embedding spaces sometimes contain semantically significant directions .for example [ 6 ] discovered that the vector defined by subtracting the point for `` man '' from the point for `` woman '' represented a kind of `` female '' direction in space .users we talked with considered the discovery of such directions interesting and worthwhile .we know of no current tools that help with this use case .the embedding projector is a web application , available as both a standalone tool and integrated into the tensorflow platform [ 1 ] .users may either upload arbitrary high - dimensional data , in a simple text format , or ( in tensorflow ) take advantage of the model checkpoint system that makes it easy to visualize any tensors as an embedding .  .]  fig .[ fig - main ] shows the main view of the web app with ( a ) _ data _ panel on the left , where users can choose data columns to color and label points , ( b ) the _ projected _ view in the center , and ( c ) the _ inspector _ panel on the right side , where users can search for particular points and see a list of nearest neighbors .the embedding projector offers three methods of reducing the dimensionality of a data set : two linear and one nonlinear .each method can be used to create either a two- or three - dimensional view .[ [ principal - component - analysis ] ] principal component analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + +  a straightforward technique for reducing dimensions is principal component analysis ( pca ) .the embedding projector computes the top 10 principal components .menus let the user project those components onto any combination of two or three .pca is often effective at supporting task 2 , examining global geometry .[ [ t - sne ] ] t - sne + + + + +  a popular non - linear dimensionality reduction technique is t - sne [ 5 ] .the embedding projector offers both two- and three - dimensional t - sne views .layout is performed client - side .because t - sne often preserves some local structure , in practice it supports both task 1 and task 2 .[ [ custom ] ] custom + + + + + +  users can construct specialized linear projections based on text searches , supporting task 3 , finding meaningful directions in space . to define a projection axis ,the user enters two search strings or regular expressions .the program computes the centroids of the sets of points whose labels match these searches , and uses the difference vector between centroids as a projection axis .for example , in the smart reply data [ 4 ] , this view uncovered a surprisingly regular relationship between phrases ending in periods versus exclamation points .   to explore a data set, users can navigate the views in either a 2d or a 3d mode , zooming , rotating , and panning using natural click - and - drag gestures . to help interpretation of the 3d mode ,the projector uses multiple redundant depth cues : changing the size of points based on distance to camera ; adding fog to fade out more distant points ; and initializing the view with an animated `` lazy susan '' motion .both modes exploit webgl to provide smooth , fluid interaction , which encourages exploration of the underlying space , supporting both task 1 and task 2 .to support task 1 , clicking on a point causes the right pane to show an explicit textual list of nearest neighbors , along with distances to the current point .the nearest - neighbor points themselves are highlighted on the projection .users sometimes wish to focus on a subset of points an interesting cluster , or perhaps the set of nearest neighbors of a given point .zooming into the cluster gives some information , but it is sometimes more helpful to restrict the view to a subset of points and perform t - sne or pca only on those points .to do so , the user can select points in multiple ways : after clicking on a point , its nearest neighbors are selected ; after a search , the points matching the query are selected ; right - clicking and dragging defines a selection sphere . after selecting a set of points ,the user can isolate those points for further analysis on their own with the `` isolate points '' button in the inspector pane on the right hand side .  in 3d mode , labels can appear as `` billboards '' flat images that always face the user . fig .[ 3d - labels ] depicts this view .when there is an image associated with a data point , we can simply use the actual image as the `` label '' .a nice example is an embedding of mnist images shown in fig .[ mnist - colorized ] where each image has a background color according to its true label .[ [ collaborative - features ] ] collaborative features + + + + + + + + + + + + + + + + + + + + + +  early in the development process we saw that users of the embedding projector wanted to share specific views of their data .indeed , previous work in visualization has shown the benefits of allowing users to share the state of a visualization [ 7 ] .  to allow easy collaboration, the embedding projector lets users save the current state ( including computed coordinates of t - sne embeddings ) as a small file . the projector can then be pointed to a set of one or more of these files , producing the panel seen in the bottom right of fig .[ fig - main ] .other users can then walk through a sequence of bookmarks .the embedding projector is a new visualization tool that helps users interpret machine learning models that rely on embeddings . unlike other high - dimensional visualization systems ,it is customized for the kinds of tasks faced by machine learning developers and researchers : exploring local neighborhoods for individual points , analyzing global geometry , and investigating semantically meaningful vectors in embedding space .the projector is part of the tensorflow platform , and seamlessly allows analysis and interpretation of tensorflow models .there are a number of directions for future work on the visualization .for example , when developing multiple versions of a model , or inspecting how a model changes over time , it could be useful to visually compare two embeddings .doing so would require nontrivial additions to the current visualizations .a second direction for future research is to make it easier for users to discover meaningful directions in the data .while the current interface makes it easy to explore various hypotheses , there may be ways for the computer to generate and test hypotheses automatically .thanks to shan carter for invaluable design contributions , and to our colleagues who provided thoughtful feedback .[ 1 ] m. abadi , a. agarwal , p. barham , e. brevdo , z. chen , c. citro , g. s. corrado , a. davis , j. dean , m. devin , et al .tensorflow : large - scale machine learning on heterogeneous distributed systems . , 2016 .[ 2 ] c. ahlberg and b. shneiderman .visual information seeking : tight coupling of dynamic query filters with starfield displays . in _ proceedings of the sigchi conference on human factors in computing systems _ , pages 313317 .acm , 1994 .

Abstract: embeddings are ubiquitous in machine learning , appearing in recommender systems , nlp , and many other applications . researchers and developers often need to explore the properties of a specific embedding , and one way to analyze embeddings is to visualize them . we present the embedding projector , a tool for interactive visualization and interpretation of embeddings .


complex networks are currently being studied across many fields of science and engineering [ 1 ] , stimulated by the fact that many systems in nature can be described by models of complex networks .a complex network is a large set of interconnected nodes , in which a node is a fundamental unit usually with specific dynamical or information content .examples include the internet , which is a complex network of routers and computers connected by various physical or wireless links ; the world wide web , which is an enormous virtual network of web sites connected by hyperlinks ; and various communication networks , food webs , biological neural networks , electrical power grids , social and economic relations , coauthorship and citation networks of scientists , cellular and metabolic networks , etc .the ubiquity of various real and artificial networks naturally motivates the current intensive study of complex networks , on both theoretical and application levels .many properties of complex networks have currently been reported in the literature .notably , it is found that many complex networks show the small - world property [ 2 ] , which implies that a network has a high degree of clustering as in some regular network and a small average distance between nodes as in a random network .another significant recent discovery is the observation that many large - scale complex networks are scale - free .this means that the degree distributions of these complex networks follow a power law form for large network size , where is the probability that a node in the network is connected to other nodes and is a positive real number determined by the given network . since power laws are free of characteristic scale , such networks are called `` scale - free networks '' [ 3 ] .the scale - free nature of many real - world networks can be generated by a mechanism of growing with preferential attachment [ 3 ] .communities are defined as collections of nodes within which connections are dense , but between which connections are sparser .there are many real - world networks which exhibit community structure , and community structures are supposed to play an important role in many real networks .for example , communities in a citation network might represent related papers on a single topic [ 4 ] ; communities on the web might represent pages on related topics [ 5 ] ; communities in a biochemical network or neuronal system might correspond to functional units of some kinds [ 6 , 7 ] ; communities also play an important role in information networks [ 8 ] .there have been several investigations into designing algorithms for detecting community structure in large - scale complex networks [ 9 - 10 ] . to study the effects of community structure on network properties and dynamics ,the modelling of real networks with community structure is very important .however , most of the existing evolving network models do not take the community structure into account . in[ 11 ] , a networked seceder model was proposed to illustrate group formation in social networks . in[ 12 ] , a bipartite growing network model for social community with group structures was proposed . in [ 13 ] , a social network model based on social distance attachment was proposed , which can exhibit community structure . in [ 14 ] , the authors proposed a growing network model with community structure .however , there is the possibility within the network model that a node belongs to a community but has no connections with nodes in this community but has connections with nodes in other communities , which is unacceptable .  in this paper, we propose an evolving network model with community structure based on the inner - community preferential attachment and inter - community preferential attachment mechanics .we use a mean - field method to analyze the degree distributions of this network model .numerical simulations are also performed to investigate the properties of this network model .some more realistic generalizations and extensions are also discussed in the the conclusions and remarks section .in this section , we describe the growing mechanics of the proposed network model . for simplicity , we consider only undirected network models in this paper .we assume there is a total of communities in the network .the proposed model is defined by the following scheme :  * step 1 - initialization * : start from a small number of fully connected nodes in each community .use inter - community links to connect each community to the other communities , so that there is a link between each community . the nodes to which the inter - community links connectare selected randomly in each community .for example , fig .1 shows an initial network with and ., width=264 ]  * step 2 - growth * : at each time step , a new node is added to a randomly selected community .the new node will be connected to nodes in the same community through _ inner - community links _ ( defined as the links that connect nodes in the same community ) , and with probability connected to nodes ( none with probability ) in the other communities through _ inter - community links _ ( defined as the links that connect nodes among different communities ) .  * step 3 - preferential attachments * :  \a ) inner - community preferential attachment : when choosing nodes in the same community ( we denote it as the community ) to which the new node connects through inner - community links , we assume that the probability that a new node will be connected to node in community depends on the inner - degree ( defined as the number of inner - links connected to node ) of that node , such that   \b ) inter - community preferential attachment : when choosing the nodes in other communities to which the new node connects through inter - community links , we assume that the probability that a new node will be connected to node in community depends on the inter - degree ( defined as the number of inter - links connected to the node ) , such that   the motivation for using the inner - community preferential attachment mechanics is the same as that in many other growing network models. there also exist inter - community preferential attachment phenomena in some real networks with community structure .for example , in scientific collaboration networks , a multidisciplinary researcher is more likely to be willing to study other unknown fields to him / her , and has stronger desire to collaborate with researchers in other fields than other single - topic researchers . in friendship networks ,an individual with many inter - community links is more likely to make friends with different kinds of people , and has higher probability of making new friends with people in other communities than other people with less inter - community links .these are all inter - community preferential attachment phenomena .after time steps , this scheme generates a network with nodes , and /2+mt+\mbox{integer}(\alpha n t) ] .if is not so small , such that $ ] for large ( usually , this is the case ) , then   the solution of this equation , with the initial condition that node in community was added to the network at time with inter - community connectivity in the sense of mathematical expectation , is similar to the above analysis of the inner - degree distribution , the inter - degree distribution can be written as predicting that the inter - degree distribution also obeys a power - law distribution with independent of and .  in this case , the total degree of node in community is it is easy to show that the total degree distribution is which indicates that the total degree also obeys a power - law distribution with .  because ( usually ) , from ( 5 ) and ( 14 ) ( or ( 12 ) ) we know that the inter - degrees of network nodes are smaller than the inner - degrees of network nodes in the sense of statistics .so , networks generated by this model will have community structure .next , we numerically study the properties of the network model .we consider a network generated by the proposed scheme with and . in the figures 3 - 5 ,the slopes of the lines were obtained by cumulative distributions of 10 runs .the inner - degree distribution of this network is shown in fig .as we can see , the inner - degree obeys a power - law distribution with ( the slope of the line is -3 ) . the inter - degree distribution is shown in fig . 4 , and the inner - degree obeys a power - law distribution with ( the slope of the line is -2.9 ) . the total degree distribution is shown in fig .5 , and we observe that the total degree also obeys a power - law distribution with .so , all these three distributions are of power - law type , indicating that the generated network is a scale - free network , and these distributions verify the above analytical results . in fig .6 we plot the time evolution of the total degree of two nodes , in which one is an initial node and the other one is added to the network at .this is in good agreement with ( 16 ) .  .as can be seen , the distribution is well approximated by a straight line with gradient .,width=264 ]  .as can be seen , the distribution is well approximated by a straight line with gradient .,width=264 ]  .as can be seen , the distribution is well approximated by a straight line with gradient .,width=264 ]   ( lower one ) . as can be seen ,the time - evolution of the degrees are well approximated by a straight line with gradient ., width=264 ]  we also calculated the diameter , the average path length and the clustering coefficient for this network .the detailed definitions of these parameters can be found in [ 1 ] .the values of these parameters are , and , respectively .so the network has small average path length and relatively large clustering coefficient ( compared to a random graph ) .in this paper we proposed an evolving network model with community structure .we have theoretically analyzed the scaling properties of the network by using a mean - field approach .the analytical and numerical results indicate that the network can produce community structure , and the inner - degree , inter - degree and total degree all obey power - law distributions , so the network has scale - free properties .the network also has small average path length and relatively large clustering coefficient ( compared to a random graph ) .note that in [ 16 ] the authors proposed a growing network model of two coupled networks , which idea is a little similar to that in the present paper , but the authors did nt consider community structure .  to make our model more understandable and to avoid unnecessarily complicated notations , we made some simplifications in our model .some more realistic cases and extensions can be considered with minor modification of our model , for example :  1 .we did nt consider the formation of new communities during the evolution of the network model . in some real world networks ,some new communities may appear during the evolution . with minor modification, our model could allow from time to time the introduction of new communities : if a node is the first of a new community , one could only set links between and the old communities at the time step of its introduction , and follow the same evolving rules in the proceeding time steps as defined in section 2 .it is easy to show that , with this modification , the main results will not change .2 .  in the preferential attachment mechanism ,if a node is not connected to a node ( in the same community or different communities ) at the beginning of its introduction , it will never be . in considering the resulting networks ,this is of no problem , but in considering the evolution process , it is somewhat unrealistic .we can overcome this problem , as follows : in each time step , besides the steps mentioned in section 2 , with probability , an existing node is selected , and we perform the same inner- and inter- community attachments of this node to all the other existing nodes that have no links with it .this will surely not change the community structure of the network model .future extensions of this work include considering other kinds of inner- and inter- linking mechanisms other than preferential attachment , such as preferential linking [ 17 ] and distance preference [ 18 ] .future extensions also include the modelling of directed and weighted network models with community structure [ 19 ] , because many real - world networks with community structures are directed and/or weighted [ 20 ] . based on this network model, we can study the effects of community structure on network dynamics , such as the stability , synchronization , disease and rumor spreading , and robustness .we can also introduce different types of dynamical node models in different communities to study the dynamical behavior of some real networks , for example biochemical networks and brain networks .the authors are grateful to the anonymous reviewers for their valuable suggestions and comments , which have led to the improvement of this paper .the authors would like to thank mrs wei huang and yigong xiao for help in calculating some of the statistical quantities .this research was supported by the key program projects of the national natural science foundation of china under grant no .70431002 , and the youth science and technology foundation of uestc under grant l08010201jx04011 .  20 s.h .strogatz , nature ( london)410 , 268 ( 2001 ) ; r. albert and a.l .barabsi , review of modern physics 74 , 47 ( 2002 ) ; s.n .dorogovtsev and j.f.f .mendes , adv . in phys .51 , 1079 ( 2002 ) ; m.e.j . newman , siam review 45 , 167 ( 2003 ) .d.j . watts , s.h .strogatz , nature 393 , 440 ( 1998 ) ; d.j .watts , _ small - worlds : the dynamics of networks between order and randomness _ , princeton , nj : princeton university press , 1999 .barabsi and r. albert , science 286 , 509 ( 1999 ) .s. redner , eur .j. b 4 , 131 ( 1998 ) .flake , s.r .lawrence , c.l .giles , f.m .coetzee , ieee computer 35 , 66 ( 2002 ) .p. holme , m. huss , h. jeong , bioinformatics 19 , 532 ( 2003 ) .o. sporns , d. r. chialvo , m. kaiser , and c. c. hilgetag , trends in cognitive science 8 , 418 ( 2004 ) . h. xie , k .- k .yan , s. maslov , cond - mat/0409087 .for a recent review , see m.e.j .newman , eur .j. b 38 , 321 ( 2004 ) .for some more recent papers see , for example , c.p .massen and j.p.k .doye , phys .e 71 , 046101 ( 2005 ) ; m. young , j. sager , g. csrdi , p. hga , cond - mat/0408263 ; j.p .bagrow , e.m .bollt , cond - mat/0412482 ; m. latapy , p. pons , cond - mat/0412368 ; a. clauset , m.e.j .newman , c. moore , phys .e 70 , 066111 ( 2004 ) cond - mat/0408187 ; j. duch , a. arenas , cond - mat/0501368 ; a. clauset , physics/0503036 ; l. da f. costa , phys .e 70 , 056106 ( 2004 ) cond - mat/0312712 .a. grnlund , p. holme , phys .e 70 , 036108 ( 2004 ) .noh , h .- c .jeong , y .- y .ahn , h. jeong , cond - mat/0412149 .m. boguna , r. pastor - satorras , a. diaz - guilera , a. arenas , phys . rev .e 70 , 056122 ( 2004 ) .m. kimura , k. saito , n. ueda , neural networks 17 , 975 ( 2004 ) .barabsi , r. albert , and h. jeong , physica a 272 , 173 ( 1999 ) .d. zheng , e. ergn , cond - mat/0112052 .dorogovtsev , j.f.f .mendes , and a.n .samukhin , phys .lett 85 , 4633 ( 2000 ) .j. jost and m.p .joy , phys .e 66 , 036126 ( 2002 ) . c. li and g. chen , modelling of weighted evolving networks with community structures , submitted . c. li and g. chen , arxiv : cond - mat/0311333 ; a. barrat , m. barthlmy , r. pastor - satorras , and a. vespignai , proc . natl .101 ( 2004 ) 3747 ; d. zheng , s. trimper , b. zheng , and p.m. hui , phys .e 67 , 040102 ( 2003 ) ; a. barrat , m. barthlmy , and a. vespignai , phys . rev .92 ( 2004 ) 228701 ; c. li , g. chen , physica a 343 ( 2004 ) 288 ; l.g .morelli , phys .e 67 , 066107 ( 2003 ) .

Abstract: many social and biological networks consist of communities - groups of nodes within which connections are dense , but between which connections are sparser . recently , there has been considerable interest in designing algorithms for detecting community structures in real - world complex networks . in this paper , we propose an evolving network model which exhibits community structure . the network model is based on the inner - community preferential attachment and inter - community preferential attachment mechanisms . the degree distributions of this network model are analyzed based on a mean - field method . theoretical results and numerical simulations indicate that this network model has community structure and scale - free properties .


the copernican principle , defined for the present paper as the principle according to which the observer should not happen to be located in any special location in the universe , is generally _ applied _ in one of three ways in twentieth century cosmology :  ( )  by assuming the cosmological principle , in which spatial sections of the universe at constant cosmological time are close to homogeneous and isotropic when averaged on `` large enough '' length scales ;  by assuming the perfect cosmological principle , in which all of space - time is close to homogeneous and isotropic , on average ; or  by assuming that the universe is very chaotic , very far from homogeneity , but that by the weak anthropic principle , we happen to live in a tiny bubble which is inflated to a locally close to flat friedmann - lematre model ;  or it is _ rejected _  ( )  by taking the `` observational '' point of view that the observer lives at the centre of a close to isotropic but radially very inhomogeneous three - dimensional space - time cone with a spherical ( ) boundary defined by the age of the local patch of the universe , so that no space outside of this sphere exists as far as science is concerned , and which happens to coincide with what one expects in case ( i ) above .case ( i ) underlies what is generally accepted as the standard hot big bang model , which can be more formally referred to as the almost friedmann - lematre model ( e.g. ) .this model may or may not have zero curvature , and it may or may not have trivial topology .numerous observations now knit together to provide a very solid foundation for the almost friedmann - lematre model , within the spatial , temporal and density limits of its validity .case ( ii ) , that of a steady state or of a quasi - steady - state model ( ) , is only defined in its general outline , and would require many more close links with recent observations in order to have predictive power .for example , the case ( i ) interpretation of quasars redshifts as primarily due to expansion of the universe , when applied to observations , implies that quasars and lyman break galaxies at high redshift trace large scale structure at ( comoving ) and that they function as a standard ruler which implies a low matter density  .this would be very difficult to explain if quasar ( and lyman break galaxy ) redshifts were non - cosmological .  case ( iii ) corresponds to the presently very popular chaotic inflation scenarios . whether or not the weak anthropic principle is consistent with the copernican principle is out of the scope of this paper .  case ( iii )is generally considered an extrapolation ( towards short cosmological times and large spatial distances ) of the almost friedmann - lematre ( fl ) model .case ( ii ) could also conceivably be defined as an extension of the fl model .the anti - copernican case ( iv ) is essentially identical to case ( i ) as far as observational analysis is concerned as long as the topology of the universe remains unmeasured . however , theoretical calculation is somewhat difficult if case ( i ) is not resorted to .if the topology of space were measured by observation ( , and see below ) , then the size of the universe would be finite and less than the horizon diameter in at least one direction , i.e. finite. models of case ( ii ) would then require serious revision and would no longer satisfy the perfect cosmological principle , and case ( iii ) models would presumably require individual `` universes '' to be born as separate space - times and might have difficulty surviving occam s razor .case ( iv ) would also be partly or totally avoidable : only the _ apparent _ space would be a sphere centred on the observer .at least some parts of the `` edge of the universe '' of the case ( iv ) interpretation would no longer be edges .on the other hand , whether or not case ( i ) would remain valid if the universe is found to be multiply connected would depend on the resolution of the ambiguity in the meanings of the words `` homogeneous '' and `` isotropic '' .the solution of the einstein - hilbert equations is only found as a local solution , so only _ local _ homogeneity and isotropy are required . in general , although friedmann - lematre models satisfy _ local _ homogeneity and isotropy , they need not satisfy _ global _ homogeneity and isotropy .observational support for local homogeneity and isotropy does not imply global homogeneity and isotropy .  in this paper , the relevance of different interpretations of these two words in the context of the global geometry of space ( curvature and topology ) is explained and illustrated by simple toy models in two dimensions . by attempting to measure the more subtle elements of inhomogeneity and anisotropy which would reveal the global shape of the universe , some of the ambiguity in deciding how to applythe copernican principle to observational cosmology could be avoidable : empirical results sometimes help resolve theoretical dilemmas.  proper distance ( , eq .14.2.21 ) in comoving units is used throughout this paper unless otherwise stated .the hubble constant is parametrised here as km s mpc values of the density parameter , , the dimensionless cosmological constant , , the dimensionless curvature , , and the curvature radius are indicated where used .when justifying the assumptions of ( i ) homogeneity and ( ii ) isotropy , these are generally interpreted to mean that  ( )  the comoving spatial density of astrophysical objects is approximately constant , and  the solid angular distribution of distant objects ( or microwave background fluctuations ) is approximately constant , after correction for the very anisotropic biases due to living in a dusty disk galaxy and due to our peculiar velocity with respect to the comoving reference frame ,  respectively .both are reasonably consistent with numerous observations , and the latter has been further reinforced recently by the only cosmological class of objects known to be isotropic _ in spite _ of galactic absorption and our peculiar velocity : the gamma ray bursts .these may in fact reveal the births of black holes , i.e. of regions of extreme curvature ( inhomogeneity ) on very small scales , that are briefly visible before being engulfed by event horizons .these two properties ( a ) , ( b ) are generally referred to as `` local homogeneity and isotropy '' .solutions to the einstein - hilbert equations are normally only found as local solutions of differential equations , and so only local assumptions of homogeneity and isotropy have any effect on friedmann - lematre models .one might also refer to these as first order homogeneity and isotropy , since only one - point statistics of density are considered .   for the earliest known article on cosmic topology , see .a recent review is that of .the multiply connected models which have been compared with observations are all fl models , i.e. they are _ locally homogeneous and isotropic_.  a simple way to illustrate multiply connected 3-manifolds which may correspond to the comoving 3-space we live in is to start with the 3-torus , .this can be thought of  ( )  _ physically _ , as a cube of which opposite faces are identified so that space is continuous , with no boundaries , but finite ,  _ observationally _ , as a tiling of euclidean space by multiple copies of the single _ physical _ `` cube '' , or  as the analogy one dimension higher of a 2-torus embedded in but endowed with an intrinsically flat metric , i.e. a metric different to the normal metric of .  for a demonstration of basic properties of locally homogeneous and isotropic manifolds ,it is useful to subtract one dimension , i.e. to consider 2-manifolds .this is the approach adopted here for pedagogical purposes . for the physical universe ,the results extrapolate to the three - dimensional case .the cube ( i ) is referred to as a _fundamental domain _ , and the space ( ii ) is referred to as the _ universal covering space _ or in a new coinage , _apparent space_. both are equivalent ways of thinking about the same manifold .the former is useful for conventional physical reasoning , according to which any particle or object only exists at one point in a space , but difficult to use for interpreting observations .the latter is useful for observational analyses , but surprising and sometimes confusing for physical reasoning , since objects `` exist '' many times in it .  in two dimensions ,the corresponding space is the flat 2-torus , of which the fundamental domain is a square ( in general a parallelogram , but a square is used for the present paper ) , and the universal covering space is .figure [ f - ptstor ] shows what such a universe would look like . for reference , the reader should remember that the horizon size for likely values of the curvature parameters is , and that objects like quasars or `` lyman break galaxies '' are seen up to typical redshifts of , which correspond to roughly half this distance .clusters of galaxies and galaxies are mostly only seen to a few .present constraints on the size of the universe ( defined here as the injectivity diameter , which is the length of the shortest spatial geodesic linking an object to an image of itself ) are , whether the universe is hyperbolic , flat or spherical .so , in order to see multiple images of a single object , catalogues of objects which are seen to typically a few tenths of would be required .this implies that effects of aging of the objects , or of viewing angles , or of movement relative to the comoving reference frame , would make multiple images of a single object appear quite different , so that proving the identity of two images would be difficult .so , the identity between multiple copies of symbols in fig .[ f - ptstor ] would not be obvious in any category of presently known objects .the model is _ locally homogeneous and isotropic _ : any study in a small region around a point is independent of the orientation of the axes of a chosen coordinate system and of the origin of that system .  _the model of fig .[ f - ptstor ] is globally homogeneous , but globally anisotropic ._  the anisotropy is shown by fig .[ f - septor ] , which shows vectors separating all pairs of images in the simulated universe .separations between multiple images of single objects correspond to multiples of the generators [ the vectors and and clearly cause clustering around these points .as pointed out by , a one - dimensional pair separation histogram should show sharp spikes corresponding to these cluster points .( if no offsets were added between multiple images , then the spikes would be dirac functions . )  the pairs of images contributing to these spikes can be referred to as _ generator pairs _ or as _ type ii pairs _ ( ) , which induce spikes in flat multiply connected spaces , but not in curved multiply connected spaces .if the axes are chosen at a different angle , then the diagram would be rotated : the pattern of spikes is not invariant ( i.e. varies ) under rotation .this proves global anisotropy .however , the diagram would be statistically identical under a shift ( without rotation ) of the origin : the space is homogeneous .many 3-manifolds are globally inhomogeneous , but the effect is observationally quite subtle .  the effect is illustrated here for the klein bottle .this is like , except that a twist is applied before identification of one pair of edges ( here , the `` vertical '' edges ) .the distribution of images is shown in fig .[ f - ptskle ] . again ,this model is _ locally homogeneous and isotropic_.  fig .[ f - sepkle ] shows that , as for the model , the klein bottle is _ globally anisotropic _ : rotation of the coordinate axes would modify the pattern of crytallographic spikes .  in order to show global inhomogeneity using `` generator pairs '' ,  ( )  knowledge of which pairs of images are due to multiple images of single objectsis required , and  this knowledge is required _ despite the fact that the crystallographic spikes corresponding to `` inhomogeneous generator pairs '' are smeared out making them very difficult to detect ._  both ( i ) and ( ii ) are , of course , easy for a simulated universe : in the observed universe they are difficult .comparison of figs [ f - septor ] and [ f - sepkle ] shows that some crystallographic spikes seem to disappear due to the twist in identifying the vertical edges of the fundamental domain .figures [ f - sepklea ] and [ f - sepkleb ] show where these spikes have `` disappeared '' to , using property ( i ) .different choices of the origin ( see figure captions ) yield different patterns .together , the two figures show how the spikes have been smeared out .hence , the klein bottle is _ globally inhomogeneous_.  one way to apply this technique observationally would be  ( )  to use properties of galaxies or quasars which uniquely label and identify individual objects , despite the large fractions of the age of the universe over which the objects age .this seems difficult given present observational limitations and known techniques , but not impossible .an alternative method would be  ( )  firstly to detect the crystallographic spikes of generator pairs which _ do _ appear , and which would imply a fundamental domain of size in the horizontal direction , and secondly to hypothesise that the real fundamental domain is just half this length .this would make predictions of which pairs of objects should correspond to those seen separately in figs [ f - sepklea ] and [ f - sepkleb ] . if the predicted multiply imaged pairs really were identical , then proving this by observation would be much easier than the `` needle in a haystack '' requirement of finding the corresponding pairs blindly as in the previous suggestion ( a ) .figs [ f - sepklea ] and [ f - sepkleb ] illustrate one way in which the copernican principle could be applied directly .one would not expect the observer to be at any special value of : the value for the toy model shown here would be surprising.   inspection of fig .[ f - ptstor ] and a little reflection show that generator pairs are not the only pairs of images which occur repeatedly in a separation vector plot ( or separation distance histogram ) .close pairs of images of _non_-identical objects within a single fundamental domain also define separation vectors which are repeated in the apparent space , though in a different way to generator pairs .these are _ local pairs _ or _type i _ pairs ( ; section 2 of ) .since there are nine copies of the fundamental domain shown in fig .[ f - ptstor ] , there are nine local pairs clustered around each of the separation vectors of modulus smaller than .this is visible as a `` graininess '' in figs [ f - septor ] and [ f - sepkle ] .  in a flat space , the same effect also occurs for pairs across many copies of the fundamental domain , but in general does not occur in curved spaces .thus the term `` local '' : it is most useful just to consider relatively close pairs of images.   noticed the effect of the local pairs in pair separation histograms , and showed how to collect these together and detect multiple connectedness .if these led to detection of multiple connectedness for a model known ( mathematically ) to be inhomogeneous , then this would enable ( a ) above to be applied : the generator pairs could be used to observationally illustrate inhomogeneity .however , local pairs in themselves would not show inhomogeneity : they are local .similarly , local -tuplets would not directly show inhomogeneity .however , since non - orientable manifolds are inhomogeneous , the discovery of significant numbers of matching -tuplets , where some of these required orientation reversals , would imply inhomogeneity .  for more details of global homogeneity and isotropy questions of 2-manifolds , and more particularly of 3-manifolds , see .the copernican principle , when used to motivate _local _ homogeneity and isotropy of the universe , does not necessarily imply _ global _ homogeneity and isotropy .if the topology of the universe is observable , then this may reveal     most techniques for measuring cosmic topology should easily lead to direct geometrical illustrations of global anisotropy if multiple connectedness is detected ( unless the 3-manifold we live is globally isotropic : the projective space , , which can be thought of by identifying all opposite points of the hypersphere , is globally both isotropic and homogeneous , even though multiply connected ) .by contrast , global inhomogeneity in a flat multiply connected manifold would be difficult to detect directly .however , prior knowledge of multiple connectedness would help in obtaining direct illustrations of global inhomogeneity .present observations imply that the observable universe is close to flat and that multiply connected flat models of size are consistent with present large angle microwave background data .the observational approach to global geometry might just possibly provide an empirical solution to an otherwise philosophical question , by showing whether not both local _ and _ global homogeneity and/or isotropy are valid assumptions ..        gomero g. i. , teixeira a. f. f. , rebouas m. j. , bernui a. , spikes in cosmic crystallography , , 1999 .hoyle f. , burbidge g. , narlikar j.v ., a quasi - steady state cosmological model with creation of matter , 437 , 1993 .lematre g. , la structure et levolution de lunivers , , ed .stoops r. , ( brussels : stoops ) , 1958 .luminet j.p ., roukema b. f. , topology of the universe : theory and observations , in , ed .lachize - rey m. , dordrecht : kluwer , p117 , arxiv : astro - ph/9901364 , 1999 .schwarzschild , k. , on the permissible curvature of space , 337 , 1900 .schwarzschild , k. , ` `` on the permissible curvature of space '' by k schwarzschild ' , 2539 , 1998 .thurston , w. p. , , ed .levy , s. , princeton , u.s.a .: princeton university press , 1997 .

Abstract: it is often stated that homogeneity and isotropy of the universe are assumptions of the almost friedmann - lematre ( fl ) model ( the hot big bang model ) , inspired from the copernican principle . however , only _ local _ homogeneity and isotropy are required by the model : multiply connected almost fl models are locally homogeneous and isotropic , but they can be _ globally _ anisotropic and/or _ globally _ inhomogeneous . toy models are used here to show how global anisotropy and/or global inhomogeneity of an almost fl model could be shown directly in observations . this approach may avoid having to make any assumptions regarding global anisotropy and inhomogeneity .