Evaluating and Optimising Models of Network Growth
This paper presents a statistically sound method for measuring the accuracy with which a probabilistic model reflects the growth of a network, and a method for optimising parameters in such a model. The technique is data-driven, and can be used for the modeling and simulation of any kind of evolving network.
The overall framework, a Framework for Evolving Topology Analysis (FETA), is tested on data sets collected from the Internet AS-level topology, social networking websites and a co-authorship network. Statistical models of the growth of these networks are produced and tested using a likelihood-based method. The models are then used to generate artificial topologies with the same statistical properties as the originals. This work can be used to predict future growth patterns for a known network, or to generate artificial models of graph topology evolution for simulation purposes. Particular application examples include strategic network planning, user profiling in social networks or infrastructure deployment in managed overlay-based services.
Evaluating and Optimising Models of Network Growth
|Richard G. Clegg|
|Department of Electrical and Electronic Engineering|
|University College London|
|Department of Electrical and Electronic Engineering|
|University College London|
|Department of Computing|
|Imperial College London|
|Department of Electrical and Electronic Engineering|
|University College London|
Categories and Subject Descriptors C.2.1 [Network Architecture and Design]: Network Topology; G.2.2 [Graph Theory]: Network Problems
Network evolution, Likelihood-based models
In recent years there has been much interest in creating simple probabilistic models which can be used to produce topologies which replicate certain statistical properties of a given target network. Many of these models depend on a procedure by which a network is progressively “grown” from a small “seed” (with a handful of links) into an artificial topology which is as large as required. If the model is successful, the artificial network will have similar properties to the original. Thus, these models rely on finding a network evolution model that produces networks which are structurally similar to the target network.
In much of the previous research in this field the usual way of achieving this is to hypothesise an evolution model for the target network, grow an artificial network of at least the same size using that model, and compare several key graph theoretical statistics with the respective ones from the target network. This is usually done multiple times, so that the expected values of the statistics can be obtained. If after this process the model is found to be unsatisfactory, it is updated accordingly and the whole process repeated.
Thus, the development of a topology evolution model following the methodology detailed above will require the construction of large numbers of “test” topologies that use tentative evolution models. Since the construction of these topologies can be computationally cumbersome if the networks in question are large, the analysis of network evolution models has not been widely adopted in practice.
The main contribution of this work is two-fold. Firstly (and most importantly), we present a set of statistics that directly measure the likelihood of a given probabilistic evolution model giving rise to a given target network – no “test” topologies need to be constructed. Secondly, we present a framework for exploratory testing and optimisation of certain (quite general) classes of network growth models.
The statistics that our technique produces are an unambiguous and statistically rigorous measure of the likelihood of the evolution of the target network arising from any particular hypothesised probabilistic model. These statistics are quick to produce (much more so than growing a test network of the same size), and could be used as a fitness function for state-space searches or genetic algorithms to automatically optimise parametrised classes of models.
We will refer to this statistical framework as FETA (Framework for Evolving Topology Analysis).
The structure of this paper is as follows. Section Evaluating and Optimising Models of Network Growth describes the FETA framework in detail. Section Evaluating and Optimising Models of Network Growth shows how the model likelihood is derived, and section Evaluating and Optimising Models of Network Growth describes the fitting procedure for optimising model parameters. Section Evaluating and Optimising Models of Network Growth describes the model fitting for the five network examples investigated in this paper. Section Evaluating and Optimising Models of Network Growth shows how the fitted models can be used to generate artificial topologies that replicate specific statistical measures of the corresponding real networks.
The problem of creating artificial topologies with the same growth dynamics as a target network is an important one. As networks grow, their statistical measures change and undesirable emergent properties may occur. A good statistical model of how a given target network grows is an important goal which has applications in many fields, but especially in the design and optimisation of distributed computation and communication systems. A tier one network provider may wish to be able to model the future growth of the AS network to predict and potentially avoid undesirable network properties, or to strategically choose its peering agreements. The owner of an online social network may wish to be able to predict, from their position in the network, consumption patterns or demographic characteristics, which users are more likely to accrue “friends” and hence influence others. This information can be used for targeted advertising, marketing or capacity planning purposes. Finally, a provider of overlay-based services (such as Skype or Akamai) may need to plan based upon the future evolution of their overlay network. As key network statistics change, they may wish to adapt their protocols, or to modify infrastructure deployment strategies accordingly.
The field of generating graphs (or networks or topologies, the words seem to be used almost interchangeably in the literature) using random processes is usually considered to begin with Erdős and Rényi [?]. An early study by Price [?] found that the degree distribution of co-authorship network of scientific papers obeyed a power law. Much later it was discovered that the Internet Autonomous System (AS) graph also follows a power law [?] and this finding was also shown to apply to a large number of other networks, including social networks, hyperlinked document networks and networks derived from biological systems. The well-known Barabási–Albert (BA) model [?] provided a seminal explanation of scaling network topologies in terms of a “preferential attachment” model where “rich get richer”: the probability of connecting to a given node is exactly proportional to its degree. This led to several papers which attempt to explain network evolution in terms of node degree and related properties such as the BA [?], ASIM [?] and AB [?] models.
Bu and Towsley [?] introduced the Generalised Linear Preference (GLP) model, which modifies the preferential attachment model by raising the degree of the node to a small power. Zhou and Mondragón [?] presented the Positive-Feedback Preference (PFP) model, which also modifies preferential attachment by raising node degree to a small power (but this power also depends on the node degree).
It has been shown that a model which faithfully reproduces the node degree distribution may not capture all the important properties of a graph [?]. To account for this, the ORBIS model [?] reproduces the statistics of subgraphs of small orders to take account of degree-degree and higher order characteristics. The ORBIS model is slightly different to the growth models which the FETA approach uses, as the model uses rewiring and rescaling rather than a hypothesised growth model.
The typical assessment of topology generation models has focused on measuring a number of statistics on the real network data. Such measures have included the number of nodes and links, average and maximum node degree, best-fit power-law exponent, rich-club connectivity, probability of nodes with low degrees (1, 2 and 3), characteristic path length, average and maximum triangle coefficient, average and maximum quadrangle coefficient, average and average and maximum betweenness (see [?] for definitions of these properties and a review of topology generation from an Internet perspective). A candidate artificial model is then tested by creating an artificial topology using the model and seeing how well the topology reproduces several statistics measured on the real data set. Occasionally, a new network statistic may be added which existing models do not reproduce and this can be used to justify a different, improved model. This approach to model testing and refining based on the generation of test topologies and the comparison of a set of statistical measures between the test topologies and the target network is characterised here as the “basket of statistics” approach.
Willinger et al [?] called for a “closing of the loop” between the discovery of “emergent phenomena” and the models which reproduce them. They emphasise the importance of a “validation” step to ensure that a particular proposed model is consistent with the real data. The evaluation framework given in section Evaluating and Optimising Models of Network Growth provides this validation, albeit at the expense of requiring data about how the network evolves rather than a static snapshot. In so doing, however, the generation of test topologies is avoided and the processing of bigger graphs or more complicated models becomes possible.
Figure Evaluating and Optimising Models of Network Growth contrasts the approach used by FETA with the “basket of statistics” approach which has been previously used. By directly assessing model likelihood, our approach short circuits the cycle of generating and measuring test networks to optimise a test model.
The Framework for Evolving Topology Analysis (FETA) allows the investigation of growth models for real networks where information is known about the order in which links were added to the network. The aim of FETA is to produce probabilistic models which fit the observed evolution of these networks. The class of models which FETA can work with includes BA [?], AB [?], GLP [?] and PFP [?].
The probabilistic models used by FETA are described in terms of two components referred to as an inner model and an outer model. It is the inner model which FETA evaluates and fits, which means that only models based on probabilistically growing networks are compatible with FETA.
The outer model chooses an operation which will make a change to the existing network. This could be add and connect a node, add a link between nodes, delete a node or delete a link between nodes. The inner model defines the probabilities for selecting the node or nodes involved in the operation.
The inner model defines the probabilities for selecting the node or nodes involved in the operation defined by the outer model. The inner model used by a given evolution model can vary, depending on whether the outer model operation is a connection to a new node or a connection between existing inner nodes.
For example, the AB model would correspond to an outer model which adds a new node and then chooses exactly three inner nodes to connect it to, along with an inner model that chooses nodes with probabilities proportional to their node degree. In PFP and GLP, the outer model is the same as with AB, but the inner model is different: the probability is proportional to the degree raised to a power.
As is common in the literature, the main focus of this paper is on the inner model. The outer models, where needed in this paper, are assumed to be of the following simple form: a new node is joined to existing nodes; following this (which can be zero) inner edges are added. The values of and are randomly selected from probability distributions empirically derived from the target network.
The framework is flexible enough to allow or disallow the possibility of multiple edges between the same node pair, and to allow or disallow nodes with connections to themselves. In this paper only undirected, connected, simple (no repeated edges and no edges from a node to itself) graphs are considered. Removal of nodes or edges is not considered in this paper.
Section Evaluating and Optimising Models of Network Growth describes the evaluation procedure of FETA, that works with any inner model which can assign probabilities to nodes or edges in a graph. It will produce statistically reliable measures of how well the model fits the observed data. Section Evaluating and Optimising Models of Network Growth describes a fitting procedure which works with a subset of inner models which combine sub-models and fit them using General Linear Models. Section Evaluating and Optimising Models of Network Growth describes how FETA is used in practice and gives information about scalability.
Consider data from an empirical network which shows how the network grows in time by the addition of nodes and edges. This growth data can be decomposed into decisions from the outer model (whether a link is between existing inner nodes or to a new node) and the choices of node (which would be controlled by the inner model).
Let be the known state of the graph at a certain time. Assume that the graph is extended by adding edges (sometimes between existing nodes and sometimes in addition to a new node) one at a time. Assume, further, that the state of the graph is known for each one of these edge additions up to some step ( is known). Let be the outer model operation (connect edge to new node or connect edge between existing nodes) for the th edge addition since . Let be the node or nodes selected by the inner model for the outer model operation . Together and define the transition between and . Conversely, if and are known then and are also known for . The best outer and inner models are those which best explain and , respectively, for the observed periods. This paper focuses on the selection of the inner model.
Let stand for all of the observed inner model choices , and let be some inner model which attempts to explain the observed inner model choices in terms of some statistical properties of the graph. At each step , maps graph properties (and, perhaps, other properties, such as whether a new node or an inner edge is being connected, or properties associated with the node but exogenous to the network topology) to probabilities.
In order to simplify the explanation, assume for remainder of this section that the outer model always involves the choice of a single existing node to connect to a new node. In this case is simply an ordered list of the nodes chosen at each observed step and is the node connected at step . Evaluation of the model is now a matter of calculating the likelihood of given the model . The larger this likelihood, the better the model fits the observed data.
Let be the probability that inner model assigns to node at step . To be a valid model should ensure that where the sum is over nodes. It follows that is the likelihood of the choice at step given model . The likelihood of all the observations given model is given by the product
It is also useful to define the log likelihood . The larger the likelihood (or log likelihood) the better the model explains .
The null model is defined as the model which gives every node in the choice set equal probability (this can also be thought of as the random model). The saturated model is a model with as many parameters as data points. In this case, ensures that for all . Hence and .
Now it is useful to define some measures of the goodness of the model using the statistic known as deviance.
The deviance of model is minus two times the log-likelihood ratio between the model and the saturated model ,
Evidently, the deviance will always be positive (or zero for the saturated model), and the smaller it is, the better the model explains the data.
The null deviance of a candidate model is given by
Thus, will always be negative if the model explains better than the null (random) model . The smaller , the better explains the choice set .
Because of the size of the data sets used in this work () then the magnitude of can be quite large and depends critically on the size of . It is useful to have a statistic which defined on a more comprehensible scale, and invariant to the size of . We present such new statistic, the per choice likelihood ratio:
The per choice likelihood ratio is the likelihood ratio between and the null model normalised by the number of choices.
The quantity is one if is exactly as good as , greater than one if it is better and less than one if it is worse. Note that and are simply different ways of looking at the model likelihood.
It should be noted though, that while a lower deviance or a higher per choice likelihood ratio always indicate a better fit, this alone does not mean a model should be preferred. The saturated model gives a perfect fit to data, but it is a useless model for practical purposes since it can only reproduce the data it has been given. What is needed is a trade off between fit to data and a parsimonious model. Adding new parameters to a model is only good if the improvement to the fit (reduction in , increase in ) justifies the extra parameter. One criteria would be Akaike’s An Information Criterion (AIC) [?] which is given by where is the number of free parameters in the model. However, given the size that typically attains in this modelling, this seems unlikely to prove a useful distinction.
An example will help comprehension. Consider an initial graph which is the two link network consisting of nodes and edges . The network grows by adding node and link and then node and link . We assume the simple outer model add one node and connect it to one existing node at every stage. The inner model must explain , and given and . The null model predicts equal probabilities ( each) for node to connect to nodes , or and equal probabilities of each, for node 5 to connect to nodes 1 to 4. Therefore, for this and the null model the likelihood is and . The null likelihood . If, on the other hand, we consider as the preferential attachment model (probability of attachment proportional to node degree) then, given the node probabilities are and given they are . The likelihoods are and giving a final likelihood . From this, deviance, null deviance and per choice likelihood ratio can be calculated. Naturally, real data sets will have many more choices and many more nodes.
The selection of edges from a set of all possible edges would present a difficult computational problem, as the set of all possible edges increases approximately as the square of the number of nodes. This can be avoided by considering the probability of choosing edge as the probability of choosing followed by , plus the probability of choosing followed by (assuming ). The second choice set can be narrowed to avoid self loops and nodes already connected to the first node if a simple graph is desired.
Separate inner models can be fitted to each type of operation for the outer model. Therefore, for example, the hypothesis that new nodes connect using preferential attachment and inner edges connect using PFP can be explored. The data set can be split into two parts, those choices associated with connecting to new nodes and those choices associated with adding edges between existing nodes. In this case, the deviance of the full inner model is the sum of the deviance of the model components, and the per choice likelihood ratio can be calculated accordingly.
The deviance and per choice likelihood ratio can determine which inner model is a better fit for a given data set. However, for parametrised models, they do not allow the automatic tuning of parameters. In this section a method is introduced based upon the statistical technique of Generalised Linear Models (GLM) which allows certain (linear) parameters to be automatically tuned for an inner model. Again, for simplicity of discussion, this section considers only inner models which connect nodes to a new node.
Consider an inner node model . It may be that the ideal model is not pure preferential attachment or PFP, but some mixture of these models. Further, it follows that probabilities may be affected other factors (both inherent in the graph topology and exogenous to the topology but available as a data input).
Let be the degree of node in graph (the graph used to make choice ). Let be the probability that model assigns to node for choice . For the null model then we have that
where is a normalising constant for a given choice (that is, it is constant for a given ) so that the probabilities sum to one over all . Similarly, for the preferential attachment model, referred to for now as , then we have that
where, again, is a normalising constant for a given . Let be the number of triangles (–cycles) node is part of in graph . Now we can consider some hypothetical model where connection probabilities depend upon the triangles,
where, again, is a constant for fixed . A model can be considered which is a linear combination of , and . Call this hypothesised model . For this model we would have that
where , and are all in the range and sum to one. These constants are the proportion of each of the models which contribute to the final model. We use GLM to find the optimal combination of parameters for a given data set. A brief summary of GLM follows.
Let be some set of observed data we desire to model. Let and (similarly defined) be sets of observed data that is to be used to explain . A relationship is hypothesised which allows to be estimated in terms of , and so on. A model is to be fitted of the form
where the are parameters (not constrained to a range this time) which give the contribution of the various components to the variable and which is an intercept parameter and is an error component. Fitting GLM can be done automatically using a statistical language such as R111http://www.r-project.org. Given observed data, this can be read into R and a GLM fitting procedure can be used to find those values which maximise the model likelihood. In addition, the fitting procedure produces the model deviance and estimates of the errors and statistical significance for each of the model parameters. If a parameter is not statistically significant it should usually be removed from the model.
Let be an indicator variable which is if and only if node was actually the node picked for choice , and otherwise. The problem of finding the best model in (1) becomes the problem of fitting the GLM,
to find the combined model that best predicts the . Thus, GLM fitting can be used to find the choice of which maximises the likelihood of this model. This is equivalent to finding the which gives the maximum likelihood for since for model , the expectation .
This will give the choice of which best combine the model components into the unified model . If the are in the range and sum to one, it can be trivially shown that is a valid probability model as long as , , are.
So, for the period between and , for each node, a data point is generated with the parameters of the graph relevant to the models, and with a or a depending on whether that node was the node actually selected as an outcome of that choice. The procedure has been tested on data from artificially generated networks and it has been found to be able to successfully recover their parameters in a wide variety of circumstances. Certain model component combinations might be problematic to fit, however. An example of this would be a model constructed from a PFP and a preferential attachment component: since these explanatory variables are very similar for most nodes, finding a satisfactory mix using GLM is usually extremely hard.
Note that only the parameters can be automatically optimised by the GLM fitting procedure. Any other parameters such as the in the PFP model must be fitted by other means, such as trying a number of parameter choices and comparing the deviance or per choice likelihood ratio.
As pointed out in remark 3, separate models can be fitted to nodes connecting to new nodes and connections to inner edges. The items of data are separated by an analysis tool and they are fitted in different GLM models. As in remark 2 fitting inner edges causes issues for the framework. The choice of inner edges is broken down into the choice of two nodes. The choice set for the second node is constrained by removing from the choice set those nodes which already have a link from the first node.
The FETA evaluation process therefore consists of hypothesising inner models which might fit the evolution of a target network and calculating their likelihood statistics as shown in section Evaluating and Optimising Models of Network Growth. The fitting procedure in section Evaluating and Optimising Models of Network Growth is used as an exploratory tool both to tune linear combinations of model components and also to provide hints as to which other components might be introduced (for example, a negative parameter will rarely produce a usable model but, for example, if the component produced a negative this suggests that the choice mechanism is avoiding nodes with a high triangle count).
The graph in figure Evaluating and Optimising Models of Network Growth shows the run time for measuring model likelihood (as described in section Evaluating and Optimising Models of Network Growth) and for creating a network file with a given number of links (using a test model which is part PFP and part random). The tests were run on a 2.66GHz quad core Xeon CPU. The plot is a log-scale showing how run time varies with network size. For 100,000 links the network creation process takes 2,631 seconds and for the likelihood estimation process takes 53 seconds. Both processes appear to scale approximately as the square of the number of links (for times under 1 second the timing information is not accurate). Neither process takes a significant amount of memory. The relative speed of the evaluation of likelihood statistics is another benefit of the FETA approach. To tune the parameters of a hypothetical parametrised model using the “basket of statistics” approach, a new network would have to be grown for every test model. This is much more computationally intensive than the calculation of likelihood statistics required by FETA.
The FETA procedure is used to create inner models for several networks of interest. Section Evaluating and Optimising Models of Network Growth fits models to a co-authorship network inferred from the arXiv database. Section Evaluating and Optimising Models of Network Growth fits models to a view of the AS network topology referred to here as the UCLA AS network and section Evaluating and Optimising Models of Network Growth fits models to a second view of the AS topology, which we refer to here as the RouteViews AS network.
Finally, section Evaluating and Optimising Models of Network Growth fits network evolution models to a network derived from user browsing behaviour, and section Evaluating and Optimising Models of Network Growth fits models to a social network derived from the popular photo sharing site Flickr.
Table Evaluating and Optimising Models of Network Growth summarises the networks considered in terms of total edges, total nodes and the edge/node ratio.
Network edges nodes edge/node arXiv 15,788 9,121 1.73 UCLA AS 93,957 29,032 3.24 RouteViews AS 94,993 33, 804 2.81 gallery 50,472 26,958 1.87 Flickr 98,931 46,557 2.13 Table \thetable: Sizes of the networks analysed
Several model components were considered in a linear combination as described in section Evaluating and Optimising Models of Network Growth. Those components are listed below, where is the probability of choosing node and is a normalising constant such that when summed over the choice set. Furthermore, is the degree of node and is the triangle count of node .
– the null model assumes all nodes have equal probability .
– the degree based preferential attachment model assumes node probability .
– the triangle count model assumes node probability .
– the singleton model assumes node probability if and otherwise.
– the doubleton model assumes node probability if and otherwise.
– the PFP model assumes node probability .
Note that the PFP model is the only one to require a parameter.
This notation allows the concise description of a linear additive model in terms of its components. For example is a model which is has a component from the singleton model (contributing of the probability) and a component from the PFP model with parameter (contributing of the probability). In the models , and there is a possibility of all nodes being assigned zero probability (if there are no singletons, doubletons or triangles respectively). In this case, is substituted for that model component. This happens on extremely few occasions and always very early in network construction. Obviously a large collection of model components could be tried but a conscious decision was taken to limit the number of possibilities for this experimentation.
For each data set, three inner models are tried:
a pure preferential attachment model,
a pure PFP model (with an optimally tuned for connections to new nodes, and another one for internal edges),
the best model found using the techniques from section Evaluating and Optimising Models of Network Growth.
Model one was picked because the preferential attachment is a reasonable baseline for improvement. Model 2 was picked because investigation showed that for almost every network the PFP model had low deviance. Model 3 was picked to show the improvement (if any) possible by using linear combinations of models.
The outer model was derived simply by calculating empirically from the network data two distributions.
the number of inner nodes each new node connects to on arrival,
the number of inner edges connected between each new node arrival.
These distributions are then used to create the outer model. This is simplistic and obviously further research is required to improve the techniques to generate this outer model.
Results are presented using the metrics from section Evaluating and Optimising Models of Network Growth: is the deviance, is the null deviance and is the per choice likelihood ratio. A better model is indicated by lower and , and by a high . The results are broken down into the contribution from the inner model to connect to new nodes and the inner model for connecting internal edges.
A publication co-authorship network was obtained from the online academic publication network arXiv. The first paper was added in April 1989 and papers are still being added to this day. To keep the size manageable, the network was produced just from the papers categorised as math. The network is a co-authorship network: an edge is added when two authors first write a paper together. In this case, because it is required that the network remains connected, edges which are not connected to the largest connected component are ignored. Multiple edges between two authors are not added. The processing of this network is far from perfect, only author names (rather than unique IDs) are matched. Inconsistent naming conventions mean some authors are recorded by first name and surname, and some by initial and surname. To avoid problems matching John Smith, J. Smith and John W. Smith, the match is on first initial and surname, though it is clear this will allow some collisions. One paper222http://arxiv.org/abs/math/0406190 was removed from analysis. The paper has sixty authors, far more than the paper with the next largest number of authors. Since each author on a paper forms a graph clique with all the other coauthors in that same paper, this paper added 1,732 links for which no arrival order significant to the evolution of the network could be found. As a size 60 clique would distort most network statistics, it was rejected as an outlier.
As described in the previous section, three models were tried, a preferential attachment model, a pure PFP model and the best model found using the fitting procedure. Model 2, the best PFP model was, for connections to new nodes, and for connections between inner edges . Model 3, the best model found, was, for new node connections, and for internal node connections the pure PFP model as in model 2 (no better model could be found).
As can be seen, the best model by all measures is model 3. It is worth noticing that the inner edge model does not perform significantly differently between the three (in any case this is the same model for 2 and 3). With such a small parameter the model is almost the same as preferential attachment (model 1). It should also be noticed that the deviance itself is hard to compare simply because it is such a large number that the relative differences seem small. For the preferential attachment model, (model 1) the new node model was actually worse than the null model and this can be seen by the fact that is positive and is less than one. Overall, the new node model appears to have made few gains relative to the new node model in all cases ( is larger for the inner edge models than the new node models) despite the simplicity of the inner edge model. This suggests that improving the new node model is the best focus for model improvements in general. The improvement in new node model from in model 2 to in model 3 seems significant and indicates that the addition of a model reflecting singletons is useful. Remembering that singletons in this case are authors with only a single other co-author, perhaps this is explained by a desire for those authors with a single co-author to collaborate with other new authors.
Model component Model 1 New node 195,000 -1,510 1.06 Model 1 Inner edge 118,000 -4,091 1.31 Model 1 Overall 312,000 -5,600 1.15 Model 2 New node 194,000 -2,450 1.10 Model 2 Inner edge 118,000 -4,170 1.32 Model 2 Overall 311,000 6,610 1.18 Model 3 New node 193,000 -3,090 1.13 Model 3 Inner edge 118,000 -4,240 1.32 Model 3 Overall 311,000 -7,340 1.21 Table \thetable: Three models tested on the arXiv network.
The data set we refer to here as the UCLA AS data set is a view of the Internet AS topology seen between January 2004 and August 2008. It comes from the Internet topology collection333http://irl.cs.ucla.edu/topology/ maintained by Oliviera et. al. [?]. These topologies are updated daily using data sources such as BGP routing tables and updates from RouteViews, RIPE,444http://www.ripe.net/db/irr.html/ Abilene555http://abilene.internet2.edu/ and LookingGlass servers. Each node and link is annotated with the times it was first and last observed during the measurement period.
As previously stated, our network growth model does not include a removal process. On the other hand, various links and nodes disappear from the UCLA data set during the time interval under analysis. To incorporate this into our modelling framework, the data is preprocessed by removing all edges and nodes which are not seen in the final sixty days of the data, so that the final state of the evolution of the network is the AS network as it is in August 2008. Edges are introduced into the network in the order of their first sighting. If this would cause the network to become disconnected, their introduction is delayed until the arrival of other links and nodes allows them to join while maintaining a connected network at all times.
The arrival order of edges is only known after timed link arrival data is available in January 2004. Furthermore, there is a period of fast discovery of nodes and edges immediately after this time where the order of edge arrival is considered to be very uncertain (since snapshots are only daily and not every link will be discovered on the first day it exists). Thus, the first days of data are considered a “warm up” period and removed from the analysis. is taken to be after this warm up period expires. The growth of the network is shown in figure Evaluating and Optimising Models of Network Growth.
Exploratory model fitting on the UCLA AS network showed that a PFP model was again favoured. Again the inner edge model seemed to have a smaller than the new node model. No model was found to be a great improvement over PFP but there was some evidence that including a small singleton component would make slight improvements.
Three models are tested on the UCLA AS network. Model 1 is pure preferential attachment. Model 2 is pure PFP with different delta parameters – for the new node model and for the inner edge model. The best model found was only slightly better than this, and it and combined PFP with a tiny amount of the singleton model The new node model was . The inner edge model was .
The results of this fitting exercise are shown on table Evaluating and Optimising Models of Network Growth. In this case, the improvement against preferential attachment was extremely marginal. It was only model 3 that showed an improvement, and this improvement was mostly in the inner edge model (indeed its new node model was worse than that of model 2). Overall, the values were relatively high indicating a good fit to the data compared with the random model.
Model component Model 1 New node 320,000 -102,000 10.6 Model 1 Inner edge 1,790,000 -402,000 5.74 Model 1 Overall 2,110,000 -504,000 6.33 Model 2 New node 319,000 -102,000 10.8 Model 2 Inner edge 1,790,000 -402,000 5.73 Model 2 Overall 2,110,000 -504,000 6.33 Model 3 New node 320,000 -102,000 10.7 Model 3 Inner edge 1,780,000 -405,000 5.82 Model 3 Overall 2,100,000 -507,000 6.41 Table \thetable: Three models tested on the UCLA AS network.
For the present paper we define the RouteViews AS data set as the view of the Internet AS topology from the point of view of a single RouteViews data collector. The raw data used to construct it comes from the University of Oregon Route Views Project[?], and it was recovered from the parsing of the routing tables obtained by running ‘show ip bgp’ on the command line of route-views3.routeviews.org and capturing the output. To construct the node and link arrival process to which we fit our evolution models we process one such table dump per day over the time interval between April 11th, 2007 and January 16th, 2009.
It is well known that an AS map obtained in such a way will not be representative of the true AS Internet topology (see [?, ?, ?, ?]). However, a validation framework like FETA should be able to discover this difference by fitting different growth models to the RouteViews AS data set and the more complete UCLA data set.
Since the basic outer models that we set out to evaluate do not have a node removal process, we consider only the addition of AS numbers and peerings to the AS map, as it is viewed from the perspective of route-views3.routeviews.org. Thus, we seek to model the cumulative AS growth process as viewed from a single BGP peer.
As with the UCLA data set, we ignore the very first tables processed, as their dynamics are not representative of the system equilibrium growth rate, and their timing information is unavailable. The growth of the network is shown in figure Evaluating and Optimising Models of Network Growth.
As before, model fitting on the RouteViews AS data set showed that a PFP model was favoured. As in the previous cases, the inner edge model seemed to have a significantly smaller than the new node model. As with the UCLA case, a small singleton component in the inner edge model yields slight improvements.
Three models are tested on the RouteViews AS data set. Model 1 is pure preferential attachment. Model 2 is pure PFP with different delta parameters ( for the new node model and for the inner edge model). Model 3 took the same new node model as model 2, but for the internal edge model it combined a pure PFP new node model with the singleton model according to .
The results are shown in table Evaluating and Optimising Models of Network Growth. As before, the improvement against preferential attachment was extremely marginal. It was only model 3 that showed improvement, as a consequence of a slightly better inner edge model. Again, values were relatively high indicating a good fit to the data compared with the random model.
Model component Model 1 New node 138,000 -45,400 12.7 Model 1 Inner edge 1,478,000 -257,000 4.36 Model 1 Overall 1,620,000 -302,400 4.81 Model 2 New node 138,000 -46,100 13.21 Model 2 Inner edge 1,480,000 -257,000 4.36 Model 2 Overall 1,620,000 -303,400 4.83 Model 3 New node 138,000 -46,100 13.21 Model 3 Inner edge 1,470,000 -264,000 4.53 Model 3 Overall 1,610,000 -310,100 5.00 Table \thetable: Three models tested on the RouteViews AS network.
As expected, the model found for the RouteViews AS data set is quite close to that one found from the UCLA data set, while still being different enough to accommodate their differing topological characteristics. Overall, the difference in the PFP/singleton mix between the best models fitted for the UCLA AS and RouteViews data sets suggests that, from the point of view of route-views3.routeviews.org, ASs with a single point of attachment to the Internet are more prone to becoming multihomed than they are from a more complete perspective of the AS topology.
The website known simply as “gallery”666http://gallery.future-i.com/ is a photo sharing website. To be able to upload pictures and have some control over the display of pictures, users have to create an account and login. From webserver logs, the path logged in users browse as they move across the network can be followed. Thus, images become nodes in the networks, and a user browsing between two photos creates a link between the two nodes that represent them. These links are overlaid for all users in order to form our network.
Model 1 is, as usual, a pure preferential attachment model . The fitting of model 2 was problematic for this network, minimising deviance for the new nodes model with the unusually low delta value of . For inner edges, the PFP model had lowest deviance. Finally, Model 3 has the new node model , and the same inner edge model as model 2 – that is, .
Table Evaluating and Optimising Models of Network Growth shows the model likelihood statistics, where the inadequacy of the proposed models to the network growth dynamics is apparent. In particular, the model to connect to new nodes was, for model 1 and model 2, worse than the null model (which connects to nodes at random). Thus, FETA allows us to discover in a straightforward way that new node connections in this network do not have a preferential attachment structure at all. We hypothesise that the peculiar new node arrival process arises from the fact that the browsing network is, uniquely amongst the networks examined here, a transient one in the sense that a link between two nodes is made by a user moving from one picture to the next – however, no permanent record of this is reflected to the user, and thus, user behaviour is not influenced by it.
Model component Model 1 New node 675,000 44,300 0.523 Model 1 Inner edge 586,000 -17,000 1.23 Model 1 Overall 1,260,000 27,000 0.815 Model 2 New node 645,000 14,000 0.810 Model 2 Inner edge 586,000 -17,200 1.30 Model 2 Overall 1,230,000 -2,750 1.02 Model 3 New node 529,000 -102,000 4.43 Model 3 Inner edge 586,000 -17,000 1.30 Model 3 Overall 1,110,000 -119,000 2.43 Table \thetable: Three models tested on the gallery user network.
The Flickr777http://flickr.com/ website allows users to associate themselves with other users by naming them as Contacts. In [?] the authors describe how they collected data for the graph made by users as they connect to other users. The first 100,000 links of this network is analysed here. The graph is generated by a web-crawling spider so the order of arrival of edges is the order in which the spider moves between the users rather than the order in which the users made the connections. Thus, the evolution dynamics of this network will be determined by the spidering code.
The analysis compares only two different network models, first a pure preferential attachment model ; second a PFP model with for the new nodes connections and for the inner edge connections. No combined model was found which improved over the PFP model.
Model component Model 1 New node 379,000 -529,000 294 Model 1 Inner edge 1,600,000 -479,000 9.83 Model 1 Overall 1,970,000 -1,010,000 27.9 Model 2 New node 352,000 -555,000 389 Model 2 Inner edge 1,590,000 -481,000 9.93 Model 2 Overall 1,945,733 -1,040,000 30.7 Table \thetable: Two models tested on the Flickr network.
Several conclusions can be drawn from the model fitting process. For most models considered in this section, providing different model components for the inner new node model and the inner edge model yields improved models. Thus, the separation of the inner model into a sub-model for connections to new nodes and a sub-model for new internal edges between existing nodes is usually productive.
Moreover, in all but one case (the Flickr data set) it was found that inner models with a higher likelihood could be obtained from a linear combination of model components. Thus, the ability to produce optimised models through the linear combination of sub-models is of use in finding improved network evolution models.
Models based upon PFP generally had high likelihoods (but this was the only parametrised model component tried, so this might be simply an issue of increased flexibility in the fitting procedure). The model parameters selected for the two different AS networks were encouragingly similar, pointing to common network evolution dynamics, but had significant differences consistent with the way their measurement characteristics.
The models explored in the the previous section have been generated purely by fitting proposed models so that their parameters best predicted the actual network growth process observed. Thus, the models were created without growing test networks, measuring statistics on them and further refining them – indeed, the models were identified without measuring any statistics about the real network.
However, it is natural to expect that if the null deviance and per choice likelihood ratio predict that model is “better” than model , this will be reflected in model growing artificial networks with a better match to the statistics of the real network than model . Here, therefore, artificial networks are generated from the seed (In the the case of the AS networks is the state of the network shortly after measurements started, while in the three remaining cases this is simply a single edge). Each of the models from the previous section and the random model are used to grow a network of the same size as the full real network, and summary statistics are compared.
The results in this section need careful interpretation. In particular, it should be remembered that the claim is not that these models are the best possible fit to the real network – in some cases, the claim is that the models tried are actually worse than simply selecting nodes at random. The fitting procedure in section Evaluating and Optimising Models of Network Growth optimises the mixture of model components (the parameters), while the evaluation procedure can optimise other model parameters (such as the PFP ) using any state space search technique. However, the models themselves need to be provided as an input, and it may be the case that no perfect model is to be found from the model components chosen.
However, independently of the precise mathematical description of the network growth models under test, a model with should be better than a random model, and the model with the highest should be the best. This is difficult to achieve using statistical network measures: saying that one model reproduces real network statistics “better” that other model is, in itself problematic. If a model scores well on three highly related statistics but extremely badly on two others, is it a good model? The “basket of statistics” does not always give an unambiguous answer to as to which model is “best”.
For this section, four statistics related to the degree distribution are used: , the maximum node degree in the network, , the proportion of nodes with degree one, , the proportion of nodes of degree 2 and , the mean square of the node degrees ( is a property of the outer model and automatically equal to that of the real network in all models here).
In addition, two further statistics are used capturing the interaction between pairs and triples of nodes. The clustering coefficient of a node is the number of 3-cycles that the node belongs to, divided by the potential number of 3-cycles between its neighbouring nodes (Obviously, nodes of degree one do not have any potential triangles and the clustering coefficient is not defined for them). In the tables in this section is the mean clustering coefficient for the graph. The assortativity coefficient is positive when nodes attach to nodes of like degree (high degree nodes attach to each other) and negative when high degree nodes tend to attach to low degree nodes. For full definitions of all these quantities see [?].
Model d=1 d=2 Real 0.314 0.237 31.3 127 0.00557 0.145 Rand 0.233 0.223 23.9 24 0.245 0.00285 1 0.483 0.215 123.5 446 -0.060 0.0154 2 0.431 0.204 39.1 97 0.152 0.00901 3 0.348 0.258 33.6 75 0.179 0.00748 Table \thetable: Summary statistics arXiv co-authorship network.
The summary statistics for the arXiv publication network are in table Evaluating and Optimising Models of Network Growth. The previous modelling in table Evaluating and Optimising Models of Network Growth rated model 1 with , model 2 with and model 3 with . If these figures are reliable, model 3 should be expected to be a better fit than model 2 which is in turn a better fit than model 1. This is certainly bourne out for the degree distribution statistics, with model 3 closest for , and and marginally worse than model 2 for . All three models generate networks which replicate badly, with model 1 being the closest. With respect to assortativity, model 1 is closest in absolute terms but it predicts a disassortative network when the actual network is assortative.
While these results are not straight-forward to interpret, the overall picture seems to confirm that model 3 reproduces the statistics of the network better than model 2, which in turn beats model 1. The relatively low value means that the models should not be a dramatic improvement upon the random baseline, and this is bourne out by the statistics (indeed, for model 1 it is arguable whether the model is even better than random).
Model d=1 d=2 Real 0.122 0.245 6,620 3,150 -0.197 0.0584 Rand 0.129 0.118 210 1,199 -0.00962 0.0173 1 0.447 0.163 2,230 4,177 -0.144 0.0190 2 0.451 0.167 3,230 5,305 -0.168 0.0148 3 0.363 0.215 3,820 6,109 -0.172 0.0121 Table \thetable: Summary statistics, UCLA AS network.
As detailed in section Evaluating and Optimising Models of Network Growth, evolution information was not known for the early part of the the UCLA AS network growth. Therefore, the first 42,000 edges were taken from the original network, and its evolution followed from this. The statistics from table Evaluating and Optimising Models of Network Growth gave for model 1 and model 2, but for model 3. This implies that model 3 should be a modest improvement on models 1 and 2. This is bourne out by the statistics in table Evaluating and Optimising Models of Network Growth for , and , but for model 3 performs the worst and is incorrect by some way. With , model 3 is again the best and relatively close to the correct value. Regarding the clustering coefficient , model 3 is best but all models are quite far away from the correct value. As predicted, model 1 and model 2 are hard to distinguish using these statistics. Overall, model 3 was best or close to best in almost all statistics measured, as the value predicts. All models would be expected to be a good improvement on the random model and this is shown in all statistics except which random gets nearly exactly.
Model d=1 d=2 Real 0.203 0.363 2,110 3,294 -0.186 0.00887 Rand 0.093 0.118 630 2,289 -0.0710 0.00266 1 0.342 0.185 2,130 4,172 -0.154 0.00631 2 0.350 0.187 2,520 4,637 -0.165 0.00590 3 0.118 0.358 2,610 4,844 -0.163 0.00443 Table \thetable: Summary statistics RouteViews AS network.
From table Evaluating and Optimising Models of Network Growth, it would be expected that model 1 () would be the same as or very slightly worse than model 2 () and model 3 () would be slightly better than either. It is worth noticing that the ratio of these figures is small and the expected improvement from 1 to 3 is slight. This hierarchy is bourne out by the statistics for nodes of degree 1 and degree 2 with model 3 being considerably better in both cases. For and , however, the expectation is reversed and for these statistics, model 1 is better. Models 2 and 3 are close to each other and the correct value for but for model 1 is better than either. In the end it is hard to say from these statistics which model is the best. The high values of do unambiguously claim that all models are superior to random by some way and this is certainly the case. The random model is the worst model for all statistics except for .
Model d=1 d=2 Real 0.0132 0.473 26.3 214 0.144 0.0829 Rand 0.217 0.117 210 30 0.283 0.000809 1 0.447 0.235 369 1,442 -0.065 0.00689 2 0.279 0.205 38.0 277 0.160 0.00992 3 0.0924 0.453 51.1 354 0.0708 0.00537 Table \thetable: Summary statistics gallery user browsing network.
The gallery likelihood table Evaluating and Optimising Models of Network Growth shows that for model 1, (worse than random), for model 2 and for model 3 . This means that, in the statistics in table Evaluating and Optimising Models of Network Growth, model 3 should outperform model 2, which itself should outperform model 1. This expectation is largely bourne out by the degree statistics, with model 1 very inaccurate for all statistics based on node degree. However, in this case, it is hard to see the very clear distinction between model 2 and model 3 which is predicted by the values. Model 3 is certainly better at predicting the number of nodes of degree one and two and does quite well with and . However, it remains hard to claim that model 3 represents the clear improvement in model accuracy that the statistic would lead us to expect.
We have a case where model 1 is expected to be worse than random and model 2 not much better. This certainly seems to match the statistics provided. The relatively poor performance of model 3 remains an anomaly of this data set.
Model d=1 d=2 Real 0.639 0.157 7,500 11,053 -0.288 0.00196 Rand 0.245 0.179 32.4 35 0.341 0.000758 1 0.560 0.172 694 1,704 -0.119 0.0216 2 0.572 0.168 1,290 3,587 -0.154 0.0107 Table \thetable: Summary statistics Flickr spider network.
As detailed in section Evaluating and Optimising Models of Network Growth, only two models were tried for the Flickr data set. Table Evaluating and Optimising Models of Network Growth gives extremely high values for both models, with model 2 being slightly better than model 1. This is definitely reflected in the statistics in table Evaluating and Optimising Models of Network Growth with model 2 being closer to the real data all statistics. The models are quite close on many statistics, but fail to predict the extremely highly connected node with degree 11,053. This may be an artifact of the browsing pattern of the spider, which may also be reflected in the value being incorrect. The high values indicate that both models should be considerably better than the random model, which is certainly true – the random model is extremely bad. These results point towards considerable structure to the network network evolution which the random model fails to capture.
None of the models tested here were perfect at reproducing the selected statistics of their respective network data sets. In the majority of cases, the best fitting models reproduced the degree distribution related metrics measured here, but finding the maximum degree was often difficult. However, one thing the modelling in this section certainly shows is the difficulty of distinguishing between models by considering a large number of, often correlated, statistics.
Obviously the models tested here could be improved. However, the network statistics measured did rank the networks in the same order as the statistics from section Evaluating and Optimising Models of Network Growth (the exception being model 3 in the gallery data which, while arguably the best model, was not better by the expected degree). This is an important confirmation of the usefulness of the likelihood statistic in assessing the fit of network evolution models. The gallery data definitely proved an exception to expectations, and this is perhaps due to the transient nature of this network as discussed in section Evaluating and Optimising Models of Network Growth.
A general conclusion of this section on the models themselves was that (apart from the gallery data), as might be expected, the PFP based models outperformed the degree based model, and the “tweaked” models from the fitting process in section Evaluating and Optimising Models of Network Growth (where better models were found) did better still in most cases.
For a given network and a given model the value of did seem (with one exception) to be an accurate predictor of how well a model would replicate the statistics of a target network. However, it is hard to see a connection between the magnitude of between networks and the success in prediction. For example, the predictions on the arXiv network seemed very good for model 3 for most statistics despite the model only having a value of 1.21.
In general, though, relative likelihood statistics for the different models was reflected in the performance at reproducing representative network statistics. Those models with higher likelihoods (lower deviances) better reproduced the statistics of the target network. This confirms the usefulness of the framework for automatic model selection.
In this paper we present FETA, the Framework for Evolving Topology Analysis. The most important contribution of FETA is a statistically rigorous and unambiguous likelihood estimate for a model of network evolution that is quick to compute and does not require the generation of test networks for its operation. The method requires a target network for which the order in which links are added is known (at least approximately) for a given period of time. Given this data, a model which purports to explain the evolution can be compared either with a second model or with the null (random) model as an explanatory model for the link and node arrivals observed in the target network. The likelihood statistics can be efficiently calculated and could be used, for example, as a fitness function for a genetic algorithm or in state space exploration for parametrised models.
A second contribution is a fitting procedure which allows the weightings of linear combinations of models to be tuned automatically to fit the target network. This is an exploratory tool and, in addition to providing the weightings which best combine the models chosen, can guide the user as to which other model components might appropriate for the target network.
Five different networks were tested, and several models were produced for each. Artificial networks were grown for each model, and for each one of these a set of summary statistics were compared against measures taken from the real target network. Models with better likelihood estimators were found to have better agreement with the statistics of the target network. This confirms that greater accuracy in terms of the likelihood estimator corresponds with a closer match to the final target network generated.
A great deal of potential future work arises from this paper. The outer model (the part of the model which selects whether to add a node or an internal edge) was not investigated in any depth. It would be useful to consider the validation and tuning of more sophisticated outer models, and which also allowed node and edge deletion.
A model form which has more promise than the linear combination of model components proposed in section Evaluating and Optimising Models of Network Growth would be one with multiplicatively combined model components (that is a model of the form ). Logistic regression would seem a promising framework for this, but nontrivial problems exist with normalisation. The evaluation framework from section Evaluating and Optimising Models of Network Growth would, however, work unchanged with this type of model.
In short, the FETA framework is promising for development in many ways. The evaluation framework fits a broad class of models of network evolution and could be a very useful tool for researchers wishing to test hypotheses. The tools and data used in this paper are freely available for download and researchers are encouraged to try them888http://www.richardclegg.org/software/FETA.
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