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it is vital to obtain a good description of a network topology because structure fundamentally affects function  @xcite . the internet contains millions of routers , which are grouped into thousands of subnetworks , called _ autonomous systems _ ( as ) , which are then glued into a global network by the border gateway protocol . effective engineering of the internet is predicated on a detailed understanding of issues such as the large - scale structure of its underlying physical topology , the manner in which it evolves over time , and the way in which its constituent components contribute to its overall function  @xcite . the recently introduced positive - feedback preference ( pfp ) model  @xcite is an internet as - level topology generator . the model uses two evolution mechanisms , namely the interactive growth and the positive - feedback preference . both mechanisms are inspired by , and coincident with , practical observations on the internet historic data . originally the phenomenological model was designed to match only the internet s rich - club connectivity  @xcite and the exact form of degree distribution , including the distribution of low degrees , the maximum degree and the degree distribution s power - law exponent  @xcite . it has been a pleasant surprise that numerical evaluation against the internet measurement date has shown that the model accurately reproduces a large set of other nontrivial topology characteristics as well , including the disassortative mixing  @xcite , the shortest path length  @xcite , short cycles  @xcite and the betweenness centrality  @xcite . the pfp model has been used in more realistic simulations of the internet , e.g.  @xcite .    as of this writing , an analytical solution of the pfp model is not available yet . in this paper we use the numerical method to analyze why and how the model is able to reproduce a fuller picture of the internet than it was designed for . in other words , we aim to investigate the exact origin of each of the topology properties captured by the pfp model . answers to these questions can be valuable for the ongoing effort on a mathematical solution of the model .    in section ii , we review the pfp model and its two mechanisms . we reflect on the intuitions underlying the design of the two evolution mechanisms . in section iii , we comparatively examine two limiting cases of the pfp model . based on detailed numerical simulations , we identify the explicit evolution dynamics that are responsible for generating the rich - club phenomenon and the degree distribution properties . we also reveal that the origin of the disassortative mixing is in fact already embedded in the model s two evolution mechanisms . furthermore we explain that the pfp model resembles the internet s shortest path length and short cycles because these two characteristics are correlated with other topology properties .    in section iv , the above results become more evident when we examine how the pfp model s topology properties react to the change of the parameters that control the model s evolution mechanisms . our investigation leads to a number of insightful discoveries . for example we find out that the rich - club connectivity is almost exclusively determined by the interactive growth mechanism . we also show that the interactive growth and the positive - feedback preference jointly contribute to the model s disassortative mixing behavior , however the two mechanisms have opposite effects on the degree distribution s power - law exponent .    in section v we conclude that this work represents a fuller and deeper understanding of the internet topology evolution dynamics . this work complements the research on evolution mechanisms and structural constrains of complex networks in general . in graph theory , degree ,  @xmath0 , is defined as the number of links a node has . degree is a local property but the distribution of degree provides a global view of a network structure . the degree distribution of the internet as - level topology is characterized by a power - law  @xcite as @xmath1 a power - law degree distribution means the majority of nodes that form a network have very few links . some nodes , however , are very well connected . barabsi and albert ( ba )  @xcite showed that a power - law degree distribution can arise from two generic mechanisms : _ growth _ , where a network `` grows '' from a small random network by attaching new nodes to @xmath2 old nodes in the existing system ( see fig.[fig : growth]a , in which @xmath3 ) ; and _ preferential attachment _ , where a new node is attached preferentially to nodes that are already well connected . the probability that a new node will be connected to a node with degree @xmath0 is given as @xmath4    during the last few years a large number of models have been proposed to resemble and explain the power - law degree distribution and other topology properties observed on various real networks  @xcite . recently zhou and mondragn  @xcite introduced the positive - feedback preference ( pfp ) model . numerical evaluation has shown that the pfp model accurately reproduces a large number of characteristics of the internet as - level topology . the model uses the following two evolution mechanisms . the interactive growth is designed to satisfy a number of observations on internet history data@xcite . firstly the internet evolution is largely due to two processes : the attachment of new nodes to the existing system , and the addition of new internal links between old nodes already present in the existing system . secondly the majority of new nodes appeared on the internet are each attached to no more than two old nodes . and thirdly the ratio of links to nodes on the internet as graph is approximately three . different from the independent growth  @xcite ( see fig.[fig : growth]b ) , in which new nodes and new internal links are added independently , the two evolution processes are interdependent in the interactive growth ( see fig.[fig : growth]c and d ) . that is to say , a new internal link always starts from an old node , we call it a _ host _ node , to whom a new node has just attached . a heuristic explanation of this interaction is that on the internet , new customers ( new nodes ) generate extra demand for service , which triggers their service providers ( host nodes ) to develop new connections to other service providers ( new internal links ) in order to accommodate the increased traffic and improve services . the interactive growth is described as such : starting from a small random graph , at each time step ,    * with probability @xmath5 $ ] , a new node is attached to a _ node , and at the same time two new internal links are added connecting the host node to two other old nodes ( see fig.[fig : growth]c ) ; * with probability @xmath6 , a new node is attached to two host nodes , and only one new internal link is added connecting one of the host nodes to another old node ( see fig.[fig : growth]d ) . numerical simulation shows that when the probability parameter @xmath7 , the interactive growth produces the best result . the pfp model uses the following nonlinear preference probability to choose old nodes for the interactive growth , @xmath8 it is called the positive - feedback preference ( pfp ) because a node s ability of acquiring new links increases as a feed - back loop of the node s degree . when the parameter @xmath9 , it is equivalent to the ba model s linear preference ( see eq.[eq : ba ] ) . numerical simulation shows that the positive - feedback preference produces the best result when the parameter @xmath10   of the positive - feedback preference . accordingly the value of parameter @xmath11 is revised as  @xmath12 . ] . the positive - feedback preference is designed to satisfy the observation  @xcite that during the internet evolution , the probability that a new node links with a low - degree node follows the ba model s linear preference , whereas high - degree nodes have a stronger ability of acquiring new links than predicted by the linear preference . a preference probability can be written as @xmath13 , where @xmath14 is a degree function . the exponential preference  @xcite , which uses the nonlinear degree function of @xmath15 , @xmath16 , also gives high - degree nodes a stronger preference than the linear preference . however as shown in fig.[fig : degreefunctions ] , only the pfp can be approximated by the linear preference for low - degree nodes . when degree increases , the pfp grows more and more rapidly and finally exceeds the exponential preference . .preference probability ratio @xmath17 when @xmath18 and @xmath19 . [ cols="^,^,^,^",options="header " , ]     table  [ table : parameter - delta ] and fig.[fig : parameter - delta ] show how topology properties of the pfp model change when the positive - feedback parameter @xmath11 increases from 0 to 0.035 while the interactive growth parameter @xmath7 . it is clear that the positive - feedback parameter @xmath11 has a fairly limited impact on the obtained rich - club exponent @xmath20 , which almost remains the same . when @xmath11 increases , greater preference is given to high - degree nodes , consequently the maximum degree increases and the network becomes more disassortative mixed . as expected , the characteristic path length decreases steadily when the network s disassortative mixing is getting stronger . when the parameter @xmath11 increases , the degree distribution power - law exponent @xmath21 slightly decreases . this is because when the positive - feedback preference gets stronger , the richest nodes attract so many new links that they suppress the degree growth of other nodes , including those with medium to high degrees . when @xmath10 ( and @xmath7 ) , the pfp model closely matches all the topology properties of the internet as graph .    as shown in fig.[fig : parameter - delta ] , by tuning the parameter @xmath11 from 0 to 0.028 , the pfp model is capable of generating a wide range of disassortative mixing networks with the value of the assortativity coefficient @xmath22 decreasing monotonically from -0.12 to -0.28 , which encompasses most technological and biological networks reported in  @xcite . notably , the pfp model achieves this while keeping the rich - club exponent and the degree distribution exponent largely unchanged . @xmath23 is the model s tipping point , beyond which the network structure changes dramatically , e.g.  the degree distribution exponent @xmath21 decreases rapidly . this is because the preferential selection becomes so biased that super - rich nodes start to emerge . these extremely rich nodes dominate the network growth and make the network grow towards a star - like structure . fig.[fig : parameter - theta][fig : parameter - alpha ] are contour plots showing how three topology properties , i.e.  the rich - club connectivity exponent @xmath20 , the degree distribution exponent @xmath21 and the assortativity coefficient @xmath22 , change when both the parameters are variables . fig.[fig : parameter - theta ] clearly shows that the value of rich - club exponent @xmath20 is sensitive to the interactive growth parameter @xmath24 and is unsensitive to the positive - feedback preference parameter @xmath11 . when parameter @xmath24 grows , the rich - club phenomenon becomes stronger . fig.[fig : parameter - gamma ] shows that the two mechanisms have opposite effects on the value of degree distribution exponent @xmath21 . in general , the exponent @xmath21 increases when parameter @xmath24 increases and parameter @xmath11 decreases . fig.[fig : parameter - alpha ] shows that the pfp model s disassortative mixing becomes stronger when either of the two parameters increases . the assortativity coefficient @xmath22 is more sensitive to parameter @xmath11 than to parameter  @xmath24 . we notice that when @xmath25 , the network s disassortative mixing actually becomes weaker . this is because , when @xmath25 , the network starts to acquire more internal links than external links , i.e.  @xmath26 ( see eq . [ eq : link - ratio ] ) . the increased new internal links among the rich nodes weaken the disassortative mixing behavior of the network . the internet has an extremely complex structure . the pfp model demonstrates a way to simultaneously reproduce many topology properties of the internet . the model achieves this by using two generic algorithms which are designed to replicate the evolution dynamics observed on the internet historic data . in this paper we have used the numerical method to investigate the success of the model .    based on detailed simulation results , we show that the rich - club phenomenon is primarily controlled by the interactive growth mechanism alone . we point out that this is because the rich - club connectivity increases with the number of new internal links which are added between rich nodes . the interactive growth also determines the probability of new nodes initial degrees and thus controls the distribution of low degrees obtained in the generated network . the pfp model s maximum degree is as large as the internet because both the interactive growth and the positive - feedback preference accelerate the degree growth rate . the interactive growth does so by the degree - leap effect , whereas the positive - feedback preference achieves so by giving stronger preference to high - degree nodes . the two mechanisms have opposite effects on the value of power - law exponent of degree distribution . the interactive growth increases the value whereas the positive - feedback preference slightly decreases the value . we have explained that the origin of the disassortative mixing has been , unintentionally , embedded in the pfp model s two evolution mechanisms , which not only enlarge the gap between the high - degree nodes and the low - degree nodes , but more critically , they increase the degree difference between end nodes of a link . our analysis suggests that the reason that the pfp model also captures other properties of the internet , such as shortest path length and triangle coefficient , is that these properties are correlated with other properties , in other words , they are constrained by other . for example the internet is small because the rich - club functions as a super traffic hub , while the disassortative mixing behavior ensures peripheral low - degree nodes always close to the hub .    by investigating the pfp model s sensitivity to the mechanism parameters , we obtain more evident details on how the two evolution mechanisms profoundly control the model s growth dynamics and therefore effectively change the generated topology properties . for example we show that by tuning the positive - feedback parameter @xmath11 , the pfp model is able to generate a wide range of disassortative mixing networks while keeping the rich - club exponent and the degree distribution exponent almost unchanged . this work represents a better understanding of the internet topology evolution dynamics . it facilitates the ongoing effort on a mathematical analysis of the pfp model . this work also complements the research on the evolution dynamics and the structural constrains of complex networks in general .
the internet structure is extremely complex . 
 the positive - feedback preference ( pfp ) model is a recently introduced internet topology generator . 
 the model uses two generic algorithms to replicate the evolution dynamics observed on the internet historic data . 
 the phenomenological model was originally designed to match only two topology properties of the internet , i.e.  the rich - club connectivity and the exact form of degree distribution . whereas numerical evaluation has shown that the pfp model accurately reproduces a large set of other nontrivial characteristics as well . 
 this paper aims to investigate why and how this generative model captures so many diverse properties of the internet . based on comprehensive simulation results , the paper presents a detailed analysis on the exact origin of each of the topology properties produced by the model . 
 this work reveals how network evolution mechanisms control the obtained topology properties and it also provides insights on correlations between various structural characteristics of complex networks .