This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Exploring the Local Connectivity Preference in Internet AS Level Topology Guoqiang Zhang
Guoqing Zhang
Institute of Computing Technology, Chinese Academy of Sciences, China Graduate University of Chinese Academy of Sciences, China Email:
[email protected]
Institute of Computing Technology, Chinese Academy of Sciences, China Email:
[email protected]
Abstract—The Internet AS level topology, upon which BGP4 runs, plays a vital role in the analysis and study of the global routing behavior. However, the study of the topology itself is not as comprehensive and extensive as the protocol. In this paper, the inherent local connectivity preference nature of the Internet AS level topology is studied from several perspectives. The presence of strong local connectivity preference in the AS graph is quantitatively verified along with the demonstration of the weakness for PFP model–which is regarded as the most accurate Internet AS level topology generator to date–to capture it. A locality-driven PFP(LDPFP) model is proposed to capture the local connectivity preference feature of the AS graph. keyword:AS graph, local connectivity preference, PFP model, LDPFP
I. I NTRODUCTION Discovery of power-law [1] in the Internet AS level topology gives rise to the flourish in topology characterization and modeling. A variety of macroscopic topological metrics have been defined to disentangle specific structural properties of the seemingly chaotic graph, among which are degree distribution, average path length, clustering [2], dissortative mixing [3] and rich-club [4]. In order to reproduce the evolution of the Internet as well as provide an accurate model for the network research community, a significant amount of Internet models or generation algorithms have been proposed in the past few years, among which transit-stub [5], Inet [6], AB [7], GLP(Generalized Linear Preference) [8], MLW(multi-localworld) [9] and PFP(Positive-feedback Preference) [10], [11] are a few distinctive ones. A model’s goodness is often evaluated as the number of properties it can reproduce. In this respect, PFP model is regarded as the most accurate AS level topology generator to date for its ability to simultaneously model a couple of characteristics, including degree distribution, maximum degree, average path length, clustering, richclub, dissortative mixing and coreness [12]. The reason why we are interested in the local connectivity feature is two folded. First, the study of a network’s routing, This work is partly supported by the National Natural Science Foundation of China under Grant No.60673168, the Hi-Tech Research and Development Program of China under Grant No.2006AA01Z207 and the ICT innovation fund under Grant No.20066033.
self-organization and invulnerability can all benefit from better understanding of how the network is locally connected. More localized action means higher adaptability to failures and changes in the network [13]. Second, it is useful in differentiating the PFP model from the AS level topology. Although the usage of macroscopic topology properties is powerful for distinguishing different categories of graphs, such as clustering for separating random networks from smallworld networks and mixing for separating social networks from information networks, it does have its limitations even when combinations of macroscopic topological properties are applied. As an example, the PFP model cannot be separated from the AS level topology by any well-known macroscopic properties today. However, the usage of local connectivity features can easily distinguish these two graphs. The rest of the paper is structured as follows. Section II gives a brief introduction to the Internet AS level topology, the PFP model and the notion of local connectivity preference. Section III details the local connectivity preference of the Internet AS level topology from different perspectives and shows the weakness of PFP model to reproduce it. A modified version of the PFP model, which we call LDPFP(LocalityDriven PFP) is set forth in section IV to enhance the PFP model to capture some of the local connectivity features while at the same time keeping the merits of PFP model. Section V summarizes our work and points out the way for future works. II. BACKGROUND A. The Internet AS Level Topology and ITDK0304 Like any other complex networks, the Internet AS level topology can be abstracted as a graph G(V, E), wherein vertexes are the collection of all operating autonomous systems(AS) and edges are peering relationships between ASes. For simplicity, we assume routes are reversible(This assumption may not hold in the real network, as the connections between ASes can fall into different commercial relationships [14]), hence, the AS level topology can be represented as a simple undirected graph. Two state-of-the-art methodologies are available for inferring the AS level topology : one is the RouteViews-like passive measurement [15], which analyzes BGP [16] tables and BGP updates accumulated from dozens of BGP border gateways to
1-4244-0353-7/07/$25.00 ©2007 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
B. PFP Model PFP model was first proposed in 2004 [10], and underwent a modification in 2006 [11]. It is a nonlinear model, and it treats the network as an interactive growth process. The two basic ingredients of this model, namely the positive-feedback preference and interactive growth, are described below: 1) Positive-feedback preference attachment The preference probability pk that a node with degree dk acquires a new link is: d1+δ ln dk pk = k 1+δ ln di i di
(1)
2) Interactive growth The growth of network is not simply the addition of nodes, but the interplay between addition of new nodes and addition of new internal edges. Specifically, at each time step: a) With probability p, add a new node N , attach N to a host node H, and generate two internal edges originating from H. b) With probability 1 − p, add a new node N , attach N to two host nodes H1 and H2 , and generate an internal edge originating from either H1 or H2 . In the above two steps, the selection probability for the endpoint of an edge all follow (1). C. Local Connectivity Preference The local connectivity preference can be described in several aspects. Possibly the mostly studied concept is the clustering properties, which is often quantitatively measured by two macroscopic metrics [2] : mean clustering(C) and clustering coefficient(C). The mean clustering metric is defined as: C= P (k)C(k) (2) k
Fig. 1: Inconsistency between C and C. In this graph, 277 C a = 364 459 = 0.793 > C b = 495 = 0.560, but 12 33 Ca = 53 = 0.226 < Cb = 71 = 0.465 is the degree-dependant where C(k) = mnn (k)/ k(k−1) 2 local clustering coefficient for nodes with degree k and mnn (k) is the average number of connections between direct neighbors of a node with degree k. The clustering coefficient is defined as: P (k)mnn (k) k(k − 1)P (k)C(k) k = k (3) C= P (k)k(k − 1)/2 k 2 − k k where k 2 is the second moment of k, and k is the mean of k. This clustering coefficient can be expressed as three times the ratio of the total number of triangles to the total number of connected vertex triples in a graph. It is more desirable to use clustering coefficient(C) than mean clustering(C) to measure a network’s macroscopic clustering feature since mean clustering simply averages over all degree-dependant local clustering, without taking into account the weights different degrees place on the network. Practically, as Fig 1 illustrates, these two measurements can easily show inconsistency. While macroscopic clustering measurements can give a rudimentary impression on how the network is clustered, they cannot reflect how the network is locally clustered. In fact, the PFP model can produce nearly equivalent clustering coefficient as the ITDK0304, but as we will show later, it behaves substantially different from the ITDK0304 when the local actions are taken into account. The local clustering is often measured by the degreedependant clustering C(k). Fig 2 presents the degreedependant clustering for ITDK0304 and PFP, clearly illus0.8
ITDK0304 PFP
0.7 0.6 0.5 C(k)
infer the AS connectivity; the other is the active measurement, which deploys multiple probes, actively probes a large set of intelligently selected IP addresses(possibly millions), collects the traceroute results and maps the router level graph to AS level graph. A widely known project of the active measurement is the Skitter project [17] initiated by the CAIDA, which deploys dozens of dedicated servers for the purpose of probing. Another project worth mention is the DIMES project [18], which takes advantage of the power of P2P for probing so that any user can volunteer to contribute to the project by simply downloading and running the DIMES client software. The passive measurement is easy to carry out, but only reflects the control plane topology. It is uncertain whether real traffic actually travels across the links. Furthermore, it is impossible to expose those unpublished private links by solely consulting the BGP information. The active measurement method, on the other hand, is more difficult to implement, but reveals the data plane topology. In this paper, we adopt the data kit ITDK0304 as the real Internet AS level topology, which is the most recent data kit published by the CAIDA’s skitter project.
0.4 0.3 0.2 0.1 0 1
10
100
1000
k
Fig. 2: C(k) for ITDK0304 and PFP
10000
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
trating the remarkable disagreement between PFP model and ITDK0304 for C(k) with small degrees. However, although local clustering can capture some local connectivity preference, it cannot tell the full story. The evolution of the Internet has its unique characteristic, where locality plays a vital role. III. L OCAL C ONNECTIVITY P REFERENCE OF THE I NTERNET AS L EVEL T OPOLOGY A. Six Regional Subgraphs The local connectivity preference feature of the Internet AS level topology, i.e, nodes in the Internet AS level topology tend to connect to locally nearby nodes, has been mentioned in previous studies, such as MLW model [9]. However, to the best of our knowledge, these discussions are all based on heuristics, and no statistics are ever presented to justify this assumption. To validate this assumption, we have extracted six subgraphs from the ITDK0304 delimited by the regional network information center, which we call ARIN, RIPENCC, APNIC, LACNIC,AFRINIC and CNNIC subgraph respectively. ARIN, RIPENCC, APNIC, LACNIC, AFRINIC are five top global Regional Information Registries(RIR) and CNNIC, the network information center of China, is under adminstration of APNIC. These subgraphs each consists of only those AS numbers allocated by the corresponding RIR. The size of each subgraph is shown in Table I B. Clustering Coefficient of the Six Subgraphs As a first attempt to show the presence of local connectivity preference, Table I lists the number of nodes, number of edges and clustering coefficient for each subgraph. It can be clearly seen from this table that as the size of the region increases, the clustering coefficient decreases accordingly, which implicitly indicates strong locality induced local connectivity preference of the graph. C. Local Connections The AS graph’s local connectivity preference can be showed by a node’s high preference to connect to nodes in the same subgraph. The degree-dependant ratio of average number of neighbors within a regional subgraph to average number of neighbors in the whole graph is calculated for all subgraphs to illustrate the locality connectivity preference. Fig 3 presents this relationship for the six subgraphs. It is evident that most of the nodes contain a considerable fraction of neighbors within their respective subgraphs, indicating the strong inclination of ASes to connect to nearby in-region ASes. However, a close look at the diagrams also shows that the ARIN subgraph behaves differently from the other five subgraphs in that the fraction of in-region connections of ARIN for median sized nodes(nodes with degree around 10) is much higher than that of other subgraphs. For nodes with degree around 10 in the ARIN, approximately 90% of the neighbors are in the ARIN itself, but for the other five subgraphs, this figure drops down to about 40% ∼ 70%.
Two possible causes can account for this phenomenon. The first possible cause may be the incompleteness of the ITDK0304 data kit itself. Due to the fact that most of the probe sites are located in USA, the likelihood for detecting those cross links between small and median sized ASes outside the ARIN is smaller compared with those inside. The second possible explanation for it can be described as follows: the ARIN serves as the core of the AS graph so that ASes– especially those median sized ones–in other regions tend to connect to a non-trivial portion of nodes in the ARIN core to acquire a relatively short average transmission path length. In order to gain a deep understanding of the locality connectivity preference in node granularity, we define a node’s dG (v) connectivity τG (v) in graph G as τG (v) = |V (G)| . If a node takes on any locality connectivity preference for a specific regional subgraph Gi , it should have τGi (v) > τG (v), where Gi is the regional subgraph to which v belongs and G is the ITDK0304 graph. We have iterated through the six subgraphs and found only 81 nodes violate this criterion, accounting for less than 1% of the total nodes. The possible reason why the violations occur is also inspected. The allocation RIR of an AS number doesn’t require the AS lie geographically in that region, for example, the AS with ASN=5669 is allocated by ARIN, but the whois database reveals that the AS name is VIA-NET-WORKS-AS PSINet Europe / VIA NET.WORKS international AS, which is physically located in Europe. Therefore, it is not surprised to find that only 9 out of its 46 connections are in ARIN, but 33 connections are with nodes in the RIPENCC subgraph, indicating its high connection preference with nodes in RIPENCC. Based on this observation, we have devised an easy yet efficient algorithm to detect this kind of ASes, which are allocated by one RIR, but operate in different regions. The algorithm is described below: 1) Perform the calculation for the violation set T . A node v ∈ Gi belongs to T if and only if it violates τGi (v) > τG (v). 2) For any node v ∈ T (v ∈ Gi ), if there exists a Gj (j = i),s.t,dGj (v) > dGi (v), put v in the result set. Among the 81 violations, we find that for 69 of them, there is a Gj , s.t, dGj (v) > dGi . The whois database is consulted to find out which one is really due to the inconsistency between the allocation RIR and its operation region. Table II lists such ASes, including the AS number, its allocation RIR, its connections with the allocation RIR subgraph, its operating region, and its connections with its operating regional subgraph. It turns out that most violations found in RIPENCC can not be ascribed to this reason. Though they satisfy the above two conditions, they are still physically located in Europe or Mideast. These nodes may be viewed as exceptional cases which violate the locality connectivity preference, but the number is so small compared with the total graph size that it can be neglected. Meanwhile, there is a potential possibility that this exception is not exception at all, but originates from the incompleteness of the ITDK0304 data kit, i.e, the loss of
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Number of Node
Number of Edges
Clustering Coefficient
CNNIC AFRINIC LACNIC APNIC RIPENCC ARIN
71 29 313 1184 3065 4101
160 37 457 2589 6692 10741
0.225 0.121 0.089 0.075 0.051 0.022
ITDK0304 PFP
9204 9204
28959 27612
0.026 0.027
ARIN
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1
10
100
1000
10000
k
(a) arin 1
LACNIC
0.9 0.8 0.7 0.6 0.5 0.4 0.3 1
10
100
k
(d) lacnic
1
RIPENCC
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 1
10
100
1000
k
(b) ripencc 1
CNNIC
0.9 0.8 0.7 0.6 0.5 0.4 0.3 1
10 k
(e) cnnic
100
fraction of connections in the APNIC subgraph fraction of connections in the AFRINIC subgraph
1
Graph Name
fraction of connections in the CNNIC subgraph fraction of connections in the RIPENCC subgraph
fraction of connections in the LACNIC subgraph fraction of connections in the ARIN subgraph
TABLE I: Clustering coefficient for PFP model, ITDK0304 and its subgraphs.
1
AFRINIC
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1
10
100
k
(c) afrinic 1
APNIC
0.9 0.8 0.7 0.6 0.5 0.4 1
10
100
1000
k
(f) apnic
Fig. 3: Percentage of neighbors within the same regional subgraph for six subgraphs
cross links between the RIPENCC ASes.
between low degree nodes than the PFP model.
D. Triangle Distribution
E. Local Connectivity Redundancies
Triangle is a specific kind of correlation, which leads to the clustering of networks. However, unlike the clustering coefficient, the correlation between triangle and degree is not well studied, especially how triangles are distributed across the network. What interests us most is whether a reasonable amount of triangles exist among low degree nodes, implying the local clustering for low degree nodes. In order to quantify this relationship, we define a k-subgraph Gk of G as: ∀v ∈ V (Gk ), dG (v) < k. In simple term, the k-subgraph of G is a subgraph of G consisting and only consisting of nodes with degrees less than k. The number of nodes,number of edges and number of triangles of Gk for ITDK0304 and PFP as a function of k are shown in Fig 4, from which we find that despite PFP model’s success in modeling the number of nodes and edges for the k-subgraph, it cannot capture the number of triangles for the k-subgraph(note that the y − axis for Fig 4(c) is in log scale). The ITDK0304 contains orders of magnitude more triangles between low degree nodes than the PFP model, indicating its higher local connectivity preference
The degree of redundancy for locally connected component is an effective measure of the network’s local robustness and routing flexibility. The redundant links can enrich the routing selection choices, resulting in more alternative paths available within a certain vicinity without troubling those hub nodes. This subsection gives an insight into the local redundancy nature of the Internet AS Graph. We order the nodes by their degrees, and remove the nodes one by one(the sequence is from high to low) until no giant component exists. We define a giant component as satisfying the following three conditions: 1) It is the largest connected component of the graph. 2 2) It should contain at least N 3 nodes of the original graph. 3) for any other connected component Gj (Gj = Gmax ), |V (Gmax )|/|V (Gj )| ≥ 5. The number of nodes and edges of the largest 10 connected components produced by ITDK0304 and PFP model in this manner are listed in Table III. It can be clearly seen that the connectivity of each connected component derived
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Total Degree
Allocation RIR
#Connections with Allocation RIR
6461
723
ARIN
273
RIPE
319
6453
252
ARIN
66
RIPE
105
Operating Regional RIR
#Connections with Operational Regional RIR
702
457
ARIN
51
RIPE
328
3491
210
ARIN
49
RIPE
110
703
126
ARIN
22
APNIC
83
6140
62
ARIN
16
LACNIC
33
22351
61
ARIN
15
RIPE
24
2686
55
ARIN
13
RIPE
29
3643
32
ARIN
10
APNIC
17
5669
46
ARIN
9
RIPE
33
26
ARIN
3
RIPE
17
8
ARIN
2
RIPE
6
10282
4
ARIN
1
RIPE
2
4004
6
ARIN
1
RIPE
4
9680
10
APNIC
1
ARIN
8
9129
8
RIPE
2
ARIN
6
9194
4
RIPE
1
ARIN
3
9500
ITDK0304 PFP
9000
8500
8000
7500
7000 1
10
100
1000
60000
ITDK0304 PFP
50000 40000 30000 20000 10000
10000
0 1
10
k
(a) nodes
100 k
(b) edges
1000
10000
number of triangles in k-subgraph
8190 13646
number of edges in k-subgraph
number of nodes in k-subgraph
TABLE II: ASes which are allocated by one RIR, but operate in different regions ASN
100000
ITDK0304 PFP
10000
1000
100
10
1 1
10
100
1000
10000
k
(c) triangles
Fig. 4: Comparison of number of nodes, edges and triangles between ITDK0304 and PFP
from ITDK0304 is much higher than PFP. A fair amount of redundant links exist in the connected components derived from ITDK0304, however, most of the connected components derived from PFP are trees. On the other hand, only 3.9% of nodes need to be pruned for the disappearance of giant component, whereas 5.7% high degree nodes need to be pruned to achieve the analogous outcome. It naturally leads to the conclusion that as a whole, the PFP model shows higher connectivity than ITDK0304, however, ITDK0304 exhibits much higher local connectivity. IV. L OCALITY-D RIVEN PFP M ODEL Inspired by the above observations, we propose a localitydriven PFP model(LDPFP), which inherits some elements of the PFP model. However, as the name implies, it also incorporates the locality information. Besides, the LDPFP model is also based on the observation that Internet AS level topology has a three-tier structure. Therefore, LDPFP can reproduce some of the local connectivity preference properties while keeping the merits of the PFP model. LDPFP is implemented within the Brite [19] framework so that it can leverage the basic utilities provided by Brite. Two
operations are at the heart of this model, namely, placement of nodes and √growth √ of network. Nodes are to be randomly placed in a 3 N × 3 N grid prior to the construction of a graph with size N . After the node placement, each node is then assigned a coordinate (x, y) in the grid. Like all other growth-based models, the network starts with a small random network, and in each time step: • with probability p, add a new node S, attach it to one host node H, and generate an internal edge originating from H. The probability pH that H is chosen follows: d1+δln dH /d(S, H)α pH = H 1+δln dV /d(S, V )α V dV
(4)
where d(S, H) denotes the Euclidean distance between node S and H. This means for two nodes with same physical distance, the one with higher degree has higher attraction, on the other hand, for two nodes with equal degrees, the closer one is more preferrable. This implication corresponds to people’s normal way of thinking. The probability that the endpoint of the internal edge is to be chosen follows (1).
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
TABLE III: Top 10 connected components after removing certain number of high degree nodes for ITDK0304 and PFP ITDK0304
PFP
rank
#node
#edge
k
#node
#edge
k
(1)
772
907
2.35
604
606
2.01
(2)
394
424
2.15
330
329
2.0
(3)
195
215
2.21
99
98
2.0
(4)
163
185
2.27
84
83
2.0
(5)
68
76
2.24
64
63
2.0
(6)
62
76
2.45
50
49
2.0
(7)
61
61
2.0
48
47
2.0
(8)
54
63
2.33
46
45
2.0
(9)
49
51
2.08
46
45
2.0
(10)
42
43
2.05
42
41
2.0
with probability q, add a new node and attach it to two host nodes H1 and H2 . The probability for H1 and H2 to be chosen follows (4). • with probability r, add a global scope internal edge. The probability that each endpoint of the global internal edge is to be chosen follows (1). • with probability 1 − p − q − r, add a local scope internal edge, which is described as follows: Randomly select a node V. If dV = 1, add a connection between V and one of its neighbor’s neighbors(i.e, nodes which are two hops away from V ), otherwise(dV > 1), add an edge between its neighbors. However, the node selection function is somewhat different from the previously mentioned ones. It favors those median sized nodes rather than high degree nodes. Suppose the median degree at the time of this step is m, then the probability for a node to be chosen is inversely proportional to |(dv − m)| + 1. This step is important for the formation of locality cluster, since the connections to be added are not at random, but between a node’s neighborhood. Numerous simulations show that when p = 0.15, q = 0.33, r = 0.25, α = 0.6, δ = 0.021, the resulting graph most accurately fits the AS graph. Fig 5 shows the cumulative degree-distribution, rich-club coefficient, degree-dependant clustering and the number of triangles for LDPFP and ITDK0304. Table IV gives the primary macroscopic properties of LDPFP and Table V presents the top 10 connected components produced in the way discussed in section III-E for LDPFP. These graphs show that LDPFP model maintains the merits of PFP model and captures some local connectivity features as well, including the degreedependant clustering, triangle distribution and local connectivity redundancy. •
TABLE IV: Major properties for ITDK0304 and LDPFP Number of Node Number of Edge Average Degree Characteristic Path Length Exponent of Power-law Clustering Coefficient Dissortative Mixing
N M k l γ C r
ITDK0304
LDPFP
9204 28959 6.3 3.11 2.22 0.026 -0.24
9204 28153 6.1 3.08 2.21 0.029 -0.27
view. The study shows that the clustering coefficient varies according to the regional subgraph size. The smaller the regional subgraph, the higher the clustering coefficient. Nearly all nodes show high preference to connect to nodes within its regional subgraph with only a few exceptions. A deep examination of the exceptions uncovers why they take place. Some exceptions result from the fact that some ASes are allocated by one RIR, but operate in different regions. We have contacted the whois database and listed such ASes. Other exceptions may be real exceptions or due to the incompleteness of the ITDK0304 data kit. The PFP model, although accurately captures nearly all macroscopic topological properties of the Internet AS graph ITDK0304, fails to capture the AS graph’s local connectivity behavior, including the degree-dependant local clustering, the number of triangles between low degree nodes and the degree of redundancy for small connected components. We propose a locality-driven PFP model(LDPFP), which captures some local connectivity features of the AS graph while still maintaining the merits of the PFP model. Looking into the future, more complete and accurate data kit is required to explain the discrepancy between the local connectivity preference patterns displayed by ARIN and the other five subgraphs. Whether the discrepancy arises from incompleteness of the data kit or is due to the core role that ARIN serves in AS graph can only be clarified then. The failure of PFP model to capture the local connectivity behavior also calls for more efforts to be devoted to the designing of an accurate Internet AS level topology generator. The LDPFP TABLE V: Top 10 connected components after removing certain number of high degree nodes for ITDK0304 and LDPFP ITDK0304
LDPFP
rank
#node
#edge
k
#node
#edge
k
(1)
772
907
2.35
265
286
2.16
(2)
394
424
2.15
113
127
2.28
(3)
195
215
2.21
73
75
2.05
(4)
163
185
2.27
63
68
2.16
V. C ONCLUSION
(5)
68
76
2.24
56
61
2.18
It is widely believed that real Internet AS graph exhibits strong local connectivity preference, i.e, node tends to connect to those nodes nearby or under the same administrative authority. After detailed analysis of the six subgraphs obtained from the CAIDA’s ITDK0304, we have corroborated this point of
(6)
62
76
2.45
52
55
2.12
(7)
61
61
2.0
52
54
2.08
(8)
54
63
2.33
51
55
2.16
(9)
49
51
2.08
46
47
2.04
(10)
42
43
2.05
42
43
2.05
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
1 Rich-club connectivity
ITDK0304 LDPFP
P(x>=k)
0.1
0.01
0.001
0.0001 1
10
100
1000
0.6 0.5
0.01
0.4 0.3
0.001
0.2 0.1 0 0.01
k
0.1
1
r/N
(a) cumulative degree-distribution 100000
1
10
100
1000
10000
k
(b) rich-club number of triangles in k-subgraph
ITDK0304 LDPFP
0.7
0.1
0.0001 0.001
10000
0.8
ITDK0304 LDPFP
C(k)
1
(c) degree-dependant clustering
ITDK0304 LDPFP
10000
1000
100
10
1 1
10
100
1000
10000
k
(d) triangles
Fig. 5: Cumulative degree distribution, rich-club, degree-dependant clustering and triangle distribution for LDPFP and ITDK0304 model proposed in this paper can reproduce some of the local connectivity features, but not all. There is still a long way to go to find a satisfactory model. R EFERENCES [1] M. Faloutsos, P. Faloutsos, and C. Faloutsos, “On power-law relationships of the internet topology,” ACM SIGCOMM Computer Communication Review, vol. 29, no. 4, pp. 251–262, 1999. [2] S.N.Dorogovtsev, “Clustering of correlated networks,” Physical Review E, vol. 69, no. 2-2, pp. 027 104–1–027 104–4, 2004. [3] M.E.J.Newman, “Assortative mixing in networks,” Physical Review Letter, vol. 89, no. 20, pp. 208 701–1–208 701–4, 2002. [4] S. Zhou and R. J.Mondragon, “The rich-club phenomenon in the internet topology,” IEEE Communications Letters, vol. 8, no. 3, pp. 180–182, Mar 2004. [5] E. W.Zegura, K. L.Calvert, and M. J.Donoboo, “A quantitative comparison of graph-based models for internet topology,” IEEE/ACM Transactions on Networking, vol. 5, no. 6, pp. 770–783, 1997. [6] http://topology.eecs.umich.edu/inet/. [7] R.Albert and A.L.Barabasi, “Topology of evolving networks:local events and universality,” Physical Review Letter, vol. 85, no. 24, pp. 5234– 5237, 2000. [8] T. Bu and T. D, “On distinguishing between internet power law topology generators,” in Proceedings of the IEEE INFOCOM 2002, vol. 2. New York: IEEE, 2002, pp. 638–647. [9] G.Chen, Z.P.Fan, and X. Li, “Modeling the complex internet topology,” Complex Dynamics in Communication Networks, pp. 213–235, 2005. [10] S.Zhou and R.J.Mondragon, “Accurately modeling the internet topology,” Physical Review E, vol. 70, no. 6-2, pp. 066 108–1–066 108–8, Dec 2004. [11] S. Zhou, “Understanding the evolution dynamics of internet topology,” Physical Review E, vol. 74, no. 1, pp. 016 124–1–016 124–11, 2006. [12] L. Alvarez-Hamelin, L. Dall’Asta, A. Barrat, and A. Vespignani, “kcore decomposition: a tool for the visualization of large scale networks,” http://arxiv.org/abs/cs.NI/0511007, 2005. [13] C. Prehofer and C. Bettstetter, “Self-organization in communication networks:principles and design paradigms,” IEEE Communications Magazine, vol. 43, no. 7, pp. 78–85, July 2005. [14] L. X. Gao, “On inferring autonomous system relationships in the internet,” IEEE/ACM Transactions on Networking, vol. 9, no. 6, pp. 733–745, Dec 2001.
[15] http://www.routeviews.org. [16] Y.Rekhter and T.Li, “A border gateway protocol 4(bgp 4),” Request for Comments 1771, Mar 1995. [17] http://www.caida.org. [18] Y. Shavitt and E. Shir, “Dimes-letting the internet measure itself,” http://www.arxiv.org/abs/cs.NI/0506099, 2005. [19] http://www.cs.bu.edu/brite/.
