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Classifying Customer-Provider Relationships in the Internet
Abstract
The problem of inferring customer-provider relationships in the autonomous system topology of the Internet leads to the following optimization problem: given an undirected graph G and a set P of paths in G, orient the edges of G such that as many paths as possible are valid, meaning that they do not contain an internal node with both incident edges on the path directed away from that node. The complexity of this problem was left open by Subramanian et al. ("Characterizing the Internet hierarchy from multiple vantage points," INFOCOM 2002).
... Le problème d'inférence des accords d'interconnexion est un problème d'optimisation industriel qui a succité beaucoup d'attention de la part de la littérature. Pour un état de l'art complet, on se reportera dans l'ordre chronologique aux travaux introductifs [60,78,173], puis aux travaux concernant la complexité du problème [14,40,46], puis aux récents travaux proposant des méthodes heuristiques de résolution [15,41,127,129,130,162,190,191]. Le problème d'inférence des accords d'interconnexion entre AS sera approfondi durant cette thèse dans le chapitre 3. Ce problème joue un rôle central dans l'étude des politiques de routage interdomaines car les différents accords influencent les procédures de mesure et d'analyse de la topologie de l'Internet en dépit des règles de routage mises en place. ...
... Le problème MaxTOR peut se voir comme un problème d'orientation partielle de graphe 6 (voir figure 3.4). Si on ne considère que les accords P 2C et C2P , le problème devient celui d'une orientation de graphe [14,46] : ...
... Les formulations étudiées dans [14,40,46] ne font pas intervenir les accords de peering. MaxTOR3-simple : affecter un label P 2C ou C2P à chaque arc du graphe tel que le nombre de chemins de P ′ 3 vérifiant l'expression régulière C2P * P 2C * soit maximum. ...
Internet supports public IP communications. It is composed of over 20 000 smaller networks called Autonomous Systems (AS). ASs belong to various administrative entities : network operators, universities, companies... Each of them has global connectivity to any other, thanks to interdomain routing handled by BGP (Border Gateway Protocol). Bilateral connections between ASs support business agreements to guarrantee Internet connectivity. During the negociation of an economical relationship between two ASs, the two administrative entities agree on network availability, price, bandwith, and ratio of the amount of trafic they send to each other. They may also restrict the set of reachable destinations they share. This economic market of network interconnections is not regulated. Typically, long-term actors take advantage of their market position to negociate their economic relationships, and large players such as content providers or access providers use providers in competition in multiple locations to minimize their cost and maximize their availability through redundancy. In interdomain topology, BGP routes are rather economical paths than shortest AS paths. Economic business agreements between ASs are private but they shape Internet routes as they define a great part of BGP route-redistribution filters. In this Ph. D. thesis, we solve several difficult problems dealing with reverse engineering of the Internet. We start by an introduction on Internet economy and interdomain routing with the BGP protocol. Then, we successively review several works respectively on : measuring the Internet topology, determining economic properties of Internet and infering Internet routing policies. We study three problems in this thesis : the measurement of a stable AS-level topology, the inference of AS business relationships and the modeling of interdomain routing at the AS-level. We propose a set of tools that bring insights on Internet economics for an operator like France Telecom. Knowledge produced by our algorithms is of great importance in negociating business relationships and in studying economic aspects of Internet interconnection markets.
... Moreover, business behavior patterns of Internet players influence directions of ISPs' collaboration and ultimately the evolution of the macroscopic infrastructure of the Internet. In this study we follow previous works [15, 26, 4, 14] in considering the following three major categories of AS relationships: customer-to-provider (c2p), peer-to-peer (p2p), and sibling-to-sibling (s2s). In the c2p category, a customer AS pays a provider AS for any traffic sent between the two. 1 In the p2p category, two ASs freely exchange traffic between themselves and their customers, but do not exchange traffic from or to their providers or other peers. ...
... If the ranks are similar, the algorithm classifies the link as p2p, otherwise it is c2p. Di Battista et al. [4] and Erlebach et al. [14] independently showed that ToR is indeed NP-complete, developed mathematically rigorous approximate solutions to the problem and proved that it is impossible to infer p2p relationships under the ToR formulation framework. For this reason, their solutions (referred to as DPP and EHS) infer c2p relationships only and ignore p2p and s2s relationships. ...
... MAX2SAT is NPcomplete , however, the semidefinite programming (SDP) ap- proach [17] delivers an approximate answer that differs from the exact answer by not more than a factor of 0.94. To reduce the objective O1 (ToR) to MAX2SAT we use the approach of DPP and EHS [4, 14]. This gives a set of xi ∨ xj clauses, where i, j ∈ E. To reduce the objective O2 to MAX2SAT, we introduce a clause xi ∨ xi for every edge i ∈ E that has an initial direction along the node degree gradient, and a clause ¯ xi ∨ ¯ xi for every edge with an initial direction against the node degree gradient. ...
Research on performance, robustness, and evolution of the global Internet is fundamentally handicapped without accurate and thorough knowledge of the nature and structure of the contractual relationships between Autonomous Systems (ASs). In this work we introduce novel heuristics for inferring AS relationships. Our heuristics improve upon previous works in several technical aspects, which we outline in detail and demonstrate with several examples. Seeking to increase the value and reliability of our inference results, we then focus on validation of inferred AS relationships. We perform a survey with ASs' network administrators to collect information on the actual connectivity and policies of the surveyed ASs. Based on the survey results, we find that our new AS relationship inference techniques achieve high levels of accuracy: we correctly infer 96.5% customer to provider (c2p), 82.8% peer to peer (p2p), and 90.3% sibling to sibling (s2s) relationships. We then cross-compare the reported AS connectivity with the AS connectivity data contained in BGP tables. We find that BGP tables miss up to 86.2% of the true adjacencies of the surveyed ASs. The majority of the missing links are of the p2p type, which highlights the limitations of present measuring techniques to capture links of this type. Finally, to make our results easily accessible and practically useful for the community, we open an AS relationship repository where we archive, on a weekly basis, and make publicly available the complete Internet AS-level topology annotated with AS relationship information for every pair of AS neighbors.
... In the Introduction we summarized the reasons why inference algorithms for computing the commercial relationships between ASes are needed. Several algorithms [10, 18, 8, 9] have been proposed in the literature for doing this. Usually, they take as input a list of AS paths and produce as output a relationship assignment. ...
... The valley-free property inspired the formulation of a combinatorial problem, which has been first presented in [18]. It has been proved [8, 9] that ToR is NP-complete. Therefore, inference algorithms are either based on heuris- tics [10, 18] or on less constrained versions of the same problem [8, 9]. ...
... It has been proved [8, 9] that ToR is NP-complete. Therefore, inference algorithms are either based on heuris- tics [10, 18] or on less constrained versions of the same problem [8, 9]. The first heuristic that has been proposed is the one by Lixin Gao [10]. ...
The Internet has recently been object of several stud-ies concerning its structural properties and the behavior of routing protocols. One of the most interesting chal-lenges is the inference of commercial relationships between Autonomous Systems. This knowledge would provide a deeper insight into the laws governing routing processes, and would constitute a useful guideline for choosing con-nection strategies and device configurations. Several algo-rithms have been proposed for inferring the relationships on the basis of the routing data of the Border Gateway Pro-tocol (BGP). This work aims at performing an analysis of the results produced by state-of-the-art algorithms, with the purpose of investigating the meaningfulness of such results. This is achieved by running the algorithms extensively on several BGP data sets and by observing how assigned rela-tionships change. Two kinds of analysis are used for doing this: one considers the relationships assigned by the same algorithm on data sets relative to different time instants; the second takes into account the relationships assigned by different algorithms on the same data set. We define a methodology and implement some tools for performing the two kinds of analysis and apply the methodology to two well known algorithms, using publicly available data sets. What comes out is that the number of AS pairs whose rela-tionship is steadily assigned never falls below 94% of the overall pairs, and that the solutions computed by the two algorithms overlap for more than 90% of the pairs. This is an evidence of the fact that the inference is well-founded, i.e., it is not heavily influenced by routing oscillations, and that following different approaches almost yields the same solution, which further validates its trustworthiness. £ Work partially supported by European Commission -Fet Open project COSIN -COevolution and Self-organisation In dynamical Net-works -IST-2001-33555, by European Commission -Fet Open project DELIS -Dynamically Evolving Large Scale Information Systems -Con-tract no 001907, by Progetto ALINWEB: Algoritmica per Internet e per il Web, MIUR Programmi di Ricerca Scientifica di Rilevante Inter-esse Nazionale, and by The Multichannel Adaptive Information Systems (MAIS) Project, MIUR Fondo per gli Investimenti della Ricerca di Base.
... Such arrangements depend implicitly on the global structure of the networks and traffic demands, but are based only on a local bid-ask type contract between two neighboring domains [26] [27]. Second, we allow the links in the network to be one of two types: customer-provider or peer-peer [10] [11] [15] [20]. Third, we include a mechanism that penalizes domains if they drop traffic emanating from one of their customers. ...
... In this case, v can refuse to transit d, save itself the transit cost, and not lose any utility since it has no customers that it would owe penalties to. In fact, this is exactly the type of demand (route) filtering that is done in the Internet today [11] [15]. Because it is always in v's interest to filter in such a case, we can simply assume that all demands with such routes have been filtered out, which is equivalent to assuming that all demands follow valid paths and that additional filtering is unnecessary. ...
... Such arrangements depend implicitly on the global structure of the networks and traffic demands, but are based only on a local bid-ask type contract between two neighboring domains [26, 27]. Second, we allow the links in the network to be one of two types: customer-provider or peer-peer [10, 11, 15, 20]. Third, we include a mechanism that penalizes domains if they drop traffic emanating from one of their customers. ...
We introduce a game theoretic model of network formation in an effort to understand the complex system of business relationships between various Internet entities (e.g., Autonomous Systems, enterprise networks, residential customers). This system is at the heart of Internet connectivity. In our model we are given a network topology of nodes and links where the nodes (modeling the various Internet entities) act as the players of the game, and links represent potential contracts. Nodes wish to satisfy their demands, which earn potential revenues, but nodes may have to pay (or be paid by) their neighbors for links incident to them. By incorporating some of the qualities of Internet business relationships, we hope that our model will have predictive value. Specifically, we assume that contracts are either customer-provider or peering contracts. As often occurs in practice, we also include a mechanism that penalizes nodes if they drop traffic emanating from one of their customers. For a natural objective function, we prove that the price of stability is at most 2. With respect to social welfare, however, the prices of anarchy and stability can both be unbounded, leading us to consider how much we must perturb the system to obtain good stable solutions. We thus focus on the quality of Nash equilibria achievable through centralized incentives: solutions created by an ""altruistic entity" (e.g., the government) able to increase individual payouts for successfully routing a particular demand. We show that if every payout is increased by a factor of 2, then there is a Nash equilibrium as good as the original centrally defined social optimum. We also show how to find equilibria efficiently in multicast trees. Finally, we give a characterization of Nash equilibria as flows of utility with certain constraints, which helps to visualize the structure of stable solutions and provides us with useful proof techniques.
... • How large are the differences between the undirected graph model and the directed graph models with respect to connectivity properties? • How do the directed graph models produced by the four algorithms proposed in Gao (2001), Subramanian et al. (2002), Di Battista et al. (2003), and Erlebach et al. (2002) compare to each other? Here, we are mainly interested in (i) comparing connectivity measures, (ii) the depth of the provider hierarchy, and (iii) the occurrence of directed customer-provider cycles in the directed graph models. ...
... @BULLET How large are the differences between the undirected graph model and the directed graph models with respect to connectivity properties? @BULLET How do the directed graph models produced by the four algorithms proposed in Gao (2001), Subramanian et al. (2002, Di Battista et al. (2003), and Erlebach et al. (2002) compare to each other? Here, we are mainly interested in (i) comparing connectivity measures, (ii) the depth of the provider hierarchy, and (iii) the occurrence of directed customer-provider cycles in the directed graph models. ...
... They also give a heuristic algorithm that infers relationships by first ranking all ASs and then applying certain rules to decide about the relationships between pairs of ASs using the rank values. Independently obtained results in Di Battista et al. (2003) and Erlebach et al. (2002) resolve the open question of Subramanian et al. (2002) and prove this inference problem to be -hard. Two heuristic algorithms for calculating approximately optimal orientations with respect to the number of valid paths are also presented in Di Battista et al. (2003) and Erlebach et al. (2002), respectively. ...
Classical measures of network connectivity are the number of disjoint paths between a pair of nodes and the size of a minimum cut. For standard graphs, these measures can be computed efficiently using network flow techniques. However, in the Internet on the level of autonomous systems (ASs), referred to as AS-level Internet, routing policies impose restrictions on the paths that traffic can take in the network. These restrictions can be captured by the valley-free path model, which assumes a special directed graph model in which edge types represent relationships between ASs. We consider the adaptation of the classical connectivity measures to the valley-free path model, where it is NP-hard to compute them. Our first main contribution consists of presenting algorithms for the computation of disjoint paths, and minimum cuts, in the valley-free path model. These algorithms are useful for ASs that want to evaluate different options for selecting upstream providers to improve the robustness of their connection to the Internet. Our second main contribution is an experimental evaluation of our algorithms on four types of directed graph models of the AS-level Internet produced by different inference algorithms. Most importantly, the evaluation shows that our algorithms are able to compute optimal solutions to instances of realistic size of the connectivity problems in the valley-free path model in reasonable time. Furthermore, our experimental results provide information about the characteristics of the directed graph models of the AS-level Internet produced by different inference algorithms. It turns out that (i) we can quantify the difference between the undirected AS-level topology and the directed graph models with respect to fundamental connectivity measures, and (ii) the different inference algorithms yield topologies that are similar with respect to connectivity and are different with respect to the types of paths that exist between pairs of ASs.
... Moreover, business behavior patterns of Internet players influence directions of ISPs' collaboration and ultimately the evolution of the macroscopic infrastructure of the Internet. In this study we follow previous works [15, 26, 4, 14] in considering the following three major categories of AS relationships: customer-to-provider (c2p), peer-to-peer (p2p), and sibling-to-sibling (s2s). In the c2p category, a customer AS pays a provider AS for any traffic sent between the two. 1 In the p2p category, two ASs freely exchange traffic between themselves and their customers, but do not exchange traffic from or to their providers or other peers. ...
... If the ranks are similar, the algorithm classifies the link as p2p, otherwise it is c2p. Di Battista et al. [4] and Erlebach et al. [14] independently showed that ToR is indeed NP-complete, developed mathematically rigorous approximate solutions to the problem and proved that it is impossible to infer p2p relationships under the ToR formulation framework. For this reason, their solutions (referred to as DPP and EHS) infer c2p relationships only and ignore p2p and s2s relationships. ...
... MAX2SAT is NPcomplete , however, the semidefinite programming (SDP) ap- proach [17] delivers an approximate answer that differs from the exact answer by not more than a factor of 0.94. To reduce the objective O1 (ToR) to MAX2SAT we use the approach of DPP and EHS [4, 14]. This gives a set of xi ∨ xj clauses, where i, j ∈ E. To reduce the objective O2 to MAX2SAT, we introduce a clause xi ∨ xi for every edge i ∈ E that has an initial direction along the node degree gradient, and a clause ¯ xi ∨ ¯ xi for every edge with an initial direction against the node degree gradient. ...
... For comparison, note that the previously adopted undirected graph model of the Internet consisted of an undirected graph with an undirected edge between two ASs if there is at least one physical link between them. Since information about the economic relationships between ASs is not publicly available (such information is often treated like a business secret), four algorithms have been proposed for inferring these relationships from BGP routing table information (Gao, 2001; Subramanian et al., 2002; Di Battista et al., 2003; Erlebach et al., 2002). However, it is not known how good the topologies produced by these algorithms are and how these topologies differ from each other. ...
... They also give a heuristic algorithm that infers relationships by first ranking all ASs and then applying certain rules to decide about the relationships between pairs of ASs using the rank values. Independently obtained results from Di Battista et al. (2003) and Erlebach et al. (2002) resolve the open question of Subramanian et al. (2002) and prove this inference problem to be N P-hard. Two heuristic algorithms for calculating approximately optimal orientations with respect to the number of valid paths are also presented by Di Battista et al. (2003) with respect to two measures. ...
... They conclude that the new algorithm proposed in the paper outperforms the algorithms from Gao (2001) and Subramanian et al. (2002). In this chapter, we compare the AS relationships computed by all four algorithms proposed by Gao (2001), Subramanian et al. (2002, Di Battista et al. (2003), and Erlebach et al. (2002) , and we try to identify important characteristics of the relationship classifications produced by these algorithms . The motivation for our investigations comes from several papers that showed the impact of BGP policies on important features of Internet routing such as path inflation and routing convergence (see Labovitz et al. (2001), and Tangmunarunkit et al. (2001a Tangmunarunkit et al. ( , 2001b Tangmunarunkit et al. ( , 2003). ...
The topology of the Internet has initially been modelled as an undirected graph, where vertices correspond to so-called Autonomous Systems (ASs),and edges correspond to physical links between pairs of ASs. However, in order to capture the impact of routing policies, it has recently become apparent that one needs to classify the edges according to the existing economic relationships (customer-provider, peer-to-peer or siblings) between the ASs. This leads to a directed graph model in which traffic can be sent only along so-called valley-free paths. Four different algorithms have been proposed in the literature for inferring AS relationships using publicly available data from routing tables. We investigate the differences in the graph models produced by these algorithms, focussing on connectivity measures. To this aim, we compute the maximum number of vertex-disjoint valley-free paths between ASs as well as the size of a minimum cut separating a pair of ASs. Although these problems are solvable in polynomial time for ordinary graphs, they are NP-hard in our setting. We formulate the two problems as integer programs, and we propose a number of exact algorithms for solving them. For the problem of finding the maximum number of vertex-disjoint paths, we discuss two algorithms; the first one is a branch-and-price algorithm based on the IP formulation, and the second algorithm is a non LP based branch-and-bound algorithm. For the problem of finding minimum cuts we use a branch-and-cut algo rithm, based on the IP formulation of this problem. Using these algorithms, we obtain exact solutions for both problems in reasonable time. It turns out that there is a large gap in terms of the connectivity measures between the undirected and directed models. This finding supports our conclusion that economic relationships need to be taken into account when building a topology of the Internet.
... The problem of inferring AS relationships was first introduced by Gao [4], who proposed an inference heuristic (LG) that identified top providers and p2p links based on AS degrees; and c2p and s2s links by assuming that every BGP path must comply with the valley-free model. This work was followed by Subramanian et al. (SARK) 1 [81], Di Battista et al. (DPP) [82], and Erlebach et al. (EHS) [83] , who developed a consistent and elegant mathematical formulation of the inference problem (SARK) and found rigorous solutions (DPP, EHS) to the problem formulated by SARK. The latter solutions (DPP, EHS) both retreated to inferring only c2p relationships after demonstrating that it was in fact impossible to rigorously infer p2p relationships within the SARK problem formulation framework. ...
... later improved by DPP [82] and independently by EHS [83]. DPP and EHS proved that ToR is NP-complete, developed rigorous approximate solutions to the problem and proved that it is not possible to infer p2p relationships in the ToR formulation. ...
... We implement our technique detailed in section 4.2.1, reusing parts of the code from [83], utilizing the LEDA v4.5 software library [92], and a publicly available SDP solver DSDP [93]. ...
The Internet is composed of thousands of diverse networks that exchange routing information using the Border Gateway Protocol (BGP). BGP is one of the most critical protocols of the Internet, since it connects these diverse networks to enable communication between remote domains. Despite its critical nature, BGP suffers from a variety of serious problems, which have triggered substantial research on developing improved versions of BGP and new routing architectures. In this dissertation, we introduce necessary tools and data-mining techniques for analyzing the present routing architecture and for evaluating new routing protocols. We focus on the problem of performing realistic BGP simulations and we first develop a BGP simulator enabling detailed and large-scale BGP simulations. Then, we introduce techniques to collect vital Internet routing data, which are essential in conducting realistic BGP simulations. Finally, we introduce models of the collected data. Ph.D. Committee Chair: George Riley; Committee Member: Doug Blough; Committee Member: Henry Owen; Committee Member: John Copeland; Committee Member: KC Claffy; Committee Member: Nick Feamster
... The authors call the problem the type-of-relationship (ToR) problem, conjecture that it is NP-complete, and provide a simple heuristic approximation. Di Battista et al. (DPP) in [1] and independently Erlebach et al. (EHS) in [2] prove that the ToR problem is indeed NP-complete. EHS prove also that it is even harder, specifically APX-complete. ...
... Even more problematic is the following dilemma. DPP (and EHS) come up with heuristics that outperform the SARK algorithm in terms of producing smaller numbers of invalid paths [1, 2]. Although these results seem to be a positive sign, closer examination of the AS relationships produced by the DPP algorithm [6] reveals that the DPP inferences are further from reality than the SARK inferences. ...
... To achieve this purpose, we start with the DPP and EHS results [1, 2] that deliver the fewest invalid paths. Suppose we have a set of BGP paths P from which we can extract the undirected AS-level graph G(V, E). ...
Recent techniques for inferring business relationships between ASs have yielded maps that have extremely few invalid BGP paths in the terminology of Gao. However, some relationships inferred by these newer algorithms are incorrect, leading to the deduction of unrealistic AS hierarchies. We investigate this problem and discover what causes it. Having obtained such insight, we generalize the problem of AS relationship inference as a multiobjective optimization problem with node-degree-based corrections to the original objective function of minimizing the number of invalid paths. We solve the generalized version of the problem using the semidefinite programming relaxation of the MAX2SAT problem. Keeping the number of invalid paths small, we obtain a more veracious solution than that yielded by recent heuristics.
... Based on AS business relationship, she proved that a reasonable AS path should follow valley-free policy. Based on this model, there have been plenty of works on inferring AS business relationship [2][3][4][5][6]. Towards a finer granularity, works [7,8] use BGP atoms to represent groups of prefixes (originated by a given AS) that receive equivalent treatment by a set of BGP routers. ...
... AS 9, AS 10, AS 11, and AS 12 are observed for 3 times. Herein, the ranking for the observed ASes is (2,3,4,6,8,9,10,11,12,5,7). When the feature size is 5, we select AS 2, 3, 4, 6, and 8. ...
... Such arrangements depend implicitly on the global structure of the networks and traffic demands, but are based only on a local bid-ask type contract between two neighboring domains [26, 27]. Second, we allow the links in the network to be one of two types: customer-provider or peer-peer [10, 11, 15, 20]. Third, we include a mechanism that penalizes domains if they drop traffic emanating from one of their customers. ...
... In this case, v can refuse to transit d, save itself the transit cost, and not lose any utility since it has no customers that it would owe penalties to. In fact, this is exactly the type of demand (route) filtering that is done in the Internet today [11, 15]. Because it is always in v's interest to filter in such a case, we can simply assume that all demands with such routes have been filtered out, which is equivalent to assuming that all demands follow valid paths and that additional filtering is unnecessary. ...
We introduce a game theoretic model of network formation in an effort to understand the complex system of business relationships between various Internet entities (e.g., Autonomous Systems, enterprise networks, residential customers). In our model we are given a network topology of nodes and links where the nodes act as the players of the game, and links represent potential contracts. Nodes wish to satisfy their demands, which earn potential revenues, but may have to pay their neighbors for links incident to them. We incorporate some of the qualities of Internet business relationships, including customer-provider and peering contracts. We show that every Nash equilibrium can be represented by a circulation flow of utility with certain constraints. This allows us to prove bounds on the prices of anarchy and stability. We also focus on the quality of equilibria achievable through centralized incentives.
... Rather, most importantly, local routing policies of independent administrative domains and their interplay based on business relations have a critical influence on route (in)stability and routing quality. As business contracts are intentionally considered to be commercial secrets, many researchers have proposed methods to elicite this crucial information indirectly (see, e.g., [12, 26, 5, 9, 6, 30, 7]). In [12], the seminal work on inferring contractual relationships on basis of sets of AS paths observable from BGP updates, heuristic approaches were devised to classify relationships into customer-to-provider, peer-to-peer, and sibling-to-sibling. ...
... But, due to their over-sensitivity to path sets, they have weaknesses in well-defined analytical test situations (such as, e.g., in [11, 13]). A purely combinatorial treatment, neither involving A nor B, was done in [5, 9, 6]. There, the authors described a linear-time algorithm solving the all-paths type-of-relationship problem , suggested in [26]: given a path set P , is there an orientation of the edges (indicating provider-to-customer or customer-to-provider relation) such that all paths of P are valley-free. ...
We contribute to the study of inferring commercial relationships between autonomous systems (AS relationships) from observable
BGP routes. We deduce several forbidden patterns of AS relationships that impose a certain type of acyclicity on the AS graph.
We investigate algorithms for solving the acyclic all-paths type-of-relationship problem, i.e., given a set of AS paths, find an orientation of the edges according to some types of AS relationships such
that the oriented AS graph is acyclic (with respect to the forbidden patterns) and all AS paths are valley-free. As possible
AS relationships we include customer-to-provider, peer-to-peer, and sibling-to-sibling. Moreover, we examine a number of problem
versions parameterized by sets K and U where K is the set of edge types available for describing explicit pre-knowledge and U is the set of edge types available for completion of partial orientations. A complete complexity classification of all 56 cases
(8 type sets for pre-knowledge and 7 type sets for completion) is given. The most relevant practical result is a linear-time
algorithm for finding an acyclic and valley-free completion using customer-to-provider relations given any kind of pre-knowledge. Interestingly, if we allow sibling-to-sibling relations for completions then most of the non-trivial
inference problems become NP-hard.
... Information about the economic relationships between autonomous systems is not publicly available. Therefore, several approaches to inferring these relationships from available topology data or AS path information have been proposed in the literature [10, 15, 8, 6]. If a communication network is represented as an undirected or directed graph in a model without routing policies, it is natural to measure the connectivity provided to an s-t-pair as the maximum number of disjoint s-t-paths or the minimum size of an s-t-cut (by Menger's theorem, these two quantities are the same). ...
... Following the terminology from [15, 6, 8], where the problem of classifying the relationships between ASs is called the Type-of-Relationship (ToR) problem, we call a simple directed graph G = (V, E) a ToR graph if G has no loops and no anti-parallel edges (i.e., (u, v) ∈ E implies (v, u) / ∈ E). In terms of the underlying motivation, a directed edge from u to v, where u, v ∈ V , means that u is a customer of v. ...
In the valley-free path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is motivated by BGP routing policies of autonomous systems in the Internet. Robustness considera- tions lead to the problem of computing a maximum number of disjoint paths between two nodes, and the minimum size of a cut that sepa- rates them. We study these problems in the valley-free path model. For the problem of computing a maximum number of edge- or vertex-disjoint valid paths between two given vertices s and t, we give a 2-approximation algorithm and show that no better approximation ratio is possible unless P = NP. For the problem of computing a minimum vertex cut that sep- arates s and t with respect to all valid paths, we give a 2-approximation algorithm and prove that the problem is APX-hard. The correspond- ing problem for edge cuts is shown to be polynomial-time solvable. We present additional results for acyclic graphs.
... While these contracts take into account global issues such as the topology of the rest of the network and traffic demands throughout the network, they are formed strictly by negotiations between two neighboring players. Another property is that two possible types of contracts can be formed: customer-provider or peer-peer [8,9,12,15]. Lastly, both models incorporate penalties such as those found in typical real world SLAs for failure to meet the conditions of a contract [19]. ...
... These are the only paths on which traffic can be routed, since any other path would contain a node v forwarding traffic between two providers, or between a peer and a provider. In our game (as well as in reality [9,12]), node v would have an incentive to not forward this traffic, since it would save on transit cost, and would not have to pay any penalties. ...
In the Internet, Autonomous Systems (ASes) make contracts called Service Level Agreements (SLAs) between each other to transit one another's trac. ASes also try to control the routing of trac to and from their networks in order to achieve ecient use of their infrastructure and to attempt to meet some level of quality of service globally. We introduce a game theoretic model in order to gain understanding of this interplay between network formation and routing. Player strategies allow them to make contracts with one another to forward trac, and to re-route trac that is currently routed through them. This model extends earlier work of (3) that only considered the network formation aspect of the problem. We study the structure and quality of Nash equilibria and quantify the prices of anarchy and stability, that is, the relative quality of a centralized optimal solution versus that of the Nash equilibria.
... The export policies are usually as follows: each of the two AS can export its internal routes and customer routes, but usually does not export its provider or peer routes. Although several algorithms3456789 have been proposed in the literature for inferring AS relationships, we find that the performances of them are more susceptible to the network topology variation, which results that the stability of those algorithms is far from satisfactory. Specifically, when we compare the inferring AS relationships of the IPv4 network with the one of the IPv6 network, we observe that quite a few of the inference results are inconsistent. ...
... If the ranks were similar, the ASs were considered as peers. Di Battista et al. [6] and Erlebach et al. [7] independently proposed a similar approach to infer AS relationships. Their algorithms were based on a reduction of ToR problem to the well-known Boolean Satisfiability Problem (SAT). ...
The commercial relationships between Autonomous Systems (ASes) are of great importance to understand the Internet reachability and calculate the AS-level paths. Several algorithms have been proposed to solve the AS relationship inference problem and applied to the data of IPv4 network. In assuming that the provider is typically larger than its customers, and the peers usually have comparable sizes, the suggested algorithms exploit the AS degree information to infer AS relationships. In analysis of the AS relationships in the IPv6 network, however, we find that quite a few of the inference results induced by the present approaches are different from the inferences in the IPv4 network. With respect to this observation, we analyze the root cause of the discrepancy and propose an algorithm which combines the AS hierarchy information, an inherent nature of the Internet structure that we can hardly neglect while analyzing the AS relationships, with the optimization model of Type-of-Relationship (ToR) problem to infer the AS relationships more realistically and stably. In this paper, we first present a methodology to classify ASes into four hierarchies, and then use the AS hierarchy information to infer AS relationships. By taking advantage of these partial AS relationship information, we introduce an improved algorithm to solve the ToR problem for the remaining AS pairs. The experimental results support our algorithm in two aspects. On one hand, the comparison with previous works in the IPv4 network shows that most of our inferring AS relationships are consistent with their inferences, while more inferences of our approach are confirmed by the export policies stored in the Internet Routing Registry (IRR) databases. On the other hand, 94.82% of our inference relationships in the IPv6 network are consistent with the inferences in the IPv4 network, which illustrates that our algorithm is more stable than previous algorithms.
... Most ASes try to hide their business relations. In the last decade researchers have introduced a number of algorithms to infer the AS relationships [8], [13], [9], [14], [15], [10], [11], [16]. These algorithms have produced conflicting results. ...
... Subramanian et al [9] formulated the ToR as an optimization problem. Two independent studies [14], [15] proved that the ToR problem is NP-hard and proposed approximate solutions by reducing the ToR to a 2SAT problem which can be solved in linear time. Dimitropoulos et al. [10] observed that the ToR formulation can result in multiple solutions without being possible to determine the best. ...
The type of business relationships between the Internet autonomous systems (AS) determines the BGP inter-domain routing. Previous works on inferring AS relationships relied on the connectivity information between ASes. In this paper we infer AS relationships by analysing the routing polices of ASes encoded in the BGP attributes Communities and the Locpref. We accumulate BGP data from RouteViews, RIPE RIS and the public Route Servers in August 2010 and February 2011. Based on the routing policies extracted from data of the two BGP attributes, we obtain AS relationships for 39% links in our data, which include all links among the Tier–1 ASes and most links between Tier–1 and Tier–2 ASes. We also reveal a number of special AS relationships, namely the hybrid relationship, the partial-transit relationship, the indirect peering relationship and the backup links. These special relationships are relevant to a better understanding of the Internet routing. Our work provides a profound methodological progress for inferring the AS relationships.
... Information about the economic relationships between autonomous systems is not publicly available. Therefore, several approaches to inferring these relationships from available topology data or AS path information have been proposed in the literature [10, 18, 8, 6, 21]. If a communication network is represented as an undirected or directed graph in a model without routing policies, it is natural to measure the connectivity provided to an s-t-pair as the maximum number of disjoint s-t-paths or the minimum size of an s-t-cut; by Menger's theorem, these two quantities are the same. ...
... Following the terminology from [18, 6, 8] , where the problem of classifying the relationships between ASs is called the Type-of-Relationship (ToR) problem, we will call a simple directed graph G = (V, E) a ToR graph if G has no loops and no anti-parallel edges, i.e., (u, v) ∈ E implies (v, u) / ∈ E. In terms of the underlying motivation, a directed edge from u to v, where u, v ∈ V, means that u is a customer of v. A path p = v 1 , v 2 , . . . ...
In the valley-free path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is motivated by routing policies of autonomous systems in the Internet. We give a 2-approximation algorithm for the problem of computing a maximumn number of edge- or vertex-disjoint valid paths between two given vertices s and t, and we show that no better approximation ratio is possible unless \PP=\NP. Furthermore, we give a 2-approximation algorithm for the problem of computing a minimum vertex cut that separates s and t with respect to all valid paths and prove that the problem is APX-hard. The corresponding problem for edge cuts is shown to be polynomial-time solvable. For the multiway variant of the cut problem, we give a 4-approximation algorithm. We present additional results for acyclic graphs
... Most ASes try to hide their business relations. In the last decade researchers have introduced a number of algorithms to infer the AS relationships [8], [13], [9], [14], [15], [10], [11], [16]. These algorithms have produced conflicting results. ...
... Subramanian et al [9] formulated the ToR as an optimization problem. Two independent studies [14], [15] proved that the ToR problem is NP-hard and proposed approximate solutions by reducing the ToR to a 2SAT problem which can be solved in linear time. Dimitropoulos et al. [10] observed that the ToR formulation can result in multiple solutions without being possible to determine the best. ...
The type of business relationships between the Internet autonomous systems
(AS) determines the BGP inter-domain routing. Previous works on inferring AS
relationships relied on the connectivity information between ASes. In this
paper we infer AS relationships by analysing the routing polices of ASes
encoded in the BGP attributes Communities and the Locpref. We accumulate BGP
data from RouteViews, RIPE RIS and the public Route Servers in August 2010 and
February 2011. Based on the routing policies extracted from data of the two BGP
attributes, we obtain AS relationships for 39% links in our data, which include
all links among the Tier-1 ASes and most links between Tier-1 and Tier-2 ASes.
We also reveal a number of special AS relationships, namely the hybrid
relationship, the partial-transit relationship, the indirect peering
relationship and the backup links. These special relationships are relevant to
a better understanding of the Internet routing. Our work provides a profound
methodological progress for inferring the AS relationships.
... AS relationships are usually subject to non-disclosure agreement and can thus only be inferred from publicly available routing data. While the state-of-the-art of inferring the global AS relationship graph has been constantly evolving [53,144,38,46,163,40,39,65,90], inference will inevitably only be able to produce an approximation of the AS graph. As with non-standard AS relations, missing or incorrect links are problematic for our graphbased algorithm, which could cause it to wrongly discard ASes as candidates. ...
... Most ASes try to hide their business relations. In the last decade researchers have introduced a number of algorithms to infer the AS relationships [6], [7], [8], [9], [10], [11], [12], [13]. These algorithms have produced conflicting results. ...
The type of business relationships between the Internet autonomous systems (AS) determines the BGP interdomain routing. Previous works on inferring AS relationships relied on the connectivity information between ASes. In this paper we thoroughly investigate the route policy between ASes, then present a method that can conduct new policies. On the basis of that, we also present a method of between AS polices verification based on AS relationships. Through this method we can find out the abnormal prefix advertise to verify the route policies. We accumulate BGP data from RouteViews, RIPE RIS and the public Route Servers in January 2013 and February 2014. Based on the routing policies extracted from data of the two BGP attributes, we obtain AS relationships for 41% links in our data, which include all links among the Tier-1 ASes and most links between Tier-1 and Tier-2 ASes. We took an experiment for our method between two ASs in AT&T, and the results are good.
... After generating a valid set of BGP paths, by using tables from Route Views at 8 hours interval during 4 months, and eliminating invalid paths under the idea of persistency and degree-valley, they infer AS relationships for this set. In order to infer s2s relationships they use RIPE, ARIN an APNIC databases, to infer c2p relationships they use parts of the code from EHS [5], LEDA v4.5 software library and DSDP v4.7 [6], and finally to infer p2p relationships they use QUALEX [7] solver to approximate the MWIS problem. ...
... Data retrieved from BGP repositories is generally analyzed to determine the structure and properties of the network of the Autonomous Systems at a specific time [13, 14, 3, 10, 12, 25]. However, a more sophisticated investigation can only be accomplished by considering a whole sequence of snapshots taken at subsequent moments. ...
In this paper we investigate to what extent the infor-mation provided by BGP routing tables about the graph of the Autonomous Systems (ASes) can be used to under-stand dynamic phenomena occurring in the network. First, we classify the time scales at which such an analysis can be performed and, consequently, the kinds of phenomena that could be anticipated. Second, we improve cutting-edge technologies used to analyze the structure of the network, most notably spectral methods for graph clustering, in or-der to be able to analyze a whole sequence of consecu-tive snapshots that capture the temporal evolution of the network. Finally, we use such tools to analyze the data collected by the Oregon RouteViews project [20] during the last few years. We confirm stable properties of the AS graph, find major trends and notice that events occurring on a smaller time-frame, like worm-attacks, misconfigura-tions, outages, DDoS attacks, etc. seem to have a very di-verse degree of impact on the AS graph structure, which suggests that these techniques could be used to distinguish some of them.
... As a consequence the work on modeling router level topologies is in preliminary stages [63] . Business relationships between ASs and BGP policies can be inferred from BGP data with the following heuristics [64,43,656667 . Development of validation techniques and inference of more complex policies, e.g., partial transit [68], are still to be explored. ...
Simulation has been the method of choice for analyzing large, complex, and highly volatile systems. One of these systems is the inter-domain routing infrastructure of the Internet. Despite the need for high quality Border Gateway Protocol (BGP) simulation tools, traditional BGP simulators have limitations either in their modeling fidelity or in their scalability. In this work we introduce BGP++, a scalable BGP simulator that employs state-of-the-art techniques to address the abstraction-scalability trade-off.BGP++ builds on high quality software in network simulation, routing and parallel-distributed simulation to deliver a detailed yet scalable implementation of BGP. Moreover, with respect to the needs of researchers and operators, BGP++ has a CISCO-like configuration language, a seamless partitioning engine for parallel-distributed simulations and a configuration toolset that expedites common simulation configuration tasks.
... She also proposes a heuristic algorithm to infer AS relationships based on the AS paths derived from BGP table. After that, many efforts have been made to more accurately define6789 and infer1011121314 AS relationships. Inspired by the AS relationship inference efforts, Mao [2] proposes an algorithm to inferring AS paths without accessing the source node. ...
Discovering the AS paths between two ASes is invaluable for a wide area of network research and application activities. The traditional techniques for path discovery require direct access to the source node. Recently, with more accurate AS relationship inferring algorithm and publicly available AS topology data, it is possible to infer AS paths without accessing the source. This paper proposes an efficient algorithm for inferring all pair shortest AS paths in a relationship annotated AS graph. The running time of the algorithm is O(NM), where N is the number of nodes and M is the number of edges in AS graph. The algorithm bases on the bread-first-search (BFS) algorithm, and experimental results show that it reduces running time dramatically compared with the existing algorithm whose running time is O(N3).
... DPP * and EHS produce the best results and even fare well in comparison to GAO (sib), which classifies roughly 1.5% of the edges as sibling edges; DPP * , EHS and GAO (sib) achieve comparable numbers of invalid paths, although DPP * and EHS do not use any sibling edges (note that sibling edges simplify the task of making paths valid). The improvement shown by DPP and EHS with respect to their preliminary implementations described in [12], [13] are due to code debugging and re-engineering. It is interesting to know how many edges of the AS graph are classified in the same way by different classification algorithms . ...
We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced by Gao [IEEE/ACM Transactions on Networking, 9(6):733-645, 2001] and Subramanian (IEEE Infocom, 2002) that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose approximation algorithms and heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments provide evidence that our algorithms perform significantly better than state-of-the-art heuristics
... Battista et al. later proved that the ToR problem is NP-complete [61] and many researchers have since focused on using approximation algorithms for MAX2SAT to provide labelings that maximize the number of valley-free paths. [61, 62, 63] These algorithms focus on correctly labeling customer-provider relationships and as discussed in 2.4.2 the results often fail relationship sanity checks (with the exception of [63]). Lixin Gao returned to the ToR problem in 2004 with Jianhong Xia to introduce a new heuristic and compare it to her own along with that proposed in [60]. ...
The Internet has developed into an important economic, military, academic, and social resource. It is a complex network, comprised of tens of thousands of independently operated networks, called Autonomous Systems (ASes). A significant strength of the Internet's design, one which enabled its rapid growth in terms of users and bandwidth, is that its underlying protocols (such as IP, TCP, and BGP) are distributed. Users and networks alike can attach and detach from the Internet at will, without causing major disruptions to global Internet connectivity. This dissertation shows that the Internet's distributed, and often redundant structure, can be exploited to increase the security of its protocols, particularly BGP (the Internet's interdomain routing protocol). It introduces Pretty Good BGP, an anomaly detection protocol coupled with an automated response that can protect individual networks from BGP attacks. It also presents statistical measurements of the Internet's structure and uses them to create a model of Internet growth. This work could be used, for instance, to test upcoming routing protocols on ensemble of large, Internet-like graphs. Finally, this dissertation shows that while the Internet is designed to be agnostic to political influence, it is actually quite centralized at the country level. With the recent rise in country-level Internet policies, such as nation-wide censorship and warrantless wiretaps, this centralized control could have significant impact on international reachability. Doctor of Computer Science Doctoral University of New Mexico. Dept. of Computer Science. Forrest, Stephanie Rexford, Jennifer Maccabe, Arthur Crandall, Jedidiah
... Its relevance is confirmed by the interest of other research groups on the same subject. In [21] analogous and independently discovered results concerning the time complexity of the general problem and the linearity in the case of all valid paths are shown. However, while such work puts more emphasys on the approximability of the problem, we focus more on the engineering and the experimentation of an effective heuristic approach. ...
The problem of computing the types of the relationships between Internet autonomous systems is investigated. We refer to the model introduced in (ref.1), (ref.2) that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments put in evidence that our heuristics performs significantly better than state of the art heuristics.
The Internet has evolved from a hierarchical and multi-tiered interconnection network to a meshed network, where autonomous systems (ASes) are interconnected with a dense topology and more and more potential paths can be used to reach a destination. However, routing policies are the key to enable these potential paths and to allow the selection of these paths. While the Gao-Rexford guidelines provide routing policies that derive a safe routing system, those potential paths are not enabled by the Gao-Rexford guidelines. A survey on deployed routing policies has shown that a significant portion of networks or ASes do not follow the Gao-Rexford guidelines completely. In this paper, we propose to broaden routing policies that enable the availability of diverse paths and provide flexibility in selecting these paths. We systematically bring out more and more flexible routing policies to implement various routing requirements. More specifically, we propose FlexibleRC, PeerBoost and hierarchical sibling routing policies and derive sufficient conditions for guaranteeing the routing system derived by these routing policies to be safe. The sufficient conditions can be verified through coordination among neighboring ASes. Thus, local coordination guarantees the routing system to be safe.
Amplification DDoS attacks inherently rely on IP spoofing to steer attack traffic to the victim. At the same time, IP spoofing undermines prosecution, as the originating attack infrastructure remains hidden. Researchers have therefore proposed various mechanisms to trace back amplification attacks (or IP-spoofed attacks in general). However, existing traceback techniques require either the cooperation of external parties or a priori knowledge about the attacker. We propose BGPeek-a-Boo, a BGP-based approach to trace back amplification attacks to their origin network. BGPeek-a-Boo monitors amplification attacks with honeypots and uses BGP poisoning to temporarily shut down ingress traffic from selected Autonomous Systems. By systematically probing the entire AS space, we detect systems forwarding and originating spoofed traffic. We then show how a graph-based model of BGP route propagation can reduce the search space, resulting in a 5x median speed-up and over 20x for 1/4 of all cases. BGPeek-a-Boo achieves a unique traceback result 60% of the time in a simulation-based evaluation supported by real-world experiments.
Research and development in global communication architectures and protocols in the Internet often require simulation studies. An authentic simulation model of the network topology is a prerequisite. On the autonomous system level, it is essential to consider different types of business relationships between connected autonomous systems as they reflect directly in connectivity rules implemented in the Border Gateway Protocol (BGP). Consequently, the possibility of routing in the Internet is limited. It has been shown by other authors that disregarding business relationships in simulations leads to substantially shorter routing paths, higher number of alternative paths and lower traffic load on particular nodes and links than in reality. Moreover, we ascertain that some communication scenarios impose availability of business relationships information as a requirement. Modelling business relationships requires a thorough analysis of the real relationships in the Internet. As the information about business contracts is generally considered confidential, recognizing true relationships is a challenging task. We give a survey of methods for their inference. It is also of high importance to study the recognized patterns of relationship links. There are some generally accepted characteristics of the Internet determined by these patterns, e.g. dominance of provider-to-customer and peer-to-peer relationships, certain form of hierarchical interconnection of autonomous systems, etc. Despite the considerable amount of work on business relationship inferences and their analysis, there are not many models which consider the findings. The majority of the proposed topology generators completely ignore them. To the best of our knowledge, there is just one exception. Basically, it is a degree-based topology generator with an extension of power-law distribution paradigm to different types of relationships. We argue that the proposed algorithm fails to adequately resemble important large-scale characteristics of the Internet, e.g. the hierarchical structure and existence of a valid routing path between each pair of autonomous systems. We propose a novel way of modelling, separating the problem of topology generation from the ascribing relationships to links. Focusing on adequate business relationship modelling, the usage of the latest state-of-the-art topology generators becomes possible, thus reducing the difficulty of the problem. We recognize five important characteristics of realistic autonomous system networks and incorporate them in the Type-of-Relationship problem in Random graphs (TRR). Development of a corresponding model is the task of our further research.
Autonomous system (AS) relationships play a critical role in data transferring and routing in the Internet. From the perspective of AS relationships, a concept of AS diversity is first proposed and the development trend of AS diversity in recent years is studied. The result shows that Internet AS diversity has significant growth trends and their distribution become more and more extensive in recent years. Moreover, AS diversity is combined with the traditional parallel-series reliability model, and then a new path diversity model based on AS relationships, named SPDSA (series-parallel path diversity system based on AS relationship) model, is presented. The relevant SPDSA measure is given at a later time. Finally, many virtual multihoming sites which depend on route views node data are simulated and dynamic experiments of AS path diversity are performed by SPDSA measure. Test results show that SPDSA measure is superior to traditional measures and it is especially effective to identify AS path diversity based on multi-homed network. Furthermore, the experiment results also show that multihoming can enhance AS path diversity in the Internet. Finally, the results indicate that though AS diversity has significant growth trends, AS path diversity reveals no major changes in recent years.
The traditional approach of modeling relationships between ASes abstracts relationship types into three broad categories: Transit, peering, and sibling. More complicated configurations exist, and understanding them may advance our knowledge of Internet economics and improve models of routing. We use BGP, traceroute, and geolocation data to extend CAIDA's AS relationship inference algorithm to infer two types of complex relationships: hybrid relationships, where two ASes have different relationships at different interconnection points, and partial transit relationships, which restrict the scope of a customer relationship to the provider's peers and customers. Using this new algorithm, we find 4.5% of the 90,272 provider-customer relationships observed in March 2014 were complex, including 1,071 hybrid relationships and 2,955 partial-transit relationships. Because most peering relationships are invisible, we believe these numbers are lower bounds. We used feedback from operators, and relationships encoded in BGP communities and RPSL, to validate 20% and 6.9% of our partial transit and hybrid inferences, respectively, and found our inferences have 92.9% and 97.0% positive predictive values. Hybrid relationships are not only established between large transit providers; in 57% of the inferred hybrid transit/peering relationships the customer had a customer cone of fewer than 5 ASes. Copyright
To study Internet's routing behavior on the granularity of Autonomous Systems (ASes), one needs to understand inter-domain routing policy. Routing policy changes over time, and may cause route oscillation, network congestion, and other problems. However, there are few works on routing policy changes and their impact on BGP's routing behaviors. In this paper, we model inter-domain routing policy as the preference to neighboring ASes, noted as neighbor preference, and propose an algorithm for quantifying routing policy changes based on neighbor preference. As a further analysis, we study the routing policy changes for the year of 2012, and find that generally an AS may experience a routing policy change for at least 20% prefixes within 6 months. An AS changes its routing policy mainly by exchanging two neighboring ASes' preference. In most cases, an AS changes a stable fraction of its prefixes' routing policy, but non-tier1 ASes may endure a large scale routing policy changing event. We also analyse the main reasons of routing policy changes, and exclude the possibilities of AS business relationship changes and topology changes.
The Internet is a complex network, comprised of thousands of interconnected Autonomous Systems. Considerable research is done in order to infer the undisclosed commercial relationships between ASes. These relationships, which have been commonly classified to four distinct Type of Relationships (ToRs), dictate the routing policies between ASes. These policies are a crucial part in understanding the Internet's traffic and behavior patterns. This work leverages Internet Point of Presence (PoP) level maps to improve AS ToR inference. We propose a method which uses PoP level maps to find complex AS relationships and detect anomalies on the AS relationship level. We present experimental results of using the method on ToR reported by CAIDA and report several types of anomalies and errors. The results demonstrate the benefits of using PoP level maps for ToR inference, requiring considerable less resources than other methods theoretically capable of detecting similar phenomena.
The contractual relationships between autonomous systems (AS) cannot be ignored in the research of large-scale communication protocols and architectures. It has been widely recognized that disregarding policy relationships leads to unrealistic routing paths in simulated communication networks and thus to inaccurate conclusions about the investigated problem. Current AS-level topology generators either completely overlook the relationships or make the annotation process inherent in topology generation. We propose a novel algorithm for annotating random graphs. Our approach differs from previous studies in focusing on the annotation process rather than on the topology generation, which enables reuse of the state-of-the-art topology generators. We identify five properties of viable annotations and formulate the problem as a type-of-relationship problem in random graphs (TRR) by analogy with the related problem of inferring AS relationships from measured routing data. We propose an annotation algorithm for solving the TRR problem by taking advantage of the stochastic properties found in the inferred annotations provided by the cooperative association for internet data analysis (CAIDA). The evaluation provides the evidence of high resemblance of our annotations to the measured ones.
The choice of the scheduling algorithm is a major design criteria for internet switches and routers. Research on scheduling algorithms has mainly focused on maximum weight matching scheduling algorithms, which are computationally very complex and on the computationally less complex maximal weight matching algorithms which require a speedup of two to guarantee the stability of the switch. For practical purposes, neither a high computational complexity nor a speedup is desirable. In this paper, we propose a specific maximal weight matching algorithm that guarantees the stability of a single switch without a speedup.
Whereas initial research has only focused on scheduling algorithms that guarantee the stability of a single switch, it is known that scheduling algorithms that guarantee the stability of individual switches do not necessarily stabilize networks of switches. Recent work has shown how scheduling algorithms for single switches can be modified in order to design scheduling algorithms that stabilize networks of input-queued switches. We apply those results to the design of the maximal weight matching algorithm proposed in this paper and show that the algorithm does not only stabilize a single switch, but also networks of input-queued switches.
The structure of the Internet is still unknown even though it provides services for most of the world. Its current configuration
is the result of complex economic interactions developed in the last 20 years among important carriers and ISPs. Although
with only some success, in the last few years some research has tried to shed light on the economic relationships established
among ASes. The typical approaches have two phases: in the first, data from BGP monitors is gathered to infer the Internet
AS-level topology graph, while in the second phase, algorithms are instantiated on this graph to derive economic tags for
all edges between nodes (i.e. ASes). This paper provides significant findings for both steps. Specifically, regarding the
second step, a small set of transit-free ASes and the lifespan of any AS paths are the input to an algorithm that we have
devised which assigns an economic tag to each AS connection. The quality of inferred tags is expressed by a weight which takes
into consideration the lifespan of the connection it refers to, as well as the outcome of the so called two-way validation
approach. Regarding the first step the paper reports another valuable contribution targeted at identifying and cleaning the
presence of fake connections induced by typos.
Classical measures of network robustness are the number of disjoint paths between two nodes and the size of a smallest cut separating them. In the Internet, the paths that traffic can take are constrained by the routing policies of the individual autonomous systems (ASs). These policies mainly depend on the economic relationships between ASs, e.g., customer-provider or peer-to-peer. Paths that are consistent with these policies can be modeled as valley-free paths. We give an overview of existing approaches to the inference of AS relationships, and we survey recent results concerning the problem of computing a maximum number of disjoint valley-free paths between two given nodes, and the problem of computing a smallest set of nodes whose removal disconnects two given nodes with respect to all valley-free paths. For both problems, we discuss NP-hardness and inapproximability results, approximation algorithms, and exact algorithms based on branch-and-bound techniques. We also summarize experimental findings that have been obtained with these algorithms in a comparison of different graph models of the AS-level Internet with respect to robustness properties.
Media distribution via podcasts is a relatively new phenomenon which follows a different paradigm compared to traditional modes of content delivery. Despite its rapidly increasing subscriber base, podcast distribution has not been measured or modeled adequately, if at all. As our contribution, we develop a measurement based profile of podcasters. This profile consists of a comprehensive and detailed analysis of interesting characteristics of podcast streams which could be used for analytical studies and simulations. We start by conducting extensive active measurements to characterize 875 popular podcast streams for over a month. The take away message from our study is that podcast traffic is significantly different from the other types of traffic such as Web traffic. For example, we find that podcast file sizes (between 2 and 35 MB) are not only significantly larger than Web files on the average, but they follow a different distribution (a bimodal Gaussian compared to a heavy tail Pareto distribution for Web files). Other interesting aspects of the podcast profile is the expected daily content download (per podcaster), in the range of 2 to 6 MB, and their content distribution patterns. We also find and quantify the heterogeneity in the intensity of content creation, since approximately 14% of podcasters contribute over 54% of files, amounting to about 30% of total byte-content.
Recent empirical studies [6] have shown that Internet topologies exhibit power laws of the form for the following relationships: (P1) outdegree of node (domain or router) versus rank; (P2) number of nodes versus outdegree; (P3) number of node pairs within a neighborhood versus neighborhood size (in hops); and (P4) eigenvalues of the adjacency matrix versus rank. However, causes for the appearance of such power laws have not been convincingly given. In this paper, we examine four factors in the formation of Internet topologies. These factors are (F1) preferential connectivity of a new node to existing nodes; (F2) incremental growth of the network; (F3) distribution of nodes in space; and (F4) locality of edge connections. In synthetically generated network topologies, we study the relevance of each factor in causing the aforementioned power laws as well as other properties, namely
In this paper we study properties of the Internet topology on the autonomous system (AS) level. We find that the normalized Laplacian spectrum (nls) of a graphpro vides a concise fingerprint of the corresponding network topology. The nls of AS graphs remains stable over time in spite of the explosive growth of the Internet, but the nls of synthetic graphs obtained using the state-of-the-art topology generator Inet-2.1 is significantly different, in particular concerning the multiplicity of eigenvalue 1. We relate this multiplicity to the sizes of certain subgraphs and thus obtain a new structural classification of the nodes in the AS graphs, which is also plausible in networking terms. These findings as well as new power-law relationships discovered in the interconnection structure of the subgraphs may lead to a new generator that creates more realistic topologies by combining structural and power-law properties.
The basic entity in complexity theory is a computational problem which, from a mathematical point of view, is simply a function F from finite binary strings to finite binary strings. To make some functions more intuitive these finite binary strings should sometimes be interpreted as integers, graphs, or descriptions of polynomials. An important special case is given by decision problems where the range consists of only two strings usually taken to be 0 or 1. A function F should be realized by an algorithm and there are many ways to mathematically formalize the notion of an algorithm. One of the first formalizations which is still heavily used is that of a Turing machine. However, since we in this paper do not deal with the fine details of the definition, the reader might as well think of a standard computer with a standard programming language. The only idealization needed is that the computer contains an infinite number of words of memory, each of which remains of bounded size. The algorithm has some means of reading information from the external world and also some mechanism to write the result. Time is measured as the number of elementary steps. A finite binary string x in the domain is simply called the input, while the output is the final result of the computation delivered to the outside world. An algorithm solves the computational problem F if, when presented x on its input device, it produces output F(x). A parameter that is important to measure the performance of the algorithm is the length of the input which simply is the number of binary symbols in x. In complexity theory the basic notion of efficiently computable is defined as computable in time polynomial in the input length. The class of polynomial-time solvable decision problems is denoted by P. Establishing that a problem cannot be solved efficiently can sometimes be done but for most naturally occurring computational problems of combinatorial nature, no such bounds are known. Many such problems fall into the
The author proves that unless NP=coR, Max Clique is hard to approximate in polynomial time within a factor n1-ε for any ε>0. This is done by, for any δ>0, constructing a proof system for NP which uses δ amortized free bits. A central lemma, which might be of independent interest, gives sufficient conditions (in the form of a certain type of agreement) for creating a global function from local functions certain local consistency conditions.
Improving and extending recent results of Matuura and Matsui, and less recent results of Feige and Goemans, we obtain improved approximation algorithms for the MAX 2-SAT and MAX DI-CUT problems. These approximation algorithms start by solving semidefinite programming relaxations of these problems. They then rotate the solution obtained, as suggested by Feige and Goemans. Finally, they round the rotated vectors using random hyperplanes chosen according to skewed distributions. The performance ratio obtained by the MAX 2-SAT algorithm is at least 0.940, while that obtained by the MAX DI-CUT algorithm is at least 0.874. We show that these are essentially the best performance ratios that can be achieved using any combination of pre-rounding rotations and skewed distributions of hyperplanes, and even using more general families of rounding procedures. The performance ratio obtained for the MAX 2-SAT problem is fairly close to the inapproximability bound of about 0.954 obtained by Håstad. The performance ratio obtained for the MAX DI-CUT problem is very close to the performance ratio of about 0.878 obtained by Goemans and Williamson for the MAX CUT problem.
The Internet consists of rapidly increasing number of hosts interconnected by constantly evolving networks of links and routers. Interdomain routing in the Internet is coordinated by the Border Gateway Protocol (BGP). BGP allows each autonomous system (AS) to choose its own administrative policy in selecting routes and propagating reachability information to others. These routing policies are constrained by the contractual commercial agreements between administrative domains. For example, an AS sets its policy so that it does not provide transit services between its providers. Such policies imply that AS relationships are an important aspect of Internet structure. We propose an augmented AS graph representation that classifies AS relationships into customer-provider, peering, and sibling relationships. We classify the types of routes that can appear in BGP routing tables based on the relationships between the ASs in the path and present heuristic algorithms that infer AS relationships from BGP routing tables. The algorithms are tested on publicly available BGP routing tables. We verify our inference results with AT&T internal information on its relationship with neighboring ASS. As much as 99.1% of our inference results are confirmed by the AT&T internal information. We also verify our inferred sibling relationships with the information acquired from the WHOIS lookup service. More than half of our inferred sibling-to-sibling relationships are confirmed by the WHOIS lookup service. To the best of our knowledge, there has been no publicly available information about AS relationships and this is the first attempt in understanding and inferring AS relationships in the Internet. We show evidence that some routing table entries stem from router misconfigurations.
Recent work has shown that the node degree in the WWW induced graph and the autonomous system (AS) level Internet topology exhibit power laws. Since then, several algorithms have been proposed to generate such power law graphs. We evaluate the effectiveness of these generators to generate representative AS-level topologies. Our conclusions are mixed. Although they (mostly) do a reasonable job at capturing the power law exponent, they do less well in capturing the clustering phenomena exhibited by the Internet topology. Based on these results, we propose a variation of the recent incremental topology generator of R. Albert and A. Barabasi (see Phys. Rev. Letters, vol.85, p.5234-7, 2000) that is more successful at matching the power law exponent and the clustering behavior of the Internet. Last, we comment on the small world behavior of the Internet topology.
The delivery of IP traffic through the Internet depends on the complex interactions between thousands of autonomous systems (AS) that exchange routing information using the border gateway protocol (BGP). This paper investigates the topological structure of the Internet in terms of customer-provider and peer-peer relationships between autonomous systems, as manifested in BGP routing policies. We describe a technique for inferring AS relationships by exploiting partial views of the AS graph available from different vantage points. Next we apply the technique to a collection of ten BGP routing tables to infer the relationships between neighboring autonomous systems. Based on these results, we analyze the hierarchical structure of the Internet and propose a five-level classification of AS. Our characterization differs from previous studies by focusing on the commercial relationships between autonomous systems rather than simply the connectivity between the nodes.
It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the Feige-Lovasz (STOC92) semidefinite programming relaxation of one-round two-prover proof systems, together with rounding techniques for the solutions of semidefinite programs, as introduced by Goemans and Williamson (STOC94). As a consequence of our approach, we present improved approximation algorithms for MAX 2SAT and MAX DICUT. The algorithms are guaranteed to deliver solutions within a factor of 0.931 of the optimum for MAX 2SAT and within a factor of 0.859 for MAX DICUT, improving upon the guarantees of 0.878 and 0.796 of Goemans and Williamson (1994)
The Internet consists of rapidly increasing number of hosts
interconnected by constantly evolving networks of links and routers.
Interdomain routing in the Internet is coordinated by the Border Gateway
Protocol (BGP). The BGP allows each autonomous system (AS) to choose its
own administrative policy in selecting routes and propagating
reachability information to others. These routing policies are
constrained by the contractual commercial agreements between
administrative domains. For example, an AS sets its policy so that it
does not provide transit services between its providers. Such policies
imply that AS relationships are an important aspect of the Internet
structure. We propose an augmented AS graph representation that
classifies AS relationships into customer-provider, peering, and sibling
relationships. We classify the types of routes that can appear in BGP
routing tables based on the relationships between the ASs in the path
and present heuristic algorithms that infer AS relationships from BGP
routing tables. The algorithms are tested on publicly available BGP
routing tables. We verify our inference results with AT&T internal
information on its relationship with neighboring ASs. As much as 99.1%
of our inference results are confirmed by the AT&T internal
information. We also verify our inferred sibling relationships with the
information acquired from the WHOIS lookup service. More than half of
our inferred sibling-to-sibling relationships are confirmed by the WHOIS
lookup service. To the best of our knowledge, there has been no publicly
available information about AS relationships and this is the first
attempt in understanding and inferring AS relationships in the Internet.
We show evidence that some routing table entries stem from router
misconfigurations
Despite the apparent randomness of the Internet, we discover some surprisingly simple power-laws of the Internet topology. These power-laws hold for three snapshots of the Internet, between November 1997 and December 1998, despite a 45% growth of its size during that period. We show that our power-laws fit the real data very well resulting in correlation coefficients of 96% or higher. Our observations provide a novel perspective of the structure of the Internet. The power-laws describe concisely skewed distributions of graph properties such as the node outdegree. In addition, these power-laws can be used to estimate important parameters such as the average neighborhood size, and facilitate the design and the performance analysis of protocols. Furthermore, we can use them to generate and select realistic topologies for simulation purposes.
We prove optimal, up to an arbitrary epsilon > 0, inapproximability results for Max-Ek-Sat for k 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-Di-Cut, and Vertex cover. For Max-E2-Sat the obtained lower bound is essentially 22/21 1.047 while the strongest upper bound is around 1.074.
We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of 1 2 for MAX CUT and 3 4 for MAX 2SAT. Slight extensions of our analysis lead to a .79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a .758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1 4 and 3 4 respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and repre...
Implementation of algorithm given in
- D R Figueiredo
- Z Ge
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Semidefinite programming solver DSDP 4
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On power-law relationships of the Internet topology
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Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems
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M. Lewin, D. Livnat, and U. Zwick. Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems. In Integer Programming and Combinatorial Optimization (IPCO), LNCS 2337, pages
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