Integrating local and partial network view for routing on scale-free networks

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1 . RESEARCH PAPER. SCIENCE CHINA Information Sciences October 2013, Vol : :10 doi: /s y Integrating local and partial network view for routing on scale-free networks TANG MingDong 1, ZHANG GuoQiang 2,3, SUN Yi 3,LIUJianXun 1, YANG Jing 4 &LINTao 5 1 Key Lab of Knowledge Processing and Networked Manufacturing, Hunan University of Science and Technology, Xiangtan, , China; 2 School of Computer Science and Technology, Nanjing Normal University, Nanjing , China; 3 Institute of Computing Technology, Chinese Academy of Sciences, Beijing , China; 4 China Mobile Research Institute, Beijing , China; 5 Institute of Acoustics, Chinese Academy of Sciences, Beijing , China Received April 13, 2012; accepted July 20, 2012; published online August 30, 2012 Abstract Traditional routing schemes, such as OSPF, optimize data plane routing efficiency by maintaining full view of the network at the control plane. However, maintaining full network view and handling frequent routing information updates are costly in large-scale complex networks, which are considered to be the root causes for the routing scalability issue. Recently, it is suggested that routing on local or partial information is plausible if slight performance degradation is acceptable. This paper proposes a routing scheme, operating on an integrated network view at each node that consists of its local neighborhood and a globally unique skeleton tree. This scheme significantly reduces storage, communication and processing costs. On scale-free networks, this benefit only comes at the cost of marginal performance degradation, which implies that it is not worthwhile to do shortest path routing based on full view of the network on scale-free networks. In contrast, the routing efficiency is severely aggravated on purely random networks, indicating the inappropriateness of this scheme and the rationality of maintaining full network view on random networks. Keywords routing, scale-free networks, complex networks, power-law Citation Tang M D, Zhang G Q, Sun Y, et al. Integrating local and partial network view for routing on scale-free networks. Sci China Inf Sci, 2013, 56: (10), doi: /s y 1 Introduction Routing is a fundamental functionality of communication networks. Usually, a routing scheme can be decomposed into two logically independent components: the control plane that exchanges network information and builds a distributed routing table at each node, and the data plane that relies on the routing table to correctly forward data packets towards their destinations. How to build the routing tables thus lies at the heart of a routing scheme. An extreme but commonly adopted approach is for every node to maintain a full view of the network through information flooding, and construct shortest paths to every other node according to this view, such as OSPF [1]. However, Corresponding author ( guoqiang@ict.ac.cn) c Science China Press and Springer-Verlag Berlin Heidelberg 2012 info.scichina.com

2 Tang M D, et al. Sci China Inf Sci October 2013 Vol :2 considering the scale of real networks, maintaining full view of the network up-to-the-minute places heavy burden on either the storage or the processing capabilities needed to handle network dynamics. It is already known that large volume as well as rapid growth rate of routing information updates is problematic to the scalability of the Internet [2,3]. Another extremity is to do routing only on local information. Examples of this kind of routing include greedy routing on navigatable small-world networks [4,5], greedy routing on pre-constructed spanning trees [6,7], random routing [8 10], and the largest degree based routing [11]. However, these routing schemes either have to program the network topology and node names according to the routing schemes in prior, or cannot guarantee the relative shortness of paths on real networks. Another category of routing schemes performs routing on partial network view [12 14]. The primary goal is to alleviate the impact of network dynamics, i.e., reducing routing information updates and cutting down the frequency of route re-computation, and meanwhile to ensure routing efficiency, i.e., relative shortness or bounded path lengths. These routing schemes can be viewed as variations of the link state routing. But unlike traditional link state routing that requires a complete network topology to be maintained at each node, these routing schemes only maintain a subgraph of the entire network topology, and compute routes based on this subgraph. Real complex networks typically show special structural properties. Most real networks, e.g., Internet and WWW, can be categorized into the so-called scale-free networks [15 18], i.e., the degree distribution p(k) follows p(k) k r. As previous work has demonstrated, although the accuracy of routing schemes should not depend on the network topology, the efficiency of these schemes is strongly tied to particular network topologies [19 22]. A natural question is: for networks with given structural properties, is it worthwhile to maintain full network view in the routing process? In other words, does the gain arising from this resource-consuming activity outweigh its cost? In this paper, we show that routing on partial network view can also utilize the structural properties of network topology to optimize its routing performance. We propose a routing scheme specialized for scale-free networks. Each node relies on a subgraph which is a union of a uniform network skeleton and its distinct neighborhood to do routing. Simulation results show that, compared with routing on full network view, the degradation of routing efficiency of this routing scheme is marginal on scale-free networks, whereas it allows significant reduction in storage, communication and computation overhead. So, maintaining full network view is not economic for scale-free networks. On the other hand, non-trivial routing efficiency degradation is observed when the scheme is applied to random networks, demonstrating that appropriateness of this scheme is topology-dependent. 2 Routing scheme Our routing scheme is built upon the concept of skeleton of complex networks proposed by Kim et al. [23]. In complex networks, the roles different nodes and edges play in forwarding network traffic may vary vastly. A simple metric that predicts the network traffic a node or edge has to handle is the betweenness δ centrality [24 26], which is defined as B(w) =Σ (w) (u,v) u v δ(u,v),whereδ(u, v) denotes the total number of shortest paths between u and v, andδ (w) (u, v) denotes the number of those shortest paths that pass through w. On scale-free networks, the betweenness centralities of either the nodes or the edges are very heterogenous. Some nodes and edges have very high centralities, meaning they are responsible for handling more traffic. It is suggested to use the spanning tree consisting of edges with large betweenness centralities as the network s skeleton [23,27], which can be viewed as the communication kernel of the corresponding network 1). It is shown that for scale-free networks, the proposed skeleton shows very similar degree distribution and joint degree distribution to the original network. However, routing solely on the skeleton tree can substantially increase lengths of some paths. To combat this problem, we also incorporate the local topological information in the routing process. Following is a 1) The skeleton tree is constructed in the following way. Initially, set the skeleton to be an empty set. Then add the edges one by one in decreasing order of betweenness values to the skeleton without causing a loop until a tree is formed

3 Tang M D, et al. Sci China Inf Sci October 2013 Vol :3 Figure 1 An illustrative example for Observation 1. detailed description of our proposed routing scheme: Routing table: each node maintains a partial view of the network, which is a union of its direct links (different from node to node) and a globally unique skeleton tree (identical to all nodes). Each node bases on this partial view to build its own shortest path routing tree to other nodes. Packet forwarding: when a packet arrives at node u, if the destination v N(u)(N(u) denotes u s direct neighbors), then the packet will be forwarded directly to v, otherwise, it will be forwarded to the next hop along the shortest path determined by its partial view. Network dynamics handling: when a link not on the skeleton tree fails, this event does not need to be propagated across the network. Only the two adjacent nodes need to adjust their routing paths to the other endpoints based on their partial views, while all other nodes remain unaffected. when a link on the skeleton tree fails, the two adjacent nodes first check whether their partial views remain connected. If at least one of the partial views remains connected, then the corresponding node chooses a candidate link that is able to connect the two disconnected components of the original skeleton tree and broadcasts this information to the whole network. Otherwise, the link failure event is flooded across the network. Upon receiving this event, each node other than the two adjacent nodes which is still able to reach all other nodes based on its partial view chooses a link that can bridge the two disjunct components of the original skeleton tree (Observation 1 guarantees that such node should exist if the whole network does not fall apart), and broadcasts the information of this candidate link to the whole network 2). Finally, each node executes a deterministic procedure in a distributed manner to agree on a link among these candidate links to be added to the skeleton tree. Such distributed yet deterministic election procedures are well equipped in the literature of distributed algorithms [29], e.g., choosing the link with the smallest node IDes. Observation 1. If a link failure on the skeleton tree does not cause a bipartition of the network as a whole, then after removing this link on each node s partial view, there is at least one node whose partial view remains connected. Proof. Suppose a link on the skeleton tree fails, then the skeleton tree falls apart into two disconnected subtrees, T l and T r (see Figure 1). Considering the left subtree T l, for any node v i V (T l ), if one of its direct neighbors is located in the right subtree T r, e.g., u j,thene(t l ) e(v i,u j ) E(T r )formsa new spanning tree. So, v i s partial view remains connected. Otherwise, for each node in V (T l ), its direct neighbors are all in V (T l ). Hence, T l is disconnected from T r, contradicting the assumption that the whole network is connected. Figure 2 gives an example to illustrate our routing scheme. Figure 2(a) and Figure 2(b) are the original 2) Mechanisms such as the exponential backoff algorithm can be used to suppress the amount of this information [28]

4 Tang M D, et al. Sci China Inf Sci October 2013 Vol :4 Figure 2 An example of the routing scheme. network and its skeleton tree respectively. Figure 2(c) (e) are the routing trees for node A, B, C respectively 3) We observe that each node bases on the skeleton tree and its neighborhood to build its distinct routing tree. We use dashed lines to denote those skeleton links that are maintained by each node but do not occur in its routing tree. If node A detects the failure of link A B, a link not on the skeleton tree, then A only has to find a new path to B based on the skeleton tree, so does B. This failure can be confined locally, and does not need to be propagated outside A, B. If, on the other hand, a skeleton tree link C E fails, then since neither C nor E s partial view remains connected, this failure event will be propagated across the whole network. After receiving this update message, A, G find their partial views disconnected; however, B, D, H and F can still reach all nodes. B decides that the link B-F can be added to the skeleton tree to form a new skeleton. This message can be broadcast across the network. Finally, a new skeleton is agreed on all nodes, as is shown in Figure 2(f). 3 Performance evaluation We tested the routing performance of our proposed routing scheme on various synthesized complex networks generated by different network models, i.e., ER model [30,31], BA model [18], and PFP model [32], and on real AS-level Internet topology. The major topological properties of the synthesized networks with 1000 nodes are shown in Table 1. Figure 3 plots the max degree, average shortest path length (ASPL) and diameter for the three kinds of synthesized networks with different sizes. The AS-level Internet topology is the ITDK0304 data kit containing 9204 nodes and edges, whose power-law exponent γ Since every routing scheme is a tradeoff between the routing efficiency and the overhead for communication, computation and storage, we present detailed analysis on this tradeoff. 3.1 Routing efficiency One major concern of routing efficiency is the cost, or simply path length of the paths taken by a given routing scheme. Stretch is usually used to evaluate the quality of paths for a given routing scheme. For a given node pair u, v, the stretch is defined as the ratio of the path length under the given routing scheme Γ to the shortest path length between them. Here we use average stretch to evaluate the routing efficiency of a given routing scheme, which is formally defined as s = 2Σ i<j d (Γ ) (i,j) d(i,j), (1) N(N 1) 3) Note that from the point of view of network structural equivalence, there are three equivalent node classes: {A, G},{C, E}, {B, D, H, F}, anda, B, C fall into different classes

5 Tang M D, et al. Sci China Inf Sci October 2013 Vol :5 Table 1 Major topological properties of the synthesized networks used in the simulation. Here, N is the number of nodes, k max is the largest degree, k is the average degree, D is the diameter, d is the average path length, C is the average clustering coefficient, and γ is the power law exponent of the BA and PFP network Network N k max k D d γ C ER BA PFP Figure 3 k max, d, andd of three kinds of synthesized networks with different sizes. (a) k max; (b) d ; (c)d. where d(i, j) is the shortest path length between i and j, andd (Γ ) (i, j) is the path length between i and j under routing algorithm Γ. Figure 4 plots the average stretch as as a function of network size on different synthesized networks for two routing schemes: one is our proposed routing scheme routing by skeleton with local links 4),and the other is routing on the skeleton tree only. Two observations can be made. First, the average stretch is relatively low on power-law networks and is stable as the network grows, whereas it is much higher on random networks, regardless of the routing scheme being used. Second, on a particular network, augmenting the routing decision with local links can significantly reduce the average stretch. On real AS graph(itdk0304), the average stretch of the Skeleton only routing scheme is 1.23, whereas if local links are incorporated, the average stretch can be reduced to Figure 5 presents the cumulative distribution of the stretch on networks listed in Table 1 (Figure 5(a)) and on real AS graph ITDK0304 (Figure 5(b)) for our proposed routing scheme. It is observed that relying on the integrated view for routing, 51% of the paths on the BA network are shortest paths, and on the PPF network, this ratio goes up to 85%. Since the PFP network is more heterogenous than the BA network, as evidenced by k max and γ, this observation also implies that this routing scheme imposes 4) Here local links refer to those direct neighboring links that are not in the skeleton

6 Tang M D, et al. Sci China Inf Sci October 2013 Vol :6 Figure 4 Average stretch of two routing schemes on different synthesized networks. Each plot line is identified by a two tuple. The first element denotes the network kind, and the second element denotes the routing scheme being used. Skeleton with local links is our proposed routing scheme, and Skeleton only is routing on the skeleton tree only. Figure 5 Stretch distribution of our routing scheme on different networks. (a) synthesized networks; (b) AS graph (ITDK0304). less impact on the quality of paths for more heterogenous networks. In sharp contrast, only less than 20% of the paths on the ER network are shortest paths, while more than 30% of the paths are two times longer than the shortest path length. For real AS Internet topology, the result is quite similar to PFP, which conforms to the reality that the PFP model accurately models the AS-level Internet topology [32]. These results show that routing on the skeleton tree augmented by local links only has slight impact on the quality of paths for scale-free networks. In contrast, the performance degrades significantly for the ER network. This reconfirms the philosophy that routing efficiency strongly depends on network topologies. Figure 6 presents the average stretch as a function of the average degree for the BA network and ER network. The networks used here are of equal number of nodes (1000 nodes), but with varying average degrees (see Figure 7 for the basic properties of these networks). For a given size, 5 instances are generated, and the results presented here are an average over 5 instances. Since networks generated by the PFP model always have fixed average degree, we omit the PFP model in this figure. It can be seen that the average stretch of this routing strategy is largely unaffected by the network density as long as it gets moderately dense enough. This property makes our proposed routing scheme more promising and attractive, because the denser the network is, the more link update information can be potentially suppressed and the fewer route re-computations needed, and consequently more gain can be expected out of applying this routing scheme. Another concern of routing efficiency is the load distribution. Acceptable routing schemes should not easily cause congestion on the network. We generate a sufficiently large number ( in our exper-

7 Tang M D, et al. Sci China Inf Sci October 2013 Vol :7 Figure 6 Average stretch of our proposed scheme on BA and ER networks with increasing density. Figure 7 k max, d and D for ER and BA networks of 1000 nodes with varying densities. (a) k max; (b) d ; (c)d. iment) of packets with random sources and destinations, and define the load imposed on a node as the number of packets passing through a node following the particular routing scheme. Figure 8 shows the load distribution of three routing schemes on the networks listed in Table 1 and on the real AS graph ITDK0304. These three routing schemes are: routing on the full network view (Full), routing on the skeleton tree only (Skeleton only), and routing on skeleton tree augmented by local links (Skeleton with local links). It is observed that on power-law networks, these three schemes show similar load distribution features. This is because on scale-free networks, even with full network view, shortest path routing itself enforces most of the traffic to go through the network skeleton, whereas on the ER network, the shortest path routing significantly outperforms the other two schemes in the sense that routing on full network view can make the traffic more balanced by routing on more links. 3.2 Overhead The overhead of a routing scheme typically involves the space needed to store the raw routing information as well as the routing table, the communication cost to exchange routing information between routers and the computation overhead to handle the dynamically received routing information. Traditional shortest path routing such as OSPF requires maintenance of the entire network topology, which has O(M) space requirement, where M is the number of links. In our routing scheme, each node v maintains at most N 1+d v 1=N + d v 2 connections, where N is the number of nodes and d v is the degree of v. Consequently, the overall space requirement of all nodes is at most Σ v V (N + d v 2) = N 2 +2M 2N, which means the average storage overhead on each node is N 2+ k, significantly smaller than maintaining the full network topology, where k is the average degree. The communication overhead and its ensuing route re-computation is believed to be one of the route causes that prohibit the scaling of a routing system. Communication overhead is mainly caused by network dynamics. Here we only consider link failures. In traditional OSPF-like routing, each time a link

8 Tang M D, et al. Sci China Inf Sci October 2013 Vol :8 Figure 8 Load distribution of three routing schemes on three different synthesized networks as well as on the real AS graph ITDK0304. (a) ER; (b) BA; (c) PFP; (d) AS graph (ITDK0304). fails, its two endpoints flood this event across the whole network so that all nodes are notified of this event and perform routing table re-computation. So, the number of messages generated per link failure is approximately 2M. In our scheme, since failures of non-skeleton links do not trigger message flooding, it is expected that the number of messages generated will be significantly reduced. Indeed, when a link on the skeleton tree fails, if the two adjacent nodes can find a link that can bridge the two disconnected components of the original skeleton tree, then the number of messages is M or 2M, depending on whether only one endpoint or both endpoints satisfy the above condition. Otherwise, if none of the two adjacent nodes partial view remains connected, then the number of messages is about (λ +2)M according to our routing scheme, where λ is the number of nodes whose partial view remain connected after the link failure 5). Figure 9 presents the simulation results of the average number of messages generated per link failure for our routing scheme and traditional OSPF-like shortest path routing scheme on both the BA and ER networks with varying densities (networks used here are the same as in Figure 6, with their basic properties presented in Figure 7). Each time, we randomly select a link, break it down, and compute the number of messages generated. For each network, we repeat the previous process for 1000 times. For a given size, we generate 5 network instances and the results presented here are the average over 5 instances. In addition, we performed the simulation on PFP networks of 1000 nodes. We generated 5 instances of the PFP network, and performed the same simulation as described above. Since PFP model has fixed degree, we don t have the plots for PFP networks with degrees other than 6. It is shown that in BA and ER networks, with traditional routing scheme, the number of messages generated per link failure grows linearly with the network density, approximately 2M = N k. On the contrary, the resulting number of messages per link failure of our routing scheme on BA network stays nearly constant as the network density increases. Under the random link failure model, the probability that a failed link is on the 5) According to our routing scheme, when none of the two adjacent nodes partial view remains connected, the link failure is first flooded across the network by the two endpoints, then the λ nodes whose partial view remain connected after the link failure will choose a candidate link and broadcast this information across the network. Hence the total number of messages is about (2 + λ)m

9 Tang M D, et al. Sci China Inf Sci October 2013 Vol :9 Figure 9 Average number of messages generated per link failure. skeleton tree is about 2/ k. On BA networks, the probability that the partial views of the two adjacent endpoints of the failed link remain connected is almost 100%, hence the average message overhead per link failure is approximately 2 k 2M =2N. However, it should be noted that when network is not dense enough, our routing scheme on ER network may generate higher volume of messages than traditional routing scheme. This is because on ER networks of low density, the probability that none of the two endpoints of the failed link finds its partial view connected cannot be neglected, e.g., 2.4% when k =4. However, k, the number of other nodes whose partial views remain connected, is very large, e.g., when k = 4. Anyway, on scale-free networks, our scheme generates much smaller number of messages per link failure than the traditional scheme, as evidenced by both the BA network and PFP network. The route re-computation overhead is simple to analyze. For traditional OSPF-like routing, in principle, each link failure event triggers route re-computation on all nodes. In comparison, for our routing scheme, only when links on the skeleton tree fails, should each node perform route re-computation. 4 Discussion and conclusion Scaling the routing system is of paramount importance for the large, complex and dynamic networks. We proposed a routing scheme that relies on both the local topology information and a globally unique skeleton tree to build the routing tables. It can substantially reduce the storage overhead and the impact of network dynamics on the routing system without sacrificing routing performance on scale-free networks. However, for purely random network, this scheme significantly degrades the routing performance, either in the sense of average stretch or the traffic balance. Hence, we argue that for scale-free networks, it is not economic to maintain full view of the network upto-the-minute while performing shortest path routing. To take the full advantage offered by full network view on scale-free networks, it is more appropriate to relax the shortest path routing to achieve good balance between other routing objectives, e.g., path quality and load balance. For example, the EFR [20] and HAP [22] routing schemes try to balance network traffic among network nodes/links on scale-free networks without significantly increasing path lengths. These routing schemes are grounded on the full network view assumption, and hence can be viewed as optimizing the routing from another angle. In contrast, unlike scale-free networks, it is rational to bear the cost of maintaining the full network view in random networks even under the shortest path routing paradigm. It should be noted that our approach focuses on constructing per-node partial topology view upon which routing is performed, whereas there are other routing schemes that focus on filtering routing updates, for example, the XL routing scheme [3]. In essence, these two schemes are orthogonal (the former focuses on the topology being used, whereas the latter emphasizes on developing filtering policies) and thus complementary to each other. Policies developed in the XL scheme can be applied to the partial

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