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Topology Selection Criteria for a Virtual Topology Controller based on Neural Memories Y. Sinan Hanay∗, Shin’ichi Arakawa†∗ and Masayuki Murata†∗ ∗ † National Institute of Information and Communications Technology, Japan Email: [email protected] Graduate School of Information Science and Technology, Osaka University, Japan Email: {arakawa, murata}@ist.osaka-u.ac.jp Abstract—This work extends a previously proposed algorithm for virtual topology reconfiguration in all optical networks. Earlier, an algorithm using auto-associative neural memories has been presented. The algorithm stores topologies by assigning equal weights. In this work, we analyzed the effect of weighing topologies differently based on maximum flow, average weighted hop and the age of topology. Although we focus on optical networks, the algorithm and the analysis we present here can be useful in other network domains, such as wireless networks. algorithm that uses neural memories [4]. In this work, we focus on selection criteria for topologies using ASB method. We present different selection criteria for assigning weights to topologies. We focus on optical networks, however, the method and analysis we present here can be applied to other network domains as well. I. I NTRODUCTION In this paper, we focus on VTR problem, which is one of the four aspects of virtual topology design along with lightpath routing, wavelength assignment and traffic routing. [5]. VTR problem is known to be NP-complete [6]. Exact solutions to VTR problem can be obtained using mixed integer linear programming (MILP), but MILP methods become intractable for more than ten nodes. However real world topologies have much more than ten nodes, for example AT&T has 154 nodes and DFN has 30 nodes [7]. Figure 1 illustrates the problem setting. There is a physical network and each node is connected through optical fiber links. In the demonstration it is assumed that two wavelengths can be carried per fiber link. The corresponding virtual topology is shown, and has five lightpaths p1−4 , p4−2 , p2−5 , p5−3 and p3−0 . ASB randomly searches for topologies and stores the found good topologies in a neural network. It is possible to modify the neural memory in a way to give different topologies different weights. We will give the weights according the following metrics. Optical networks have become the primary selection to transmit data over long distances due to the high efficiency of fiber technology over copper wires. In addition, optical networks utilize wavelength-division multiplexing (WDM) which allows carriage of multiple channels over a single fiber. This allows a flexibility in construction of “virtual” (or logical) topologies on top of the physical layer. These topologies are router-level (i.e. intra-AS) topologies, that a network operator can optimize based on traffic demand. Virtual Topology Design (VTD) problem is to find a topology that satisfies some performance (e.g., minimizing maximally loaded link) or a resource metric (e.g., minimizing the number of transceivers). In a physical topology, the edges are optical fiber cables, and in virtual topology the edges are called lightpaths. VTD flow starts with choosing the lightpaths that have to be constructed. Next, a controller runs a routing algorithm in the physical layer to see if the network resources allow the establishment of the chosen lightpaths. Each lightpath is assigned a unique wavelength. However, with wavelength converters, a lightpath can be carried by different lightpaths. Since network traffic may change over time, a reconfiguration of the topology may be needed to meet the performance goals. This second problem of reconfiguration is called virtual topology reconfiguration (VTR). Extra resources may be added or some resources maybe unavailable, and in such a situation a network operator may want to change the goal from performance maximization to resource minimization in order to reduce the energy consumption. VTR has attracted much focus in the last two decades. Methods based on mixed-integer linear programming, heuristics and genetic algorithms were also presented [1]–[3]. Attractor Selection Based (ASB) VTR is a previously proposed II. P ROBLEM S TATEMENT • • • The average traffic weighted hop of the topology The capacity the topology The recency of the topology The first criteria mentions that the topologies having less average weighted hop, should be assigned more weights. We find the capacity of the topology by running a max-flow algorithm. The recency of topology is related to self similarity of the Internet traffic. Assuming that Internet traffic is periodic, we can give more weight to the older topologies, projecting that they are more likely correspond to the future traffic. III. P RELIMINARIES This section presents ASB and auto-associative memories briefly. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: Fig. 1. A virtual network topology example. In the physical topology, each physical link is a fiber, and each fiber can carry two wavelengths. The virtual topology, corresponding to the wavelength assignment, has 5 lightpaths. 12: 13: 14: 15: 16: A. Attractor Selection Based Topology Control 17: ASB utilizes neural networks. Neural networks are useful tools, when a pattern exists in the data. Here, we assume that self-similarity of Internet traffic can be considered as a pattern, and we randomly search for network topologies, when good topologies are found we add them in a neural memory. After some time, the traffic characteristic will change and there may be other topologies that can serve that traffic better. After some period of time, the first topology found will be highly likely to serve again. At each time step, the topology is updated according to a system dynamics given by Equation 1. Algorithm 1 presents the overview of ASB. One key part with ASB is selection of topologies. First, it measures the utilization of the maximally loaded link umax , and based on this it calcualtes the performance metric VG . Then by ComputeW eightM atrix it adds the found topology to memory using auto-associative memories, which has been explained in the next section in detail. On line 4, ASB updates its system state corresponding to each lightpath variable by the following equation [4]: ⎞ ⎤ ⎡ ⎛ n % dxi (1) = ⎣f ⎝ wij xj ⎠ − xi ⎦ VG + N (0, 1) dt j=1 +, * +, * auto-associative memory random walk where N (0, 1) is the standard normal random variable. Here VG is performance metric, which is VG = 1 umax (2) The system dynamics shown in Eq. (1) consist of two components: auto-associative memory and random walk. ASB uses binary auto-associative memories, an extension of ASB that uses multistate memories was also proposed [8]. When the topology performs well, VG is high, the auto-associative memory part steers the topology selection; however when VG is low, the new topologies are searched randomly. procedure ASB( time t ) VG ← 1/umax ComputeW eightM atrix(VG, t) ComputeExpression() U pdateLightP ath() procedure C OMPUTE W EIGHT M ATRIX( VG , time t) if (VG (t − 1) < 0.5 & VG (t) > 0.5) then for i ← 1, n do for j ← 1, n do W[i, j] −= Hebb(i, j, Ak ) ◃ Eq. 6 for i ← 1, n × (n − 1) do Ak [i] = LightP ath[i] ◃ Update attractors for i ← 1, n do for j ← 1, n do W[i, j] += Hebb(i, j, Ak ) k = (k + 1) mod numberOf Attractors 18: 19: 20: 21: 22: function C OMPUTE E XPRESSION ◃ by Eq. 1 for i ← 1, n × (n − 1) do for j ← 1, n × (n − 1) do x[i] += ComputeDeltaExp(i, j) 23: 24: 25: 26: 27: 28: 29: procedure U PDATE L IGHT PATH for i ← 1, n × (n − 1) do if (x[i] > 0.5 & CanEstablish(i)) then EstablishLighpath(i) ◃ LightPath[i]=1 else if (IsEstablished(i) & x[i] < 0.5) then RemoveLighpath(i) ◃ LightPath[i]=0 Algorithm 1. ASB method B. Auto-associative Memories The main function of auto-associative memories is to correct noisy inputs, or find the closest matching stored pattern for a given input. Auto-associative memories are neural memories, and the queries are done by a matrix multiplication [9]. In that sense the neural memories are different that computer memories where values are read from bitcells. The output O for an input I is calculated by O = sgn(I W) (3) Here O and I are row vectors, sgn() is the sign function given by ⎧ ⎪ ⎪ ⎨ 1 if x ≥ 0 sgn(x) = (4) ⎪ ⎪ ⎩ −1 if x < 0 In ASB, the set of stored topologies are called “attractors” as the system state is attracted towards this topologies. That is, at any point, system tries to converge one of the stored topologies. These stored topologies are found by random walk, and they satisfy the performance criteria. A topology Ti for and N node network can be described by a binary vector of size N 2 − N . If there is a lightpath between two pairs, corresponding bit in Ti is set to 1, otherwise 0. However, in neural memories it is better to code values bipolar as (1, -1) rather than (1, 0) [10]. Thus, we adopted this notation in our work. Thus, an attractor is defined as a k by N 2 − N matrix consists of topologies, where k is the number of topologies that can be stored. For this A, the weight matrix W is calculated by W = A⊤ A (5) Here, W is the auto-correlation matrix of A, which is based on Hebbian learning [11]. We analyzed the effects of different learning algorithms previously [12]. However we use Hebbian learning in order to keep the discussion simpler. Hebbian Learning: ASB’s auto-associative memory uses Hebbian learning to store and read the elements. In our case, virtual topologies are stored in a auto-associative memory of which weight matrix is constructed using Hebbian learning. Weight matrix can be constructed, using Hebbian learning weight update rule below: ∆wi,j = αli lj (6) The learning rate is denoted by α, it was kept constant equal to 1 in previous works [4], [8], [12]. The weight matrix can be thought of summation of several weight matrices that corresponds to each topology. Each topology becomes W = α1 W1 + α2 W2 + .. + αN WN (7) The weight matrix can also be constructed using pseudoinverse [10], [13]. Even though, pseudoinverse matrix performs better recognizing noisy inputs, it is shown that the calculation of pseudoinverse matrix is two order of magnitude slower than that of auto-correlation matrix [12]. In order to clarify the discussion so far, let assume we want to store two topologies T1 and T2 . For a running example, let T1 = [1 1 -1 -1] and T2 = [-1 -1 1 -1]. Then the attractor matrix A is represented as A = [T1 ; T2 ], which is 2 3 1 1 -1 -1 A= -1 -1 1 -1 and by Equation 5, the weight matrix corresponding to this A becomes ⎡ ⎤ 2 2 -2 0 ⎢ 2 2 -2 0 ⎥ ⎥ W=⎢ ⎣ -2 -2 2 0 ⎦ 0 0 0 2 Then, outputs can be calculated by Equation 3 as sign(T1 W ) = T1 and sign(T2 W ) = T2 . It is straightforward to check 6 7 T1 W = 6 6 -6 2 6 7 sign(T1 W) = 1 1 -1 1 Let assume that a noisy input Tnoisy is given as 6 7 Tnoisy = -1 -1 -1 -1 , (8) Note that this noisy input has a hamming distance of 1 to T1 and 2 to T2 . Thus, it is expected that the autoassociative memory has to return T1 for this noisy input. It is straighforward to check sign(Tnoisy W ) = T1 holds. In other words, when Tnoisy is presented, the memory returns T1 . However, the memory cannot correct every possible noisy input. For example, [1 1 -1 1] cannot be corrected. Even though this input has a hamming distance of 1 to T1 , it is exact complement of T2 . This is a well known issue with auto-associative memories [10]. In the construction of weight matrix, both topologies have equal weights. The construction of weight matrix can also be thought as sum of the weight matrix belonging to each topology. W1 = T1⊤ T1 W2 = T2⊤ T2 W = α1 W1 + α1 W1 (9) (10) (11) In our example, we took α1 and α2 as 1. However, if we take α1 >> α2 then the W will be biased toward T1 . With such a modification, the second noisy input, which we presented above, can be corrected with this new weight matrix. IV. C RITERIA FOR T OPOLOGY S ELECTION As discussed in the previous section, we refer stored topologies as “attractors”. Previously, each attractor has same weights. However, some attractors may perform better than others. We present the following weight selection strategies for evaluating topologies. Our motivation in selecting different weights for different attractors is based on a few factors. First, the attractor update strategy is not optimal (line 8), and it is hard to devise an optimal update. For example, an attractor that is found when the traffic load is low might not be as efficient as an attractor that is found when the network is highly loaded. Also, a topology can be more efficient intrinsically as we show in terms of max-flow capacity in Section V-B. 1) F low: Each topology has a different maximum flow, thus each attractor can be assigned a different weight. We calculated the maximum flow of the topologies using EdmondsKarp max-flow algorithm [14]. α(Ti ) = C × max f low (12) where C is normalizing constant. The equation above shows that any topology Ti has a weight proportional to its maximum flow. The top 10% of the loaded links were taken, and the maximum flow between the corresponding node pairs were calculated. 2) Recent: Under the assumption that the internet traffic is self-similar in small time-scales [15], we gave higher weights to recently found topologies. α(Ti ) = 2 × α(Ti−1 ) (13) Here, one important aspect is the time between rounds. That is, Ti − Ti−1 . 3) W eightedHop: Average weighted hop is a commonly used metric in evaluating performance of VTR algorithms. When an attractor is found, its average weighted hop is calculated and we associate this metric with is attractor weight by the following equation. α(t + 1) = C × 1 wh (14) where wh is the traffic weighted average hop, which is desired to be lower. V. S IMULATIONS This section presents the comparison of the four methods using an optical network simulator, which has been implemented in C#. A. Simulation Setting The physical network we simulated consists of 100 nodes. We used the the log-normal traffic, that was explained [16]. Dijsktra’s shortest path algorithm was used for lightpath and traffic routing. We assumed that the nodes are equipped with wavelength converters. We run each method with varying number of stored topologies as: 2,4 and 8. Each method was run 36 times. Each run was run for 400 rounds, and the performance metric was summed for 400 rounds. B. Simulation Results performance Figure 2 shows the simulation results. F low outperforms all the other methods, however W eightedHop and Recent failed to show any significant improvement over ASB. The results look promising, as F low’s median is 34% higher than that of ASB. The Z score for difference between F low and ASB was found to be 24.84, which gives a p-value of less than 0.00001. Thus we conclude that F low improves the performance, and the improvement is from 23% to 29% with 99% confidence interval. The results for Recent and W eightedHop is insignificant. We noticed no effect of the number of stored patterns between 4 and 8, but they are marginally better than size of 2. However, this depends on the characteristics of traffic matrix. We are currently analyzing how to set the weights effectively. 400 350 300 250 200 150 100 50 0 Flow Recent WeightedHop ASB Fig. 2. The median for F low is 34% higher than that of ASB, while W eightedHop is 5% lower than ASB. VI. C ONCLUSION VTR problem has generated huge interest in the last decade. ASB method is one highly adaptive VTR method. ASB uses neural memories to store topologies. However, the topology selection is purely based on the congestion faced with the topology. In this work, we analyzed the topology storing strategies of ASB. We proposed new selection metrics based on average weighted hop, max-flow and age of the found topology. We showed that topology selection based on F low ensures the selection of highly efficient topologies, by improving ASB 26% on average. 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