Static and Dynamic Properties of Small-world Connection Topologies Based on Transit-stub Networks

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1 Static and Dynamic Proerties of Small-world Connection Toologies Based on Transit-stub Networks Carlos Aguirre Fernando Corbacho Ramón Huerta Comuter Engineering Deartment, Universidad Autónoma de Madrid, 2849 Madrid, Sain Many real comlex networks are believed to belong to a class called small-world (SW) networks. SW networks are grahs with high local clustering and small distances between nodes. A standard aroach to constructing SW networks consists of varying the robability of rewiring each edge on a regular grah. As the initial substrate for the regular grah some secific toologies are usually selected such as ring-lattices or grids. However, these regular grahs are not suitable for modeling certain hierarchical toologies. A new regular substrate is roosed in this aer. The roosed substrate resembles toologies with certain hierarchical roerties more accurately. Then, different dynamics insired by networking rotocols are used to characterize dynamical roerties of a network. Measuring transmission times and error rates lead us to consider networks with SW features as the most reliable and fastest, regardless of the routing olicies.. Introduction In the ast few years, growth in the field of communication networks [2, 3, 4], multi-agent systems [5], architectures [7], and comlex systems have generated many studies focused on network toologies. In this framework grahs reresent the most adequate abstract reresentation of a network, where each node reresents a router or host and each edge reresents a connection between routers or hosts. There are many investigations that deal with grahs that model networks [8, 9, ]. A common feature in all these studies is the set of grahs subjected to study such as star, ring, grid, regular, and random networks. Recently the use of transit-stub networks [] has been roosed as an imrovement for Electronic mail address: Also Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA, Comlex Systems, 4 (23) 28; 23 Comlex Systems Publications, Inc.

2 2 C. Aguirre, F. Corbacho, and R. Huerta modeling real communication networks. Transit-stub networks cature most of the relevant toological characteristics in terms of the following metrics: diameter, average degree, average ath length, number of connected comonents, and number of biconnected comonents. Many interesting roerties arise when the network toology itself is an intrinsic arameter that can be modified using some secific rules. These methods allow slowly varying the metrics of the grahs in such a way that the dynamical behavior of any given roblem can be carefully analyzed. In [3] a method to study the dynamic behavior of networks when the network is shifted from a regular, ordered network to a random one is roosed. The method is based on random rewiring with a fixed robability for every edge in the grah. We obtain the original regular grah for, and a random grah for. This method shows that the characteristic ath length (i.e., the average distance between nodes measured as the minimal ath length between them) decreases with increasing values of much more raidly than does the clustering coefficient (i.e., the average number of neighbors of each node that are neighbors between them). It was found that there is a range of values for where aths are short but the grah is highly clustered. The toologies in this range are known as small-world (SW) toologies. SW toologies resent some very interesting features that make them suitable for efficient transmission of commodities [3, 4, 5]. Moreover, they aear in many real life networks [6, 7], the World Wide Web [8], as a result of natural evolution [9], or as result of a learning rocess [2]. In [6], for examle, it is shown that on SW networks coherent oscillations and temoral coding can coexist in synergy in a fast time scale on a set of couled neurons. As the initial substrate for the generation of SW, the use of a ringlattice or a grid is usually roosed [4, 2]. They are used because these grahs are connected, resent a good transition from regular to random, and there are no secific nodes on them. The substrate selected in [3] resents a single biconnected comonent, indeendent of the number of nodes in the grah and does not resent a hierarchical structure. In many networks there are secial nodes that connect backbones with subnetworks. Furthermore, in [], the grahs generated are organized following a clear hierarchical structure and resent a high number of biconnected comonents. Formally, a biconnected comonent is a maximal set of edges such that any two edges in the set lie on a common single cycle. Intuitively, a biconnected comonent is a subnetwork connected by a set of oututs to the network. The number of biconnected comonents in the stub-domain grahs grows with the number of nodes in the grah. Realistic models of network toologies are necessary in order to obtain correct behavior for most kinds of algorithms and olicies over these hierarchical networks [2]. These models and methods should also scale to increasingly larger networks because Comlex Systems, 4 (23) 28

3 Small-world Connection Toologies 3 communication networks continue to grow in size and imortance. Our objective is to construct regular grahs that can resemble the toology of hierarchical networks but are able to shift from regular to random in the same way ring-lattices do. We also want to study some well known routing olicies over this new kind of regular grah. In this work we resent a theorem of existence along with a method for building regular grahs with a high number of biconnected comonents. In these grahs the number of biconnected comonents is a function of the number of nodes in the grah and the number of neighbors of each node. We rovide an analytic exression for the number of biconnected comonents, the characteristic ath length, and the cluster coefficient. Next, we investigate the behavior of some networking insired dynamics using this transit-stub based network. In articular, we analyze by means of intensive comuter simulations the dynamical behavior of information broadcasting, multicasting, and unicasting on a set of grahs ranging from transit-stub regular to random. The structure of the article is as follows. First, we show the existence of regular grahs with several biconnected comonents and resent an algorithm for building this tye of grah. Additionally we rovide an analytic exression for the characteristic ath length and cluster coefficient for this tye of grah. Second, we study several static arameters for this kind of grah and investigate toological metrics when the grah is shifted from a regular to a random situation. Then, we comare these metrics with the results obtained by the regular-lattice substrate. And finally, we analyze by means of comuter simulations the behavior of information broadcast, multicast, and unicast over both tyes of substrates to determine those grahs that achieve minimum transmission error and delivery delay. 2. Biconnected small-worlds This section describes an algorithm to generate a SW grah with a higher number of biconnected comonents than do lattices. In the literature about SW, the ring-lattice is used as the initial grah substrate due to the following advantages. Ring substrates are connected. They have a good transition from large to small. They have no secial nodes such as trees or stars do. The ring-lattice substrate has some characteristics that make it unsuitable for modeling some secific tyes of networks such as hierarchical multi-agent networks or the Internet. These networks have secial nodes, the ones that connect stubs with backbones, and resent a number of biconnected comonents greater than one. However, the number Comlex Systems, 4 (23) 28

4 4 C. Aguirre, F. Corbacho, and R. Huerta of biconnected comonents for any grah obtained from the ring-lattice substrate by the Watts Strogatz method is close to one. In this section we develo a constructive method for building a finite regular grah with multile biconnected comonents that we call a transit-stub regular grah. This grah is the initial substrate for our SW model of hierarchical networks. It is not clear a riori that such grahs exist and, if they exist, it needs to be roved that they can be built algorithmically for given values of n and k. For a given grah G, we use n, G as the number of nodes in the grah, k i as the number of neighbors that node i has, and we use k to denote n i k i /n, that is, the average number of neighbors for each node. Remember that these grahs must be connected and have a good transition from large to small in the same way that ring-lattice substrates have. Here we resent a theorem that establishes the existence of regular finite grahs with several biconnected comonents. The theorem is constructive, that is, it roves existence by building such a grah. The algorithm builds a regular ring-lattice and then attaches an almostregular stub to every node in the central ring. This algorithm roduces a k-regular grah with n nodes and a number of biconnected comonents higher than one. We show that for fixed k the number of biconnected comonents grows linearly with n. The seudocode of the algorithm is given in aendix A. We now state our theorem. Theorem. Provided that k is odd and (k 3) divides n, there is a regular grah with n nodes, k neighbors er node, and 2n/(k 3) biconnected comonents. Proof. The roof is constructive. First we build a central ring-lattice with n/k 3 nodes and k neighbors er node. Then we build a subgrah for each node in the ring-lattice with k 2 nodes where each node connects to k neighbors, excet for one of them, which connects only to k neighbors (this is ossible since k is odd, and therefore k 2 is also odd). This last node connects to its corresonding node in the ring-lattice. We call this secial node (i.e., the node in the subgrah) the stub node. For convenience, we also call the subgrah joint to the node in the ring-lattice the stub grah, such that every node in the ring-lattice belongs to one single stub grah. The revious algorithm generates a regular grah with n nodes and k neighbors er node. The number of biconnected comonents can be calculated by observing that the central ring is one biconnected comonent. Every stub generates a new biconnected comonent since it is only connected by a single node to the central ring. The edges that connect the stubs to the central ring generate another biconnected comonent. Finally, 2n/(k 3) biconnected comonents exist in the grah. Comlex Systems, 4 (23) 28

5 Small-world Connection Toologies 5 Figure. Nodes of stub grahs for k 3 left, and k 4 right. Note that, by our definition, the two nodes at the extremes of each base do not belong to the stub grah but are deicted here for clarity. A similar theorem can be established when k is even. Theorem 2. Provided that k is even and (k 3) divides n, there is a regular grah with n nodes, k neighbors er node, and n/(k 3) biconnected comonents. Proof. The roof is again constructive. First we build a central ringlattice with n/k 3 nodes and k 2 neighbors er node. Then we build a subgrah for every node in the ring-lattice with k 2 nodes where each node connects to k neighbors, excet for two of them, which connect only to k neighbors (this is ossible since k is even, and therefore k 2 is also even). These nodes connect to their corresonding node in the ring-lattice. Figure shows how the stub grahs are built for k even and odd. 3. Characteristic ath length and cluster coefficient scaling In this section we give analytical exressions for both the cluster coefficient and the characteristic ath length for regular transit-stub grahs. 3. Cluster coefficient Intuitively the cluster coefficient is the average number of neighbors of each node that are neighbors between them. More recisely, for a vertex v let us define (v) as the subgrah comosed by the neighbors of v (without v itself). Let us define the cluster coefficient for a given node v as: C v E( (v)) 2 k v (k () v ), Comlex Systems, 4 (23) 28

6 6 C. Aguirre, F. Corbacho, and R. Huerta d c b Figure 2. Distribution of nodes in a stub for k 3. a where E( (v)) is the number of edges in the neighborhood of v. The cluster coefficient for a given grah G can now be defined as C n i C i. (2) n In order to obtain an exression for the cluster coefficient for regular odd transit-stub grahs, let us observe that there are four tyes of nodes in each stub. These are distributed in the following way (as shown in Figure 2). One node in the central ring, node a. One stub node in each stub, node b. k nodes in each stub connected with the stub node, node c. Two nodes in each stub not connected with the stub node, node d. We count the nodes by tye to obtain C a 3 k 3 4 k, C b k 3 k, (k 3)(k ) 2 C c, C (k)(k ) d k k. (3) Then we weight them by averaging over all vertices in the stub: C k 3 C a C b C c C d 3 4 k 3 k k 3 k which can be simlified to C 4 (k ) k 3 (k 3)(k ) 2 (k)(k ) 2 k k, (4) 5 k 33 k 2 3k. (5) Note that the transit-stub grahs differ from the most clustered ones only by an amount O(/k). Furthermore, when n, k can be made Comlex Systems, 4 (23) 28

7 Small-world Connection Toologies 7 arbitrarily large without violating the sarseness condition (n k). This makes a clear difference with resect to ring-lattices, where C 3 (k 2) 4 (k ). In this case, we have that C 3/4 as k, which will never reach the otimal value of the cluster coefficient. 3.2 Characteristic ath length The characteristic ath length of a grah indicates how far the nodes are from each other. For a vertex v let us define its characteristic length as L(v) n i d(v, i), (6) n where d(v, i) indicates the length of the shortest ath connecting v and i. Using L(v) we define the characteristic length over a grah as L n i L(i). (7) n In order to estimate the characteristic ath length of a transit-stub regular grah we calculate the average distance between the nodes in the same stub d l, the average distance between nodes in different stubs d g, and two length scales: L l as the characteristic ath length for nodes in the same stub, and L g as the characteristic ath length between stubs. This method for estimating the characteristic ath length of a regular grah is similar to the method followed in [2]. In each stub we have (k 3)(k 2)/2 ossible connections organized as follows. Hence: Two of them with a distance of 3 (these corresond to connections between a and d nodes). There are k airs of nodes with a distance of 2 (a, c and d, b airs). There are k(k 3)/2 airs of nodes with a distance of. d l L l 2 (k 2)(k 3) k(k 3) 2 3 2(k ) 2 2(k 5) (k 2)(k 3). (8) We can see that d l when k. If we think of each stub grah as a suer-vertex of the central ring, the average distance between the nodes of different stubs d g is determined by L g and L l. In [2] it is shown that Comlex Systems, 4 (23) 28

8 8 C. Aguirre, F. Corbacho, and R. Huerta a regular ring-lattice with n nodes and k neighbors has the following value: L n(n k 2) 2k(n ). (9) In the transit-stub grah the central ring that connects the stubs is a regular ring-lattice with n r n/(k 3) and k r k where n r and k r reresent the values of n and k for the central ring. Hence, we have L g n k 3 n k 3 (k ) 2 2(k ) n k 3 n n k 3 k 3 2(k )(n k 3). () A ath from a node v in one stub to a node u in another stub consists of three comonents: the edges contained in the stub with v, the edges in the central ring, and the edges in the stub with u. We add u these terms to obtain n 4(k 5) d g 2L l L g 2 (k 2)(k 3) that, for n k, we have d g 2k n k 3 k. k 3 n (k ) 2 k 3 2(k ) n k 3 () There are N l n(k 2)/2 airs of nodes that are in the same stub and N g n(n k 3)/2 airs of nodes that are in different stubs. We average them to calculate L 2 n(n ) (N l d l N g d g ) k 2 n (n k 3) n k 3 k 2k(n ) 2k n k (2) k 3 where we assume that n k, that is, sarseness. Note that this transit-stub network has a shorter characteristic ath length than the characteristic ath length of ring-lattices. In a random grah with fixed k it can be shown that L scales as log(n) and C scales to as n tends to infinity [27]. This means that the transit-stub grahs have a different scaling regime than the random grah model. This makes us exect that at some oint, when we shift from these regular models to a random grah, there must be a hase change both in the value of L and C. If this hase change is roduced at different values of for L and C we can build SW grahs using this regular substrate, and therefore, SW grahs exist for these models. Furthermore, if the value of that changes the hase of L is small, the SW grahs generated from regular transit-stub grahs will have a high Comlex Systems, 4 (23) 28

9 Small-world Connection Toologies 9 number of biconnected comonents. This is because each rewired edge will decrease the number of biconnected comonents at most by one. This means that we need to rewire a minimum of 2n/(k 3) edges for k odd and n/(k 3) edges for k even in order to reduce the number of biconnected comonents to one. 4. Static metric behavior In order to study the static metric behavior of our grahs, we follow a modified version of the standard rocedure described in [3]. Our method consists of rewiring iteratively every edge with robability. Once an edge is evaluated, it is not reconsidered. Other rocedures for building SW grahs have been described in [22, 23, 24]. However these methods increase the number of edges and/or nodes in the grah or can only be alied to ring-lattice substrates. Our method maintains a constant number of nodes and edges and can be alied to any kind of substrate. Figure 3 dislays the rocess of shifting a regular transit-stub grah to random. When no changes are made over the substrate and the grah remains unchanged. As increases shortcuts aear in the grah. In the limit of a random grah is fully develoed. The metrics used to classify the grahs are the characteristic ath length L, the characteristic cluster C [2], the number of biconnected comonents B [25], and the average edge euclidean length D. These arameters are the most extensively studied in the existing literature. The diameter of the grah is tightly related to L (being in fact an uer bound) [27]. Therefore we consider L because it rovides more secific information about the distance of the nodes in the grah. In the same way, the length diameter [] is an uer bound of D. In Figure 4 we show the behavior of L and C in a ring-lattice and in a regular transit-stub grah for both k even and odd when they are shifted (a) (b) (c) Figure 3. Transit-stub grahs for n 2 and k 3. (a) Regular transit-stub grah for. (b) SW transit-stub grah for.2. (c) Random grah for. Comlex Systems, 4 (23) 28

10 C. Aguirre, F. Corbacho, and R. Huerta,8,6,4,2 L()/L() Ring C()/C()!5!4!3!2!,8,6,4,2 Stub Odd C()/C() L()/L()!5!4!3!2!,8,6,4,2 Stub Even L()/L() C()/C()!5!4!3!2! Figure 4. L (squares) and C (triangles) for ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. from a regular to a random situation. Note that all of them resent a clearly visible SW area (i.e., low L and high C). However the decrease of L in the transit-stub models is smoother because the characteristic aths are shorter for regular transit-stub substrates than for ring-lattice ones. In the ring-lattice substrate the value of the characteristic ath L is very close to the maximum for any regular grah and has a very fast descent. On the other hand, when we calculate the characteristic cluster C we do not find significant differences between these tyes of substrates. The number of biconnected comonents in a grah is a useful measure of connectedness or edge redundancy. Formally seaking a biconnected comonent is a set of edges such that any two edges in the set lie on a common single cycle. A biconnected comonent cannot be disconnected by removing a single node or a single edge. Furthermore, each air of nodes in a biconnected comonent is connected by at least two aths. Intuitively a biconnected comonent is a subgrah connected to the rest of the grah by articulation nodes (see [25] for more information about biconnectivity). Figure 5 dislays the number of biconnected comonents in the grah as a function of. Observe that the number of biconnected comonents is reduced to a very low number as increases in an initially regular transit-stub grah substrate. However it remains almost constant when a ring-lattice grah is emloyed. Therefore, we can see that the main difference between the regular and the transit-stub grah is found in the number of biconnected comonents. Notice also that in both models, there is a small increase in the number of biconnected comonents in the random grah area. Finally, for a given embedding of a grah G into an euclidean sace we can assign a length to each edge in the grah. This length is the euclidean distance between the two vertices connected by the edge. As the grah becomes more random, the edges get longer in the euclidean sace. In Figure 6 we lot the average length D of the edges for both Comlex Systems, 4 (23) 28

11 Small-world Connection Toologies,,8,6 B,4,2,98 Ring!5!4!3!2! Stub Odd!5!4!3!2! Stub Even!5!4!3!2! Figure 5. Number of biconnected comonents for ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transitstub grahs (k even) for n 2992 and k 8 (the average of exeriments is lotted). 8 Ring 2 Stub Odd 2 Stub Even D !5!4!3!2!!5!4!3!2!!5!4!3!2! Figure 6. Average euclidean edge length for ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. The nodes of the ring-lattice lie over a circumference with 6 units of radium. In the case of the transit-stub grahs the backbone nodes are over a circumference of 578 units of radium, each stub has unit of radium and is 2 units away from its corresonding node in the backbone (the average of exeriments is lotted). kinds of substrates when they are shifted from regular to random, that is, D E i d (i), (3) E where d (i) is the euclidean length of edge i. The embeddings selected are such that the nodes of the ring-lattice lie over a circumference with 6 units of radium. In the case of the transit-stub grahs the backbone nodes are over a circumference of 578 units of radium, each stub has unit of radium and is a distance of 2 units from its corresonding node in the backbone. Notice that both substrates behave in a similar way. We can now define the euclidean ath length H (v) for a vertex v as H (v) n i H (v, i), (4) n Comlex Systems, 4 (23) 28

12 2 C. Aguirre, F. Corbacho, and R. Huerta H Ring Stub Odd Stub Even !5!4!3!2!!5!4!3!2!!5!4!3!2! Figure 7. Normalized characteristic euclidean ath length for ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. The circumferences and distances are the same as Figure 6 (the average of exeriments is lotted). where H (v, i) is the total euclidean length of the edges that lie in the shortest ath between vertices v and i with the embedding. Now, the characteristic euclidean ath length for a given grah G and a euclidean embedding can be defined as H (G) n i H (i). (5) n The average euclidean ath length of a grah is a measure of how far the nodes are in terms of the euclidean distance, instead of the number of hos. In contrast with the average edge euclidean length, observe in Figure 7 the different qualitative behavior in the characteristic euclidean ath length for both kinds of substrate. The ring-lattices dislay an average euclidean ath length with a lateau in the SW area. The highest value for the average euclidean ath length is reached in the regular area, and the lowest value is reached in the random area. For transitstub grahs, the minimum value is reached near the random area ( but < ), meanwhile the highest euclidean ath length is observed in the SW area. The embeddings considered are the same utilized in the calculus of H. This measure is imortant when the cost of the network is considered as, for examle [4, 5], or when the error rate in the transmission of information ackages deends on the distance from source to destination. If each edge has a robability which reflects how rone it is to transmission error, longer links tend to roduce more transmission errors. In our model, the robability of error in an edge is linearly roortional to its length. This is due to the fact that if is the robability of error by unit of length, the robability m of error in an edge of length m units of length is given by: m ( ) m m if. (6) Comlex Systems, 4 (23) 28

13 Small-world Connection Toologies 3 From the definition of H it is clear that the average number of corruted ackages due to transmission errors should be roortional to H when the routing rotocol transorts ackages from a source node to a destination node using the shortest ath between both nodes. 5. Dynamic metrics We consider two metrics for evaluating the efficiency of information transmission over the network. One of the them is the error rate, which may determine the degree of redundancy to include in messages. The second one is the transmission time. Forecasting transmission times of the Internet has been studied using neural networks and ergodic techniques [3]. As ointed out in [3, 4], transmission times should decrease with the grah characteristic ath length. We define the transfer time as the average time a acket takes to reach the target starting from a given source node in several exeriments. We comute it by the number of time stes it takes to deliver a set of ackets to the given set of target agents in the network averaged by the number of nodes in the target set and the number of ackets sent to each agent, that is, T t l(n ), (7) where t is the number of time stes until the system stos, l is the number of ackets, and n is the number of nodes in the target set. The error rate increments with the length of the edges and with the characteristic ath length of the grah. The other metric we comute is the average error rate that we define as the number of ackets that become corruted on their way through the network. We use the Hamming distance between the original ackages and the ackages at each node: E n i 2 l v f (H( v, iv )), (8) l(n ) where iv is the ackage v at node i, H is the usual Hamming distance, and f (x) is the ste function f (x) if x, otherwise. (9) The node is assumed to be the source. The error rate would deend essentially on the characteristic ath length and on the number of longrange connections resent in the grah. Comlex Systems, 4 (23) 28

14 4 C. Aguirre, F. Corbacho, and R. Huerta To better understand the balance between time delay and error rate we arbitrarily roose analyzing an average of both normalized measures: P 2 T T min E E min, (2) T max T min E max E min where T min, T max are the minimum and maximum values of T and E min, E max are the minimum and maximum values of E. 6. Dynamical simulations over the networks Comarisons based on toological metrics rovide differences between grah tyes. However, it does not mean much unless a link to a function is established. Therefore, we study how different static toologies affect erformance in some networking insired dynamics, such as broadcast, multicast, and unicast of information ackages. As ointed out in [], erformance arameters associated with a telecommunication network warrants a meaningful model for the comlexity of the communication system. A relevant modeling of such comlexity should include stochasticity of interacting resources and in the flow of information between the nodes of the network. Furthermore, some of these arameters are fuzzy in nature [26]. An accurate modeling of the Internet, or even smaller communications systems, is not an easy task and is out of the scoe of this work due to the high heterogeneity of the elements resent in the network and the high number of variables that affect the erformance of a modern communication network. In our model, a set of qualitative roerties found in communication networks has been added to each of the agents in the network. In fact, all sorts of additional information about the network can be added to the toological structure by associating information with the nodes and edges. For instance, nodes may be assigned numbers reresenting their buffer caacity, that is, the number of ackages that they can hold in the queue (including a sto symbol). The queue has a saturation limit beyond which new incoming ackages are discarded. An edge may also have values of several tyes; including costs, such as the roagation delay on the link; and constraints, such as the bandwidth of the link. In our exeriments most of the arameters were constant between the different networking roblems (queue sizes, number of ackages er file, lifetime of each ackage, etc.). This means that some of the arameters have no influence on the result of the exeriments for a articular roblem but does allow us to maintain some consistency between exeriments. For our simulations we assume that the substrates are embedded in the usual two-dimensional euclidean sace. The nodes of the ring-lattice substrate lie over a circumference with 6 units of radium. In the case of the transit-stub grahs we assume that the backbone nodes are over Comlex Systems, 4 (23) 28

15 Small-world Connection Toologies 5 a circumference of 578 units of radium. Each stub has unit of radium and is 2 units away from its corresonding node in the backbone. These distances (but not the toology) corresond with the measures of the Sanish Research network RedIris [33]. We also assume that the robability of corruting a ackage er unit length is Simulations based on broadcasting This section resents a dynamics insired by simlified broadcasting using the selective diffusion algorithm described in [32] to understand the dynamic advantages of different substrate-based network toologies. Broadcast is the rocess of sending a ackage or a set of ackages to all ossible destinations in the network. Broadcast is a costly rocess in terms of the amount of ackages that are introduced in the network. Moreover, broadcast is frequently used both as stand-alone rocesses (e.g., military networks, live TV and radio broadcast, etc.) and as a art of more comlex routing algorithms (e.g., OSPF [35]). In our simulations each node maintains a finite queue of ackages. In the l first time instants l ackages of a file containing routing information are sent by one fixed agent using selective diffusion to the whole set of nodes in the network. Each ackage has an unique identification number and a time-to-live (ttl) counter that decreases by one for each ho the ackage crosses. The ackage also maintains a list of visited nodes in order to trace the route the ackage has followed and to effectively imlement the selective diffusion. When the ttl counter becomes, the ackage is considered obsolete and removed from the network. In addition, the ackage can become corruted at any bit, including ttl, route, or identification number. For convenience, the ackets are of fixed size. The following haens at each time instant for each node. The first valid ackage is obtained from its queue and the ackage is sent by selective diffusion. A ackage becomes obsolete if there is another coy of the ackage in the queue with a higher ttl or its ttl takes the value. If the destination node is congested (i.e., its queue is full) the ackage is removed from the network reflecting the congestion roerties of the network. At every time instant all obsolete ackages are removed from the queues. This allows queues to maintain the most recent version of the ackage. At each crossed ho, the ackage decrements its ttl and udates its route register. The ackage has a maximum number of hos that corresonds to its ttl, if the maximum is reached (i.e., ttl ), the ackage is removed from the network. At each ho the ackage has a robability of becoming corruted as it crosses links with a certain robability of corruting ackages. Comlex Systems, 4 (23) 28

16 6 C. Aguirre, F. Corbacho, and R. Huerta Ring Stub Odd Stub Even T !5!4!3!2!!5!4!3!2!!5!4!3!2!,3,3,3,2 E,,2,,2, P!5!4!3!2!,8,6,4,2!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2! Figure 8. T, E, and P for broadcast in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, queue size =, and ttl 38. Results are averaged over exeriments. The system stos when the whole set of nodes owns a coy of the comlete file or there are no more ackages to deliver in the agent s queue. When the system stos the transfer time T and the average error rate E are comuted. We have carried out our simulations for each of the different toologies obtained by alying the modified Watts Strogatz method in ringlattice and transit-stub substrates. All the metrics are averaged over different exeriments. In Figure 8 we dislay the transfer time T, the error rate E, and the time/error average P for the three substrates. The left column shows the dynamic metrics values for the ring-lattice, the central and right columns show the dynamic metrics for the transit-stub grahs. We can observe that, in both substrates, the average time T decreases when grows (i.e., when the grah becomes more random). The reduction in the average time rate is due to shorter aths that are tyical of random grahs (see Figure 4). On the other hand, the error rate E behaves differently in the case of ring-lattices than for transit-stub grahs. In the case of ring-lattices, the error increases with the value Comlex Systems, 4 (23) 28

17 Small-world Connection Toologies 7 of ; meanwhile, the error decreases with in the case of transit-stub substrates. In both substrates the error rate is related with the behavior of the characteristic euclidean ath length. If an average measure of the two normalized arameters P is used as in [4, 5], an otimal oint for building networks with an otimal error-time delay ratio would be those corresonding to the random area for transit-stub grahs. Meanwhile the otimal ratio for ringlattices is when the grah is in the SW area. Note that we do not intend to establish a comarison between ring-lattice and transit-stub networks, but, given a secific substrate and an embedding, we want to determine the best erformance. 6.2 Simulations based on multicast In the ast few years, esecially with the emergence of multimedia over the Internet and distributed database systems, there has been an increase in the number of alications that need to establish communications between grous of hosts. Secific algorithms have been develoed for multicast, and even some secific networks with their own toologies as, for examle, MBONE [34], have been develoed for multicast services. Multicast requires the existence and management of grous of nodes. Each node of the network can join one or several grous or can be searate from any grou to serve as a mere transort node. The creation and management of grous are indeendent tasks of the ackage delivery rocess over the network and have their own algorithms and rotocols. For this reason, as a ste rior to each multicast simulation, two grous comosed of % of the nodes in the network are created. The nodes that form a grou are selected at random with uniform robability over the whole set of nodes in the network. A node can belong to none, one, or several grous. The nodes that do not belong to any grou work as transort nodes. We assume that the members of each grou do not change once they are established. In our simulations we have imlemented a multicast routing algorithm based on sanning trees [32]. A rerequisite to carrying out the simulations is the calculation of the sanning trees that originate in the source node. As in section 6., each node maintains a queue of ackages. In the l first time stes, 2l ackages of two different files are sent by a source agent. Each file consists of l ackages and is addressed to one of the two grous in the network. Each ackage has a file number, an identification number inside the file, and a ttl counter. The ackage also maintains a list of visited nodes and when the ttl counter becomes, the ackage is considered obsolete. As in the revious exeriment, the ackage can become corruted at any bit, including ttl, route, file, or identification number. Fixed-size ackages are again used. Comlex Systems, 4 (23) 28

18 8 C. Aguirre, F. Corbacho, and R. Huerta The following haens at each time instant for each node. The first valid ackage is obtained from its queue and the grou to which the ackage is addressed is checked. The node sends the ackage only to neighbor nodes that belong to the grou to which the ackage is addressed, or are on the way to nodes of that grou, avoiding the neighbor node that sent the ackage. When the destination node is congested, the ackage is lost on its way to that node. At every time instant all obsolete ackages are removed from the queues. At each crossed ho, the ackage decrements its ttl and udates its route register. The ackages have some given robability of becoming corruted as they cross links. The system stos when the whole set of nodes execting files (i.e., the nodes belonging to the two grous) owns a coy of the corresonding file or files or there are no more ackages to deliver. When the system stos, the average error and transfer time are comuted. As in section 6. the Watts Strogatz method has been alied in ringlattice and transit-stub substrates. All the metrics are averaged over different exeriments. Multicast results for T, E, and P are lotted in Figures 9 and. As an illustration of the consistency of the results we can observe in Figure the same qualitative results when the number of ackages and ttl are increased. Multicast resents its worst results both for transfer time and error rate in the regular area ( ). Otimal oints are obtained for the SW area in the case of ring-lattices and in the random area for transfer-stub grahs. This is consistent with the broadcast results due to the fact that both are minimal ath based algorithms. 6.3 Point-to-oint transort (unicast) In unicast, or oint-to-oint transmission, a ackage has to cross a number of intermediate machines (routers) in order to go from the source to the destination. Multile routes or aths of different length are ossible. In this framework, routing algorithms lay an imortant role in the efficient transmission of ackages. In the Internet, for examle, each router obtains routing information (link state, toology, congestions, etc.) from other routers. Routing information can be rovided by routers in the same subnetwork or, for border network routers, from the neighbor subnetwork [35]. The information rovided by the routers allows maintaining the best ossible route at each moment. In Comlex Systems, 4 (23) 28

19 Ring Stub Odd Stub Even T !5!4!3!2!!5!4!3!2!!5!4!3!2!,3,3,3,2 E,,2,,2, P!5!4!3!2!!5!4!3!2!,8,8,8,6,6,6,4,4,4,2,2,2!5!4!3!2!!5!4!3!2!!5!4!3!2!!5!4!3!2! Figure 9. T, E, and P for multicast with sanning trees routing in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, queue size =, and ttl 38. Results are averaged over exeriments. Ring Stub Odd Stub Even T !5!4!3!2!!5!4!3!2!!5!4!3!2!,3,3,3,2 E,,2,,2, P!5!4!3!2!,8,6,4,2!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2! Figure. T, E, and P, for multicast with sanning trees routing in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, queue size =, and ttl 75. Results are averaged over exeriments.

20 2 C. Aguirre, F. Corbacho, and R. Huerta our model, each agent maintains a routing table based on minimal distance to the other agents in the network. Note that in reality the routing rocess does not follow minimal aths [36]. However, minimal distance routing can be considered as a lower bound of routing algorithms. In our unicast exeriment, each node maintains a queue of ackages. In the l first time stes, one node selected at random sends a file of l ackages to each of two different randomly selected destination nodes. Each ackage has an origin and destination number, a file number, an identification number inside the file, and a ttl counter. The ackage also maintains a list of visited nodes and when the ttl counter becomes, the ackage is considered obsolete. As in the revious exeriment, the ackage can become corruted at any bit, including destination, origin, ttl, route, file, or identification number. Fixed-size ackages are again used. The following haens at each time instant for each node. The first valid ackage is obatined from its queue and the node to which the ackage is addressed is checked. The node sends the ackage to a neighbor node that is closer to the destination address. When the destination node is congested, the ackage is lost on its way to that node. At every time instant all obsolete ackages are removed from the queues. At each crossed ho, the ackage decrements its ttl and udates its route register. Packages have a robability of becoming corruted as they cross links. Unicast results for T, E, and P are lotted in Figures and 2. Both substrates resent a very similar qualitative behavior with resect to the transfer time. The transfer time in unicast is clearly related to the characteristic ath of the substrate. Both substrates now resent different characteristics in the error rate. The error rate increases with in the case of ring-lattices, whereas in transit-stub grahs the error rate has a minimum in the random region, reaching its worst values in the regular extreme of ( ). The otimal time/error average is in the SW area for ring-lattices and in the random area for transit-stub grahs. 6.4 Multiho networks simulations Multiho networks have a small number of neighbors er node and have been mainly used for comuter clusters and lightwave networks. Some very secific toologies have been studied for multiho networks, such as the toroidal and diagonal mesh [37] or the Manhattan street [38]. Comlex Systems, 4 (23) 28

21 Ring Stub Odd Stub Even T !5!4!3!2!!5!4!3!2!!5!4!3!2!,4,2,3 E,2,,2, P!5!4!3!2!!5!4!3!2!,8,8,8,6,6,6,4,4,4,2,2,2!5!4!3!2!!5!4!3!2!!5!4!3!2!!5!4!3!2! Figure. T, E, and P for unicast with minimal distance routing in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, queue size =, and ttl 38. Results are averaged over exeriments. Ring Stub Odd Stub Even T !5!4!3!2!!5!4!3!2!!5!4!3!2!,3,3,3,2 E,,2,,2, P!5!4!3!2!,8,6,4,2!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2! Figure 2. T, E, and P for unicast with minimal distance routing in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, queue size =, and ttl 75. Results are averaged over exeriments.

22 22 C. Aguirre, F. Corbacho, and R. Huerta Deflection routing or hot otato routing is a oular unicast routing strategy, secially used over multiho networks. Deflection routing is a bufferless routing algorithm. Packages are sorted by a deflection criterion as, for examle, the time the ackage is in the network, or the distance to its destination. Packages with higher riority are routed to the otimal links. Store and forward routing is another algorithm used over multiho networks. Store and forward is a buffered algorithm and all ackages are routed over the shortest ath. In this exeriment, l ackages of a file are sent by a source node to a random destination node. Each ackage has an identification number, a destination address, and a ttl counter. As in the revious sections, the ackage maintains a list of visited nodes and when the ttl counter becomes, the ackage is considered obsolete. Package corrution is also ossible. For deflection routing the highest riority is given to ackages that have low ttl. This is the otimal criteria as shown in [37, 39]. At each time instant, the algorithm works as follows. Each node gets every valid ackage from its queue. If the ackage has a high ttl, it is sent to a random neighbor. If the ttl of the ackage is below a given threshold (in our exeriments we take this threshold to be the diameter of the ring-lattice of size n), then it is sent to the following node by using the shortest ath to the destination node. Deflection routing results for T, E, and P are lotted in Figure 3. As exected, the ring-lattice resents the worst transfer time, but there are no big differences in resect to the transfer time of the transit-stub grahs. This is due to the fact that transfer time is mainly determined in both substrates for the random walk art in the route of each ackage. Some differences are found in the error rate. In the case of ring-lattices, the error decreases until the minimum value is situated in the SW region, then the error rate increases to a similar value observed in the regular ring substrate. For transit-stub grahs, the error rate remains almost constant during the SW area, increasing only when is close to. Note also that in both substrates the otimal time/error average is obtained in the SW area. Deflection routing is not a minimal distance algorithm, so we cannot exect the error rate to follow the behavior of the characteristic euclidean ath length. In this algorithm, the error rate is determined mainly by the random walk art of each route. We now analyze store and forward routing. This routing algorithm erforms the following at each time instant. Each node gets the first valid ackage from its queue and sends it to the following node using the shortest ath to the destination node of the ackage. Comlex Systems, 4 (23) 28

23 Small-world Connection Toologies 23 T E P Ring Stub Odd Stub Even !5!4!3!2!!5!4!3!2!!5!4!3!2!,5,8,2,4,7,5,6,3,5,,2,4,,5,3,2!5!4!3!2!!5!4!3!2!!5!4!3!2!,8,6,4,2!5!4!3!2!,8,6,4,2!5!4!3!2!,8,6,4,2!5!4!3!2! Figure 3. T, E, and P for the deflection routing olicy in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, and ttl 38. Results are averaged over exeriments. As in the revious exeriments, node congestion, obsolete ackages, corrution of ackages, and ttl countdown have been imlemented. Store and forward results for T, E, and P are lotted in Figure 4. When very few ackages l are resent in the network the algorithm shows a high velocity, with a very small (close to the otimal) transfer time. The transfer time results resond to the exected values (decrease in and higher T values for the ring-lattice substrate). The error rate increases with in the ring-lattice substrate and decreases for transitstub grahs, resembling the lot of the characteristic euclidean ath length. Note that the otimal time/error average is obtained in the SW area for the ring-lattice substrate, while the transit-stub grahs reach the otimal value in the random area. 7. Conclusions and discussion We have develoed a method for building initial regular substrates by considering toological metrics different from the usual ones, as for examle, the number of biconnected comonents. We studied a set of networking insired dynamics over these new toologies by comaring them with the same dynamics over the usual small-world (SW) substrate. Comlex Systems, 4 (23) 28

24 24 C. Aguirre, F. Corbacho, and R. Huerta T E,2 P 2 5 5,4,3,!5!4!3!2!!5!4!3!2!,8,6,4,2 Ring!5!4!3!2! !5!4!3!2!,4,3,2, Stub Odd!5!4!3!2!,8,6,4,2!5!4!3!2! Stub Even !5!4!3!2!,4,3,2,!5!4!3!2!,8,6,4,2!5!4!3!2! Figure 4. T, E, and P for the store and forward routing olicy in ring-lattices with n 2992 and k 8, transit-stub grahs (k odd) with n 3 and k 9, and transit-stub grahs (k even) for n 2992 and k 8. l, queue size =, and ttl 38. Results are averaged over exeriments. In our simulations we study a very simlified deterministic model based on a set of homogeneous agents that interact following olicies insired by some of the dynamics found in intercommunication networks. The results obtained could be very far from the erformance arameters given in real communication systems such as the Internet. However, we think that these results can ossibly be aliyed to small, homogeneous systems and can show that, in these situations, the underlying network toology heavily determines the dynamic behavior of the network. We conclude with the following remarks. Regular grahs exist that resent a high number of biconnected comonents. These networks better resemble hierarchical networks than structures like stars, rings, or grids. These new transit-stub networks resent a similar SW area when they are shifted from a regular to a random situation in a way similar to ring-lattice substrates. However, transit-stub networks resent a slower descent in their characteristic ath. The selected substrate changes the values of the dynamical metrics of a set of well known internetworking insired dynamics over a network of agents. Comlex Systems, 4 (23) 28

25 Small-world Connection Toologies 25 The SW area is an otimal region for time/error averages for ring-lattice grahs over several internetworking insired dynamics. There is one excetion, the deflection routing algorithm, because this arameter is controlled mainly by a random walk art of the route. Acknowledgments We thank the MCyT (BFI 2-5). (RH) was also funded by DE-FG3-96ER492 and (CA) was suorted by ARO-MURI grant DAA during a four month stay at UCSD. We also thank Lev Trimsing for useful discussion. Aendix Pseudocode to generate regular multi-biconnected grahs for k odd. Procedure CreateRegularStubsOdd(n,k) //n number of nodes in the grah //k neighbors er node // Build the central ring For j = to n/(k+3) For l = to (k-)/2 // bidirectional edge AddEdge (j, (j+l) mod (n/k+3)) // Attach a stub to each // node of the central ring For j = to n/k+3 AddStubOdd(j) End Procedure CreateRegularStubs Procedure AddStubOdd(x) // x = node to attach the stub // for each node in the stub For j = n/(k+3) + x*(k+2) to n/(k+3) + x*(k+2) + k + // number of neighbors For v= to (k-)/2 // check cyclic connections in the stubs if j+v < n/(k+3) + x*(k+2) + k + 2 ind = j+v else ind = (j+v)-(k+2) Comlex Systems, 4 (23) 28

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