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An efficient organization mechanism for spatial networks
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Physica A 366 (2006) 608–618 www.elsevier.com/locate/physa
An efficient organization mechanism for spatial networks Fei Liu, Qianchuan Zhao Department of Automation, Center for Intelligent and Networked Systems, Tsinghua University, Beijing 100084, China Received 6 April 2005; received in revised form 21 September 2005 Available online 21 November 2005
Abstract Spatial networks, also known as random geometric graphs, are random graphs with certain distance metric, in which each node is connected to some others within its neighborhood disc. Due to the rapid increase of network scales, the design of spatial networks becomes increasingly challenging. Inspired by the recently discovered small-world topology in relational networks, we identify an efficient organization mechanism for spatial networks, which we believe is useful for spatial network design. Two types of discs are introduced. Edges in the large discs are counterparts of ‘‘shortcuts.’’ We find that such ‘‘two-radius’’ spatial networks exhibit small characteristic path length, yet with low cost. This mechanism is applied to broadcasting protocol design for wireless ad hoc/sensor networks. r 2005 Elsevier B.V. All rights reserved. Keywords: Statistical physics of complex networks; Spatial network design; Small world; Optimization; Wireless ad hoc/sensor networks
1. Introduction Popular network models can be classified as relational and spatial networks. The constructions of relational networks do not depend upon any external metric of distance [1], while spatial networks are networks with certain metric, such as Euclidean distance [2]. The design of networks becomes more and more challenging than before. Global optimization seems impossible due to the rapid increase of the network scales. The success of existing large-scale man-made networks depends heavily on the decentralizing nature of the rules in their organization mechanisms. The entire network should be efficient both globally and locally. It should be efficient on a global scale as its function is usually realized by the joint coordination among many nodes. Its measurement is the typical separation between two vertices in the graph. Meanwhile, the network should be efficient on a local scale. The measurements include the clustering/cliquishness of a local neighborhood (first introduced in Ref. [5]), and the cost to build a neighborhood (first introduced in Ref. [4]). It is found in Refs. [3–5] that, to design an efficient organization mechanism for both relational and spatial networks, small degrees of separation, high clustering, and low cost are all required. One of the most important discoveries for decentralized organization mechanism for large-scale networks is the small-world topology. By analogy with the small-world phenomenon in social networks, Watts and Strogatz [5] have shown that many real relational networks possess small-world characteristics, that is, small Corresponding author. Tel.: +86 10 62783612; fax.: +86 10 62786911.
E-mail addresses: [email protected] (F. Liu), [email protected] (Q. Zhao). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.10.022
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typical separation between two vertices (measured by characteristic path length) and high clustering (measured by clustering coefficient). Note that the Watts–Strogatz model assumes the sparseness of the edges. Whether the small worlds still emerge in spatial networks? On the one hand, as Watts analyzed in Ref. [6], spatial networks admit no small-world class, unless some edges have long ranges (shortcuts) that are comparable to the network size. On the other hand, according to Refs. [7,8], after relaxing the strict definition of spatial networks, i.e., affecting the disc-like neighborhoods by adding ‘‘shortcuts’’ with probabilities depending on inter-node distance, small worlds can be introduced. Based on these two observations, it is natural to ask, when the definition is yet strictly obeyed, whether spatial networks present similar behaviors like the small worlds. In fact, obeying this definition is meaningful in both theoretical (e.g. relational vs. spatial networks) and practical (e.g. wireless systems with omni antenna) perspectives. For spatial networks, in reality, the clustering tends to be consistently high due to locality of the edges [11,12]. Meanwhile, motivated by applications, we regard the node operational cost as another indispensable local metric for network design. In wireless applications, the cost turns out to be the power consumption of wireless devices, and the fact that the battery power is not unlimited should be taken into account. In this paper, we are focusing on whether there exist some efficient organization mechanisms for spatial networks so that they have both small typical separation and low cost (measured by Node Degree, as defined later. It is a kind of measure for cost or energy consumption in wireless applications, as will be explained in Section 5). Note the conclusion of Watts on spatial networks was made based on the use of the clustering coefficient as the only efficiency measurement on a local scale. If we also take into account the node degree that measures the number of neighbors of a typical node in the graph, the key concept of shortcuts may still be applicable for spatial networks. This paper shows that in this sense a counterpart of shortcuts does exist for spatial networks. Observe that if we connect a node to all neighbors in a large neighborhood disc centered at this node in a random geometric network, then the appearance of such discs provides shortcuts to reduce the characteristic path length efficiently. Observe also that such large discs increase the cost, namely increase the node degree. The new efficient organization mechanisms we find for spatial networks are based on the above two observations and can be regarded as an extension of the small-world topology for relational networks. The idea is to construct random geometric networks by connecting most of the nodes with ‘‘nonshortcuts’’ and adding ‘‘shortcuts’’ connections randomly with small probability. The ‘‘nonshortcuts’’ connections are corresponding to edges in small discs, which result with large separation and low cost; and the ‘‘shortcuts’’ are corresponding to edges in large discs. Because there are two types of discs in the networks we call them ‘‘tworadius’’ spatial networks. We will show in this paper they exhibit small separation, high clustering, and low cost. The rest of this paper is organized as follows. In Section 2, we propose three spatial network models. Based on these models, in Section 3, collective dynamics of the two-radius spatial networks are presented. Then we provide analysis from the perspective of optimization in Section 4, and apply our results to wireless ad hoc/ sensor networks in Section 5. Section 6 concludes the paper.
2. Three spatial network models A spatial network is a graph where each node (vertex) is assigned random coordinates in a d-dimensional box of unit volume, and each edge is added to connect pairs of nodes which are close to each other by metric. Here 2-dimensional spatial networks with Euclidean distances are considered. In the square area ½0; 1 ½0; 1, N nodes are assigned random coordinates ðxi ; yi Þ ði ¼ 1; 2; . . . ; NÞ, thus with density r ¼ N. Note that no periodic boundary conditions are used here. We assume that the neighborhood ranges of all nodes are circles, that is, each node only ‘‘communicate’’ with some others within its neighborhood disc. Let ri denote the disc radius of node i. For any other node j , if jðxj ; yj Þ ðxi ; yi Þjori is true, it defines a directed link i ! j, and j is then i’s outgoing neighbor. If the link j ! i also exists, then the two nodes are connected via a bidirectional link i2j and are able to directly arrive back and forth. In the following part, to explore efficient behaviors in spatial networks, three models are presented: model I, uniform small radius; model II, uniform large radius; and model III, two-radius model, viz. small/ large radii.
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2.1. Model I: uniform small radius The most widely used structural design of spatial networks is to utilize the same radius. On the one hand, the uniform radius should be large enough to make the network strongly connected; on the other hand, the uniform radius in this model should be as small as possible to reduce the number of edges. Accordingly we choose the uniform small radius to be near the minimal radius at which the network is just strongly connected. Itp was shown in Refs. [9,10] that if N nodes are placed in a disc of unit area and each node’s radius chosen as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðlog N þ gN Þ=ðpNÞ , then the resulting network is asymptotically connected with probability one as N! if and only if the sequence gN ! þ1 . If we set gN ¼ log N ! þ1 , then the lower bound pþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 2 log N=ðpNÞ. According to this, we choose uniform small radius in this model I of order rI
ðlog NÞ0:5 N 0:5
(1)
to make the whole network just strongly connected. 2.2. Model II: uniform large radius In this model, a large radius rII is uniformly assigned to each node, and it should obey the following two assumptions: (a) 0orII prmax , where rmax is the maximum radius at which the network is just fully connected; (b) rII 4rI and rII =rI ! 1 as N ! 1, such that the range of rII is enough larger than that of rI . Accordingly we choose the large radius of order rII
ðlog NÞ1:5 N 0:5
(2)
such that rII =rI ¼ log N ! 1. Note rII can be chosen of other orders, e.g. rII 1= log N, and rII c ðrI ocprmax Þ, as long as they follow the two assumptions. 2.3. Model III: two-radius model, viz. small/large radii In this model, the small radius rs is set to be rI as in model I, and the large radius rl to be rII as in model II. Then rs and rl are randomly assigned to each node in the network with probability 1 p and p ð0opo1Þ, respectively. Note that not all the links remain bidirectional in this model, since small-radius node i, for instance, can be within the range of some large-radius node j while node j may lie beyond the range of node i. Illustrations for models I–III are presented in Fig. 1. 3. Collective dynamics of the two-radius spatial networks The two-radius spatial networks are to interpolate between low-connectivity and high-connectivity spatial networks, compared with the Watts–Strogatz’s relational model between regularity and randomness. Starting from a low-connectivity network generated by model I, we alter each node’s small radius rI to the large radius rII at random with the probability p. Once a node’s radius is increased, one-directed edges from this node to those within its range will be added to the graph if they do not originally exist. This construction allows us to ‘‘tune’’ the graph between low (p ¼ 0, as in model I) and high connectivity (p ¼ 1, as in model II), and thereby to examine the intermediate region 0opo1, as in model III. Two statistics of interest are examined to quantify network efficiency. They are the characteristic path length L and the node degree k which have most relevance from practical point of view. It should be noted that our network model is different from those of Refs. [3–5] in that our model is directed graph. We justify the use of these measures in detail below. The first statistic is the characteristic path length L, defined as the average number of edges of the shortest one-directed path over all pairs of source/destination nodes in the graph.
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Fig. 1. Illustrations for network models I–III of 100 nodes: (a) rI ¼ 0:15; (b) rII ¼ 0:45; (c) rs ¼ 0:15, rl ¼ 0:45, p ¼ 0:10; (d) the outgoing degree distribution of (c).
L measures the typical separation between two vertices in the graph, and it is then a global property (because, in general, determining the shortest path length between any two nodes requires information about the entire graph). The second statistic is the node degree k, which is a measure of the local graph structure. Specifically, for a node v, the degree kv counts the number of vertices in the subgraph of v’s immediate neighborhood. k is the average of kv over all vertices in the graph. k measures the cost to build a neighborhood in un-weighted graphs, and see Section 5 for the variant of k in weighted graphs. In directed graphs, kin and kout , which, respectively, count the numbers of vertices in ingoing and outgoing neighborhoods, are actually equal to each other; so either of them can be utilized as the degree measure. The clustering coefficient C [5] is another local metric. It should be noted, however, that C will not be the focus of our work, due to the high clustering nature of spatial networks. For node v, C v counts the number of edges in the subgraph of v’s immediate neighborhood. (In the literature, C v is always normalized by kv ðkv 1Þ=2, since at most kv ðkv 1Þ=2 edges can exist in v’s neighborhood.) As far as spatial networks are concerned, C remains consistently large due to locality of the edges. It has been proposed in Refs. [11,12] that, for 2-dimensional spatial networks with uniform radius (such as model I), the clustering coefficient has a high pffiffiffi asymptotic value C ¼ 1 3 3=ð4pÞ 0:59 in the limit of N ! 1. In our simulations, the resulting clustering coefficients for models I–III are always large, see Table 1 for typical results.
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Table 1 Structure properties of the two-radius spatial networks of size N ¼ 1000 CI
C II
C III
LI
LII
LIII
kI
kII
kIII
0.605
0.727
0.642
12.7
1.82
3.73
9.71
342
42.8
Here rI ¼ 0:057, rII ¼ 0:398, p ¼ 0:10.
1
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0 10-3
10-2
p
10-1
100
Fig. 2. Collective dynamics of the two-radius spatial networks in terms of the characteristic path length and the node degree: rI ¼ 0:057, rII ¼ 0:398, N ¼ 1000.
3.1. Small separation with low cost Start from the uniform-radius spatial networks. The characteristic path length L for these networks is inversely proportional to the uniform radius r approximately, that is L
const: , r
(3)
where the constant is the average Euclidean distance between two arbitrary points on a unit plane. Meanwhile, the node degree k for these networks can be calculated as k ¼ r pr2
(4)
when periodic boundaries are adopted. Even if this condition is not satisfied in our models, Eq. (4) can still be approximately used. Then we consider the two limiting cases of the two-radius spatial networks. Substitute the small and large radii from Eqs. (1) and (2) into Eqs. (3) and (4), we find that L LI N 0:5 =ðlog NÞ0:5 and k kI log N as p ! 0, while L LII N 0:5 =ðlog NÞ1:5 and k kII ðlog NÞ3 as p ! 1. Further we get LI =LII log N ! 1 and kI =kII 1=ðlog NÞ2 ! 0 as N ! 1. Thus network model I at p ¼ 0 is a sparsely connected, large world, whereas model II at p ¼ 1 is a densely connected, small world. These limiting cases might lead one to suspect that small k is always associated with large L, and large k with small L. On the contrary, during a broad interval of p, kðpÞ is almost as small as kI yet LðpÞ5LI , as revealed in Table 1 and Fig. 2. Here, rI ¼ 0:057 and rII ¼ 0:398 are chosen near the orders of Eqs. (1) and (2) for N ¼ 1000. The data shown here are averages over 20 random realizations of the radius-increase process. Note that the data in Fig. 2 have been normalized by the values LI and kII , that is to say, L=LI and k=kII þ kI =kII þ 1 are, respectively plotted.
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These efficient behaviors of the two-radius spatial networks result from different functional forms between kðpÞ and LðpÞ. On the one hand, the introduction of a few nodes with the large radius only increases kðpÞ linearly with regard to p, that is kðpÞ ¼ ð1 pÞ kI þ p kII .
(5)
When the majority of nodes take the small radius, namely for small p, kðpÞ remains practically unchanged as compared with kI . So the two-radius spatial networks with small p have nearly the same small amount of edges as the lower bound at which the network is just strongly connected, and it is ‘‘economical’’ to build such networks. On the other hand, significant drop in LðpÞ can be caused by this introduction of a few large-radius nodes, as shown in Fig. 2. Like the shortcuts in the small-world networks, the long-range connections of these large-radius nodes can provide direct ‘‘channels’’ for those nodes that would otherwise be far apart with each other. For small p, each large-radius node has a highly nonlinear effect on L, ‘‘facilitating’’ communications not just for this large-radius node itself, but for all the source nodes that should pass through this node to reach their destinations. These large-radius nodes can actually play the role as ‘‘hubs’’, i.e., whenever one node wants to reach its destinations far apart, large-radius nodes can be utilized as relay. As a result, in the tworadius spatial networks, those small amount of large-radius nodes effectively play the main role of reducing L, while all the other small-radius nodes play the role of keeping small k, resulting in small separation with low cost in spatial networks. Note that, to check the robustness of these results, we have tested different small radii which are near the lower bound for connectivity, and many different large radii whose ranges are enough larger than those of the small radii, for different network sizes. They all give qualitatively similar results. In summary, together with the high clustering nature of spatial networks, our observations are that the tworadius structures possess both small separation and low cost provide an efficient organization mechanism for the networks. These observations differ somewhat from the small-world effects, where small separation and high clustering are observed for sparse (low cost) relational networks. 4. From the perspective of optimization The purpose to discuss optimization is due to the interesting finding that, the phenomenon that large-radius and small-radius nodes coexist in the graph emerges during the procedure of tuning individual radius in order to optimize the weighted sum of L and k. We use the centralized simulated annealing (SA) algorithms to search the optimal spatial network design in terms of L and k. (Note that, in order to restrict the searching space into the sub-space of spatial networks, an ad-hoc SA should be utilized. Details can be found in Krause et al.’s work [12].) The weighted sum of L and k is used as the objective function f :
x 1x Lþ k, aðNÞ bðNÞ
(6)
where x is the weight factor, aðNÞ and bðNÞ are to normalize L and k respectively. The resulting optimized structure of size N ¼ 100 is given in Fig. 3. (In fact, the centralized procedure is too time consuming to be practically carried out for networks of even larger scales.) It is shown in Fig. 3(b) that the majority of nodes only provide a few outgoing links to close neighbors, i.e., they take small radii. While some nodes can cover many other nodes as direct outgoing neighbors, such as the white nodes in Fig. 3(a), i.e., they take large radii. The emergence of this coexistence in the optimal structure is meaningful, leading one not to uniformly utilize the same radius for all nodes in the network so as to improve L and k. The advantage of our two-radius model is that it is a totally decentralized scheme, yet can achieve fairly good overall performance in both measures L and k. First of all, the two-radius model is naturally decentralized and scalable. In this model, the ranges of rI and rII are determined only by the size of the network, N. The choice for each node’s radius is carried out according to the probability p without requiring coordination with its neighborhood. (Note that N and p both can be predetermined.) As a result, the tworadius design can be easily applied to large-scale networks. Moreover, in terms of L and k, the two-radius model always outperforms the model that utilizes the same radius for all nodes in the network. Consider the uniform model of medium radius rm (rI orm orII ).
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Fig. 3. The optimized results with regard to function f ðÞ using the SA algorithm: (a) structure; (b) outgoing degree distribution. Here, N ¼ 100, a ¼ 1:0, b ¼ 4:6, and x ¼ 0:5.
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Fig. 4. Comparisons between the uniform-radius and the two-radius spatial networks, in terms of both the characteristic path length and the node degree: (a) rI ¼ 0:057, rII ¼ 0:285; (b) rI ¼ 0:07, rII ¼ 0:35; (c) rI ¼ 0:14, rII ¼ 0:70.
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It interpolates between low-connectivity and high-connectivity spatial networks like the two-radius model. Its two limiting cases, i.e., rm ! rI and rm ! rII , are, respectively equivalent to the two limiting cases of the tworadius model. During the intermediate region, that is, as rm increases from rI to rII , we find Lðrm Þ decreases yet kðrm Þ increases. While in the two-radius model, as p increases from 0 to 1, LðpÞ decreases yet kðpÞ increases as well. Due to these two similar trends, it is natural to ask, in terms of L and k, which mechanism is more efficient during this intermediate region. In our experiments, we find that, the two-radius model outperforms the uniform medium-radius model over the whole intermediate region. The results are presented in Fig. 4 using Pareto frontier techniques [13] to evaluate the multi-objectives L and k. Three sub-figures representt results under three typical sets of different small/large radii. In every sub-figure, the circles from top left to bottom right represent results of the two-radius structures as p increases from 0 to 1. The dash lines from top left to bottom right represent results of the uniform-radius structures as the radius increases from the lower bound (i.e., the radius at which the network is just strongly connected) to the upper bound (i.e., the radius at which the network is just fully connected). Note that the performance improvement by the two-radius model still emerges even if the small and large radii do not follow the orders in Eqs. (1) and (2). For example, in Fig. 4(c), the small radius is much larger than the lower bound in Eq. (1); however, the performance of the two-radius model is still better than that of the uniform medium-radius model. 5. Application The purpose of this section is to show the practical implications of our two-radius model. The application in wireless ad hoc/sensor networks [14] is investigated here. The topology of static ad hoc/sensor networks can be viewed as spatial graphs, where distributed nodes communicate with each other via a wireless medium. Network-wide broadcasting [15], in which packets from a source node are required to spread over the network by multi-hop relay, is a normally used building block for many other network layer protocols of ad hoc/sensor networks. An efficient broadcasting scheme is to spread the message over the network with low average reception time and low transmission power. From the characteristic path length to the average reception time: In ad hoc/sensor networks, broadcasting always chooses the shortest path, because it chooses every possible path in parallel. For simplicity we assume each node has one-unit processing/waiting time. The average reception time, which measures the average time for broadcasting from a randomly chosen source node to a random destination node, is equal to the characteristic path length as discussed in Section 3. From the node degree to the transmission power: On the one hand, for the transmission power, we assume an rq energy loss due to channel transmission, and the exponent q typically falls into the region 2pqp4 [16]. On the other hand, the variant of the node degree k for weighted graphs can also follow an rq ð2pqp4Þ increase. As in Eq. (4), k follows an r2 increase for un-weighted graphs. In weighted graphs, however, we define the variant of k as the sum of edge weights over all outgoing edges of a typical node, that is Z r kweight 9 gðlÞ r 2pl dl, (7) 0
where the weight g for each edge is a function of the Euclidean distance l of the edge. If we set g ¼ 1, then kweight r2 , as is the case for un-weighted graphs. If g ¼ l q2 , then kweight rq , that is, kweight can be viewed as the counterpart of the transmission power. The broadcasting procedure depends on two things: (1) how messages are transmitted in one ‘‘hop’’, i.e., from one node to its immediate neighbors; (2) how messages are relayed over ‘‘multi-hop’’ to far away nodes. All existing schemes pay attention to the second problem and try to reduce the percentage of retransmission nodes during the message relay over multi-hop. Uniform radius in one hop is generally used. Here we try to investigate the first problem and introduce a ‘‘two-radius’’ broadcasting. With no loss of generality, we use one of the existing schemes, distance-based scheme [15], as the relay scheme. In our experiments, N nodes are randomly deployed on a D D square area, a message of k-bit from a random source node is required to spread over the network. The small radius rs and the large radius rl are chosen of the orders in Eqs. (1) and (2), which should be multiplied by D. rs and rl are randomly assigned to each retransmission node as the one-hop size with probability 1 p and p, respectively. According to the
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distance-based relay scheme, signal strength of received message is used to estimate the relative distance between neighboring nodes, and only those receivers that is far away from the transmitters are required to retransmit. During a fixed time slot, each recipient node may receive several copies of the same message. If all the relative distances measured by the copies’ signal strengths are above a threshold th, the node should retransmit; else, it need not retransmit. Here we define a constant y as the ratio of the threshold distance th to the transmission range r, thus in the two-radius broadcastings, the threshold distance th for small-radius and large-radius nodes is y rs and y rl , respectively. We assume a simple power consumption model proposed in Refs. [17,18]. Each transmitter in this model dissipates energy to run the radio electronics and the power amplifier. To transmit a k-bit message over a distance r, the power consumption of the transmitter is E T ðk; rÞ ¼ k E elec þ k amp;q rq ,
(8)
where we set E elec ¼ 50 nJ=bit, amp;q¼2 ¼ 100 pJ=bit=m2 , and amp;q¼4 ¼ 0:013 pJ=bit=m4 . 1
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p Fig. 5. Efficient behaviors of the two-radius broadcasting in terms of the average reception time and the transmission power: D ¼ 500 m, rs ¼ 28:5 m, rl ¼ 199 m, N ¼ 1000, y ¼ 0:75, and k ¼ 2000.
Average Reception Time
2-radius uniform radius
101
100 Transmission Power (J) Fig. 6. Comparisons between the uniform-radius broadcastings and the two-radius broadcastings. Here q ¼ 2.
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It is observed in our experiments that, the two-radius broadcastings present efficient behaviors like the behaviors in the two-radius structures: during a broad interval of p, the transmission power is almost as small as that of the uniform short-radius protocols, while the average reception time is far less than that of the uniform short-radius protocols, as revealed in Fig. 5. The data shown here are averages over 50 random realizations, and have been normalized. Note that, the path-loss exponent q is, respectively set to be 2 and 4, and the efficient behaviors of the two-radius broadcasting emerge in both situations. That is because, no matter how the exponent q is set, the transmission power is naturally a local metric which increases linearly with regard to p after introducing a few large-radius nodes (as is similar to the case of the node degree k in Eq. (5)). Besides, we find that, the two-radius protocols always outperform the uniform-radius protocols in terms of the average reception time and the transmission power, as presented in Fig. 6. It should be pointed out that, the two-radius model may result in uni-directional links between large-radius and small-radius nodes, and these asymmetric links can cause problems in communication protocols, especially those without global view of the network. For broadcasting protocol, however, it is sufficient to use such uni-directional links since feedback messages are usually not required. 6. Conclusion In this work, our main objective is to investigate the efficient two-radius mechanism for spatial networks. We find that, after adopting the two-radius pattern and therefore introducing a new spectrum between low and high connectivity, spatial networks possess small typical separation with low cost during a broad interval of p. Like shortcuts in small-world networks, those small amount of large-radius nodes effectively ‘‘facilitate’’ global communications; meanwhile, all the other small-radius nodes keep their local cost at low level. Together with the high clustering nature of spatial networks, the two-radius spatial networks with both small separation and low cost provide an efficient organization mechanism for network design. Our observations differ somewhat from the small-world effects, where small separation and high clustering are observed for sparse (low cost) relational networks. We compare the two-radius model to the uniform medium-radius model and the centralized model, in terms of global separation and cost. We find that the two-radius model is better than the uniform medium-radius model. Moreover, the decentralized two-radius model is scalable, while the centralized model is not. Finally, we try to apply the two-radius design to the broadcasting protocols for wireless ad hoc/sensor networks, and find that the two-radius broadcastings present efficient behaviors like the behaviors in the two-radius structures. We hope that, our work will stimulate further studies on the two-radius spatial networks, including comparisons with other spatial network models. We also hope that our observations so far will provide potential applications in other networked systems. Acknowledgements This work was supported in part by NSFC (Grant nos. 60074012, 60274011, 60574067), National Key Project of China, Fundamental Research Funds from Tsinghua University, NCET program (NCET-04-0094) of China and Funds from National Key Lab for Power Systems in Tsinghua University. The authors would like to thank the reviewers whose comments and suggestions materially improved the presentation of this paper. They would also like to thank Mr. Xiaoying Liang, Mr. Song Yang, and Mr. Yongcai Wang for helpful discussions. References [1] [2] [3] [4] [5]
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F. Liu, Q. Zhao / Physica A 366 (2006) 608–618 D.J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness, Princeton University Press, NJ, 1999. A. Helmy, IEEE Commun. Lett. 7 (2003) 490. J.M. Kleinberg, Nature 406 (2000) 845. P. Gupta, P.R. Kumar, Critical power for asymptotic connectivity in wireless networks, in: W.M. McEneany, G. Yin, Q. Zhang (Eds.), Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, Birkhauser, Boston, MA, 1998, p. 547. F. Xue, P.R. Kumar, Wireless Networks 10 (2004) 169. J. Dall, M. Christensen, Phys. Rev. E 66 (2002) 016121. W. Krause, I. Glauche, R. Sollacher, M. Greiner, Physica A 338 (2004) 633. M. Kaisa, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Dordrect, 1999. I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, IEEE Commun. Mag. August (2002) 102. Y.C. Tseng, S.Y. Ni, Y.S. Chen, J.P. Sheu, Wireless Networks 8 (2002) 153. J.W. Mark, W. Zhuang, Wireless Communications and Networking, Prentice-Hall, Englewood Cliffs, NJ, 2003. W.B. Heinzelman, A.P. Chandrakasan, H. Balakrishnan, in: Proceedings of the 33rd Hawaii International Conference on System Sciences, 2000. W.B. Heinzelman, A.P. Chandrakasan, H. Balakrishnan, IEEE Trans. Wireless Commun. 1 (2002) 660.
Physica A 366 (2006) 608–618 www.elsevier.com/locate/physa
An efficient organization mechanism for spatial networks Fei Liu, Qianchuan Zhao Department of Automation, Center for Intelligent and Networked Systems, Tsinghua University, Beijing 100084, China Received 6 April 2005; received in revised form 21 September 2005 Available online 21 November 2005
Abstract Spatial networks, also known as random geometric graphs, are random graphs with certain distance metric, in which each node is connected to some others within its neighborhood disc. Due to the rapid increase of network scales, the design of spatial networks becomes increasingly challenging. Inspired by the recently discovered small-world topology in relational networks, we identify an efficient organization mechanism for spatial networks, which we believe is useful for spatial network design. Two types of discs are introduced. Edges in the large discs are counterparts of ‘‘shortcuts.’’ We find that such ‘‘two-radius’’ spatial networks exhibit small characteristic path length, yet with low cost. This mechanism is applied to broadcasting protocol design for wireless ad hoc/sensor networks. r 2005 Elsevier B.V. All rights reserved. Keywords: Statistical physics of complex networks; Spatial network design; Small world; Optimization; Wireless ad hoc/sensor networks
1. Introduction Popular network models can be classified as relational and spatial networks. The constructions of relational networks do not depend upon any external metric of distance [1], while spatial networks are networks with certain metric, such as Euclidean distance [2]. The design of networks becomes more and more challenging than before. Global optimization seems impossible due to the rapid increase of the network scales. The success of existing large-scale man-made networks depends heavily on the decentralizing nature of the rules in their organization mechanisms. The entire network should be efficient both globally and locally. It should be efficient on a global scale as its function is usually realized by the joint coordination among many nodes. Its measurement is the typical separation between two vertices in the graph. Meanwhile, the network should be efficient on a local scale. The measurements include the clustering/cliquishness of a local neighborhood (first introduced in Ref. [5]), and the cost to build a neighborhood (first introduced in Ref. [4]). It is found in Refs. [3–5] that, to design an efficient organization mechanism for both relational and spatial networks, small degrees of separation, high clustering, and low cost are all required. One of the most important discoveries for decentralized organization mechanism for large-scale networks is the small-world topology. By analogy with the small-world phenomenon in social networks, Watts and Strogatz [5] have shown that many real relational networks possess small-world characteristics, that is, small Corresponding author. Tel.: +86 10 62783612; fax.: +86 10 62786911.
E-mail addresses: [email protected] (F. Liu), [email protected] (Q. Zhao). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.10.022
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typical separation between two vertices (measured by characteristic path length) and high clustering (measured by clustering coefficient). Note that the Watts–Strogatz model assumes the sparseness of the edges. Whether the small worlds still emerge in spatial networks? On the one hand, as Watts analyzed in Ref. [6], spatial networks admit no small-world class, unless some edges have long ranges (shortcuts) that are comparable to the network size. On the other hand, according to Refs. [7,8], after relaxing the strict definition of spatial networks, i.e., affecting the disc-like neighborhoods by adding ‘‘shortcuts’’ with probabilities depending on inter-node distance, small worlds can be introduced. Based on these two observations, it is natural to ask, when the definition is yet strictly obeyed, whether spatial networks present similar behaviors like the small worlds. In fact, obeying this definition is meaningful in both theoretical (e.g. relational vs. spatial networks) and practical (e.g. wireless systems with omni antenna) perspectives. For spatial networks, in reality, the clustering tends to be consistently high due to locality of the edges [11,12]. Meanwhile, motivated by applications, we regard the node operational cost as another indispensable local metric for network design. In wireless applications, the cost turns out to be the power consumption of wireless devices, and the fact that the battery power is not unlimited should be taken into account. In this paper, we are focusing on whether there exist some efficient organization mechanisms for spatial networks so that they have both small typical separation and low cost (measured by Node Degree, as defined later. It is a kind of measure for cost or energy consumption in wireless applications, as will be explained in Section 5). Note the conclusion of Watts on spatial networks was made based on the use of the clustering coefficient as the only efficiency measurement on a local scale. If we also take into account the node degree that measures the number of neighbors of a typical node in the graph, the key concept of shortcuts may still be applicable for spatial networks. This paper shows that in this sense a counterpart of shortcuts does exist for spatial networks. Observe that if we connect a node to all neighbors in a large neighborhood disc centered at this node in a random geometric network, then the appearance of such discs provides shortcuts to reduce the characteristic path length efficiently. Observe also that such large discs increase the cost, namely increase the node degree. The new efficient organization mechanisms we find for spatial networks are based on the above two observations and can be regarded as an extension of the small-world topology for relational networks. The idea is to construct random geometric networks by connecting most of the nodes with ‘‘nonshortcuts’’ and adding ‘‘shortcuts’’ connections randomly with small probability. The ‘‘nonshortcuts’’ connections are corresponding to edges in small discs, which result with large separation and low cost; and the ‘‘shortcuts’’ are corresponding to edges in large discs. Because there are two types of discs in the networks we call them ‘‘tworadius’’ spatial networks. We will show in this paper they exhibit small separation, high clustering, and low cost. The rest of this paper is organized as follows. In Section 2, we propose three spatial network models. Based on these models, in Section 3, collective dynamics of the two-radius spatial networks are presented. Then we provide analysis from the perspective of optimization in Section 4, and apply our results to wireless ad hoc/ sensor networks in Section 5. Section 6 concludes the paper.
2. Three spatial network models A spatial network is a graph where each node (vertex) is assigned random coordinates in a d-dimensional box of unit volume, and each edge is added to connect pairs of nodes which are close to each other by metric. Here 2-dimensional spatial networks with Euclidean distances are considered. In the square area ½0; 1 ½0; 1, N nodes are assigned random coordinates ðxi ; yi Þ ði ¼ 1; 2; . . . ; NÞ, thus with density r ¼ N. Note that no periodic boundary conditions are used here. We assume that the neighborhood ranges of all nodes are circles, that is, each node only ‘‘communicate’’ with some others within its neighborhood disc. Let ri denote the disc radius of node i. For any other node j , if jðxj ; yj Þ ðxi ; yi Þjori is true, it defines a directed link i ! j, and j is then i’s outgoing neighbor. If the link j ! i also exists, then the two nodes are connected via a bidirectional link i2j and are able to directly arrive back and forth. In the following part, to explore efficient behaviors in spatial networks, three models are presented: model I, uniform small radius; model II, uniform large radius; and model III, two-radius model, viz. small/ large radii.
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2.1. Model I: uniform small radius The most widely used structural design of spatial networks is to utilize the same radius. On the one hand, the uniform radius should be large enough to make the network strongly connected; on the other hand, the uniform radius in this model should be as small as possible to reduce the number of edges. Accordingly we choose the uniform small radius to be near the minimal radius at which the network is just strongly connected. Itp was shown in Refs. [9,10] that if N nodes are placed in a disc of unit area and each node’s radius chosen as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðlog N þ gN Þ=ðpNÞ , then the resulting network is asymptotically connected with probability one as N! if and only if the sequence gN ! þ1 . If we set gN ¼ log N ! þ1 , then the lower bound pþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 2 log N=ðpNÞ. According to this, we choose uniform small radius in this model I of order rI
ðlog NÞ0:5 N 0:5
(1)
to make the whole network just strongly connected. 2.2. Model II: uniform large radius In this model, a large radius rII is uniformly assigned to each node, and it should obey the following two assumptions: (a) 0orII prmax , where rmax is the maximum radius at which the network is just fully connected; (b) rII 4rI and rII =rI ! 1 as N ! 1, such that the range of rII is enough larger than that of rI . Accordingly we choose the large radius of order rII
ðlog NÞ1:5 N 0:5
(2)
such that rII =rI ¼ log N ! 1. Note rII can be chosen of other orders, e.g. rII 1= log N, and rII c ðrI ocprmax Þ, as long as they follow the two assumptions. 2.3. Model III: two-radius model, viz. small/large radii In this model, the small radius rs is set to be rI as in model I, and the large radius rl to be rII as in model II. Then rs and rl are randomly assigned to each node in the network with probability 1 p and p ð0opo1Þ, respectively. Note that not all the links remain bidirectional in this model, since small-radius node i, for instance, can be within the range of some large-radius node j while node j may lie beyond the range of node i. Illustrations for models I–III are presented in Fig. 1. 3. Collective dynamics of the two-radius spatial networks The two-radius spatial networks are to interpolate between low-connectivity and high-connectivity spatial networks, compared with the Watts–Strogatz’s relational model between regularity and randomness. Starting from a low-connectivity network generated by model I, we alter each node’s small radius rI to the large radius rII at random with the probability p. Once a node’s radius is increased, one-directed edges from this node to those within its range will be added to the graph if they do not originally exist. This construction allows us to ‘‘tune’’ the graph between low (p ¼ 0, as in model I) and high connectivity (p ¼ 1, as in model II), and thereby to examine the intermediate region 0opo1, as in model III. Two statistics of interest are examined to quantify network efficiency. They are the characteristic path length L and the node degree k which have most relevance from practical point of view. It should be noted that our network model is different from those of Refs. [3–5] in that our model is directed graph. We justify the use of these measures in detail below. The first statistic is the characteristic path length L, defined as the average number of edges of the shortest one-directed path over all pairs of source/destination nodes in the graph.
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Fig. 1. Illustrations for network models I–III of 100 nodes: (a) rI ¼ 0:15; (b) rII ¼ 0:45; (c) rs ¼ 0:15, rl ¼ 0:45, p ¼ 0:10; (d) the outgoing degree distribution of (c).
L measures the typical separation between two vertices in the graph, and it is then a global property (because, in general, determining the shortest path length between any two nodes requires information about the entire graph). The second statistic is the node degree k, which is a measure of the local graph structure. Specifically, for a node v, the degree kv counts the number of vertices in the subgraph of v’s immediate neighborhood. k is the average of kv over all vertices in the graph. k measures the cost to build a neighborhood in un-weighted graphs, and see Section 5 for the variant of k in weighted graphs. In directed graphs, kin and kout , which, respectively, count the numbers of vertices in ingoing and outgoing neighborhoods, are actually equal to each other; so either of them can be utilized as the degree measure. The clustering coefficient C [5] is another local metric. It should be noted, however, that C will not be the focus of our work, due to the high clustering nature of spatial networks. For node v, C v counts the number of edges in the subgraph of v’s immediate neighborhood. (In the literature, C v is always normalized by kv ðkv 1Þ=2, since at most kv ðkv 1Þ=2 edges can exist in v’s neighborhood.) As far as spatial networks are concerned, C remains consistently large due to locality of the edges. It has been proposed in Refs. [11,12] that, for 2-dimensional spatial networks with uniform radius (such as model I), the clustering coefficient has a high pffiffiffi asymptotic value C ¼ 1 3 3=ð4pÞ 0:59 in the limit of N ! 1. In our simulations, the resulting clustering coefficients for models I–III are always large, see Table 1 for typical results.
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Table 1 Structure properties of the two-radius spatial networks of size N ¼ 1000 CI
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kIII
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0.727
0.642
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Fig. 2. Collective dynamics of the two-radius spatial networks in terms of the characteristic path length and the node degree: rI ¼ 0:057, rII ¼ 0:398, N ¼ 1000.
3.1. Small separation with low cost Start from the uniform-radius spatial networks. The characteristic path length L for these networks is inversely proportional to the uniform radius r approximately, that is L
const: , r
(3)
where the constant is the average Euclidean distance between two arbitrary points on a unit plane. Meanwhile, the node degree k for these networks can be calculated as k ¼ r pr2
(4)
when periodic boundaries are adopted. Even if this condition is not satisfied in our models, Eq. (4) can still be approximately used. Then we consider the two limiting cases of the two-radius spatial networks. Substitute the small and large radii from Eqs. (1) and (2) into Eqs. (3) and (4), we find that L LI N 0:5 =ðlog NÞ0:5 and k kI log N as p ! 0, while L LII N 0:5 =ðlog NÞ1:5 and k kII ðlog NÞ3 as p ! 1. Further we get LI =LII log N ! 1 and kI =kII 1=ðlog NÞ2 ! 0 as N ! 1. Thus network model I at p ¼ 0 is a sparsely connected, large world, whereas model II at p ¼ 1 is a densely connected, small world. These limiting cases might lead one to suspect that small k is always associated with large L, and large k with small L. On the contrary, during a broad interval of p, kðpÞ is almost as small as kI yet LðpÞ5LI , as revealed in Table 1 and Fig. 2. Here, rI ¼ 0:057 and rII ¼ 0:398 are chosen near the orders of Eqs. (1) and (2) for N ¼ 1000. The data shown here are averages over 20 random realizations of the radius-increase process. Note that the data in Fig. 2 have been normalized by the values LI and kII , that is to say, L=LI and k=kII þ kI =kII þ 1 are, respectively plotted.
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These efficient behaviors of the two-radius spatial networks result from different functional forms between kðpÞ and LðpÞ. On the one hand, the introduction of a few nodes with the large radius only increases kðpÞ linearly with regard to p, that is kðpÞ ¼ ð1 pÞ kI þ p kII .
(5)
When the majority of nodes take the small radius, namely for small p, kðpÞ remains practically unchanged as compared with kI . So the two-radius spatial networks with small p have nearly the same small amount of edges as the lower bound at which the network is just strongly connected, and it is ‘‘economical’’ to build such networks. On the other hand, significant drop in LðpÞ can be caused by this introduction of a few large-radius nodes, as shown in Fig. 2. Like the shortcuts in the small-world networks, the long-range connections of these large-radius nodes can provide direct ‘‘channels’’ for those nodes that would otherwise be far apart with each other. For small p, each large-radius node has a highly nonlinear effect on L, ‘‘facilitating’’ communications not just for this large-radius node itself, but for all the source nodes that should pass through this node to reach their destinations. These large-radius nodes can actually play the role as ‘‘hubs’’, i.e., whenever one node wants to reach its destinations far apart, large-radius nodes can be utilized as relay. As a result, in the tworadius spatial networks, those small amount of large-radius nodes effectively play the main role of reducing L, while all the other small-radius nodes play the role of keeping small k, resulting in small separation with low cost in spatial networks. Note that, to check the robustness of these results, we have tested different small radii which are near the lower bound for connectivity, and many different large radii whose ranges are enough larger than those of the small radii, for different network sizes. They all give qualitatively similar results. In summary, together with the high clustering nature of spatial networks, our observations are that the tworadius structures possess both small separation and low cost provide an efficient organization mechanism for the networks. These observations differ somewhat from the small-world effects, where small separation and high clustering are observed for sparse (low cost) relational networks. 4. From the perspective of optimization The purpose to discuss optimization is due to the interesting finding that, the phenomenon that large-radius and small-radius nodes coexist in the graph emerges during the procedure of tuning individual radius in order to optimize the weighted sum of L and k. We use the centralized simulated annealing (SA) algorithms to search the optimal spatial network design in terms of L and k. (Note that, in order to restrict the searching space into the sub-space of spatial networks, an ad-hoc SA should be utilized. Details can be found in Krause et al.’s work [12].) The weighted sum of L and k is used as the objective function f :
x 1x Lþ k, aðNÞ bðNÞ
(6)
where x is the weight factor, aðNÞ and bðNÞ are to normalize L and k respectively. The resulting optimized structure of size N ¼ 100 is given in Fig. 3. (In fact, the centralized procedure is too time consuming to be practically carried out for networks of even larger scales.) It is shown in Fig. 3(b) that the majority of nodes only provide a few outgoing links to close neighbors, i.e., they take small radii. While some nodes can cover many other nodes as direct outgoing neighbors, such as the white nodes in Fig. 3(a), i.e., they take large radii. The emergence of this coexistence in the optimal structure is meaningful, leading one not to uniformly utilize the same radius for all nodes in the network so as to improve L and k. The advantage of our two-radius model is that it is a totally decentralized scheme, yet can achieve fairly good overall performance in both measures L and k. First of all, the two-radius model is naturally decentralized and scalable. In this model, the ranges of rI and rII are determined only by the size of the network, N. The choice for each node’s radius is carried out according to the probability p without requiring coordination with its neighborhood. (Note that N and p both can be predetermined.) As a result, the tworadius design can be easily applied to large-scale networks. Moreover, in terms of L and k, the two-radius model always outperforms the model that utilizes the same radius for all nodes in the network. Consider the uniform model of medium radius rm (rI orm orII ).
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Fig. 4. Comparisons between the uniform-radius and the two-radius spatial networks, in terms of both the characteristic path length and the node degree: (a) rI ¼ 0:057, rII ¼ 0:285; (b) rI ¼ 0:07, rII ¼ 0:35; (c) rI ¼ 0:14, rII ¼ 0:70.
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It interpolates between low-connectivity and high-connectivity spatial networks like the two-radius model. Its two limiting cases, i.e., rm ! rI and rm ! rII , are, respectively equivalent to the two limiting cases of the tworadius model. During the intermediate region, that is, as rm increases from rI to rII , we find Lðrm Þ decreases yet kðrm Þ increases. While in the two-radius model, as p increases from 0 to 1, LðpÞ decreases yet kðpÞ increases as well. Due to these two similar trends, it is natural to ask, in terms of L and k, which mechanism is more efficient during this intermediate region. In our experiments, we find that, the two-radius model outperforms the uniform medium-radius model over the whole intermediate region. The results are presented in Fig. 4 using Pareto frontier techniques [13] to evaluate the multi-objectives L and k. Three sub-figures representt results under three typical sets of different small/large radii. In every sub-figure, the circles from top left to bottom right represent results of the two-radius structures as p increases from 0 to 1. The dash lines from top left to bottom right represent results of the uniform-radius structures as the radius increases from the lower bound (i.e., the radius at which the network is just strongly connected) to the upper bound (i.e., the radius at which the network is just fully connected). Note that the performance improvement by the two-radius model still emerges even if the small and large radii do not follow the orders in Eqs. (1) and (2). For example, in Fig. 4(c), the small radius is much larger than the lower bound in Eq. (1); however, the performance of the two-radius model is still better than that of the uniform medium-radius model. 5. Application The purpose of this section is to show the practical implications of our two-radius model. The application in wireless ad hoc/sensor networks [14] is investigated here. The topology of static ad hoc/sensor networks can be viewed as spatial graphs, where distributed nodes communicate with each other via a wireless medium. Network-wide broadcasting [15], in which packets from a source node are required to spread over the network by multi-hop relay, is a normally used building block for many other network layer protocols of ad hoc/sensor networks. An efficient broadcasting scheme is to spread the message over the network with low average reception time and low transmission power. From the characteristic path length to the average reception time: In ad hoc/sensor networks, broadcasting always chooses the shortest path, because it chooses every possible path in parallel. For simplicity we assume each node has one-unit processing/waiting time. The average reception time, which measures the average time for broadcasting from a randomly chosen source node to a random destination node, is equal to the characteristic path length as discussed in Section 3. From the node degree to the transmission power: On the one hand, for the transmission power, we assume an rq energy loss due to channel transmission, and the exponent q typically falls into the region 2pqp4 [16]. On the other hand, the variant of the node degree k for weighted graphs can also follow an rq ð2pqp4Þ increase. As in Eq. (4), k follows an r2 increase for un-weighted graphs. In weighted graphs, however, we define the variant of k as the sum of edge weights over all outgoing edges of a typical node, that is Z r kweight 9 gðlÞ r 2pl dl, (7) 0
where the weight g for each edge is a function of the Euclidean distance l of the edge. If we set g ¼ 1, then kweight r2 , as is the case for un-weighted graphs. If g ¼ l q2 , then kweight rq , that is, kweight can be viewed as the counterpart of the transmission power. The broadcasting procedure depends on two things: (1) how messages are transmitted in one ‘‘hop’’, i.e., from one node to its immediate neighbors; (2) how messages are relayed over ‘‘multi-hop’’ to far away nodes. All existing schemes pay attention to the second problem and try to reduce the percentage of retransmission nodes during the message relay over multi-hop. Uniform radius in one hop is generally used. Here we try to investigate the first problem and introduce a ‘‘two-radius’’ broadcasting. With no loss of generality, we use one of the existing schemes, distance-based scheme [15], as the relay scheme. In our experiments, N nodes are randomly deployed on a D D square area, a message of k-bit from a random source node is required to spread over the network. The small radius rs and the large radius rl are chosen of the orders in Eqs. (1) and (2), which should be multiplied by D. rs and rl are randomly assigned to each retransmission node as the one-hop size with probability 1 p and p, respectively. According to the
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distance-based relay scheme, signal strength of received message is used to estimate the relative distance between neighboring nodes, and only those receivers that is far away from the transmitters are required to retransmit. During a fixed time slot, each recipient node may receive several copies of the same message. If all the relative distances measured by the copies’ signal strengths are above a threshold th, the node should retransmit; else, it need not retransmit. Here we define a constant y as the ratio of the threshold distance th to the transmission range r, thus in the two-radius broadcastings, the threshold distance th for small-radius and large-radius nodes is y rs and y rl , respectively. We assume a simple power consumption model proposed in Refs. [17,18]. Each transmitter in this model dissipates energy to run the radio electronics and the power amplifier. To transmit a k-bit message over a distance r, the power consumption of the transmitter is E T ðk; rÞ ¼ k E elec þ k amp;q rq ,
(8)
where we set E elec ¼ 50 nJ=bit, amp;q¼2 ¼ 100 pJ=bit=m2 , and amp;q¼4 ¼ 0:013 pJ=bit=m4 . 1
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p Fig. 5. Efficient behaviors of the two-radius broadcasting in terms of the average reception time and the transmission power: D ¼ 500 m, rs ¼ 28:5 m, rl ¼ 199 m, N ¼ 1000, y ¼ 0:75, and k ¼ 2000.
Average Reception Time
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100 Transmission Power (J) Fig. 6. Comparisons between the uniform-radius broadcastings and the two-radius broadcastings. Here q ¼ 2.
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It is observed in our experiments that, the two-radius broadcastings present efficient behaviors like the behaviors in the two-radius structures: during a broad interval of p, the transmission power is almost as small as that of the uniform short-radius protocols, while the average reception time is far less than that of the uniform short-radius protocols, as revealed in Fig. 5. The data shown here are averages over 50 random realizations, and have been normalized. Note that, the path-loss exponent q is, respectively set to be 2 and 4, and the efficient behaviors of the two-radius broadcasting emerge in both situations. That is because, no matter how the exponent q is set, the transmission power is naturally a local metric which increases linearly with regard to p after introducing a few large-radius nodes (as is similar to the case of the node degree k in Eq. (5)). Besides, we find that, the two-radius protocols always outperform the uniform-radius protocols in terms of the average reception time and the transmission power, as presented in Fig. 6. It should be pointed out that, the two-radius model may result in uni-directional links between large-radius and small-radius nodes, and these asymmetric links can cause problems in communication protocols, especially those without global view of the network. For broadcasting protocol, however, it is sufficient to use such uni-directional links since feedback messages are usually not required. 6. Conclusion In this work, our main objective is to investigate the efficient two-radius mechanism for spatial networks. We find that, after adopting the two-radius pattern and therefore introducing a new spectrum between low and high connectivity, spatial networks possess small typical separation with low cost during a broad interval of p. Like shortcuts in small-world networks, those small amount of large-radius nodes effectively ‘‘facilitate’’ global communications; meanwhile, all the other small-radius nodes keep their local cost at low level. Together with the high clustering nature of spatial networks, the two-radius spatial networks with both small separation and low cost provide an efficient organization mechanism for network design. Our observations differ somewhat from the small-world effects, where small separation and high clustering are observed for sparse (low cost) relational networks. We compare the two-radius model to the uniform medium-radius model and the centralized model, in terms of global separation and cost. We find that the two-radius model is better than the uniform medium-radius model. Moreover, the decentralized two-radius model is scalable, while the centralized model is not. Finally, we try to apply the two-radius design to the broadcasting protocols for wireless ad hoc/sensor networks, and find that the two-radius broadcastings present efficient behaviors like the behaviors in the two-radius structures. We hope that, our work will stimulate further studies on the two-radius spatial networks, including comparisons with other spatial network models. We also hope that our observations so far will provide potential applications in other networked systems. Acknowledgements This work was supported in part by NSFC (Grant nos. 60074012, 60274011, 60574067), National Key Project of China, Fundamental Research Funds from Tsinghua University, NCET program (NCET-04-0094) of China and Funds from National Key Lab for Power Systems in Tsinghua University. The authors would like to thank the reviewers whose comments and suggestions materially improved the presentation of this paper. They would also like to thank Mr. Xiaoying Liang, Mr. Song Yang, and Mr. Yongcai Wang for helpful discussions. References [1] [2] [3] [4] [5]
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