GENERATING STRUCTURE SPECIFIC ... - Semantic Scholar

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Advances in Complex Systems, Vol. 13, No. 2 (2010) 239–250 c World Scientific Publishing Company DOI: 10.1142/S0219525910002517

GENERATING STRUCTURE SPECIFIC NETWORKS

∗ , A. JONSSON† and J. LENNARTSSON‡ ˙ N. HAKANSSON

Systems Biology Research Centre, Sk¨ ovde University, Box 408, 541 28 Sk¨ ovde, Sweden ∗[email protected] †[email protected] ‡[email protected] ¨ § and U. WENNERGREN¶ T. LINDSTROM IFM Therory and Modeling, Link¨ oping University, 581 83 Link¨ oping, Sweden §[email protected] ¶[email protected] Received 17 September 2009 Revised 8 January 2010

Theoretical exploration of network structure significance requires a range of different networks for comparison. Here, we present a new method to construct networks in a spatial setting that uses spectral methods in combination with a probability distribution function. Nearly all previous algorithms for network construction have assumed randomized distribution of links or a distribution dependent on the degree of the nodes. We relax those assumptions. Our algorithm is capable of creating spectral networks along a gradient from random to highly clustered or diverse networks. Number of nodes and link density are specified from start and the structure is tuned by three parameters (γ, σ, κ). The structure is measured by fragmentation, degree assortativity, clustering and group betweenness of the networks. The parameter γ regulates the aggregation in the spatial node pattern and σ and κ regulates the probability of link forming. Keywords: Network; spectral; assortativity; fragmentation; clustering; betweenness centralization; spatial network; network algorithm.

1. Introduction Representing complex systems as networks is common in many subject fields, for example biology, social science and economy. Examples of networks are ecological food webs, metapopulations, regulatory protein networks, disease transmission networks, networks of movie actors and collaboration networks of scientists. Networks † Corresponding

author. 239

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can be described and categorized according to different network measures [29]. Statistics for a variety of empirical investigated networks can be found in [21]. A well-studied group of networks are the random networks [3]. A random network is a graph with nodes randomly linked together. It means that a link between two nodes is completely independent of the presence of other links. Random graphs are frequently used in discrete mathematical problems but also in models of various real-world problems, especially epidemiological ones [22]. However, investigations of real-world networks show that their structure often is widely different from a random graph with Poisson distributed node degrees [22]. Thus, there is a great risk of missing important features if random graphs are used for network modeling. The concept of random graphs has therefore been developed to also include other degree distributions, for example exponential or power law distributions [28]. Many real-world networks have been classified as scale-free networks or smallworld networks [4, 30]. The method for classification of scale-free networks is however discussed and questioned [16]. In a scale-free network, the degree distribution of the nodes follows a power law. A common method of generating scale free graphs is by preferential attachment, that is networks are generated by attaching nodes at random to previously existing nodes with a probability proportional to the node degree [30]. A consequence of this method is a high proportion of disconnected graphs [1]. Uncorrelated scale-free networks can however be generated without preferential attachment by the uncorrelated configuration model (UCM) developed from the classical configuration models (CM) [9]. The CM models design network with an a priori set degree distribution and have also been developed to control degree–degree dependent correlations and/or degree dependent clustering [24, 27, 31]. Some of the real-world networks are embedded in a spatial setting, for example communication networks [14] and disease transmission networks [15], but the majority of network generation models do not take spatial aspect into consideration [11]. A way to consider the spatial aspects is to use distance selection when distributing the links, that is letting the probability of a link depend on the distance between the nodes, this is used for example in [5, 6, 15]. A model that uses an adaptation-migration scheme to create networks with the properties of mostly short-distance links but still triangles at large distances is suggested in [17]. Here, we present a new model, the spectral network algorithm, with linkattachment design based on geographical distances instead of a prefixed degree distribution. The algorithm uses spectral methods, which are powerful tools for analyzing, comparing and generating graphical patterns [25]. It is similar to Keeling’s [15] in the sense that it also uses a spatial pattern for organizing the nodes and distance selection to distribute the links. Other models of geographical networks have been reviewed by Hayashi [13]. He categorizes them into three classes, networks with disadvantaged long-range links, networks embedded in lattices and networks created by space-filling. The spectral network algorithm fits in to the first class since it can be used to create networks with few long-range links.

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We evaluate the algorithm by exploring the parameter intervals of fragmentation index, degree assortativity, clustering coefficient and group betweenness centralization index. These measures are defined as a single numerical value for the entire network. The intervals indicate what kind of networks that can be generated by the algorithm. Furthermore, we discuss the algorithm in relation to three general criteria: (i) The algorithm shall create a wide range of network structures. (ii) The algorithm parameters can be deduced from real-world settings. (iii) The methodology to obtain a specified network structure shall be straightforward. 2. Method 2.1. Network generation algorithm To generate networks with different structures we have developed a two-step algorithm. First, a spatial node pattern is generated (we will refer to this as the landscape) with spectral methods including fast Fourier transformation (FFT). The technique is similar to that presented in [12]. See also [18]. Links between nodes are added using a probability distribution function. The probability of forming a link depends on the Euclidean distance between the nodes in the landscape. The algorithm requires, apart from order (number of nodes) and link density (the number of links divided by the number of possible links), three parameters (γ, σ, κ) that are used for tuning the network to the desired structure. The parameter γ regulates the spatial aggregation of nodes. The parameters σ and κ are scale and shape parameters, respectively, regulating how the probability of links decreases with distance between nodes. Programs for generating the networks were written in MATLAB (version 7.5.0). 2.1.1. The landscape algorithm in more detail The algorithm starts by generating a matrix, Lrand , with random values from a Gaussian distribution (1). This matrix will be transformed to a spatial node distribution or landscape. The mean and standard deviation of the Gaussian distribution does not affect the properties of the network, because the absolute values in the matrix are not used (see below). The matrix Lrand is transformed to the frequency domain using fast Fourier transformation (2). The amplitudes of the function in the frequency domain is scaled (3), to give the power spectral density function the frequency pattern of 1/f γ -noise (4). The degree of aggregation is determined by the parameter γ. The scaled matrix Lconverted is finally obtained by inverse transforming the scaled function back to the spatial domain by inverse fast Fourier transformation (5). Lrand (x, y) = ∼ N (0.5, 2), X = FF T (Lrand ),

(1) (2)

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Xscaled (f ) =

|X(f )| , γ f2

(3)

P SDF = |Xscaled (f )|2 ∝ Lconverted = IFFT (Xscaled ).

1 , fγ

(4) (5)

The continuous matrix Lconverted is digitalized to a node landscape, L, with the number of nodes equal to the order of the network (n). The nodes are placed at the n elements in Lconverted with highest values. Hence, the absolute values of Lconverted are not used and the variance and mean of the starting Gaussian distribution will not affect the positions of the nodes. The size of the starting matrix √ √ will however affect the landscape. If the size of the matrix is close to n× n, almost every element will be assigned a node and these will be regularly distributed in the landscape. This will make network structures impossible to √ tune with √ the√node √ arrangement. We therefore recommend a matrix of size 6n× 6n to 20n× 20n. 2.1.2. Distribution of links The number of links required to receive a requested link density, is added to the network one at a time. The links are drawn randomly with probability for a link between node i and j to be drawn, P (lij ), is given by the generalized Gaussian distribution in Eq. (6). See also [18]. P (lij ) = Ke

−

“d

ij a

”b

.

(6)

The parameter dij is the Euclidean distance between node i and j. Periodic boundaries (also used in landscape modeling [32]) were used to avoid edge effects. This means that the left edge is considered connected to the right edge and the upper edge connected to the lower edge. The constant K normalizes the distribution so that the probability of all possible links sums to 1. Two specific nodes can only have one link between them and therefore the probabilities for already drawn links are zero and the normalization constant K is recalculated for every draw of a link. The parameters a and b regulate kurtosis (κ) and standard deviation (σ) of the probability function. A high kurtosis means that the distribution has a higher peak at distances close to zero and a fatter tail. A low kurtosis means that the probability distribution has similar values for distances from zero up to some distance controlled by σ. Because the nodes are distributed in two dimensions, it is more relevant to also calculate κ and σ in two dimensions. The reader should be aware that all values of κ and σ refer to two-dimensional distributions. To obtain a distribution with requested κ and σ, a and b are calculated from Eqs. (7) and (8). The parameter Γ is the gamma function. Γ 6b Γ 2b (7) κ= 2 , Γ 4b

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Γ 2b . a=σ Γ 4b

243

(8)

Some well-known distributions are special cases of this distribution, i.e. uniform distribution, (κ = 43 as b approaches ∞), Gaussian distribution (κ = 2) and exponential distribution (κ = 10 3 ). Random networks [7] can be generated by using uniform distribution and a standard deviation equal to the diagonal of the matrix. 2.2. Testing the performance of the algorithm To test the performance of our method, networks with different values for the parameters γ, σ, κ, n and link density were generated (Table 1) and network measures were compared. A total of 420 combinations were tested and 10 replicates were made for each combination. 2.2.1. Network measures studied In addition to link density and order, which are specified, four measures were calculated: fragmentation index [8], degree assortativity [23], clustering coefficient [30] and group betweenness centralization index, GBCI [29]. The results were analyzed according to our criteria for a good algorithm. Fragmentation index measures to what extent the network is disconnected and values ranges from 0 to 1. It measures the proportion of node pairs that are unreachable. A network with at least 90% of the nodes in the largest connected component has a fragmentation index smaller than 0.2. Assortativity ranges from −1 to 1. Assortativity close to 1 means that nodes with equal degree are often connected, and assortativity of −1 means that nodes with different degree are often connected. If the chance of two nodes being connected is independent of node degrees, assortativity is 0. Clustering coefficient for the network is calculated as the mean clustering coefficient over the nodes and ranges from 0 to 1. Clustering coefficient for a node is the number of edges that exists between neighbors of a node, divided by all possible links between the neighbors. Table 1. γ 0.0 0.5 1.0 1.5 2.0

σ 0.01D a

The parameters and values used. κ

n

Link density

10/3 500 1% 0.05D 2 1000 5% 0.10D 10% 0.15D 0.20D 0.25D 0.30D √ aD = 2 × 100 is the diagonal of the landscape.

Landscape size 100 × 100

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GBCI also ranges from 0 to 1, and measures whether nodes have similar betweenness (GBCI = 0) or varying betweenness (GBCI = 1). Betweenness measures how important nodes are, in the sense that a node with high betweenness has many shortest paths between a pair of nodes passing that node. GBCI was measured for the largest connected component, because otherwise there do not exist paths between all node pairs. Networks with fragmentation index larger than 0.2 were disregarded. About 5% of the networks were disregarded. The Python library NetworkX 0.36 was used to calculate betweenness and clustering coefficient. Programs to calculate fragmentation index, GBCI and assortativity were implemented in Python 2.4.1. A 5-way ANOVA with categorical explanatory variables link density, γ, σ, κ, n and the considered network measure as dependent was done. Mean sum of squares (MS) from the ANOVA were used for determining the importance of the different tuning parameters. The function anovan with model type “interaction” in MATLAB was used to do the ANOVA. 3. Results 3.1. Range of network structures All combinations of order and mean degree can be attained by setting n and link density appropriately. Fragmentation index is best tuned by a combination of parameters. See Table 2. For most of the parameter combinations, our method gives highly connected networks (i.e. more than 90% of nodes in the largest connected component). The parameter combination most likely to create fragmented networks is low σ, high γ and low mean degree (mean number of links per node). For parameter combinations with σ ≥ 0.05D (D is the diagonal of the landscape), only 32 out of 3600 networks had fragmentation index > 0.2. Among the networks with σ = 0.01D, 138 out of 600 networks had fragmentation index > 0.2, and 120 of those had γ ≥ 1.0. Notice that it is possible to create networks with low mean degree which are highly connected. More than 85% of the networks with mean degree 5 are highly connected. The algorithm was less successful for generation of dissassortative (negative assortativity) networks. It could obtain values from −0.10 up to 0.85. Assortativity is best tuned by σ (61% of MS) and γ (17% of MS) (Table 2). There is a clear positive relationship between γ and assortativity. In Fig. 1, assortativity for link density of 10% is plotted. For other link densities, the behavior is similar. The obtained values for γ = 0.5 lies between those of γ = 0 and γ = 1.0. However, there is little difference between the results of γ = 1.5 and γ = 2.0, indicating that higher values of γ does not affect the network assortativity. For all γ > 2.0, the node landscape is effectively the same, consisting of one tightly aggregated cluster. The γ parameter can be used to regulate assortativity, given a σ. For the clustering coefficient, the algorithm can generate a large range of values (from 0.01 to 0.80, Fig. 2). Table 2 show that σ is the most important parameter when tuning clustering coefficient (64% of MS). For a specific link density,

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Generating Structure Specific Networks Table 2.

Fragmentation index

245

Percent of mean sum of squares after an ANOVA (categorical variables). Obtained values

Link density

n

γ

σ

κd

Interaction

0.00–0.97

29%

12%

7%

20%

0%

32% 7% (density-n) 14% (density-σ) 4% (γ-σ)

Assortativity

−0.10–0.85

16%

1%

17%

61%

1%

1%

Clustering coefficient

0.01–0.80

33%

0%

0%

64%

0%

2%

GBCIc

0.00–0.48

35%

22%

8%

19%

0%

16% 6% (density-n) 4% (density-σ)

c Notice

that the ANOVA for GBCI has 170 missing replicates, however MS can still give an indication on which parameters are likely to influence the measure. d The small effect of κ can depend on the fact that we used a small interval of κ. The exponential and Gaussian distribution are similar compared to for example uniform distribution (not used here).

1

Assortativity

0.8

γ=0.0 γ=1.0 γ=2.0

0.6 0.4 0.2 0 0.01 0.05 0.1 0.15 0.2 0.25 0.3 σ: standard deviation of dispersal kernel

Fig. 1. Degree assortativity for different σ for 10% link density. Bars indicate + and − one standard deviation. Each mean is calculated over 40 values made up from the 4 combinations of the other variables (n and κ) and 10 replicates of each.

clustering coefficient from the link density up to about 0.6 is possible by changing σ. Expected clustering coefficient for a random network is equal to the link density in question [30]. Group betweenness centralization index were only measured for the networks with at least 90% of nodes in the largest connected component. This, of course, affects the replicates situation in the experiment, but almost none of the networks with σ > 0.05D were that fragmented. Values for GBCI from 0.00 up to 0.48 were found. Most networks had very low GBCI values. 92% of the networks got GBCI less than 0.05. Only 48 networks had GBCI larger than 0.2. The parameter combination expected to give high GCBI were high γ and low σ and low mean degree. See Table 3. Notice that the combination expected to give highest GBCI

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Clustering coefficient

1 link density=1% link density=5% link density=10% link density=1% link density=5% link density=10%

0.8 0.6 0.4 0.2

0 0.01 0.05 0.1 0.15 0.2 0.25 0.3 σ: standard deviation of dispersal kernel Fig. 2. Mean clustering coefficient for different σ. Bars indicate + and − one standard deviation. Each mean is calculated over 200 values made up from the 20 combinations over the other variables (n, κ and γ) and 10 replicates of each. Table 3. σ 0.1 0.1 0.1 0.5 0.5 0.5 0.5

n — 500 500 500 500 1000 500

Combinations of parameters that can result in higher GBCI.

Mean degree

γ

Mean GBCI

Std GCIB

Networks with fragmentation index < 0.2

≤10 25 25 5 5 10 25

≥1.5 2.0 1.5 2.0 1.5 2.0 2.0

0.28 0.30 0.13 0.30 0.24 0.14 0.17

— 0.26 0.06 0.10 0.09 0.07 0.09

1/80 2/20 19/20 2/20 12/20 18/20 18/20

is also the combination resulting in very fragmented networks. A high kurtosis should increase the probability to get a connected network despite a low σ. In the experiment, we got a minimal difference with 87 networks with fragmentation index > 0.2 for κ = 2 compared to 83 for κ = 10 3 . The most important tuning parameter to influence GBCI is σ (19% of MS). See Table 2. 3.2. Example, generating a network with requested properties To indicate how to set the parameters, we consider the train routes network analyzed in [26]. See also [21]. It has 587 nodes and 19603 links, a clustering coefficient of 0.69 and assortativity −0.033. The link density is 11% and n = 587. A matrix size of 100 × 100 is suitable. Because the assortativity should be close to zero, the lowest gamma possible γ = 0 will be best. By setting σ = 0.05D (see Fig. 2) the clustering coefficient will be close to the requested value. Unfortunately, this low σ will also increase assortativity (see Fig. 1). However the variance is much larger for assortativity, so keeping σ close to 0.05D and generating a number of replicates is recommended.

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4. Discussion Wide ranges of network structures have been reported and relevant parameter intervals may differ between subject fields. For example, see [21]. The spectral method presented here generates a fairly large set of networks with different structures. We will discuss the method in the light of our three criteria. We will also compare our method with Keeling’s focal point method and other algorithms for generating complex networks and discuss possible future improvements of the spectral network algorithm. We find that the method can generate a large range of networks, including random networks. All combinations of number of nodes and link density are possible because they are set from the beginning. Fragmentation, assortativity, clustering coefficient and betweenness centralization are tuned by only three parameters. It is possible to both generate fragmented and highly connected landscape. The algorithm generated clustering coefficient up to 0.6–0.7 and the lowest value depends on link density and is approximately equal to the current link density. GBCI spanned from 0 to 0.5. The aggregation parameter makes it possible to change assortativity, after setting the clustering coefficient. Comparing the networks generated by the algorithm with the collection of real networks in [21], we find that the algorithm is well-suited for social networks and less suited for biological networks with large negative assortativity. Reported assortativity varies between −0.37 and +0.37 and negative assortativity was not obtained with the spectral network algorithm. The same problem exists in Keeling’s focal point algorithm [2]. The collection of real world network characteristics include clustering coefficients varying from near zero up to 0.9 which the algorithm is able to follow. The reported mean degrees varies between 1.44 and 113 while the order spans from 92 to 47,000,000. Link densities were never higher than 12% and often lower than 1% for the real networks. All link densities and mean degrees are possible with our method since they are set from start. For networks formed in a spatial setting, the input parameters can be estimated from the underlying mechanisms. The two parameters of the probability distribution for link forming can be estimated from, for example, transport or dispersal data [10, 19]. Also, the spatial arrangement of nodes can be estimated by calculating the R-spectrum [20] and estimating γ with linear regression. Depending on the purpose of a network generating algorithm, and the constraints one wishes to apply, different methods may be appropriate. Considering distribution of links among nodes, we can distinguish between two different ways of modeling. Either the degree distribution of links among nodes is set a priori like the CM models, or a distance function is used for the distribution of links among nodes where nodes are placed in a spatial setting like the spectral network algorithm. When working with analytical studies of networks, a controlled degree distribution may be the most proper method. To mimic a real-world network in a spatial setting with Euclidean distances, social distances or other distance measures, a distance

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selection link distribution method such as the spectral network algorithm method is more adapted and straight-forward. Besides link distributions, it is useful to be able to specify other topological characters of networks. The CM algorithms has been developed to make it possible to obtain a specified degree-dependent clustering and assortativity (degree–degree correlation) respectively [24, 27, 31]. In the same spirit, the spectral network algorithm has the potential to generate networks in spatial settings with specified networks structures such as fragmentation, positive assortativity, clustering coefficient and group betweenness centrality. The spectral network algorithm mostly resembles the algorithm created by Keeling [15] since spatial patterns are used for organizing the nodes and probability density functions to distribute the links. However, there are differences. By varying both kurtosis and variance of the displacement kernel, we can change the relationship between probability for long-range and short-range links in a more flexible way. Because the number of links is explicitly set, the relation between probability for near and distant links can be kept even for large mean degrees. This is not possible by Keeling’s algorithm where absolute probabilities are used. That is the probability for a link between two nodes is min(1, H · P (di,j )) and P (di,j ) is given by a Gaussian distribution and H is a constant. To increase mean degree, H must be increased. The method has large potential to be developed further according to specific needs. We have chosen only one simple spectral density function with a power-law relationship between amplitudes and frequencies. With a spectral density function that amplifies one of the low frequencies (but not the lowest), the important feature of Keeling’s method with several tightly aggregated node clusters can be obtained. Other shapes of spectral density functions will produce yet other spatial patterns which may yield an even wider range of network structures. References [1] Abramson, G. and Kuperman, M., Social games in a social network, Phys. Rev. E 63 (2001) 030901. [2] Badham, J., Hussein, A. and Stocker, R., Parameterisation of keeling’s network generation algorithm, Theoretical Population Biology 74 (2008) 161–166. [3] Barab´ asi, A.-L. and Albert, R., Emergence of Scaling in Random Networks, Science 286 (1999) 509–512. [4] Barab´ asi, A. L. and Bonabeau, E., Scale-free networks, Sci. Am. 288 (2003) 60–69. [5] Barthelèmy, M., Crossover from scale-free to spatial networks, EPL (Europhysics Letters) 63 (2003) 915–921. [6] Bogu˜ n´ a, M., Pastor-Satorras, R., D´ıaz-Guilera, A. and Arenas, A., Models of social networks based on social distance attachment, Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 70 (2004) 056122. [7] Bollobas, B., Random Graphs (2nd edn.) (Cambridge University Press, 2001). [8] Borgatti, S., The Key Player Problem in Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers, eds. Breiger, R., Carley, K. and Pattison, P. (National Academy of Sciences Press, 2003).

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249

[9] Catanzaro, M., Bogu˜ n´ a, M. and Pastor-Satorras, R., Generation of uncorrelated random scale-free networks, Phys. Rev. E 71 (2005) 027103. [10] Clark, J. S., Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, The American Naturalist 152 (1998) 204–224. [11] Gastner, M. T. and Newman, The spatial structure of networks, The European Physical Journal B — Condensed Matter and Complex Systems 49 (2006) 247–252. [12] Halley, J. M., Hartley, S., Kallimanis, A. S., Kunin, W. E., Lennon, J. J. and Sgardelis, S. P., Uses and abuses of fractal methodology in ecology, Ecology Letters 7 (2004) 254–271. [13] Hayashi, Y., A review of recent studies of geographical scale-free networks, IPSJ Digital Courier 2 (2006) 155–164. [14] Hayashi, Y. and Matsukubo, J., Geographical effects on the path length and the robustness in complex networks, Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 73 (2006) 066113. [15] Keeling, M., The implications of network structure for epidemic dynamics, Theoretical Population Biology 67 (2005) 1–8. [16] Keller, E. F., Revisiting “scale-free” networks, Bioessays 27 (2005) 1060–1068. [17] Lambiotte, R., Blondel, V. D., de Kerchove, C., Huens, E., Prieur, C., Smoreda, Z. and Van Dooren, P., Geographical dispersal of mobile communication networks, Physica A: Statistical Mechanics and Its Applications 387 (2008) 5317–5325. [18] Lindstr¨ om, T., H˚ akansson, N., Westerberg, L. and Wennergren, U., Splitting the tail of the displacement kernel shows the unimportance of kurtosis, Ecology 89 (2008) 1784–1790. [19] Lindstr¨ om, T., Sisson, S. A., N¨ oremark, M., Jonsson, A., and Wennergren, U., Estimation of distance related probability of animal movements between holdings and implications for disease spread modeling, Preventive Veterinary Medicine (in press). [20] Mugglestone, M. A. and Renshaw, E., Spectral tests of randomness for spatial point patterns, Environmental and Ecological Statistics 8 (2001) 237–251. [21] Newman, M., The structure and function of complex networks, SIAM Review 45 (2003) 167–256. [22] Newman, M. E., Strogatz, S. H. and Watts, D. J., Random graphs with arbitrary degree distributions and their applications., Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 64 (2001) 026118. [23] Newman, M. E. J., Assortative mixing in networks, Phys. Rev. Lett. 89 (2002) 208701. [24] Pusch, A., Weber, S. and Porto, M., Generating random networks with given degreedegree correlations and degree-dependent clustering, Phys. Rev. E: Stat. Nonlin. Soft Matter Phys. 77 (2008) 017101. [25] Schabenberger, O. and Gotway, C. A., Statistical Methods for Spatial Data Analysis (Texts in Statistical Science Series), Chap. 2.5 (Chapman & Hall/CRC, 2004), ISBN 1584883227, pp. 62–78. [26] Sen, P., Dasgupta, S., Chatterjee, A., Sreeram, P. A., Mukherjee, G. and Manna, S. S., Small-world properties of the Indian Railway network, Physical Review E 67 (2003) 036106. [27] Serrano, M. A. and Bogu˜ n´ a, M., Tuning clustering in random networks with arbitrary degree distributions, Physical Review E 72 (2005) 036133. [28] Strogatz, S. H., Exploring complex networks, Nature 410 (2001) 268–276. [29] Wasserman, S. and Faust, K., Social Network Analysis: Methods and Applications (Structural Analysis in the Social Sciences), Chap. 5 (Cambridge University Press, 1994), ISBN 0521387078, pp. 188–192.

April 29, 2010 14:53 WSPC/169-ACS

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00251


[30] Watts, D. J. and Strogatz, S. H., Collective dynamics of “small-world” networks, Nature 393 (1998) 440–442. [31] Weber, S. and Porto, M., Generation of arbitrarily two-point-correlated random networks, Phys. Rev. E: Stat. Nonlin. Soft Matter Physics 76 (2007) 046111. [32] Westerberg, L. and Wennergren, U., Predicting the spatial distribution of a population in a heterogeneous landscape, Ecological Modelling 166 (2003) 53–65.