Quantifying soil complexity using network models of soil porous structure

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1 Nonlin. Processes Geophys., 0, 4 45, 03 doi:0.594/npg Author(s) 03. CC Attribution 3.0 License. Nonlinear Processes in Geophysics Quantifying complexity using network models of porous structure M. Samec, A. Santiago, J. P. Cárdenas, R. M. Benito, A. M. Tarquis 3, S. J. Mooney 4, and D. Korošak,5 Faculty of Civil Engineering, University of Maribor, Smetanova ulica 7, 000 Maribor, Slovenia Grupo de Sistemas Complejos, Departamento de Física, Universidad Politécnica de Madrid, 8040 Madrid, Spain 3 CEIGRAM ETSIA, Universidad Politécnica de Madrid, 8040 Madrid, Spain 4 Environmental Sciences Section, School of Biosciences, Sutton Bonington Campus, University of Nottingham, Loughborough, Leics LE 5RD, UK 5 Institute of Physiology, Faculty of Medicine, University of Maribor, Maribor, Slovenia Correspondence to: M. Samec (marko.samec@um.si) Received: 3 July 0 Revised: 6 December 0 Accepted: 9 December 0 Published: 5 January 03 Abstract. This paper describes an investigation into the properties of spatially embedded complex networks representing the porous architecture of systems. We suggest an approach to quantify the complexity of structure based on the node-node link correlation properties of the networks. We show that the complexity depends on the strength of spatial embedding of the network and that this is related to the transition from a non-compact to compact phase of the network. Introduction One of great challenges in science today (Lal, 007) is to understand how s behave as a complex system (Crawford, 00). Over the last decade, network theory has become a much discussed cross-disciplinary research field contributing to social, biological and information sciences (Barabasi, 009, 0), demonstrating that quite diverse systems might share similar topological organization. Whilst scientists have studied the complexity of for decades (Berkowitz and Ewing, 998), it is only recently that networks have been considered as complex systems. Specifically two approaches to describe structure as complex networks of s have recently been proposed (Mooney and Korošak, 009; Cárdenas et al., 00), both emphasizing the role of the spatial embedding of the network. Soil structure was described as a complex network of s using a spatially embedded varying fitness network model (Mooney and Korošak, 009) or heterogeneous preferential attachment scheme (Santiago et al., 008). Both approaches reveal the apparent scale-free topology of s a power-law distribution P (k) k γ of the degrees k of nodes as a consequence of a heterogeneous distribution. Methodology Soil networks are spatial networks (Barthelemy, 0), i.e. complex networks nodes (s) embedded in. Often the interaction T ij between distant nodes i and j is modelled using a gravity model: T ij = s i s j f (d ij ) () where f describes the effect of spatial embedding of the network. In the threshold network (Mooney and Korošak, 009) model, f = dij m and the nodes are linked if T ij > θ. Here, the non-dimensional parameter m measures the importance of the distance, and the threshold value θ controls the number of links in the network. In the growing network mechanism extended preferential attachment rule (Santiago et al., 008), the probability to connect depends on the s i and the distance between the s i and j to the power of m. In the threshold model the exponent is for strong enough embedding m > D/(α ) given by γ = + m(α )/D (Masuda et al., 005; Yakubo and Korošak, 0), where D is the embedding or fractal dimension, while Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.

2 4 M. Samec et al.: Quantifying complexity using network models of porous structure in the growing network model multi (Bianconi and Barabasi, 00) was found γ = + /w (Cárdenas et al., 00), where w is the normalized fitness of the nodes. The analysis of the scale-free network embedded in fractal (Yakubo and Korošak, 0) has shown that we can distinguish three phases of embedded network: (i) noncompact phase for m < m c0 = D/(α ) y =, (ii) intermediate phase for m c0 < m < m c = (D + )/(α ), and (iii) compact phase for m > m c. Here, we first show using structure data from -D X-CT image that both methods of network construction, in which we can tune the network heterogeneity, predict topologically similar networks of. Furthermore, we calculate the assortativity r (Newman, 00) as a function of network heterogeneity and find that the crossover from disassortative to assortative network configuration might depend on fractal properties (Rieu and Sposito, 99; Dimri, 000) obtained from multifractal analysis of structural images derived from X-ray computed tomography (Mooney and Morris, 008). Finally, we use the entropy of node-node link correlations (Claussen, 007) to quantify the complexity of porous architecture allowing us to relate the exponent of the distribution and the complexity of the network of porous structure. This is a computationally simple network complexity measure that is sensitive to the network structure. The complexity measure is defined as the entropy of node-node link correlations that are given the matrix elements c k,k counting the number of links between nodes degrees k and k in the network: c k,k = N i,j= N i,j= a ij δ ki kδ kj k k = k a ij δ ki kδ kj k k = k, () where a ij are the elements of the adjacency matrix. The probability that a randomly chosen edge links two nodes the degree difference k, b k is then constructed from the nodenode link correlation matrix c k,k : b k = kmax k k= c k,k+ k kmax k=0 kmax k k= c k,k+ k. (3) Here the denominator is equal to M (total number of edges) and the numerator gives the number of edges connecting node pairs the degree difference k for any two k, k. Finally, the complexity is defined as an entropy measure of b k and measures how widely the degree differences of connected node pairs in the network are distributed: k max h = b k logb k. (4) k= Fig.. Network representation of structure obtained by Figure the evolving : Network network representation (EN) of model (top) structure andobtained static by network the evolving (SN) network (EN) model model (bottom) (top) and static fromnetwork the same (SN) set model of (bottom) data derived from the from same images set of data ofderived from images structure of and structure and equal equal model model parameters: parameters: equal equal number number of links of and m links and m. The area of node corresponds to the node degree,. The area of node corresponds to the node degree, i.e. the number of links attached. The i.e. the number of links attached. The data for the network construction included s and positions of the s detected from data for the network construction included s and positions of the s detected from D X-CT scanned and analysed images. Though here rendered ly, positions of -D X-CT scanned and analysed images. Though here rendered nodes ly, not corresponding to positions actual positions of theof nodes the s not in corresponding the analysed images to (for a the networks actual positions overlayed over of the s images insee the (Mooney analysed and images Korošak, (for 009)), networks both networks actually overlayed show over similar topology images, see the Mooney well connected and Korošak, nodes (larger 009), dots) both not directly linked, networks but preferably actually linked showthrough similar nodes topology small degrees the(smaller well-connected dots). This type of network nodes structure (largerindicates dots) not that directly both networks linked, are but correlated. preferably linked through nodes small degrees (smaller dots). This type of network structure indicates that both networks are correlated. 3 Results and discussion In Fig. we display examples of networks representing the structure. The input data for both network models were the geometric centres of positions (nodes of the network) and s (measured as area) obtained from -D X-CT image. The number of s obtained from image analysis was of the order of N 0 3. The upper network in Fig. is obtained using the growing network mechanism extended preferential attachment rule (Santiago et al., 008), while the lower network in Fig. results from the threshold model (Mooney and Korošak, 009). In all analysed samples the were found to follow a power law W (s) s α the exponent <α<. Both methods lead to scalefree organization of networks for small enough m (of the order of or less, in non-compact phase) the degree distribution P (k) k γ, and to more compact, homogenous Nonlin. Processes Geophys., 0, 4 45, 03

3 M. Samec et al.: Quantifying complexity using network models of porous structure 43 Table h hof of of their Table :Complexity Complexity ofseveral several their Table ::Complexity h of several of their Table. Complexity h of several of their at m = mc /. Pore is isblack.! /.Pore Pore black.! /.!Pore at at!at! =!=!=!!/. is black. is black. α h!!! h hh.4,4,4.853,4,853,853,853.5,5, ,5.6,6, ,6 3,4877 3,4877 3, Cumulative degree evolving (EN) Figure :Fig. Cumulative degree of evolving of network (EN)network model (left) and static 3 (left)(right) and static (SN) (right)of for and m. The Fig. 3. Results of the multifractal analysis of the binary image. network model (SN) model modelsnetwork for small and model large values the small parameter Figure 3: Results of the multifractal analysis of the binary image. Fractal dimension as a 3 function of scale parameter q is shown. large values of the parameter m. The exponent of the scalefractal dimension as a function of scale parameter q is shown. exponent of the scale-free degree distribution obtained for small m (straight line in free degree distribution obtained for small m (straight line in plots) plots) is!is=γ.=the power-laws (straight and cumulative.. fits Thearefits are power laws lines) (straight lines) andpoisson cumulative Poisson. organization for large m as shown in Fig. by the cumulative degree for m and m. Multifractal analysis of the same -D images (Stanley and Meakin, 988; Perrier et al., 006) was performed, which yielded D = Dq (q = 0).74 shown in Fig. 3. In an attempt to correlate the properties of the threshold network structure the porous structure as obtained from -D image, we considered the assortativity r (Newman, 00), which is the Pearson s correlation coefficient of the degrees of adjacent nodes, of the threshold network as a function of the parameter m. From the result displayed in Fig. 4, we see that there is a crossover from disassortative (r < 0) to assortative (r > 0) network structure at approximately mc 4.3. This change is a consequence of progressively more homogeneous network structure increasing m. In geographical scale-free networks this crossover was found to occur when the degree exponent was γ = 3 (Morita, 006). Equating the coefficients for the growing network and threshold network models gives the crossover parameter mc = D/(α ) = mc0 > mc indicating that the crossover from disassortative to assortative network occurs well in in the compact phase of the network. The degree exponent in the growing network model exhibits multi γ = + /w, so a homogenous network normalized fitness sharply distributed around w = will have γ = 3. To further explore the effect of the network parameter m on node correlations, we calculated the node-node link correlations cij (Claussen, 007) defined as links from node degree ki to node degree kj. We expect a network a large h to have many links that connect nodes of various degrees, while networks low h will have mostly links connecting node pairs of almost the same degree resembling a lattice-like network or a network close to a complete graph. The correlation matrices for a smaller geographical threshold network are shown in Fig. 5 for m (a), m = (b) and m (c). Again, we illustrate that, the increasing value Nonlin. Processes Geophys., 0, 4 45, 03

4 44 M. Samec et al.: Quantifying complexity using network models of porous structure a)! Fig. Figure 4. 4: Pearson s Pearson's coefficient r applied rto applied the links in to the the network links as a in measure the network of degree- as degree correlations. r is shown as a function of parameter m in the static network model. We a measure of degree degree correlations. r is shown as a function of observe parameter a crossover mfrom in the disassortative static network (! < 0) to assortative model. (! > 0) network structure at 3 We observe a crossover!! = 4.3. from disassortative (r < 0) to assortative (r > 0) network structure at m c = 4.3. b)! = of the parameter m, the network changes from disassortative (nodes degrees preferably connected) to assortative type. To quantify the complexity h of the structure represented the network, we computed the entropy of the normalized diagonal sums of the correlation matrix c ij. Finally, we have calculated the complexity of several of their at m = m c / (Table ). In the following we discuss two possible geophysical implications of our findings that may help4 to elucidate the role of the threshold network model parameter m: the of fractures in rocks (Berkowitz et al., 000) and the influence of the network properties on biological 5 invasion in c)! (Perez-Reche et al., 009, 0). The study of fracture networks in rocks (Berkowitz et al., 000) showed that the length distribution of fractures in rocks scales as n(l) l a. Here a is the fracture length distribution exponent. For a > D + the properties are independent of system ; for a < D + they depend on the system, and a = D + signals the connectivity threshold (D is here the fractal dimension of the problem). In experiments they find that the range of a is < a < 3. If we compare a to the exponent of length distribution function for scale-free networks embedded in fractal (Yakubo and Korošak, 0), a model we used to describe networks above, we have a = D + m(α ). Immediately we get that the connectivity threshold 6 a =+D corresponds exactly to m c = D/(α ), the suspected crossover from disassortative to assortative organization of the network. Furthermore, we have the lower bound 7 a = occurring Figure 5: Correlation network. matrices for a smaller geographical threshold network Fig. 5. Correlation matrices for a smaller geographical threshold at m c0 = D/(α ), i.e. at the boundary between the noncompact and the intermediate phase of the network structure. Nonlin. Processes Geophys., 0, 4 45, 03

5 M. Samec et al.: Quantifying complexity using network models of porous structure 45 The embedding parameter m of the threshold network model controls the compactness of the network organization. For values m < m (non-compact and intermediate phase) the network exhibits long range connections between s, while in the compact phase m > m the s in the network are mostly connected to spatially nearest neighbours. Recently, the impact of the structural heterogeneity of networks on microbial spread in s has been investigated using network models (Perez- Reche et al., 009, 0), and the idea of long range bridges that link distance s has been introduced to explain the strong effect of the channel heterogeneity on microbial invasion in s. The channel was found to be correlated the arc length of the channel between two s i and j as R ij L β ij (Perez-Reche et al., 009). It was also suggested that the channel depends on the radii of s i and j : R ij R i R j (Li et al., 986). Bringing these observations together gives the expression for the channel R ij R i R j L β ij that is similar in structure to the interaction T ij (see Eq. ) used to construct the network model. However, the implications of the possible connection between the spatial embedding of the network and the invasion of microorganisms in s remain to be further investigated. In conclusion, networks obtained from two models demonstrate their scale-free structure, but characteristics such as assortativity or node-node link correlations that depend on the strength of spatial embedding of the network. For the evolving network model (Santiago et al., 008, Cárdenas et al., 00, 0), it was shown that the exponent of the asymptotic degree distribution is in the interval < γ 3. Comparing the two network models, this finding sets the upper limit to the strength of embedding in the threshold network model m m c = D/(α ) indicating that networks are predominantly disassortative. 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