Chaos, Solitons & Fractals 69 (2014) 100–106

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Evaluating the transport in small-world and scale-free networks R. Juárez-López a,b, B. Obregón-Quintana a,c, R. Hernández-Pérez d, I. Reyes-Ramírez a, L. Guzmán-Vargas a,⇑ a Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Instituto Politécnico Nacional, Av. IPN No. 2580, L. Ticomán, México D.F. 07340, Mexico b Petróleos Mexicanos, Av. Prol. de Juárez, Comalcalco, Tabasco 86388, Mexico c Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Mexico d SATMEX, Av. de las Telecomunicaciones S/N, CONTEL Edif. SGA-II, México, D.F. 09310, Mexico

a r t i c l e

i n f o

Article history: Received 14 March 2014 Accepted 10 September 2014 Available online 8 October 2014

a b s t r a c t We present a study of some properties of transport in small-world and scale-free networks. Particularly, we compare two types of transport: subject to friction (electrical case) and in the absence of friction (maximum flow). We found that in clustered networks based on the Watts–Strogatz (WS) model, for both transport types the small-world configurations exhibit the best trade-off between local and global levels. For non-clustered WS networks the local transport is independent of the rewiring parameter, while the transport improves globally. Moreover, we analyzed both transport types in scale-free networks considering tendencies in the assortative or disassortative mixing of nodes. We construct the distribution of the conductance G and flow F to evaluate the effects of the assortative (disassortative) mixing, finding that for scale-free networks, as we introduce different levels of the degree–degree correlations, the power-law decay in the conductances is altered, while for the flow, the power-law tail remains unchanged. In addition, we analyze the effect on the conductance and the flow of the minimum degree and the shortest path between the source and destination nodes, finding notable differences between these two types of transport. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, researchers from different disciplines have shown an increasing interest in the study of complex networks [1,2,4,3]. In particular, several approaches to classify and characterize complex networks have been proposed, which aim to help in the understanding of the networks operation and organization under different conditions [5,6], and very recently researchers have addressed the multiplex character of real-world systems [7]. As it is well known, a network is comprised of a set of nodes or vertices, and the set of links or edges that interconnect

⇑ Corresponding author. E-mail address: [email protected] (L. Guzmán-Vargas). http://dx.doi.org/10.1016/j.chaos.2014.09.007 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.

the nodes. The links can have a given direction and are known as directed; and in some cases they represent also a certain intensity in the connection which leads to networks with weighted links. The degree of a node is defined as the number of links that fall on it, when the links are not directed. However, in a directed network, the degree can be either interior (links coming into the node) or exterior (links coming out of the node). Moreover, the degree distribution PðkÞ, where k is the degree, allows describing the network connectivity, i.e., this distribution is obtained from plotting the frequencies against the degree, and characterizes the network; thus, the structure of PðkÞ provides information on how the links are distributed [8,9]. For instance, it is known that random networks possess a characteristic connectivity scale [4], meaning that most of the

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nodes have an average number of links, that is described by a Poisson-like distribution. In many cases, real networks exhibit the small-world property, that is, the average path length is small and the clustering coefficient (local structure) is high [10]. A representative small-world model was proposed by Watts and Strogatz [10], which interpolates between a regular clustered network and a random graph; for intermediate configurations the small-world feature is observed. Moreover, there are networks where the distribution has no characteristic scale so they are called scale-free networks for which a significant number of nodes coexist with few links and few highly connected nodes known as hubs. For these networks, the degree disk tribution is given by a power law: PðkÞ  k [9]. Recent studies on the transport properties in complex networks have reported that scale-free networks display better transport conditions than random networks, due to the presence of hubs [11,12]. Transport in non regular media, such as complex networks, provides an approach to the exploration of transport in many real conditions from electrical networks to the Internet [13–17]. Transport within a network consists of sending an entity from a specific node called the origin or source to another node called destination or sink. This problem can be stated as a flow problem to find the paths from source to destination for which it is possible to send as much flow as possible while satisfying capacity constraints on the links and flow conservation at the intermediate nodes [18,19]. Moreover, transport in many real situations involves the presence of friction. These cases can be modeled using analogies with electrical systems: a positive potential is assigned to the origin node while a zero potential is assigned to the destination node, and the links are considered as resistors. Based on the law of conservation of electrical charge, it is possible to estimate the current flow from the origin to the destination. On the other hand, a complex network can be classified according to the bias or the degree–degree correlations, i.e., if there is a bias in the connectivity between nodes with high or low degree, then the network has assortative or disassortative mixing [20]. Here, we are interested in evaluating the transport properties in smallworld networks and the effect of assortative (disassortative) mixing on the transport in scale-free networks. Particularly, we focus on comparing the transport (with and without friction) in clustered and declustered smallworld networks, by calculating the average conductance and the average flow on local and global scales; we find that for clustered networks and for both transport types, the best trade-off between local and global levels is observed for configurations with small-world topology, while for declustered networks, the transport improves globally and it is independent of the rewiring parameter. Besides, we also compare the effect of degree–degree correlations on the conductance and flow distributions in scale-free networks for three specific configurations. We observe a significant difference between the distributions for the three levels of assortative mixing. Our quantitative analysis also permits to test the effect on the conductance and the flow of the minimum degree and the shortest path between the source and destination nodes, finding notable differences between these two types of transport. The

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paper is organized as follows: In Section 2, a brief description of small-world and scale-free network models is given. Next, Section 3 describes the way the degree–degree correlations are introduced. The results and discussion are given in Section 4. Finally, in Section 5 some conclusions are presented. 2. Small-world and scale-free networks Regular networks are the simplest model to describe the relationship between nodes since all nodes have the same degree; however, the model is not always appropriate to study real networks. An important model that interpolates between a regular and a random network is the Watts–Strogatz model (WS) [10]: starting with N nodes arranged in a ring with links to its k next–nearest neighbors (a k-regular network), small-world configurations can be created through a rewiring process that with some probability p reassigns links (when p is large enough, this rewiring process leads to a random network), creating shortcuts between distant sections of the ring. Thus, for p ¼ 0 we have the case of a clustered WS network whereas for intermediate values of p, it is observed a high average clustering coefficient and short average path length, the main feature of the small-world network. A declustered WS model was proposed by Vragovic´ et al. [21] and consists in starting with a regular ring with next–nearest neighbor connections and adding links from each site to only its nth neighbors [21]. In this way, the initial configuration has zero clustering coefficient. In our study, we consider that shortcuts are created with a random rewiring process as in the ordinary WS model. We notice that in the declustered model proposed in [21], the rewiring procedure considers only the more distant neighbors of a given site, while nearest-neighbor links are kept unchanged. In our case, we consider clustered and declustered WS networks with the same size and equal number of initial edges. Moreover in many real systems the description of the connectivities is given by a power-law/scale-free distribution. An illustrative model to generate a scale-free network is that of Barabási–Albert where starting from a set of nodes with certain links, new nodes join according to the so-called preferential attachment process: nodes with more links are more likely to link new nodes [9]. The emerging network has a degree distribution of the k power-law type PðkÞ  k where the exponent, k, depends on the type of network under consideration [1]. A more appropriate model to generate scale-free networks with a random mixing and a defined exponent, consists of using the Molloy–Reed algorithm on a set of N nodes [22]: in which, ki copies for each node i are generated, where the probability of having a degree equal to ki satisfies k Pðki Þ  ki . These copies of nodes are randomly linked, without repeating links and avoiding self-loops. 3. Degree–degree correlations Many complex networks exhibit tendencies in the assortative mixing between nodes. It has been observed in many cases that high-degree nodes tend to connect

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high-degree nodes. Such networks are said to display assortative mixing [20,23]. In contrast, when high-degree nodes tend to connect with low-degree nodes, it is said that the network has disassortative mixing. Several parameters and quantities have been proposed to estimate the tendencies in assortative mixing. In 2002, Newman proposed the r-parameter defined as the Pearson correlation coefficient of degree between pairs of connected nodes, which is observed to lie within the interval 1 6 r 6 1 [20]. For r > 0, the network displays assortative mixing patterns, while for r < 0, it exhibits disassortative mixing tendencies. Alternatively, other authors have proposed P 0 0 the measure knn ¼ k0 kPðk jkÞ, where Pðk jkÞ is the conditional probability that an arc coming out of a node 0 of degree k points to a node of degree k , for estimating the presence of correlations in a network [23]. When knn is an increasing (decreasing) function of k, the network shows assortative (disassortative) mixing. Moreover, Doyle et al. proposed the quantity S to capture the tendencies in assortative mixing in a very direct way [24]; defined as P S ¼ i;j kj ki , where the sum operates over all pairs of nodes that are linked. Thus a high value of S denotes that high degree nodes are linked to each other and a low value indicates that high degree nodes are connected with low degree nodes. For the present study, we generate scale-free networks with exponent k ¼ 2:5 using the Molloy–Reed method described above [22]. We use the criterion proposed by Doyle et al. [24] to generate three configurations: (i) assortative Smax (ii) disassortative Smin and (iii) neutral Srand , as follows: first, the nodes are sorted from highest to lowest degree. Then, to generate Smax we start from the node with the highest degree and subsequently create links to nodes in descending order until the degree is exhausted. In contrast to generate Smin , we start from the node with highest degree and links to nodes are created according to the ascending order. Finally, to generate Srand , the nodes are randomly paired according to the degree of each node. In all cases, the repetition of links and the network disconnectedness shall be avoided. 4. Electric transport and max-flow To study some transport properties we calculated the conductance G and maximum-flow F between two nodes A and B in both clustered/declustered Watts–Strogatz and scale-free networks, with different assortative mixing. The Kirchhoff’s laws are used to obtain G, imposing a potential V A ¼ 1 to the source node while the destination node is fixed to zero potential V B ¼ 0. We also assume that all arcs have unitary resistance. We solve the system of linear equations to determine the value of the potential V i in all nodes of the network. To estimate the total current entering A and leaving B, the currents from A to the nearest neighbors are added up, finding that the total current is numerically equal to the conductance G [11]. For the case of frictionless transport, we use linear programming techniques [18] to calculate the maximum static flow (stable flow state) between nodes A and B; while the flow conservation law is used for intermediate nodes [25]. Moreover, the capacity of the edges shall not be exceeded. In particular, in the present work the edges have

unitary capacity and the flow sent through any path is an integer amount of flow, and therefore, the flow is determined by the number of edge-disjoint paths (paths that do not share any edges) from the source A to the destination B [12]. In our analysis we calculate the global conductance (G) and maximum global flow (F) as the average between all pairs of nodes, that is,

Gg ¼

X 1 Gij NðN  1Þ i–j

ð1Þ

X 1 F ij : NðN  1Þ i–j

ð2Þ

and

Fg ¼

We also compute both quantities locally, considering only the neighborhood of the node i; Di , i.e., over all pairs of neighboring nodes of the node in question, including itself, and then averaging over all nodes in the network,

X 1X 1 Gst Gl ¼ N i ki ðki þ 1Þ s–t2D

ð3Þ

i

and

X 1X 1 Fl ¼ F st ; N i ki ðki þ 1Þ s–t2D

ð4Þ

i

where ki represents the degree of node i. 4.1. Results for Watts–Strogatz networks Fig. 1a shows the results of the network transport based on the clustered and declustered WS model. In our simulations we consider N ¼ 512 nodes, the number of neighbors k ¼ 4 for clustered networks, and k ¼ 4 and n ¼ 3 for the declustered model. For the electrical case and clustered WS model (Fig. 1a), the values for local and global average conductance are very different for regular settings (low p), observing that Gl > Gg , which indicates that the transport is higher locally. By increasing the probability p; Gg increases while Gl decreases, revealing that the appearance of shortcuts allows the transport to increase globally while is locally low. In addition, for random networks (high values of p), it is observed that Gl < Gg . It is noteworthy that for configurations that are neither regular nor completely random, both Gl and Gg have high values and, the absolute difference jDGj ¼ jGl  Gg j has a minimum around p  0:2 (see inset of Fig. 1a). From this we can conclude that transport within initial clustered networks with small-world topology is good both globally and locally. Moreover, for the declustered networks, the global conductance exhibit an increasing trend as the rewiring probability increases, whereas, the local conductance remains almost constant for different values of p, indicating that the presence of shortcuts improves the transport on global scale but not locally. When observing the absolute difference, jDGj ¼ jGl  Gg j, for declustered networks (inset of Fig. 1a), the minimum is located around p  0:04, indicating that both quantities are similar for regular configurations. On the other hand, the maximum flow case for both clustered

R. Juárez-López et al. / Chaos, Solitons & Fractals 69 (2014) 100–106

(a)

(b)

Fig. 1. (a) Simulation results for the average global (Gg ) and local (Gl ) conductance for different values of the rewiring probability p in the clustered (declustered) Watts–Strogatz model. For initial clustered WS networks, Gg (open diamonds) and Gl (open circles) follow opposite trends for increasing p, and they intersect at p  0:2 (see inset), suggesting that for a small-world configuration both conductances reach intermediate values and the best trade-off between local and global levels. For initial non-clustered configurations, Gg (closed diamonds) increases as p increases, while Gl (closed circles) remains constant. The inset shows the absolute difference jDGj ¼ jGl  Gg j for both clustered (dash line) and declustered (dash-dotted line) models. (b) Results for the maximum-flow vs. p. For clustered and declustered WS models, both the local and global flow decrease for increasing p, with the local flow decaying faster, suggesting that for random networks the capacity for frictionless flow is higher locally (the inset shows the absolute difference between the two flows). The results shown in (a) and (b) are average values obtained in 10 independent realizations.

and declustered WS models displays that both local and global medium flows decrease with the rewiring probability p (Fig. 1b), such that the overall flow decreases faster while the local flow reaches a higher value. In contrast to the electrical case and for clustered networks, the absolute difference jDFj ¼ jF l  F g j shows an increasing behavior as p increases (see inset of Fig. 1b), indicating that both quantities are quite similar at low p (regular network), whereas for random configurations (p  1) they exhibit a different average value with F l > F g . 4.2. Results for scale-free networks Given a scale-free network with degree distribution satk isfying, PðkÞ  k , we manipulate the assortative mixing

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(a)

(b)

(c)

Fig. 2. Representative cases of scale-free networks with different levels of assortative mixing: (a) Smin , (b) Srand and (c) Smax . For the Smin configuration the nodes with high degree connect with low degree nodes, leading to ‘‘bottlenecks’’ in the communication with distant nodes. For Smax , high degree nodes connect between them, increasing the number of alternate paths to communicate two given nodes; whereas, for Srand , the assortativity is random and therefore there are no tendencies.

introducing tendencies to generate networks matching the different cases Smax ; Smin and Srand (see Fig. 2 for representative cases); and we proceed with computing both G and F for various configurations. In order to study the transport behavior, we compute the cumulative

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Smin Srand Smax

20

*

distribution functions (CDF) HðG > gÞ and HðF > f Þ which measure the probability of having conductances (or flow) greater than a given value. Fig. 3 presents the results for a scale-free network (k ¼ 2:5) with different levels of assortative mixing. Note that for the Srand case, the CDF in the first decade, 1 < G < 10, is well fitted by a power-law of the form HðGÞ  Ga , with a ¼ 2k  2 [11,12]. As can be seen, difference in the distributions for large conductances emerge when tendencies in the assortative mixing are introduced. For the Smax case, it is observed that the probability of having large values of the conductance (G > 10) is greater than for the random case; while in the intermediate conductances range (1 < G < 10), the probability for the random case is higher, indicating that the increase in the probability for large conductances is penalized with a decreasing of probability for low values of G. However, a higher probability for large conductances represents a higher transport capacity for networks with the Smax configuration. In contrast, for the Smin case, the probability of intermediate conductances (1 < G < 10), is very close to the random case, while the probability for large conductance (G > 10) is smaller than Srand and Smax . For very small conductances range (G < 1), Smax shows a higher probability than the other two configurations (see inset of Fig. 2). Therefore, we can conclude that the transport capacity is affected by tendencies in the assortative mixing, being the random case the one with the best balance, despite the fact that the probabilities of the networks with Smax have a higher probability for large conductances. To further explore the effects on transport of the assortative mixing, we use the backbone picture proposed by López et al. [11], that is, we assume that the conductance, G, between node A and B, can be expressed as 1=G ¼ 1=GA þ 1=GB þ 1=GTB , where GA (GB ) and GTB represent the conductance of the neighborhood of node A (B), and the transport backbone, respectively. Following López et al., and assuming that GA ¼ ckA and GB ¼ ckB , with c a constant, kA ; kB the degree of nodes A and B, respectively, we obtain

mean conductance G

104

~0.4

~0.5

linear regime

15 quadratic regime

~0.3

10

5

0

10

20

30

40

degree kA=kB Fig. 4. Average conductance G vs. degree kA for the symmetric case kA ¼ kB . We observe two regions, for small values (kA  14), the quadratic behavior is relevant, while for kA > 14, a linear behavior is observed. For the linear regime the backbone conductance is quite large and the conductance between source–target nodes is dominated by the node’s degree with different values of the constant c=2 (slope).

1=G ¼ ðkA þ kB Þ=ðckA kB Þ þ 1=GTB . In order to estimate the effect of the assortative mixing over the transport backbone, for simplicity, we consider the symmetric case kA ¼ kB , and thus G ¼ ckA =ð2 þ ckA =GTB Þ. Next, we consider first-order approximation of the term ckA =GTB in thepffiffiffibinomial 1 ½2 þ ckA =GTB  , leading to G  ckA ½1=2  ckA =ð 2GTB Þ. To test the prediction of this approximation, we plot G vs. kA (Fig. 4) for the symmetric case (kA ¼ kB ) in our simulation data. We identify two regimes, for small values, kA  14, the quadratic term seems to be relevant, while for kA > 14, the linear behavior is quite dominant. For the quadratic regime, both conductances for Smin and Srand show the qualitative behavior predicted by the first-order approximation, revealing that the backbone’s conductance influences the global electric transport. Surprisingly, for Smax , the conductance is smaller than the other two configurations, which indicates that for low degree

0

Smin Srand Smax

10

α~2.5

0

10

-2

10

-3

10

-4

10

-1

10

-2

10

-3

10

-4

10

-5

10

0

1

10

10

-5

10

0

10

1

10

2

10

conductance, G Fig. 3. The CDFs for the conductance G for the three assortative mixing cases Smax ; Srand and Smin . They display differences that depend on the tendencies in the mixing. The CDF for Srand is consistent with a power law HðGÞ  G2k2 , where k is the exponent of the degree distribution of the k network, PðkÞ  k . (Inset) Probability density function of the cases described in the main frame. These results were obtained from 10 independent simulations for networks with N ¼ 512 nodes.

Smin Srand Smax

-1

10

α~3

-2

10

Probability density h(F)

-1

Cumulative distribution, H(F)

α~3

Probability density h(G)

Cumulative distribution, H(G)

10

-3

10

-4

10

0

10

-2

10

-4

10

-6

10

0

10

1

10

2

10

-5

10

0

10

1

10

2

10

flow F Fig. 5. CDFs for the flow F for the different configurations of assortative mixing: Smax ; Srand and Smin . In the three cases, the CDF is consistent with a power law behavior HðFÞ  F 2k2 , where k is the exponent of the degree distribution of the network. (Inset) Probability density function for the cases described in the main frame.

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60

(a)

Smin Srand Smax

3

mean conductance

mean conductance

4

2

1

0

Increase min(kA,kB) 1

2

3

4

5

6

7

Smin Srand Smax

40

20

0

0

(a)

8

0

20

shortest path length (l) 80

5

Smin Srand Smax

mean flow

min(kA,kB)=5 4

min(kA,kB)=4 3

min(kA,kB)=3

60

80

(b) Smin Srand Smax

60

mean flow

(b)

40

minimum degree (kA,kB)

40

20

2

min(kA,kB)=2 1

0

0

2

4

6

8

shortest path length (l) Fig. 6. Mean conductance and mean flow vs. the length of the shortest path ‘AB , for different values of the minimum degree between the source– destination pair and different levels of assortative mixing. (a) For the electrical case, the mean conductance decreases with increasing ‘AB . (b) For the maximum flow case, the transport capacity remains almost constant for different values of ‘AB , mainly in the Smin case.

source–target nodes, the number of parallel paths is reduced dramatically, even below of the Smin and Srand cases. The best fit for the linear regime leads to the values of the slope cmax =2 ¼ 0:531  0:009; crand =2 ¼ 0:333  0:005 and cmin =2 ¼ 0:395  0:014, respectively. We notice that according to these results, conductances for the three configurations exhibit a linear behavior with different grow rates in terms of the degree. Remarkably, the slope for Smax is around 0.5 and it is bigger than the slope for the other two configurations, indicating that the transport improves at better rates for Smax as source–target degree increases, whereas for Srand and Smin , this rate is reduced. On the other hand, the effect of tendencies in the assortative mixing on the flow behavior in a scale-free network is analyzed, founding that the probabilities are close for low flow values, while for high flow values there is a slight separation between the probabilities (see Fig. 5). This discrepancy is related to the availability of edge-disjoint paths whose number is related to tendencies in assortative mixing, i.e., the number of parallel routes in Smin configurations is reduced due to the increased presence of bottlenecks.

0

20

40

60

80

minimum degree (kA,kB) Fig. 7. Mean conductance vs. minimum degree ðkA ; kB Þ for different levels of assortative mixing. (a) For the electrical case, the mean conductance is greater for Smax than for both Smin and Srand for large values of the minimum degree; while for lower values of the minimum degree the mean conductances are similar. (b) For the maximum flow case, the three cases overlap, indicating that the transport capacity improves with increasing values of the minimum degree. The flow for the Smin case is lower than for both the Smax and the Srand configurations, since the Smin network promotes the appearance of bottlenecks, in contrast to the other two cases, where alternatives paths appear more frequently.

Let us now explore some differences between the two types of transport we have discussed so far. As it has been reported on [11,17], the electrical transport is strongly influenced by the average distance between the source and the destination due to the presence of friction, while for the flow transport capacity is not determined by the distance and depends only on the independent route. Fig. 6 shows the results of conductance and flow performance as a function of the shortest path (‘AB ) between nodes A and B, for different values of the minimum degree of both source and destination nodes. It is observed that the conductance decreases with the path length while its average value increases for larger values of the minimum degree (see Fig. 6a). Note for the Smin and Srand cases, the average conductance reaches significantly high values for ‘AB ¼ 1. In contrast, Fig. 6b shows the results for the flow, which is maintained approximately constant for a range of path lengths and for several values of the minimum

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degree of the source–destination node pair, indicating that the flow is approximately independent of the path length. Continuing with the comparison of the two types of transport, we computed the mean conductance and the mean flow as a function of the minimum degree of the pair of source–destination nodes. For the electrical case (Fig. 7a), the mean conductance shows different slopes corresponding to the three cases of tendencies in the assortative mixing. In particular, note that for large values of the minimum degree, the Smax configuration displays high values in the mean conductance; while for the Smin case the conductance is lower; with the Srand in between. When the mean flow case is analyzed with respect to the minimum degree (see Fig. 7b), we find that the three configurations overlap except for deviations for large values of minðkA ; kB Þ, confirming that the transport is not strongly affected by the assortative mixing. 5. Conclusions We have analyzed some transport properties in clustered/declustered small-world and scale-free networks. After comparing electrical transport and maximum flow in networks based on the WS model, we find that the clustered small-world configurations are characterized for displaying the best trade-off for the local and global conductances, while for regular networks the local conductance is larger than the global one; and for the random configuration is the reversed. Therefore, we can conclude that the small-world networks are transport efficient in both the local and global scales. For declustered networks, the transport with friction globally improves as the rewiring parameter increases and locally remains constant, whereas for friction-less transport, globally and locally decreases. Moreover, after analyzing the effect of the assortative mixing on the transport capacity, we find that the maximum assortative mixing increases the probability for having large conductance values, but this advantage is penalized by reducing the probability of having low conductances when comparing to the random case for assortative mixing. For the minimum mixing case, there is not an improvement for low conductances, but the probabilities of having large conductances are lower than for the other assortative configurations. Comparing both types of transport leads to remarkable differences, i.e., the transport with friction is influenced by the tendency in the assortativity, such that the positive (negative) assortativity lead to large (low) conductances. Moreover, the transport capacity under friction decreases when the mean distance between the source and destination nodes increases; while the flow is independent of this distance. This is related to

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