Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance

Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance

Abstract

Many real networks are embedded in space, where in some of them the links length decay as a power law distribution with distance. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations for model networks embedded on lattices of dimensions and . We evaluate the dimension from the power law scaling of (a) the mass of the network with the Euclidean radius and (b) the probability of return to the origin with the distance travelled by the random walker. Both approaches yield the same dimension. For networks with , is infinity, while for , obtains the value of the embedding dimension . In the intermediate regime of interest , our numerical results suggest that decreases continously from to , with for close to . Finally, we discuss the scaling of the mass and the Euclidean distance with the topological distance (minimum number of links between two sites in the network). Our results suggest that in the intermediate regime , and do not increase with as a power law but with a stretched exponential, and , where . The parameters and are related to by , such that . For , increases exponentially with , as known for , while is constant and independent of . For , we find power law scaling, and , with . For networks embedded in , we find the expected result, , while for networks embedded in we find surprisingly, that although , and , in contrast to regular lattices.

1

1 Introduction

It has been realized in the last decades that a large number of complex systems are structured in the form of networks. The structures can be man-made like the World Wide Web and transportation or power grid networks or natural like protein and neural networks [2, 6, 1, 3, 7, 8, 9, 11, 12, 5, 14, 13, 4, 19, 15, 16, ?, 10]. When studying the properties of these networks it is usually assumed that spatial constraints can be neglected. This assumption is certainly correct for networks like the World Wide Web (WWW) or the citation network where the real (Euclidean) distance does not play any role, but it may not be justified in networks where the Euclidean distance matters [20]. Typical examples of such networks include the Internet [6, 11], airline networks [21, 22], wireless communication networks [23], and social networks (like friendship and author networks) [24, 25], which are all embedded in two-dimensional space (surface of the earth), as well as protein and neural networks [26], which are embedded in three dimensions.
To model these networks, two network classes are of particular interest: Erds-Rnyi (ER) graphs [28, 27] and Barabasi-Albert (BA) scale free networks [29]. In ER-networks, the distribution of the number of links per node (degree-distribution) is Poissonian with a pronounced maximum at a certain -value, such that nearly each node is linked to the same number of nodes. In BA networks, the distribution follows a power law , with typically between 2 and 3. Here we focus on ER-type networks embedded in one- and two-dimensional space. We actually use a degree distribution that is close to a delta function (as the case in simple lattices). We found that the results are the same for both kinds of distributions. We follow Refs. [31, 30, 32] and assume that nodes are connected to each other with a probability , where is the Euclidean distance between the nodes. The choice of a power law for the distance distribution is supported from findings in the Internet, airline networks, human travel networks and other social networks [22, 25, 33]. Our model of embedding links of length , chosen from Eq. (1), in a - dimensional lattice can be regarded as a generalization of the known Watts Strogatz (WS) model [1, 3]. In the WS model links of any possible lengths with the same probability are added in the lattice system which corresponds to the case of Eq. (1). Other methods for embedding networks in Euclidean space have been proposed in [34, 35, 36, 37].
It has recently been shown that spatial constraints are important and may alter the dimension and therefore the topological properties of the networks (likethe dependence of the mean topological distance on the system size) as well as their robustness [31, 30]. Here we are interested in studying how in these model networks the spatial constraints quantified by the distance exponent modify the scaling relations between mass (number of nodes), Euclidean distance and topological distance . Our earlier study on ER networks embedded in a square lattice (with dimension ), indicate that by varying the exponent one can actually change continuously the dimension of the network, from for to for [32]. In the present manuscript we present further extensive numerical simulations for that support this claim as well as simulations in linear chains () that suggest analogous conclusions. In we find that for the system behaves like an infinite dimensional network (as the original ER-network). When continuously increasing the dimension becomes finite for and approaches for . Since the dimension of a system plays a critical role in many physical phenomena like diffusion, percolation and phase transition phenomena, our results are important for understanding and characterizing the properties of real world networks.
Our manuscript is organized as follows. In Section 2, we discuss the characteristic distances in the spatially constrained networks. In Section 3 we describe the method to generate the spatial network models. In Section 4 we present our numerical results for the dimension , for networks embedded in linear chains and in square lattices, that we obtain from the scaling relation of the mass and the distance . In Section 5 we present our numerical results for the dimension , that we obtain from the scaling relation of the probability of return to the origin of a diffusing particle and its distance . In Section 6, we discuss the scaling of the mass and the Euclidean distance with the topological distance . The conclusions in Section 7 summarize our main results.

Figure 1: Illustration of ER networks embedded in linear chains (top) and square lattices (bottom), for various distance exponents .

2 Characteristic distances

First we estimate how the characteristic distances, in a network of nodes, depend on its linear size , on and on the embedding dimension . We normalize the distance distribution such that , which yields

(1)

From we obtain and the related length scales . The maximum distance is determined by . The results for and are

(2)

and

(3)

Accordingly, for all length scales ( and ) are proporional to , the spatial constraints are weak and the system can be regarded as an infinite dimensional system. On the other hand, for , and tend to zero in the asymptotic limit. In this case, we expect that the physical properties of the network are close to those of regular lattices of dimension . However, large finite size effects are expected for close to where decays only very slowly to zero. In the intermediate -regime , scales as , while tends to zero in the asymptotic limit. In this regime our simulation results (Chap. 4) suggest intermediate behavior represented by a dimension between and infinity that changes with .

3 Generation of the networks

The nodes of the network are located at the sites of a -dimensional regular lattice, in our case a linear chain of length () or a square lattice of size (). We assign to each node a fixed number of links (in most cases, ). Actually this network is a random regular (RR) network since all nodes have the same degree. It is expected (and we have also verified it numerically) that both networks, ER and RR with the same spatial constraints, are in the same universality class.
To generate the spatially embedded networks, we use the following iterative algorithm: (i) We pick a node randomly and choose, for one of its available links, a distance from the given probability distribution , Eq. (1). It is easy to see that the distance can be obtained from random numbers chosen from the uniform distribution, by

(4)
Figure 2: The distance distribution for ER-networks embedded (a) in linear chains where and (b) in square lattices where , when (circle), (diamond), (triangle up), (triangle left) and . The numbers denote the slopes of , which are identical to the anticipated ones. For the same set of parameter as in (a) and (b), the panels (c) and (d) show the degree distribution , which is for and otherwise.

(ii) We consider all nodes between distance and from node , that are not yet connected to node . Without loss of generality, we choose for the linear chain and for the square lattice. (iii) We pick randomly one of these nodes . If node has at least one available link, we connect it with node . If not, we do not connect it. Then we return to (i) and proceed with another randomly chosen node. At each step of the process, either 2 or zero links are added. For generating the network, we have typically performed trials. Due to the generation process, the nodes of the final network do not all have exactly the same degree, but the degree follows a narrow distribution with a mean slightly below . Figure 1 illustrates the ER networks embedded in =1 and =2 for . Figure 2 shows the actual narrow degree distribution as well as obtained in the simulations.

4 The dimension of the networks

For determining the dimensions of the spatially embedded networks, we follow the method developed by Daqing et al [32]. We use the fact that the mass (number of nodes) of an object within an hypersphere of radius scales with as

(5)

where the exponent represents the dimension of the network. When using this relation without taking into account the way the nodes are linked, one trivially and erroneously finds that the dimension of the network is identical to the dimension of the embedding space.

Figure 3: Illustration of the shells (square), (circle), (triangle up), (diamond), (triangle left), (triangle down) and (triangle right) for ER networks embedded in a square lattice with (left panel), and the mass as function of within this shells (right panel).

To properly take into account the connectivity, when considering the dimension of the network, we proceed as follows (see

Figure 4: (a) The mass as function of the relative distance for ER networks embedded in linear chains with , for the system sizes with (from left to right). The straight lines are best fits to the data that yield the dimension of the network. (b) The same as panel (a), but for and .

Fig 3): We choose a node as origin and determine its nearest neighbors (referred to as shell 1) and their number , the number of second nearest neighbors , and so on. Next we measure the mean Euclidean distance of the nodes in shell from the origin and determine the number of nodes within shell . To improve the statistics, we repeat the calculations for many origin nodes and then average and . To reduce finite size effects, we do not choose the origin nodes randomly in the underlying lattice, but from a region with radius around the central node. From the scaling relation between the average and the average , Eq. (5), we determine the dimension of the network.
Figure 4 shows the results for networks embedded in linear chains, for distance exponents between 1.25 and 2.5 . In (a), we consider networks with fixed and different system sizes (, and ), while in (b) we consider networks with a fixed size and various values (). In both panels, we have plotted as a function of , where is the mean distance, see Eq. (2).
Figure 4a shows that for in the interesting regime between and , the curves for different collapse nicely (For transparency, the curves (except ) have been shifted along the -axis by a factor of  and  ). From the slopes of the straight lines, we obtain the dimension , and . For , the data starts to overshoot above some crossover value that increases with the system size and thus can be regarded as a finite size effect. To understand the reason for this crossover note that a node close to the boundary has a considerably higher probability to be linked with nodes closer to the center of the underlying lattice. As a consequence, for large shell numbers , the mean Euclidean distance of the nodes from the origin node will be underestimated and thus the mass within large Euclidean distances overestimated. This effect is most pronounced in the linear chain, for intermediate -values, and gives rise to the overshooting of for between 2 and 2.5, where . For and , the total number of nodes in the spatially constrained network is well below , since the network is separated into smaller clusters. For larger -values, this effect is less likely to appear. Figure 4b shows that the dimension of the networks does not depend on their average degree. The curves collapse for different , and thus give rise to the same dimensions. This indicates the universality feature of the dimension.

Figure 5: The same as Fig 4, but for ER networks embedded in a square lattice, the system sizes are with (from left to right).

Figure 5 shows the corresponding results for networks embedded in square lattices (), again for 6 exponents between 1.25 and 2.5 , three network sizes (), and three values (). From the slopes of the straight lines we obtain , , and . For above 4, is close to , as expected. The figure confirms that the finite size effects in are considerably less pronounced than in , contrary to the intuition, since the linear size of the underlying embedding lattice is considerably higher in than in . As in , the dimensions are independent of the mean degree of the networks.

Figure 6: The difference between network dimension and embedding dimension as a function of for (circles) and (triangles).

.

Figure 6 summarizes our results for the dimensions of the spatially embedded networks in the intermediate regime between and , where the dimension is supposed to bridge the gap between for the unconstrained case below and for the highly constrained case above . The figure shows as a function of the relative distance exponent for both considered lattices. The figure shows that in both cases, the curves approximately collapse to a single line which can be represented by

(6)

where . According to Eq. (6), diverges for approaching the critical relative distance exponent .

5 The probability of return to the origin

The network dimension plays an important role also in physical processes such as diffusion [38, 40, 39]. The probability that a diffusing particle, after having traveled steps, has returned to the origin, is related to the root mean square displacement of the particle by [32, 40, 41]

(7)
Figure 7: (a) The probability that a diffusing particle is at its starting site, after travelling an average distance , as a function of the relative distance for ER networks embedded in linear chains with , for the system size with (from left to right). The straight lines are best fits to the data that yield the dimension of the network. (b) The same as panel (a), but for ER networks embedded in a square lattice, the system size is with (from left to right). Note that the values of obtained here are almost the same as those obtained by direct measurements in Figs. 4 and 5.

To derive Eq. (7) one assumes that the probability of the particle to be in any site in the volume is the same. As a consequence, , which leads to Eq. (7). Figure 7 shows as a function of in and 2, for the same -values as in Figs. 4 and 5. For convenience, we show only the results for the largest system size, for and for . To obtain , we averaged, for each value of , over diffusing particles and 50 network realizations. From the straight lines in the double-logarithmic presentations of Figure 7 we obtain the dimension of the networks, which are listed in the figure. The dimensions obtained in Figure 7 agree very well with those obtained by direct measurements in Figs. 4 and 5.

6 The topological dimension and the dimension of the shortest path

In order to find how scales with the Euclidean distance , we determined in Sect. 4 how and scale with the topological length , and obtained the dimension from . In this section, we discuss explicitely how and depend on .
It is well known that for regular lattices as well as for fractal structures, and scale with as power laws,

(8a)
(8b)
Figure 8: The mass (left column) and the relative distance (right column) as function of ( is the topological distance and ) for ER networks embedded in linear chains with , for the system sizes with . The straight lines are best fits to the data with slopes and respectively.
Figure 9: The same as Fig 8, but for ER networks embedded in a square lattice, the system sizes are with .
Figure 10: The mass (left column) and the relative distance (right column) as a function of the topological distance for ER networks embedded in linear chains with , for the system sizes with . The straight lines are best fits to the data that yield the topological dimension and the dimension of the shortest path . Note that the slopes below the crossover in (b), (c), (e) and (f) of and vs are the same. This yields for all range of as indeed seen in Fig. 4.
Figure 11: The same as Fig 10, but for ER networks embedded in a square lattice. The system sizes are with . The lines in (c) and (f) demonstrate for comparison slopes 2 and 1 respectively.

where is the topological (”chemical”) dimension and is the dimension of the shortest path, see e.g., [42, 43]. For regular lattices of dimension , and . Thus we expect that for , the power law relations (8) hold.
For the network has no spatial constraints and it is known that the mean topological distance between 2 nodes on the network scales with the network size as [5]. This represents the small world nature of random graphs. Since plays the role of the mass of the network, it follows that increases exponentially with , i.e. . We expect that this relation holds for where and are both proportional to the linear scale of the network, see Eqs. (2) and (3). Since for we expect power law relations (8), we conjecture that in the intermediate regime , will increase slower than exponential and faster than a power law, via a stretched exponential,

(8i)

This function can bridge between the exponential behavior for and the power law for . For approaching from above, should approach 1, while for approaching from below, should approach 0, consistent with a power law. The conjecture, Eq. (8i) is supported by earlier numerical simulations [30] where it was found that in the intermediate regime, scales as , leading to . On the basis of numerical simulations it was estimated [30], that in and in , which actually can be combined into a single equation, , when the relative distance exponent is introduced. Thus our conjecture (8i) becomes

(8j)

where the prefactor may depend on and . To test this hypothesis, we have plotted, in Figs. 8, a, b, c and Figs. 9 a, b, c , versus , in a semi-logarithmic fashion. The relative distance exponents are 0.5, 1.25 and 1.75 in both cases. The lattice sizes are the same as in Figs. 4 and 5. For where the spatial constraints are irrelevant, we find , in agreement with (8j). In the intermediate regime we find that , with = 0.93 () and 0.43 (), also in agreement with (8j). Accordingly, in the intermediate -regime, scales with the topological distance as a stretched exponential which serves as a ”bridge” between the exponential behavior for and the anticipated power law behavior for well above .
Now the question arises how the power law in Eq. (5) that describes the scaling of with and the stretched exponential in Eq. (8j) that describes the scaling of with , can be simultaneously satisfied. The only way to fulfill both equations is, that also is a stretched exponential with the same in the intermediate regime i.e.,

(8k)

and the ratio between the prefactors and should yield the dimension of the network. This is since . Figs. 8 e, f and 9 e, f support the assumption (11). The prefactor is obtained from the slopes of the straight lines in the figures and indeed the values of are found to be identical to the values of the dimensions we obtained in the previous section. For below (see Figs. 8d and 9d), is independent of and (see Figs. 8a and 9a).
For , we expect that and follow power laws, such that we can determine, from a double logarithmic plot, the chemical dimension and the dimension of the shortest path, . Figures 10 and 11 show that this is the case. But surprizingly, for (but close to 2), the values of and do not agree with the values for the corresponding regular lattices. For , we obtain in and in , significantly higher than the corresponding values and in regular lattices. Furthermore, the dimension of the shortest path is considerably smaller than in regular lattices (), in and in . Since , the dimension of the network for is simply , which yields in and in , in agreement with our results of Figs. 4 - 7. For above we expect that and accept the values of the corresponding regular lattices. Figure 10 shows that this is indeed the case in , with a pronounced crossover behavior for and 2.5. The crossover point decreases with increasing . In , in contrast, for and the dimensions do not seem to reach their anticipated values and , even though was obtained for both values. Figure 11 does not suggest that this is a finite size effect since a bending down for larger system sizes cannot be seen similar to that in . However, we cannot exclude the possibility that at very large system sizes that right now cannot be analyzed with the current state-of-the-art computers, there will be a crossover towards the anticipated values of and .

7 Summary

In summary, we studied the effect of spatial constraints on complex networks where the length of each link was taken from a power law distribution, Eq. (1), characterized by the exponent . Spatial constraints are relevant in all networks where distance matters, such as the Internet, power grid networks, and transportation networks, as well as in cellular phone networks and collaboration networks [6, 11, 20, 23, 24, 25]. Our results suggest that for below the embedding dimension , the dimension of the network is infinite as in the case of netwoks that are not embedded in space (represented by ). For between and 2, the dimension decreases monotonically, from to . Above 2, . We also studied how the mass and the Euclidean distance scale with the topological distance . For below , increases exponentially with , while does not depend on . For between and 2, both the mass and the Euclidean distance increase with as a stretched exponential, with the same exponent but different prefactors in the exponential. The ratio between these two prefactors yields the dimension of the embedded network. Exactly at , the exponent becomes zero and and scale with as power laws, defining the exponents and , respectively similar to fractal structures [42, 43]. While the dimension is equal to , surprisingly and do not have the values and that are expected for regular lattices. This effect seems to hold in also for values somewhat greater than 2.
Our results have been obtained for a nearly -functional degree distribution, but we argue that they are valid for any narrow degree distribution, like Possonian, Gaussian or exponential degree distribution since all those networks are expected to be in the same universality class. For power law degree distributions (scale free networks [29]), there may be differences for small values of , since it is known that nonembedded random graphs and scale free networks are in different universality classes [44, 45]. In the relevant intermediate regime (), we cannot exclude the possibility that the dimensions do not depend on the degree distribution. Indications are from measurements of the dimension of the airline network and the Internet [32]. Both are scale free networks, with close to 3 (airline network) and close to 2.6 (Internet). For the airline network, is close to 3, while for the Internet, is close to 4.5. These values are consistent with those obtained here for the ER-networks, with the same -values. We have assumed a power law distribution, Eq. (1), for the link length. Other distributions are possible, for example an exponential distribution which holds for the power grid and ground transportation networks [20]. This case is equivalent to , since we have a finite length scale and thus the dimension of the network is expected to be the same as the dimension of the embedding space .
A power law distribution of Euclidean distances appears also in other physical systems where the present results may be relevant. For example, model systems where the interactions between particles decay as have been studied extensively for many years, for recent reviews on the statistical physics and dynamical properties of these systems, see [46, 47]. Magnetic models on lattices with long range bonds whose lengths follow a power law distribution have also been studied, see e.g., [48]. In Levy flights and walks, the jump lengths follow a power law distribution. For reviews see [53, 54, 39]. Finally, it has been found that a power law distribution of link lengths with or (depending on the type of transport) is optimal for navigation [49, 50, 51, 52, 19].

AB and SH are grateful to the Deutsche Forschungsgemeinschaft for financial support. DL thanks the National Science Foundation of China (NSFC) for financial support.

References

Footnotes

  1. : New J. Phys.

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