Biased random walk in spatially embedded networks with total cost constraint

Physica A xx (xxxx) xxx–xxx

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Biased random walk in spatially embedded networks with total cost constraint Rui-Wu Niu, Gui-Jun Pan ∗ Faculty of Physics and Electronic Technology, Hubei University, Wuhan 430062, People’s Republic of China

highlights • This paper studies random walk with a bias in spatial network with total cost restriction. • This paper finds that the best optimal transport is obtained with an exponent α = d + 1 for all p. • The special phenomena can be possibly explained by the theory of information entropy.

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Article history: Received 14 December 2015 Received in revised form 14 March 2016 Available online xxxx Keywords: Spatial network Navigation Information entropy

abstract We investigate random walk with a bias toward a target node in spatially embedded networks with total cost restriction introduced by Li et al. (2010). Precisely, The network is built from a two-dimension regular lattice to be improved by adding long-range shortcuts with probability P (rij ) ∼ rij−α , where rij is the Manhattan distance between sites i and j, and α is a variable exponent, the total length of the long-range connections is restricted. Bias is represented as a probability p of the packet or particle to travel at every hop toward the node which has the smallest Manhattan distance to the target node. By studying the mean first passage time (MFPT) for different exponent log ⟨l⟩, we find that the best transportation condition is obtained with an exponent α = d + 1(d = 2) for all p. The special phenomena can be possibly explained by the theory of information entropy, we find that when α = d + 1 (d = 2), the spatial network with total cost restriction becomes an optimal network which has a maximum information entropy. In addition, the scaling of the MFPT with the size of the network is also investigated, and finds that the scaling of the MFPT with L follows a linear distribution for all p > 0. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Many real complex networks can be geographically represented or spatially embedded [1], such as social networks [2,3], the global airport network [4,5], wireless communication networks [6], physical systems [7] and also the network of activity in the brain [8,9], and so on. Both theoretical and empirical studies have revealed that spatially embedded networks structure and dynamics on networks exhibit interdependent relationships with each other, which is actually an important and fundamental problem in the field of complex networks [1,10,11]. In recent studies, it has been shown that the navigability of the spatially embedded networks can be influenced by the geometrical structure of networks [12–19]. Generally speaking, navigation in complex networks with global information is the most efficient way to transport a particle from the source to the target. But in reality, it is hard to get the global

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Corresponding author. Tel.: +86 027 88662552. E-mail address: [email protected] (G.-J. Pan).

http://dx.doi.org/10.1016/j.physa.2016.05.024 0378-4371/© 2016 Elsevier B.V. All rights reserved.

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information from the networks. Because of this, a practical scheme is proposed, and has caught many scientists’ attention, which is navigation with local information. First, Kleinberg established a model to study optimal navigation with local knowledge in a small-world network by using greedy algorithm [12]. Kleinberg considered an L × L square lattice, where each node is connected with its neighbors and randomly generates a long-range connection with a probability P (rij ) ∼ rij−α , where rij is the Manhattan distance between sites i and j, and α is a variable exponent, and found that the small-world feature of the network can only be efficiently accessed if the exponent α = d [12]. Later, Roberson et al. studied the navigation problem in fractal small-world networks [13], where they proved that α = d is also the optimal power-law exponent in the fractal case. And then Cartozo et al. used dynamical equations to study the process of Kleinberg navigation [14,15]. They provided an exact solution for the asymptotic behavior of such a greedy algorithm as a function of the dimension d of the lattice and the power-law exponent α . Thadakamalla et al. proposed several decentralized search algorithms, including greedy algorithm, on the spatial scale-free networks, and found that the spatial scale-free network is searchable for a wide range of parameter space [16]. Moreover, M. Boguñá et al. showed that specific structure features of many complex networks support efficient transportation without global information, such as greedy algorithm, and indicated that real networks may have hidden metric space that undiscovered [17]. Recently, based on Kleinberg’s spatial networks, Li et al. proposed a cost constraint on the total length of the additional links [18,19]. They found an interesting phenomena that the best transport condition is obtained with a power-law exponent α = d + 1 for both local and global navigation. Here comes the question, is the optimal transport condition also α = d + 1 if some nodes or all nodes of the networks hold null information about the structure of the network? In order to answer this question, we consider a biased random walk navigation strategy, the bias is represented as a probability p of the packet or particle to travel at every hop toward the node which has the smallest Manhattan distance to the target node. That is, navigating with a probability p to follow the greedy algorithm and 1 − p to follow random walk algorithm. This condition can help us to optimize the navigation in a spacial network such as social network and airline network. By investigating the MFPT for different exponent α , we find that the best transportation condition is obtained with an exponent α = d + 1 for all p. The special phenomena can be possibly explained by the theory of information entropy, we find that the spatial network with total cost restriction is an optimal network with a maximum information entropy when α = d + 1.

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2. Model

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2.1. Generating the network

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Based on Kleinberg’s spatial networks, Li et al. propose a spatial network model with total cost restriction [18,19]. Fig. 1 shows a typical spatial network with total cost restriction, which is a regular two-dimensional square lattice with N = L × L node, where L is the linear size of the lattice, each node is connected with its four nearest neighbors. In the model, pairs of nodes i and j are randomly chosen to receive long-range connections with probability P (rij ) ∼ rij−α , where rij is the Manhattan distance between nodes i and j. And the total length of the long-range connections is restrict by Λ = CL × L. The probability P (rij ) that nodes i and j will have a long range connection can be mapped on a density distribution p(r ), where r = rij . The number of nodes separated by a lattice distance r from a given node in a d-dimensional lattice is proportional to r d−1 . L Thus the distance distribution of the long-range connections p(r ) ∼ r −α r d−1 can be normalized as 1 p(r )dr = 1, which yields

p(r ) =

  (d − α)

  1 r −α r d−1 , ln L

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r −α r d−1 ,

α ̸= d, (1)

α = d.

And then, the distance r can be obtained from random numbers o < u ≤ 1 chosen from the uniform distribution, by

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1 Ld−α − 1

r =

[1 − u(1 − Ld−α )]1/(d−α) , Lu , α = d .

α ̸= d,

(2)

The network model can be generated following algorithm in Refs. [19,20]: (i) Creating a regular d-dimensional lattice with N nodes with each node connected to its 2d nearest neighbors. (in this paper, d = 2). (ii) We randomly chose a node i to create a long-range connection. The length of the long-range connection r (1 < r ≤ L) is randomly selected using Eqs. (2). We consider all Nr nodes on the Manhattan distance S = [r ] (if r − [r ] > 0.5, then Q4 S = [r ] + 1, and if r − [r ] ≤ 0.5, then S = [r ]) from node i, that are not yet connected to node i. (iii) We randomly pick one node j from the Nr nodes, and then connect nodes i and j. (iv) Return to step (ii), until the total length of the long-connections reaches the preset cost Λ, e.g. Λ = L × L. This algorithm ensures that the desired distribution function p(r ) ∼ r −α r d−1 is fulfilled.

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Fig. 1. In a two-dimensional space, each node i has four short-range connections to its nearest neighbors (a, b, c and d). A long-range connection is added to a randomly chosen node j with probability proportional to r −α .

2.2. Biased random walk in spatially embedded networks

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The navigation strategies depend on the knowledge of the entire network topology which is possessed by any node. If each node possesses null information of the network topology, the navigation strategy is considered as random walks, but the random navigation scheme usually exhibits low efficiency in searching out a short path to target, especially in large-scale networks. When each element of the system has a full view of the global network topology, finding short routes to target destinations is a well-understood computational process, the highest efficiency can be achieved with the total navigation time equal to the topological shortest path length. If nodes only possess local information, the optimal navigation strategy based on local information would be considered, such as the greedy algorithms. In this paper, we combine random navigation scheme and greedy algorithms by introducing a bias p. Precisely, the source node i and target node j are randomly selected, packet or particle to travel at every hop toward the node which has the smallest Manhattan distance to the target node by probability p, and toward a random neighbor by probability 1 − p. Consequently, for p = 1 the packet transports by using greedy algorithm as same as Kleinberg navigation [12,18], while for p = 0 it performs a stochastic random walk.

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3. Results In order to explore the influence of parameter p on the navigability of the network, we study the mean first passage time (MFPT) for different exponent α and the scaling of the MFPT with the size of the network L. The MFPT is the time required for a walker to reach a given target point for the first time. And each data point is the average result of 100 000 total runs over different pairs of node i and j and different network realizations. In Fig. 2, we plot the relation between the MFPT ⟨l⟩ and exponent α for p = 0, and find that the best optimal value is α = 3 for fully random walk navigation. It has been shown that the best optimal value is α = 3 when p = 1 [18,19], so we speculate that α = 3 should be best optimal value for all p, which will be verified as follows. Fig. 3 shows the relations between the MFPT ⟨l⟩ and exponent α for p > 0. In Fig. 3(a), we find that for a large p, the minimum ⟨l⟩ is achieved at α = 3. We find that for α > 3, from Figs. 2 and 3(b), ⟨l⟩ increases much faster for p = 0 than p = 0.1 when increasing α . The reason of this is that, when α > 3 the network has more shortcuts and the lengths of the shortcuts have a relatively small value, and the number of long shortcuts decreases rapidly because of the cost constraint. So that for p = 0 fully random walk condition, the navigation time increases rapidly with α and when greedy algorithm involved (for example p = 0.1) the navigation time increases relatively slow. And in Fig. 3(b), we also observed a small drift of the optimal value from α = 3 for a small p. The possible reason is the effect of finite size of the network. Indeed, the results presented in Fig. 4 clearly indicate that, in a large restricted system, navigate with a small probability to follow the greedy algorithm and large probability to follow random walks, the optimal MFPT is close to α = 3. To further illustrate how the probability p affects the navigability, we investigate the scaling of the MFPT with system linear size L. We assume that ⟨l⟩ vs. L follows a power law with exponent δ , i.e., ⟨l⟩ ∼ L−δ . First, the condition p = 0 is considered. In Fig. 5(a) we plot the data assuming this function in a double logarithmic plot for different α . As can be seen, which fits quite well as a straight line. Indeed, we plot in Fig. 5(b) the successive slopes δ obtained from the plot of log ⟨l⟩ vs.

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Fig. 2. For p = 0, navigation in a spatially embedded networks with total cost constraint (L = 512, Λ = L × L) follows fully random walks. A minimum MFPT ⟨l⟩ is observed at α ≈ 3. Each data point is a result of 100 000 simulations.

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Fig. 3. Navigation in spatially embedded networks with total cost constraint with different p (L = 512 , Λ = L × L). (a) For p ≥ 0.4, the optimal MFPT is achieved at α = 3. (b) For p < 0.4, the optimal MFPT is achieved at α > 3. Each data point is a result of 100 000 simulations.

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log L, we see that δ keeps almost a constant value, and δ ≈ 2. That is, when p = 0, the scaling of ⟨l⟩ with L follows a power

Q6 law distribution for different α , and the power law exponent δ is independent on α .

Next, we investigate the scaling of the MFPT ⟨l⟩ with system linear size L when p > 0 and α = 3. In Fig. 6(a), ⟨l⟩ as a function of L in a double logarithmic is reported. And the plot of the successive slopes δ obtained from log ⟨l⟩ vs. log L is shown in Fig. 6(b). We find that when p > 0, the scaling of ⟨l⟩ with L follows a power law distribution, and for p > 0.1 the power exponent δ ≈ 1, that is, the scaling of ⟨l⟩ with L follows a nearly linear distribution. When p is very small, it is observed that in Fig. 7, the scaling exponent δ has a sudden change from δ ≈ 2 to δ ≈ 1. Finally, we investigate the scaling of ⟨l⟩ with L for different p > 0 under the none optimal condition. In Fig. 8, we report ⟨l⟩ as a function of L in a double

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Fig. 4. MFPT ⟨l⟩ as a function of α for p = 0.2. We find that the optimal MFPT is achieved at α = 3 for large network size. Each data point is a result of 100 000 simulations. (Λ = L × L).

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Fig. 5. The scaling of MFPT ⟨l⟩ with system linear size L for p = 0 in different α . (a) ⟨l⟩ as a function of L in a double logarithmic plot, (b) successive slopes δ obtained from log ⟨l⟩ vs. log L taken from (a). In (b) δ keeps roughly constant and the value is 2. We find that for different α , the scaling of ⟨l⟩ with L follows a power law distribution. Each data point is a result of 100 000 simulations. (Λ = L × L).

logarithmic plot, and the plot of the successive slopes δ obtained from log ⟨l⟩ vs. log L for p = 0.1. In Fig. 9, the scaling of ⟨l⟩ with L for p = 0.6 is investigated. We also find that the scaling of ⟨l⟩ with L follows a nearly linear distribution. In fact, it has been shown that the scaling of ⟨l⟩ with L follows a nearly linear distribution in different α when p = 1 [18,19], here, our results show that for p > 0, the scaling of ⟨l⟩ with L follows a nearly linear distribution in different α .

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Fig. 6. The scaling of the MFPT ⟨l⟩ with system linear size L for α = 3 in different p. (a) ⟨l⟩ as a function of L in a double logarithmic plot, (b) successive slopes δ obtained from log ⟨l⟩ vs. log L taken from (a). In (b) δ keeps roughly constant and the value is about 1. We find that for p > 0, the scaling of ⟨l⟩ with L follows a nearly linear distribution. Each data point is a result of 100 000 simulations. (Λ = L × L).

Fig. 7. The relationship between scaling exponent δ of ⟨l⟩ ∼ L−δ and bias parameter p, where p is probability that the packet follows the greedy algorithm for navigation. When p = 0, the scaling exponent δ ≈ 2, while for p > 0 the value of δ has a sudden change from δ ≈ 2 to δ ≈ 1. Each data point is a result of 100 000 simulations. 1

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4. Discussion In the spatial networks with total cost constraint, the optimal transport condition is α = d + 1, regardless the strategy used for navigation, being based on local or global knowledge of the network structure [18,19]. Here, we find that the optimal transport condition is also α = d + 1 for any biased random walk. All these results indicate that the optimal spatial networks

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Fig. 8. The scaling of MFPT ⟨l⟩ with system linear size L for p = 0.1 in different α . (a) ⟨l⟩ as a function of L in a double logarithmic plot, (b) successive slopes δ obtained from log ⟨l⟩ vs. log L taken from (a). In (b) δ keeps roughly constant and the value is 1. We find that for different α , the scaling of ⟨l⟩ with L follows a nearly linear distribution. Each data point is a result of 100 000 simulations. (Λ = L × L).

should own a special structure property. To investigate the phenomena theoretically, we study the relationship between information entropy S and the network structure exponent α . The information entropy is a concept being used to measure the quantity of information and diversity as proposed by Shannon, see e.g., Ref. [21]. In this context, a higher information Entropy represents an efficient collecting of the information on the networks. We will show that a spatial network with α = d + 1 is an optimal network with a maximum information entropy which benefits nodes in collecting maximal information. For a given network, the information that node i brings to j can be evaluated by considering the information of node i, all its neighbors, neighbors of neighbors and so on. Thus, the information that j collected can be expressed by the sequence of nodes as illustrated in Fig. 10 and the entropy of the whole sequence measures the amount of information. The information entropy Sj of node j is defined as follows [22]. Sj = −

N 

qm log qm

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m=1

where qm denotes the frequency of node m in the information sequence of node j (see Fig. 10) and N is the size of the network. When m is not in the whole sequence of j, qm = 0, and we define qm log qm = 0. So that, the information entropy S of the network can be defined as the average of the information entropy of all nodes, that is,

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N

S=

1  N j =1

Sj .

First, we study the relationship between information entropy S and structure exponent α for the spatial network with cost Λ = CL ∗ L. In Fig. 11, we find that when we consider two layers of neighbors of nodes to calculate the information entropy, the optimal value of α is α ≈ 4. If we consider three layers of neighbors of nodes to calculate the information entropy, we can find that the optimal value of α is nearly α ≈ 3.7. If we consider more layers of neighbors to calculate the information entropy, the optimal value of α is getting close to α = 3.

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Fig. 9. The scaling of MFPT ⟨l⟩ with system linear size L for p = 0.6 in different α . (a) ⟨l⟩ as a function of L in a double logarithmic plot, (b) successive slopes δ obtained from log ⟨l⟩ vs. log L taken from (a). In (b) δ keeps roughly constant and the value is 1. We find that for different α , the scaling of ⟨l⟩ with L follows a nearly linear distribution. Each data point is a result of 100 000 simulations. (Λ = L × L).

Fig. 10. The neighbors and neighbors of neighbors of node 1. Nodes 2 and 3 are the neighbors of node 1. The size of the network is n = 9 and the information sequence is {2, 3, 4, 5, 6, 6, 7} and the frequencies of all nodes are q2 = q3 = q4 = q5 = q7 = 71 , q6 = 27 , q1 = q8 = q9 = 0. If one node is reached several times from different nearest neighbors (such as node 6 that can be reached through node 2 and 3), it will appear in the node sequence the same number of times.

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Fig. 11. The relationship between information entropy S and structure exponent α for a L × L (L = 512) spatial network with cost (Λ = L ∗ L). (a) Two layers of neighbors of nodes are considered to calculate the information entropy, the optimal value of α is α ≈ 4. (b) Three layers of neighbors of nodes are considered, the optimal value of α is nearly α = 3.7. (c) Six layers of neighbors of nodes are considered, the optimal value of α is nearly α = 3.5. (d) At last, ten layers of neighbors of nodes are considered, we can find that the optimal value of α is almost α = 3. Above all, when we consider more layers of neighbors to calculate the information entropy, the optimal value of α is getting close to α = 3.

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Fig. 12. The relationship between information entropy S and structure exponent α for a L × L (L = 512) spatial network with different costs Λ = CL ∗ L. (a) (b) (c) and (d) are the information entropies calculated by considering 2, 3, 5 and 6 layers of neighbors, respectively. From (a) (b) (c) and (d), we can see that he optimal value of α is getting close to α = 3 when considering a higher cost Λ. And when the cost Λ and the number of layers get larger, the peak near α = 3 becomes more sharp.

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In addition, we also study the relationship between information entropy S and structure exponent α for the spatial networks with different costs Λ = CL ∗ L. Fig. 12(a–d) are the information entropy calculated by considering 2, 3, 5 and 6 layers of neighbors respectively. In Fig. 12(a), we can see that the optimal value of α gradually approaches to 3 and reaches α = 3 when the cost Λ is high enough, although only 2 layers of neighbors are considered. In Fig. 12(a–d), we find that if more layers of neighbors and more high cost Λ are considered simultaneously, the optimal value of α is getting rapidly close to α = 3. And when the cost Λ and the number of layers get larger, for the peak near α = 3 becomes more sharp. All results show that the spatial network is an optimal network with a maximum information entropy when α = 3.

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5. Conclusions

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We investigate random walk with a bias toward a target node in spatially embedded networks with total cost restriction. The bias is represented by the parameter p, which is the probability that the packet follows the greedy algorithm for navigation. We study the MFPT for different exponent α and the scaling of the MFPT with the size of the network. We find the best transportation condition is obtained with an exponent α = d + 1 for all p. In addition, for all values of α , the scaling of MFPT with L follows a power law distribution with power exponent δ ≈ 2 when p = 0, and the scaling of the MFPT with L follows a linear distribution for all p > 0. The special phenomena can be possibly explained by the theory of information entropy, that is, the spatial network with total cost restriction is an optimal network with a maximum information entropy when α = d + 1. And this conclusion is only a possible explanation and needs further investigation.

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Acknowledgment

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This work was supported by the Education Foundation of Hubei Province through Grant No. (D20120104). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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