Introduction

As an emerging science, complex networks have witnessed substantial progress in the past years1. One of the ultimate goals in the study of complex networks is to uncover the influences of various structural properties on the function or dynamical processes taking place on them. Among different dynamical processes, random walks lie at the core, since they are a fundamental mechanism for a wealth of other dynamic processes, such as navigation2, search3,4 and cooperative control5. Except for the importance in the area of network science, random walks also provide a paradigmatic model for analyzing and understanding a large variety of real-world phenomena, for example, animal6 and human7 mobility. Thus far, random walks have found numerous applications8 in many aspects of interdisciplinary sciences, including image segmentation9, community detection10,11, collaborative recommendation12 and signal propagation in proteins13 to name a few.

A highly desirable quantity for random walks is first passage time (FPT)14, defined as the expected time for a random walker going from a starting node to a given target averaged over all possible trajectories. The mean of FPTs over all starting nodes to the target is called mean first passage time (MFPT), which is an important characteristic of random walks due to the first encounter properties in numerous realistic situations. In the past years, the study of MFPT has triggered an increasing attention from the scientific community15,16. One focus of theoretical activity is to develop general methods to efficiently compute MFPT17,18,19,20. Another direction is to unveil how the behavior of MFPT is affected by different structural properties of the underlying systems, such as heterogeneity of degree21 or strength22, fractality23 and modularity24.

Previous studies proposed several frameworks for evaluating MFPT and uncovered the discernible effects of some nontrivial structural aspects on the target search efficiency measured by MFPT. However, most existing works ignore the impact of link reciprocity, the tendency of node pairs to form mutual connection in directed networks, on the behavior of random walks, despite the fact that reciprocity is a common characteristic of many realistic networks25, such as the World-Wide Web26, e-mail networks27,28 and world trade web29. In addition to binary networks, the nontrivial pattern reciprocity is also ubiquitous in real-life systems described by weighted networks30,31,32. It has been shown the ubiquitous link reciprocity strongly affects dynamical processes in binary networks, for example, spread of computer viruses28 or information33 and percolation34. By contrast, the influence of reciprocity on dynamical processes in weighted networks has attracted much less attention, although it is suggested that reciprocity could play a crucial role in network dynamics. In particular, the lack of analytical results in this field limits our understanding of the impact of weight reciprocity on the function of weighted networks32.

In this paper, we propose a weighted directed scale-free network through replacing each edge in the previous binary network35,36 by double links with opposite directions and different weights. In the weighted network, the link weights are adjusted by a parameter characterizing the weight reciprocity of network. We then study random walks in the weighted network in the presence of a perfect trap at the central large-degree node. During the process of random walks, the transition probability is dependent on the weight parameter. We derive two formulas for the MFPT to the target by using two disparate approaches, the results of which completely agree with each other. We also determine all the eigenvalues and their multiplicities of the fundamental matrix characterizing the random-walk process and show that the largest eigenvalue has the same leading scaling as that of the MFPT. The obtained results demonstrate that the behavior of MFPT to the trap depends on the weighted parameter, implying a drastic influence of the weight reciprocity on random walks defining on weighted networks.

Results

Network models and properties

Before introducing the weighted directed network with scale-free fractal properties. We first give a brief introduction to a binary scale-free fractal network, which has the same topology as the weighted network.

Model and properties of binary network

The binary network is constructed in an iterative way35,36. Let Fg (g ≥ 0) represent the network after g iterations (generations). For g = 0, F0 is an edge linked by two nodes. In each successive iteration g ≥ 1, Fg is constructed from Fg−1 by performing the following operations on every existing edge in Fg−1 as shown in Fig. 1: two new nodes (called external nodes) are firstly created and attached, respectively, to both endpoints of the edge; then, the edge is broken, another new node (referred to as an internal node) is placed in its middle and linked to both endpoints of the original edge. Figure 2 illustrates the first several construction processes of the network. The structure of Fg is enciphered in its adjacency matrix Ag, the entries Ag(i, j) of which are defined by Ag(i, j) = 1 if two nodes i and j are adjacent in Fg, or Ag(i, j) = 0 otherwise.

Figure 1
figure 1

Illustration of construction of the binary network.

The next generation is obtained from current generation by replacing each edge with the cluster on the right-hand side of the arrow, where w is a new internal node, while x and y are external nodes.

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Figure 2
figure 2

Iterative growth processes for the first several generations.

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The particular construction of the network allows to calculate exactly its relevant properties. At each generation gi (gi ≥ 1), the number of newly created nodes is . Let be the set of nodes generated at iteration gi, then can be further classified into two sets and satisfying , among which is the set of external nodes and is the set of internal nodes. We use |Ω| to stand for the cardinality of a set Ω. Because , it is easy to derive and . We represent the set of nodes in Fg as Λg. Hence, the number of nodes and edges in Fg is and Eg = Ng − 1 = 4g, respectively. Let ki(g) denote the degree of an arbitrary node i in Fg that was generated at generation gi (gi ≥ 0), then ki(g + 1) = 2 ki(g). Hence, after each new iteration the degree of every node doubles.

This resultant network displays the remarkable scale-free37 and fractal38 features as observed in diverse real-life systems. It has a power law degree distribution with an exponent 3 and its fractal dimension is 2.

Model and properties of weighted directed network

The above introduced binary network Fg can be extended to a weighted directed network with nonnegative and asymmetrical edge weights. Let denote the weighted directed network corresponding to Fg. Both and Fg have an identical topological structure. The only difference between and Fg is that every undirected edge in Fg is replaced by two directed edges with opposite directions and distinct positive weights. We use Wg to represent the nonnegative and asymmetrical weight matrix for such that Wij(g) > 0 if and only if there is a directed edge (arc) pointing to node j from node i. The weight of each arc in the weighted directed network is defined recursively in the following way. When g = 0, has two nodes, denoted by a and b and the weights of arcs and are defined to be Wab(0) = Wba(0) = 1. When g ≥ 1, by construction, Fg is obtained from Fg−1 by substituting each undirected edge e(u, v) in Fg−1 with two undirected edges e(u, w) and e(w, v) and generating two additional nodes, x and y, attaching to u and v, respectively. The weights of resultant arcs in are defined as: Wuw(g) = Wuv(g − 1), Wvw(g) = Wvu(g − 1), Wwu(g) = Wwv(g) = 1, Wxu(g) = 1, Wyv(g) = 1, Wux(g) = θWuv(g−1) and Wvy(g) = θWvu(g−1). Here θ is a tunable positive real number, that is, θ > 0. The weight parameter is of paramount importance since it characterizes the weight reciprocity of network . When θ = 1, reduces to Fg and the weights in two directions between any pair of adjacent nodes are completely reciprocated; when θ ≠ 1, the weights are non-reciprocated30,31,32: the larger the deviation of θ from 1, the smaller the level of weight reciprocity.

In fact, the introduced weight parameter θ acts as a similar role of energetic funnel in dendrimers controlling the MFPT to the center39,40,41. It has been shown theoretically39,42 that the MFPT of Cayley trees as models of dendrimers can be influenced by both geometrical and energetic features. However, for complex systems with scale-free and fractal properties, it is still not well-understood how to superimpose an energetic funnel on their topological architecture, so that the MFPT depends on the interplay between structural and energetic properties. The main purpose of this work is to fill the gap, uncovering the collective impacts of topological and energetic aspects on dynamical processes, especially MFPT of random walks.

In undirected weighted networks43, node strength is a key quantity characterizing the property of a node. Here we extend the definition of strength of a node to the directed weighted network by defining the out-strength and in-strength of node i in as and , respectively. For , we can obtain the out-strength for an arbitrary node i that entered the network at generation gi (gi ≥ 0). If i was an external node when it entered the network, ; otherwise, if i was an internal node when it was born, . Therefore, after each new iteration, the out-strength of a node increases by a factor of θ. It is easy to obtain that the node out-strength in obeys a distribution of power law form with the exponent being . Note that in some realistic networks, the node strength also display a broad distribution43.

Formulation of biased walks in the weighted directed network

After introducing the construction and properties of the weighted directed network , we now define and study biased discrete-time random walks performing . Let denote the transition probability that a particle jumps from node i to its neighboring node j per time step. Note that rij(g) constitutes an entry of transition matrix Rg = (Sg)−1Wg, where Sg is the diagonal out-strength matrix of , with the ith diagonal entry of Sg being .

In this paper, we focus on a specific case of biased random walks, often called trapping problem, in in the presence of a trap placed at the central hub node, i.e., the internal node generated at the first iteration. To facilitate the description of the following text, all the Ng nodes in are labeled sequentially as 1, 2, …, Ng − 1, Ng as follows. For , the newly generated internal node is labeled 1, the initial two nodes in are labeled as 2 and 3, while the two new external nodes are labeled by 4 and 5. For each new iteration gi > 1, we label consecutively the new nodes born at this iteration from to , while we keep the labels of those nodes created before iteration gi unchanged.

For the trapping problem, what we are concerned with are the trapping time and the average trapping time. Let represent the trapping time for a particle initially placed at node i (i ≠ 1) in to arrive at the trap node for the first time, which is equal to the FPT from the i to the trap. The average trapping time, 〈T〉g, is actually the MFPT to the trap, defined as the mean of over all non-trap initial nodes in network Fg:

Below we will show how to compute the two quantities and 〈T〉g.

For , it obeys the relation

which can be recast in matrix form as:

where is an (Ng − 1)-dimensional vector, e = (1, 1, …, 1)T is the (Ng−1)-dimensional vector of all ones and is a matrix of order Ng−1, which a submatrix of Rg and obtained from Rg by deleting the first row and the first column corresponding to the trap. From equation (3) we have

where I is the (Ng−1) × (Ng−1) identity matrix, matrix is the fundamental matrix44 of the addressed trapping problem. Equation (4) implies

where Kg(i, j) is the ijth entry of matrix Kg, representing the expected number of visitations to node j by a particle starting from node i before being absorbed by the trap. Plugging equation (5) into equation (1) yields

Equation (6) indicates that the computation of MFPT 〈T〉g can be reduced to finding the sum of all entries of the corresponding fundamental matrix. A disadvantage of this method is that it demands a large computational effort when the network is very large. However, equation (6) provides exact results for 〈T〉g that can be applied to check the results for MFPT obtained using other techniques. Next we analytically determine the closed-form expression for MFPT 〈T〉g using an alternative approach, the results of which are consistent with those of equation (6). It should be noted that MFPT for other deterministic scale-free unweighed networks have been previously addressed45,46,47.

Exact solution to the MFPT 〈T〉g

The particular selection of trap location and the specific network structure allow to determine exactly the MFPT 〈T〉g for arbitrary g. In order to obtain a close-form expression for 〈T〉g, we first establish the dependence of on iteration g. For a node i in , at iteration g + 1, its degree doubles, increasing from ki(g) to 2ki(g). All these 2ki(g) neighboring nodes are created at iteration g + 1, among which one half are external nodes with a single degree and the other half are internal nodes with degree 2.

We now consider the trapping problem in . Let A be the FPT for a particle starting from node i to any of its ki(g) old neighbors, that is, those nodes adjacent to i at iteration g; let B (resp. C) be the FPT for a particle staring from any of the ki(g) internal (resp. external) neighbors of i to one of its ki(g) old neighbors. Then the FPTs obey relations:

Eliminating B and C in equation (7), we obtain A = 4(θ + 1). Therefore, when the network grows from iteration g to iteration g + 1, the FPT from any node to another node increases by a factor of 4(θ + 1). Hence, hold for any g, which is a useful for deriving the exact expression for MFPT.

Having obtained the scaling dominating the evolution for FPTs, we continue determining the MFPT 〈T〉g. For this purpose, we introduce two intermediary quantities for any n ≤ g: and . Then,

By definition, . To find , it is necessary to explicitly determine the quantity . To this end, we define two additional quantities for n ≤ g: and . Obviously, . Thus, in order to find , one may alternatively evaluate and .

We first establish the relationship between and . By construction (see Fig. 1), at a given generation, each edge connecting two nodes u and v will give rise three new nodes (w, x and y) in the next generation. The two external nodes x and y are separately attached to u and v, while the only internal node w is linked simultaneously to u and v. For any iteration g, the FPTs for the three new nodes satisfy: , and . Therefore, . Summing this relation over all old edges at the generation before growth, we find that for all n ≤ g, always holds. In this way, the issue of determining is reduced to finding that can be obtained as follows.

For an arbitrary external node iext in , which is created at generation g and attached to an old node i, we have , a relation valid for any node pair containing an old node and one of its new external adjacent nodes. By applying relation to two sum (the first one is over a given old node and all its new adjacent external nodes, the other is summing the first one over all old nodes), we obtain

From equation (9), one can derive the recursive relation

Considering the initial condition , equation (10) is solved to yield

Because and , we have

Inserting equation (12) into equation (8) leads to

Using , equation (13) is solved to get

Then, the rigorous expression for the MFPT 〈T〉g of the weighted directed network is

We have checked the analytical solution in equation (15) against extensive numerical results obtained from equation (6), see Fig. 3. For different θ and g, both the analytical and numerical results are in full agreement with each other, indicating that the explicit expression in equation (15) is correct. In addition, for the particular case θ = 1, the network is reduced to Fg and equation (15) recovers the result23 previously obtained for Fg. This also validates equation (15).

Figure 3
figure 3

MFPT 〈T〉g as a function of g for different networks with various θ.

The filled symbols are the data obtained by direct calculation from equation (6); while the empty symbols are those exact analytical values given by equation (15).

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We proceed to express 〈T〉g in terms of the network size Ng, in order to uncover how 〈T〉g scales with Ng. From Ng = 4g + 1, we have g = log4(Ng − 1). Then,

For a very large network (i.e., Ng → ∞), the leading term of 〈T〉g can be represented as:

Equation (17) shows that for the directed weighted network , the MFPT 〈T〉g behaves as a power-law function of the network size Ng, with the exponent η(θ) = 1 + log4(θ + 1) increasing with the weight parameter θ. Thus, the weight reciprocity has an essential effect on the efficiency of the trapping problem, measured by the MFPT.

Eigenvalues of the fundamental matrix

We now study the eigenvalues of the fundamental matrix Kn of the trapping problem addressed above. We will determine all the eigenvalues of the fundamental matrix as well as their multiplicities. Moreover, we will show that the largest eigenvalue has the same leading scaling as that of 〈T〉g. To attain this goal, we introduce matrix Pg defined by . Let λi(g) and σi(g), where i = 1, 2, …, Nn−1, denote the eigenvalues of Pg and Kg, such that and . Then, the one-to-one relation λi(g) = 1/σi(g) holds. Thus, to compute the eigenvalues of matrix Kn, we can alternatively determine the eigenvalues for Pg. In the sequel, we will use the decimation method48,49 to find all the eigenvalues of matrix Pg.

Full spectrum of fundamental matrix

The decimation procedure48,49 makes it possible to obtain the eigenvalues for related matrix of the current iteration from those of the previous iteration.

We now consider the eigenvalue problem for matrix Pg+1. Let α denote the set of nodes in network and β the set of nodes created at iteration g + 1. Suppose that λi(g + 1) is an eigenvalue of Pg+1 and is an eigenvector associated with λi(g + 1), where uα and uβ correspond to nodes belonging to sets α and β, respectively. Then, eigenvalue equation for matrix Pg+1 can be represented in a block form:

where Pα,α and Pβ,β are the identity matrix.

Equation (18) can be expressed as two equations:

Equation (20) implies

provided that λi(g + 1) ≠ 1. Inserting equation (21) into equation (19) yields

In this way, we reduce the problem of determining the eigenvalue λi(g + 1) for matrix Pg+1 of order 4g+1 to finding the eigenvalue problem of matrix Pα,βPβ,α with a smaller order 4g.

We can prove (see Methods) that

where Ig is the identity matrix of order 4g, identical to that of Pg. Equation (23) relates the product matrix Pα,βPβ,α to matrix Pg. Therefore, the eigenvalues of matrix Pg+1 can be expressed in terms of those of matrix Pg.

We next show how to obtain the eigenvalues of Pg+1 through the eigenvalues of Pg. According to equations (22) and (23), we can derive

Hence, if λi(g) is an eigenvalue of Pg associated with eigenvector ua, equation (24) indicates

Solving the above quadratic equation in the variable λi(g + 1) given by equation (25), one obtains the two roots:

and

Equations (26) and (27) relate λi(g + 1) to λi(g), with each eigenvalue λi(g) of Pg giving rise two different eigenvalues of Pg+1. As a matter of fact, all eigenvalues of the Pg+1 can be obtained by these two recursive relations. In Methods, we determine the multiplicity of each eigenvalue and show that all the eigenvalues can be found by equations (26) and (27).

Since there is a one-to-one relation between the eigenvalues of Pg and the fundamental matrix Kg, we thus have also found all the eigenvalues of Kg.

The largest eigenvalue of fundamental matrix and MFPT

In the above, we have determined all eigenvalues for the inverse Pg of the fundamental matrix Kg and thus all eigenvalues of Kg. Here we continue to estimate the greatest eigenvalue σmax(g) of the fundamental matrix Kg, which actually equals the reciprocal of the smallest eigenvalue for matrix Pg, denoted by λmin(g). Below we will show that in a large network the leading behavior of the MFPT 〈T〉g for trapping in and the reciprocal of λmin(g) is identical, that is, 〈T〉g ~ 1/λmin(g) = σmax(g).

We begin by providing some useful properties of eigenvalues for matrix Pg. Assume that Δg is the set of the 4g eigenvalues of matrix Pg, namely, . According to the above analysis, Δg can be categorized into two subsets and satisfying , where consists of all eigenvalues 1, while contains the rest eigenvalues. Thus,

These 2 × 4g−1 eigenvalues are labeled sequentially by , since they provide a natural increasing order of all eigenvalues for Pg, as will be shown.

The remaining 2 × 4g−1 eigenvalues in set are all determined by equations (26) and (27). Let be the 4g−1 eigenvalues of matrix Pg−1, arranged in an increasing order . Then, for each eigenvalue λi(g − 1) of Pg−1, equations (26) and (27) produce two eigenvalues of Pg, which are labeled as λi(g) and :

and

Plugging each eigenvalue of Pg−1 into equations (26) and (27) generates all eigenvalues in .

It is easy to see that λi(g) given by equation (29) monotonously increases with λi(g − 1) and belongs to interval (0, 1), while provided by equation (30) monotonously decreases with λi(g − 1) and lies in interval (1, 2). Thus, provide an increasing order of all eigenvalues for matrix Pg.

We hasten to estimate λmin(g) of matrix Pg. From the above arguments, the smallest eigenvalue λmin(g) must be the one generated from λmin(g − 1) through equation (29):

Using Taylor's formula, we have

Considering , equation (32) is solved to yield

Thus,

which, together with equation (15), means that and 〈T〉g have the same dominating term and thus identical leading scaling.

Discussion

Real-life weighted networks exhibit a rich and diverse reciprocity structure. In this paper, we have proposed a scale-free weighted directed network with asymmetric edge weights, which are controlled by a parameter characterizing the network reciprocity. We then studied random walks performed on the network with a trap fixed at the central hub node. Applying two different approaches, we have evaluated the MFPT to the trap. Moreover, based on the self-similar architecture of the network, we have found all the eigenvalues and their multiplicities of the fundamental matrix describing the random-walk process, the largest one of which has the same leading scaling as that of the MFPT. The obtained results indicate that the MFPT scales as a power-law function of the the system size, with the power exponent increasing with the weight parameter, revealing that the reciprocity has a significant impact on dynamical processes running on weighted networks. Finally, it should be mentioned that although we have only studied a particular network, our methods can also be applied to other self-similar networks, obtaining analogous results. Thus, our work deepens the understanding of random-walk dynamics in complex systems and opens a novel avenue to control random walks in a weighted network by changing its reciprocity.

Methods

Proof of equation (23)

In order to prove equation (23), we rewrite Pα,β and Pβ,α in the block form as

and

respectively. In equations (35) and (36), Eg = 4g is the number of edges in Fg; Ui (1 ≤ i ≤ Eg) is a 4g × 3 matrix describing the transition probability from the 4g non-trap nodes of Fg to the three nodes newly generated by the ith edge of Fg; similarly, Vi (1 ≤ i ≤ Eg) is a 3 × 4g matrix indicating the transition probability from the three new nodes created by the ith edge to those 4g old non-trap nodes belonging to Fg. Then,

which completes the proof of equation (23). Note that in equation (37), li and ri are the two endpoints of the ith edge of Fg; εi is a vector having only one nonzero element 1 at ith entry with other entries being zeros; ai and bi are two entries of Pg corresponding to edges (li, ri) and (ri, li), respectively.

Alternative proof of equation (23)

Equation (23) can also be proved using another technique. Assume that Rg = Pα,βPβ,α and . In order to prove , it suffices to show that the entries of Rg are equal to their counterparts of Qg. For matrix Qg, it is easy to see that its entries are: for i = j and otherwise. If Pg+1(i, j) denotes the (i, j) entry of matrix Pg+1, the entries of Rg(i, j) of matrix Rg can be evaluated by distinguishing two cases: i = j and i ≠ j.

For the case of i = j, the diagonal element of Rg is

where the relation is used. In equation (38), i ~ z indicates that two nodes i and z are adjacent in network Fg+1.

For the other case of i ≠ j, the non-diagonal element of Rg is

which, together with (38) proves equation (23).

Multiplicities of eigenvalues

By numerically computing the eigenvalues for the first several iterations, we can observe some important phenomena and properties about the structure of the eigenvalues. When g = 1, the eigenvalues of P1 are and , both of which have a multiplicity of 2. When g = 2, P2 have 16 eigenvalues: eigenvalue 1 with degeneracy 8 and 4 two-fold other eigenvalues generated by and . When g ≥ 3, all the eigenvalues Pg can be put into two classes. The first class includes eigenvalue 1 and those generated by 1, which display the following feature that each eigenvalue appearing at a given iteration gi will continue to appear at all subsequent generations greater than gi. The second class contains those eigenvalues generated by the two eigenvalues and of P1. Each eigenvalue in this class is two-fold and each eigenvalue of a given iteration gi does not appear at any of subsequent iterations larger than gi. For the two eigenvalue classes, each eigenvalue (other than 1) of current generation keeps the multiplicity of its father of the previous generation.

Using the above-observed properties of the eigenvalue structure, we can determine the multiplicities of all eigenvalues. Let Mg(λ) denote the multiplicity of eigenvalue λ of matrix Pg. We first determine the number of eigenvalue 1 for Pg. To this end, let r(X) denote the rank of matrix X. Then

For g = 1, M1(1) = 0; for g = 2, M2(1) = 8. For g ≥ 2, it is obvious that r(Pg+1 − Ig+1) = r(Pα,β) + r(Pβ,α), where r(Pα,β) and r(Pβ,α) can be determined in the following way.

We first show that Pβ,α is a full column rank matrix. Let

where Mi is the column vector of Pβ,α representing the ith column of Pβ,α. Let . Suppose that ϕ = 0. Then, we can prove that for an arbitrary ki, ki = 0 always holds. By construction, for any old node i ∈ α, there exists a new leaf node l ∈ β attached to i. Then, for , only Mi,l ≠ 0 but all Mx,l = 0 for x ≠ i. From ϕl = 0, we have ki = 0. Therefore, r(Pβ,α) = 4g. Analogously, we can verify that Pα,β is a full row rank matrix and r(Pα,β) = 4g.

Combining the above results, the multiplicity of eigenvalue 1 of Pg is

We proceed to compute the multiplicities of other eigenvalues generated by 1 that are in the first eigenvalue class. Since every eigenvalue at a given iteration keeps the multiplicity of its father at the preceding iteration, for matrix Pg, the multiplicity of each first-generation descendant of eigenvalue 1 is 2 × 4g−2, the multiplicity of each second-generation descendant of eigenvalue 1 is 2 × 4g−3 and the multiplicity of each (g − 2)nd generation descendant of eigenvalue 1 is 2 × 4. Moreover, we can derive that that the number of the ith (0 ≤ i ≤ g − 2) generation distinct descendants of eigenvalue is 2i, where 0th generation descendants refer to the 2 × 4g−1 eigenvalues 1 themselves. Finally, it is easy to verify that the number of all the eigenvalues in the second eigenvalue class is 4 × 2g−1. Hence, the total number of eigenvalues of matrix Pg is

indicating that all the eigenvalues of Pg are successfully found.