An evolving network model with information filtering and mixed attachment mechanisms

Journal Pre-proof An evolving network model with information filtering and mixed attachment mechanisms Xikun Huang, Ruqian Lu

PII: DOI: Reference:

S0378-4371(19)31910-7 https://doi.org/10.1016/j.physa.2019.123421 PHYSA 123421

To appear in:

Physica A

Received date : 14 March 2019 Revised date : 19 August 2019 Please cite this article as: X. Huang and R. Lu, An evolving network model with information filtering and mixed attachment mechanisms, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123421. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Journal Pre-proof

An evolving network model with information filtering and mixed attachment mechanisms Xikun Huanga,b,∗, Ruqian Lua a Academy

of Mathematics and Systems Science & Key Lab-MADIS�Chinese Academy of Sciences, Beijing 100190, China of the Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

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Abstract

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In this paper, we propose an evolving network model with information filtering and mixed attachment mechanisms. We theoretically analyze the in-degree distribution of networks generated by the proposed model in two special cases, and prove that the in-degree distribution is power-law if the new coming vertex does not filter information. Otherwise, the in-degree distribution becomes complex which has a transition between exponential and power-law scaling. Numerical simulations are consistent with analytical results. In addition, we analyze how model parameters influence the topology of the network by means of numerical simulations, and compare values of various measures of networks generated by the proposed model with that of networks generated by uniform attachment model, Barabási-Albert model and copying model. It shows that our model can generate networks with more diverse topological features. Keywords: Network generative model, Information filtering, Preferential attachment, Copying, In-degree distribution

1. Introduction

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Modeling complex networks have been in the forefront of network science research for almost two decades. Empirical studies on real-life networks such as social networks, biological networks, citation networks, and the World Wide Web have shown that many networks exhibit common topological features[1–6]. In particular, two important characteristics that many real networks share are small-world behavior [7] and power-law degree distribution [8]. A great deal of network models have been formulated to capture the properties of real networks [7– 18]. One of the earliest models is the classical Erdős-Rényi(ER) random graph model [9], where edges are distributed randomly among a fixed number of vertices. However, the degree distribution of the network generated by the ER model is Poisson distribution which is not consistent with that of many real networks. In 1999, Barabási and Albert [8] proposed a growing network model, also called BA model, which can generate networks with power-law degree distribution. The BA model has two important ingredients: growth and preferential attachment. Growth means that network continuously expands by the addition of new vertices with several edges attached to it, and preferential attachment means that new added vertex tends to connect to high-degree vertices rather than low-degree vertices [19]. Despite its great success in explaining power-law degree distribution of real networks, the BA model possesses several limitations. Indeed, the BA model is strictly topological as the node degrees are the only metrics that drive network evolution, and it does not consider the intrinsic properties of nodes [12]. Santiago and Benito [12, 20] presented a framework for the extension of the BA model to incorporate the influence of element properties. In their model, node properties are described by fixed states in an arbitrary space and ∗ Corresponding

author Email address: [email protected] (Xikun Huang)

Preprint submitted to Physica A

November 4, 2019

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the attachment probabilities are biased according to an affinity function, and they proved that the degree densities exhibit a richer scaling behavior compared with their homogeneous counterparts. There are many specific affinity functions which consider different factors, such as the aging effect of vertices [21], the local world effect of new added vertices [11, 15, 22–24]. Actually, there is an implicit assumption in the BA model that the new added vertex knows the degree values of all existing vertices, and this assumption does not hold in many large real networks [22, 23]. For example, in social networks, it is impossible for a person to know all other people’s number of friends when he or she makes friends in a new city. Indeed, it is more reasonable to assume that the vertices only own partial information of the entire network, or they filter information according to their particular “interests” when they choose targets. Several models considered this phenomenon [11, 15, 22–24]. Mossa et al. [22] proposed a model which incorporates information filtering, and they found that the in-degree distribution decays as a power law with an exponential truncation. Li and Chen [11] proposed a local-world evolving network model, where the new vertex will firstly select Mt vertices randomly from the existing network as its “local world”, and then it only connects to m vertices that are all selected from its local world with preferential attachment rule. Li et al. showed that the degree distribution of the network generated by their proposed model represents a transition between exponential and power law. The above-mentioned models successfully incorporate information filtering or local world effect, but they have ignored the fact that the new vertex may connect to vertices that are not in its local world even the new vertex does not own the information of those vertices. In a social network for instance, it is common that our friends will introduce their friends, who are strangers for us, to us. Similar phenomenon can also be found in citation networks where researchers may refer those papers that appeared in papers they have read. We can mimic this behavior by using copying mechanism [10]. Kleinberg et al. [10] proposed a model which integrates copying mechanism when they studied the World Wide Web, where a new vertex chooses an old vertex as its prototype vertex and copies some out-neighbors of this prototype vertex to be its own out-neighbors. In this paper, inspired by above thoughts, we propose an evolving network model with information filtering and mixed attachment mechanisms, and we call it IFMAM model. Information filtering means that the new vertex will firstly choose some vertices from the existing network based on node properties as its local world. And then the new vertex can connect to vertices chosen from its local world with preferential attachment rule, or it can copy behaviors of an existing vertex which is chosen randomly from its local world. As a result, on the one hand, the model incorporates the influence of node intrinsic properties when the new vertex chooses its local world. On the other hand, it is possible for the new vertex to connect to any vertex of the existing network, even those that are not in its local word. Several models such as uniform attachment model [19] and the BA model [8] are limiting cases of the proposed model. If the new vertex selects all existing vertices as its local world, we prove that the in-degree distribution is a power-law distribution, and the power-law exponent has range (3/2, ∞). If the new vertex selects a fixed number of existing vertices as its local world, we also derive the asymptotic solution of the in-degree distribution, which is complex and has a transition between exponential and power-law scaling. Additionally, we numerically explore how model parameters influence the topology of the network, and calculate various measures of networks generated by the IFMAM model, such as clustering coefficient and average path length. It shows that the proposed model can generate networks with small-world behavior. We also compare values of these measures with that of networks generated by uniform attachment model, the BA model and the copying model, and it shows that the IFMAM model can generate networks with more diverse structural properties. The rest of the paper is organized as follows: The IFMAM model is introduced and analyzed in Section 2. Properties of complex networks generated by the IFMAM model are analyzed in Section 3. Finally, we conclude in Section 4. 2. The Model

In this section, we firstly introduce the IFMAM model which considers information filtering and combines preferential attachment and copying mechanisms. And then we analyze the in-degree distribution of networks generated by the proposed model. Before going into further detail, we give notations used in this paper in Table 1. 2

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of

Meaning Graph at time t Vertex set at time t Number of vertex at time t In-degree of vertex v at time t Degree of vertex v The exponent of power-law distribution p(x) ∼ x−α Number of edges added at each time step Copy factor Vertex set of the local world of the vertex vt Size of Lt , i.e., Mt = |Lt | Size of Lt when Mt is a constant Initial attractiveness of each vertex The probability that the new added vertex is connected to vertex v at time t + 1. The proportion of vertices whose in-degree are k when the size of graph is Nt

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Notation Gt Vt Nt Ivt Dv α m γ Lt Mt M a Πt (v) pk (Nt )

Table 1: Notations used in this paper

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2.1. Model Description We consider directed graph model and assume that all vertices have same out-degree. If there is an edge originating from vertex u and ends at vertex v, then we call v is an out-neighbor of u, so each vertex has exactly m out-neighbors. The model is described by a graph process {Gt }t≥0 . Start with a directed graph G0 consisting of m0 vertices where every vertex connects to other m < m0 vertices randomly. For t ≥ 1, the graph Gt is constructed from Gt−1 by following steps: 1. Add a new vertex vt and m new edges originating from vt , into Gt−1 . The other ends of new edges are selected by following rules described in step 2 and 3. 2. Select Mt old vertices randomly from Vt−1 as the local world of vt , using Lt to represent this local world. Randomly select a vertex ut from Lt as the prototype vertex of vt . 3. Consider ut ’s out-neighbors one by one. With probability γ, let vt connect to this out-neighbor. With probability 1 − γ, let vt connect to a vertex w in Lt selected by preferential attachment rule; i.e., I t−1 +a the probability that w is selected in this case is ∑ w (I t−1 +a) , which depends on its in-degree, and

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z∈Lt

z

a represents initial attractiveness of each vertex and remains constant during the evolution of the network.

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It is worth mentioning the relationship between our model and the heterogeneous preferential attachment(HPA) models that proposed by Santiago and Benito [12]. Santiago and Benito [12, 20] proposed a general class of heterogeneous preferential attachment models that the attachment probabilities are biased according to an affinity function which is related to vertices properties. Indeed, the local world effect can be regarded as one kind of HPA models in the sense that new added vertices choose their local worlds based on vertices properties, and Mt determines the extent of this kind of influence. Besides, our model incorporates copying mechanism which makes it possible for new added vertices to connect to vertices outside of their local worlds, and γ determines the extent of this kind of influence. Several previous models are limiting cases of undirected versions of the proposed model. The proposed model becomes the BA model [8] when γ = 0, a = m, Mt = Nt . When γ = 0, a = m, Mt = m, our model is same as model A in [19], where the preferential attachment rule in the BA model is replaced by uniform attachment, and we call it uniform attachment model. And when Mt = m for all t, the preferential attachment selection in Lt will be equivalent to random selection from Vt to some extent, because vertices in Lt are chosen randomly from Vt and the out-degree of the new vertex is m. In this case, the IFMAM model is similar to copying model. 2.2. Analysis In this subsection, we use the method in Newman’s book [5] to derive the in-degree distribution of the network generated by the IFMAM model. Let Nt denote the size of Gt and Ivt denote the in-degree of 3

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vertex v after time t. Then at time t + 1, an old vertex v may be chosen as the new coming vertex vt+1 ’s out-neighbor by two possible ways. One is that vertex v is an out-neighbor of vt+1 ’s prototype vertex and is γI t chosen with probability mNvt . The other is that v is in the local world of vt+1 and is chosen with preferential attachment rule. As a result, the probability that the new vertex vt+1 is connected to v is γIvt Mt It + a ∑ v t + (1 − γ) mNt Nt u∈Lt (Iu + a)

(1)

of

Πt (v) =

Below we discuss the in-degree distribution of the network in two cases. One is that the new vertex selects all existing vertices as its local world, i.e., Mt = Nt . The other is that the sizes of local worlds of each vertex are same, i.e., Mt = M , where M is a constant.

Πt (v) =

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2.2.1. Case A: Mt = Nt If Mt = Nt , equation (1) becomes

γIvt Ivt + a + (1 − γ) mNt Nt (m + a)

(2)

When the size of graph is Nt , for all vertices whose in-degree are k, the expectation of gotten new edges is

γk k+a 1−γ k+a ] = [γk + (1 − γ)m ]pk (Nt ) + mNt Nt m + a m+a

Pr e-

Nt × pk (Nt ) × m × [

(3)

where pk (Nt ) is the proportion of vertices whose in-degree are k when the size of graph is Nt . And the master equation for in-degree distribution is (Nt + 1)pk (Nt + 1) = Nt pk (Nt ) + [γ(k − 1) + (1 − γ)m

k−1+a k+a ]pk−1 (Nt ) − [γk + (1 − γ)m ]pk (Nt ) (4) m+a m+a

This equation is true except for k = 0. When k = 0 the second term of the right part of the equation disappears and the new added vertex’s in-degree is zero. So the equation becomes

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(Nt + 1)p0 (Nt + 1) = Nt p0 (Nt ) + 1 −

(1 − γ)ma p0 (Nt ) m+a

(5)

Let t → ∞ and pk ≡ pk (∞) we have

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pk = [γ(k − 1) + (1 − γ)m

k+a k−1+a ]pk−1 − [γk + (1 − γ)m ]pk m+a m+a

p0 = 1 −

(1 − γ)ma p0 m+a

(6)

(7)

where k ≥ 1. Rearrange above equations we have

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pk =

(m + a)γ(k − 1) + (1 − γ)m(k − 1 + a) pk−1 (m + a)(1 + γk) + (1 − γ)m(k + a)

(8)

m+a m + a + (1 − γ)ma

(9)

p0 =

The asymptotic solution of equation (8) and (9) is

B(k + 1, m+a+(1−γ)ma ) 1 m+a m+aγ pk = m(1−γ)(a−1)−(m+a)γ k m + aγ B(k, 1 + ) m+aγ

4

(10)

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where B(x, y) is Beta function. The derivation details are given in Appendix A. Using the property of Beta function B(x, y) ≈ x−y Γ(y) f or x → ∞ and x ≫ y When k is large enough, the in-degree distribution behaves like

∼k

−α

(k + 1)−[ k −[1+

m+a+(1−γ)ma ] m+aγ

m(1−γ)(a−1)−(m+a)γ m+aγ

where

m+a m+γ

(12)

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α=1+

(11)

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pk ∼ k

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So we get that when new vertices select all old vertices as their local worlds, the in-degree distribution m+a has a power-law form with exponent α = 1+ m+γ , which has range (3/2, ∞). The copying factor only affects the power law exponent, and the larger the copying factor is, the more likely huge hub vertices emerge. Note also that we have mentioned in Section 2.1 that the undirected version of the proposed model is same as the BA model when γ = 0, a = m and Mt = Nt . In this case, our derivation is consistent with that of the BA model as α = 1 + m+m m+0 = 3. We also carry out numerical simulations to verify our derivation. Figure 1 shows the relationship between the power-law exponent and parameters of the IFMAM model. We use the method proposed by Clauset et al. [25] to compute the power-law exponent. It can be seen that numerical results are coincident with theoretical results. 2.7

numerical simulation theoretical calculation

2.3

2.5

2.25

2.4

,

,

2.2

2.15

2.3

2.1

2.2

2.05

2.1

3

numerical simulation theoretical calculation

2.6

numerical simulation theoretical calculation

2.9 2.8 2.7 2.6

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2.5 2.4 2.3 2.2 2.1

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0.2

0.3

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0.5

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(a) power-law exponent versus copy fac- (b) power-law exponent versus out- (c) power-law exponent versus initial attor degree tractiveness

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Figure 1: Comparison between the analytical results (lines) derived from Eq.(12) and the simulation results (circles) for different parameter values. The quantity shown is the exponent α of the power law in-degree distribution of the resulting network. (a) N = 10000000, m = 3 and a = 1. (b) N = 10000000, γ = 0.2 and a = 1. (c) N = 10000000, γ = 0.1 and m = 3.

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2.2.2. Case B: Mt is a fixed constant Now we consider the situation that the sizes of local worlds of each vertex are same, i.e., Mt = M , and set a = 1 for simplicity. When the size of graph is Nt , for all vertices whose in-degree are k, the expectation of gotten new edges is k+1 ]pk (Nt ) t u∈Lt (Iu + 1)

Nt × pk (Nt ) × m × Πt (v) = [γk + (1 − γ)mM ∑

Then we can get the master equation. For k ≥ 1

5

(13)

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(Nt + 1)pk (Nt + 1) = Nt pk (Nt ) k ]pk−1 (Nt ) t u∈Lt (Iu + 1) k+1 − [γk + (1 − γ)mM ∑ ]pk (Nt ) t u∈Lt (Iu + 1)

And for k = 0

(14)

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+ [γ(k − 1) + (1 − γ)mM ∑

1 ]p0 (Nt ) (15) t u∈Lt (Iu + 1) ∑ It is worthy to note that we cannot get the exact value of the sum term u∈Lt (Iut + 1). By using similar tricks in [15], we can approximate the sum ∑ term. If we know that Lt contains a vertex whose in-degree is k, then the sum term can be approximated as u∈Lt (Iut + 1) = k + (M − 1)m + M . Substitute approximations into equation (14) and (15) , for k ≥ 1 we have

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(Nt + 1)p0 (Nt + 1) = Nt p0 (Nt ) + 1 − [(1 − γ)mM ∑

(Nt + 1)pk (Nt + 1) = Nt pk (Nt )

k ]pk−1 (Nt ) k − 1 + (M − 1)m + M k+1 ]pk (Nt ) − [γk + (1 − γ)mM k + (M − 1)m + M

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+ [γ(k − 1) + (1 − γ)mM

And for k = 0 we have

(Nt + 1)p0 (Nt + 1) = Nt p0 (Nt ) + 1 − [(1 − γ)mM Let t → ∞ and pk ≡ pk (∞), then for k ≥ 1

(17)

k k+1 ]pk−1 − [γk + (1 − γ)mM ]pk (18) k − 1 + (M − 1)m + M k + (M − 1)m + M

And for k = 0

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pk = [γ(k − 1) + (1 − γ)mM

1 ]p0 (Nt ) (M − 1)m + M

(16)

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p0 = 1 − [(1 − γ)mM

Rearrange above equations we get pk =

γ(k − 1) + (1 − γ)mM k−1+(Mk−1)m+M 1 + γk + (1 − γ)mM k+(Mk+1 −1)m+M p0 =

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1 ]p0 (M − 1)m + M

(19)

pk−1

1 + (M − 1)m + M (1 − γ)mM

(20)

(21)

The asymptotic solution of equation (20) and (21) is complex, and we can only write it using Beta function as equation (22), and the derivation details are given in Appendix B. The asymptotic solution is pk =

√

B(k, M m+M γ−mγ+ 2γ B(k,

▽+1

√ M m+M γ−mγ+ △ 2γ

+ 1)B(k, M m+M γ−mγ− 2γ + 1)B(k,

√ √

▽+1

M m+M γ−mγ− △ 2γ

+ 1)

+ 1)

1 + (M − 1)m + M k + (M − 1)m + M × γk(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) (M − 1)m + M 6

(22)

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where △ = M 2 m2 + 2M 2 mγ + M 2 γ 2 − 2M m2 γ + 2M mγ 2 − 4M mγ + m2 γ 2 and ▽ = M 2 m2 + 2M 2 mγ + M 2 γ 2 − 2M m2 γ + 2M mγ 2 − 8M mγ + 2M m − 2M γ + m2 γ 2 + 2mγ + 1. 1

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It is not easy to analyze the above solution completely. But when k ≫ Mγm we have pk ∼ k −(1+ γ ) , and this is the same result as copying model [5]. In addition, we plot the in-degree distribution of networks generated by the IFMAM model with different M and γ values. And we compare the results obtained by theoretical calculation with numerical simulation in Figure 2. It shows that the in-degree distribution has a transition between exponential and power-law scaling. When M and γ are both small, the in-degree distribution is similar to exponential distribution. The reason for this is that when the local world of the new added vertex is very small, it is likely that many old vertices with large in-degree are not included in this local world. Besides, small copying factor makes the new vertex tends to select targets from its local world. As a result, huge hub vertices can hardly emerge. As M gets larger, the new added vertex selects more vertices as its local world, and old vertices with large in-degree are more likely selected as the other end of new links based on preferential attachment and copying mechanisms.

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3. Property of networks generated by the IFMAM Model

In this section, we numerically calculate several measures of networks generated by the IFMAM model. And the measures include average path length, clustering coefficient, number of triangles, effective diameter, spectral radius, and assortativity. In addition, we compare values of these measures with that of networks generated by uniform attachment model, BA model and copying model. Every edge is treated as undirected when calculating these measures using Python language package NetworkX [26]. In addition, we assume the sizes of local worlds of each vertex are same, and analyze how the parameters γ and M influence the topology of networks. 3.1. Average path length

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Average path length [27] is the average number of steps along the shortest paths for all possible pairs of network vertices. Most real networks have small average path length [7]. In Figure 3, we plot average path length of networks generated by the IFMAM model with different parameter values. It shows that the average path length increases only with the logarithm of network size. In addition, the larger the copying factor and size of the local world, the smaller the average path length. One possible reason for this is that large γ and M lead to the emergence of huge hub vertices, which play the role of bridges that connect many vertices with low in-degrees. We also compare the average path length of networks generated by our model with that of networks generated by copying model [10], BA model [8] and uniform attachment model [19]. Figure 6 shows that our model can generate networks which have almost the same average path length as copying model, BA model and uniform attachment model. On the other hand, the IFMAM model can generate networks with smaller average path length than others.

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3.2. Clustering coefficient

The local clustering coefficient Cv for the vertex v is defined as Cv =

|Γv | 1/2Dv (Dv − 1)

where |Γv | is the number of actual edges existing in the network connecting v’s neighbors, and Dv is the degree of vertex v [28, 29]. Figure 4 shows the results of degree-averaged clustering coefficient distribution for different values of copying factor γ and local world size M . It can be seen that there is an inverse relationship 7

10 0

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10 0

numerical simulation theoretical calculation

numerical simulation theoretical calculation

10 -4

10 -4

P(k)

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10 -2

P(k)

10 -2

10 -6

10 -8

10 -8

10 -10 10 0

10 1

10 2

k

(a) 10

10

Pr e-

10 -6

10 -10 10 0

10 3

10 1

10 2

10 3

k

(b)

10 0

0

numerical simulation theoretical calculation

numerical simulation theoretical calculation

-2

10 -2

10 -4

P(k)

10 -6

10 -8

10 1

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10 -10

10 -12 10 0

10 -6

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P(k)

10 -4

10 2

10 3

10 4

10 -8

10 -10 10 0

10 5

10 1

10 2

10 3

10 4

10 5

k

k

(c)

(d)

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Figure 2: Comparison between the analytical results (dashed lines) derived from Eq. (22) and the simulation results (circles) with different parameter values. The quantity shown is the in-degree distribution of the resulting network. And the values of input parameters are: (a) N = 10000000, M = 3, m = 3, γ = 0.001 (b) N = 10000000, M = 3, m = 3, γ = 0.1, (c) N = 10000000, M = 10, m = 3, γ = 0.5, (d) N = 10000000, M = 100, m = 3, γ = 0.5. The agreement is good.

8

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.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

4.5

3.5

3

2.5

3.5

3

2.5

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2 10 2

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1.5 10 2

M = 10 M = 100 M = 1000

4

Average path length

Average path length

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Network size

10 4

Network size

(a)

(b)

Figure 3: The average path length versus network size with the given values of input parameters: (a) M = 10, m = 3, a = 1 and γ = 0.1, 0.3, 0.5, 0.7 and 0.9. (b) M = 10, 100 and 1000, m = 3, a = 1 and γ = 0.3.

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3.3. Other properties

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between the clustering coefficient and the degrees of vertices, and this coincides with the results observed on real networks [30]. In addition, the degree-averaged clustering coefficient increases when the copying factor γ or local world size M gets large, irrespective of degrees. The reason is that when vertices frequently copy the behaviors of their prototype vertices, their neighbors are more likely linked together since they have common neighbors. Besides, larger local world size reduces the possibility of random selection. The global clustering coefficient of a network is defined as the average local clustering coefficient of all vertices in the network [7]. In Fig 5, we plot global clustering coefficient of networks generated by the IFMAM model with different parameter settings. It shows that the global clustering coefficient is positively correlated to M and γ when the network size is fixed, which can be attributed to the positive relationship between local clustering coefficient and those parameters. As the network size increases, the global clustering coefficient decreases firstly and then reaches a stable value gradually. We also plot the clustering coefficient of networks generated by our model, copying model, BA model and uniform attachment model in Fig6. It shows that the IFMAM model can generate networks which has similar clustering coefficient as the three models. In addition, our model can generate networks with larger clustering coefficient.

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The above results show that our model can indeed generate networks with small-world property, i.e., short average path length and high clustering coefficient [7]. Besides above mentioned properties, we also calculate other measures including number of triangles [16], effective diameter [4], spectral radius [31], and assortativity [32], which are plotted in Figure 7 and Figure 8. And here we just explain the results briefly. Figure 7 and Figure 8 show that our model can generate diverse networks by tuning parameter γ and M . In addition, we compare values of these measures with that of networks generated by copying model, BA model and uniform attachment model in Figure9. It shows that the IFMAM model can generate networks which has similar triangle counts, effective diameter, and spectral radius as other three models. On the other hand, the IFMAM model can generate networks with larger triangle count and spectral radius, and smaller effective diameter. But it can only generate networks with disassortative mixing. 4. Conclusions

In this paper we have proposed an evolving network model, which we call the IFMAM model, that considers information filtering and mixed mechanisms. We have showed that several previous models are 9

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10 0

10 -2

10 -3

10 -4 10 0

10 1

10 2

10 3

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10 -1

M=10 M=100 M=1000

10 -2

p ro

.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

Degree-averaged clustering coefficient

Degree-averaged clustering coefficient

10 0

10 -3

10 -4 10 0

10 4

Degree

10 1

10 2

10 3

10 4

Degree

(b)

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(a)

Figure 4: Distribution of degree-averaged clustering coefficient with the given values of input parameters: (a) M = 100, m = 3, a = 1 and γ = 0.1, 0.3, 0.5, 0.7 and 0.9. (b) M = 10, 100 and 1000, m = 3, a = 1 and γ = 0.3.

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6000

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M = 10 M = 100 M = 1000

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Clustering coefficient

0.8

Clustering coefficient

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.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

0.3 0.25 0.2 0.15 0.1 0.05 0

10000

0

Network size

2000

4000

6000

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10000

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Network size

(a)

(b)

Figure 5: The clustering coefficient versus network size with the given values of input parameters: (a) M = 10, m = 3, a = 1 and γ = 0.1, 0.3, 0.5, 0.7 and 0.9. (b) M = 10, 100 and 1000, m = 3, a = 1 and γ = 0.3.

10

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M=3 M = 10 M = 100 copying BA UA

5

M=3 M = 10 M = 100 Copying BA UA

0.45 0.4 0.35

Clustering coefficient

Average path length

4.5

0.5

4

3.5

3

0.3 0.25 0.2

of

5.5

0.15 0.1

2.5 0.05

10 3

0

10 4

p ro

2 10 2

0

Network size

2000

4000

6000

8000

10000

Network size

(a)

(b)

Pr e-

Figure 6: Average path length and clustering coefficient of different size of networks generated by the IFMAM model, copying model, BA model and uniform attachment model(UA). We consider three parameter settings for the IFMAM model: M = 3, 10 and 100 when m = 3, a = 1, and γ = 0.3. We set copy factor γ = 0.3 for copying model, and out-degree m = 3 for uniform attachment model and BA model. (a) Average path length. (b) Clustering coefficient. 18000 16000 14000 12000 10000 8000 6000 4000

M = 10 M = 100 M = 1000

5000

Number of triangles

Number of triangles

6000

.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

4000

3000

2000

1000

2000 0 4000

6000

Network size

(a)

Effective diameter

5 4.5 4 3.5 3 2.5 2 10 2

Jo

5.5

10 3

0

10000

urn

6

8000

0

2000

4000

6000

8000

10000

Network size

(b) 5.5

.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

M = 10 M = 100 M = 1000

5

Effective diameter

2000

al

0

4.5

4

3.5

3

2.5 10 2

10 4

Network size

10 3

10 4

Network size

(c)

(d)

Figure 7: Various topological features versus network size with the given values of input parameters. (a)(c): M = 10, m = 3, a = 1 and γ = 0.1, 0.3, 0.5, 0.7 and 0.9. (b)(d): M = 10, 100 and 1000, m = 3, a = 1 and γ = 0.3.

11

.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

90 80

50

M = 10 M = 100 M = 1000

40

Spectral radius

Spectral radius

70

p ro

60

100

60 50 40 30 20

30

20

Pr e-

10

10

0

0 0

2000

4000

6000

8000

10000

0

(a) 0

-0.3

al 0

2000

4000

6000

10000

8000

(b) M = 10 M = 100 M = 1000

-0.2

-0.25 -0.3 -0.35 -0.4

urn

-0.8

8000

-0.15

-0.4

-0.7

6000

-0.1

Assortativity index

-0.2

-0.6

4000

-0.05

.=0.1 .=0.3 .=0.5 .=0.7 .=0.9

-0.1

-0.5

2000

Network size

Network size

Assortativity index

of

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-0.45

10000

0

Network size

2000

4000

6000

8000

10000

Network size

(c)

(d)

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Figure 8: Various topological features versus network size with the given values of input parameters. (a)(c): M = 10, m = 3, a = 1 and γ = 0.1, 0.3, 0.5, 0.7 and 0.9. (b)(d): M = 10, 100 and 1000, m = 3, a = 1 and γ = 0.3.

12

6.5

2500

M=3 M = 10 M = 100 Copying BA Random

5.5

1000

p ro

1500

M=3 M = 10 M = 100 Copying BA UA

6

Effective diameter

2000

Triangle counts

of

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5

4.5

4

3.5 500

0 0

2000

4000

6000

Network size

(a) 40

8000

2.5 10 2

10000

10 3

10 4

Network size

(b)

0.3

M=3 M = 10 M = 100 Copying BA UA

35

30

0.2 0.1

Assortativity

0

25

20

al

Spectral radius

Pr e-

3

15

5 0

2000

urn

10

4000

6000

8000

-0.1 -0.2

M=3 M = 10 M = 100 Copying BA UA

-0.3 -0.4 -0.5

10000

0

Network size

2000

4000

6000

8000

10000

Network size

(c)

(d)

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Figure 9: Various measures of different size of networks generated by the IFMAM model, copying model, BA model and uniform attachment model(UA). We consider three parameter settings for the IFMAM model: M = 3, 10 and 100 when m = 3, a = 1, and γ = 0.3. We set copy factor γ = 0.3 for copying model, and out-degree m = 3 for uniform attachment model and BA model. (a) Number of triangles. (b) Effective diameter. (c) Spectral radius. (d) Assortativity.

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p ro

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limiting cases of the proposed model. If the new vertex select all existing vertices as its local world, we prove that the in-degree distribution is a power-law distribution, and the power-law exponent has range (3/2, ∞). If the new vertex select a fixed number of existing vertices as its local world, we also derive the asymptotic solution of the in-degree distribution, which is a little complex. Theoretical derivations were in good consistent with numerical simulations. In order to investigate the properties of networks generated by the IFMAM model, we have calculated several network metrics including average path length, clustering coefficient, triangle count, effective diameter, etc. At the same time, we have tried to explain how model parameters influence the topology of the network. By comparing the IFMAM model with previously mentioned models, we have shown that the IFMAM model are more powerful in the sense that it could generate networks with more diverge topological features. Even though the proposed model are more powerful than previous models, it also has limitations. One of the limitations is that we just consider two cases of the size of new vertex’s local world. In the future, we can consider models that allowing different vertices may have different sizes of local worlds based on their capabilities. Furthermore, the future research can also focus on generating networks that similar to given real-life networks, and we plan to solve this problem by parameter estimation in the future. 5. Acknowledgments

Pr e-

This work has been supported by National Key Research and Development Program of China under grant 2016YFB1000902, NSFC Project (Grant Nos. 61232015, 61472412, 61621003 , 61872352).

Appendices

A. Derivation of the in-degree distribution when Mt = Nt

In this section, we give the derivation of the asymptotic solution of the in-degree distribution when Mt = Nt . From equation (8) and (9), we have

m(1−γ)(a−1)−(m+a)γ m+aγ k + m+a+(1−γ)ma m+aγ

k+

urn =

p0 =

Jo

And

al

(m + a)γ(k − 1) + m(1 − γ)(k − 1 + a) pk−1 (m + a)(1 + γk) + (1 − γ)m(k + a) (m + aγ)k + a(1 − γ)(a − 1) − (m + a)γ = pk−1 (m + aγ)k + m + a + (1 − γ)ma

pk =

=

pk−1

m+a m + a + (1 − γ)ma m+a m+aγ m+a+(1−γ)ma m+aγ

14

(A.1)

(A.2)

Journal Pre-proof

For general k , by iteration we have m(1−γ)(a−1)−(m+a)γ ] · · · [1 m+aγ [k + m+a+(1−γ)ma ] · · · [1 m+aγ

m+a = m + aγ Γ(k 1 m+a = k m + aγ =

1 m+a k m + aγ

+ +

Γ( m+a+(1−γ)ma ) m+aγ + 1 + m+a+(1−γ)ma ) m+aγ

m(1−γ)(a−1)−(m+a)γ m+a ] m+aγ m+aγ m+a+(1−γ)ma m+a+(1−γ)ma ] m+aγ m+aγ m(1−γ)(a−1)−(m+a)γ ) m+aγ m(1−γ)(a−1)−(m+a)γ ) m+aγ

Γ(k + 1 + Γ(1 +

)Γ(k + 1) Γ( m+a+(1−γ)ma m+aγ m+a+(1−γ)ma Γ(k + 1 + ) m+aγ

Γ(k + 1 + Γ(k)Γ(1 +

of

[k +

m(1−γ)(a−1)−(m+a)γ ) m+aγ m(1−γ)(a−1)−(m+a)γ ) m+aγ

B(k + 1, m+a+(1−γ)ma ) m+aγ m(1−γ)(a−1)−(m+a)γ B(k, 1 + ) m+aγ

When x → ∞ and x ≫ y, Beta function has the property

(A.3)

p ro

pk =

B(x, y) ≈ x−y Γ(y)

Pr e-

So when k is large enough, the in-degree distribution behaves like pk ∼ k −1 ∼k where

(k + 1)−[

k −[1+

m+a+(1−γ)ma ] m+aγ

m(1−γ)(a−1)−(m+a)γ m+aγ

]

(A.4)

−α

α=1+

m+a m+γ

(A.5)

B. Derivation of the in-degree distribution when Mt = M

pk =

al

In this section we try to find the asymptotic solution of equation (20) and (21). γ(k − 1)(k − 1 + (M − 1)m + M ) + (1 − γ)mM k k + (M − 1)m + M pk−1 (1 + γk)(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) k − 1 + (M − 1)m + M

(B.1)

pk = □ where

urn

We try to find the relationship between (1 + γk)(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) and γk(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) since we want to get the format γ(k − 1)(k − 1 + (M − 1)m + M ) + (1 − γ)mM k k + (M − 1)m + M pk−1 γk(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) k − 1 + (M − 1)m + M γk(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) (1 + γk)(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) γk 2 + (M m + M γ − mγ)k + M m − M mγ = 2 γk + (M m + M γ − mγ + 1)k + M − m + 2M m − M mγ

(B.2)

Jo

□=

(B.3)

Factor the expression we have

γk 2 + (M m + M γ − mγ)q + M m − M mγ √ √ M m + M γ − mγ − △ M m + M γ − mγ + △ )(k + ) = γ(k + 2γ 2γ 15

(B.4)

Journal Pre-proof

where △ = M 2 m2 + 2M 2 mγ + M 2 γ 2 − 2M m2 γ + 2M mγ 2 − 4M mγ + m2 γ 2 And (B.5)

of

γk 2 + (M m + M γ − mγ + 1)k + M − m + 2M m − M mγ √ √ M m + M γ − mγ + ▽ + 1 M m + M γ − mγ − ▽ + 1 = γ(k + )(k + ) 2γ 2γ where

▽ = M 2 m2 + 2M 2 mγ + M 2 γ 2 − 2M m2 γ + 2M mγ 2 − 8M mγ + 2M m − 2M γ + m2 γ 2 + 2mγ + 1

p ro

By iteration, we have

(1 − γ)mM k + (M − 1)m + M p0 γk(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) (M − 1)m + M 1 + (M − 1)m + M k + (M − 1)m + M =■ γk(k + (M − 1)m + M ) + (1 − γ)mM (k + 1) (M − 1)m + M

pk = ■

■=

=

Pr e-

where

√ √ M m+M γ−mγ+ △ △ )(i + M m+M γ−mγ− ) 2γ 2γ √ √ M m+M γ−mγ+ ▽+1 ▽+1 )(i + M m+M γ−mγ− ) i=1 (i + 2γ 2γ √ √ M m+M γ−mγ+ ▽+1 M m+M γ−mγ− ▽+1 B(k, + 1)B(k, + 1) 2γ 2γ √ √ M m+M γ−mγ+ △ M m+M γ−mγ− △ B(k, + 1)B(k, + 1) 2γ 2γ k ∏

We use

(i +

(B.6)

(B.7)

Γ(x + n) = (x + n − 1)(x + n − 2) · · · x Γ(x)

al

and B(x, y) =

References

urn

to derive Eq. (B.6).

Γ(x)Γ(y) Γ(x + y)

Jo

[1] R. Albert, H. Jeong, A.-L. Barabási, Internet: Diameter of the world-wide web, nature 401 (1999) 130. [2] S. Dorogovtsev, J. Mendes, Evolution of networks: from biological nets to the Internet and WWW, Oxford University Press, 2003. [3] S. Li, C. M. Armstrong, N. Bertin, H. Ge, S. Milstein, M. Boxem, P.-O. Vidalain, J.-D. J. Han, A. Chesneau, T. Hao, et al., A map of the interactome network of the metazoan c. elegans, Science 303 (2004) 540–543. [4] J. Leskovec, J. Kleinberg, C. Faloutsos, Graph evolution: Densification and shrinking diameters, ACM Transactions on Knowledge Discovery from Data (TKDD) 1 (2007) 2. [5] M. Newman, Networks: An Introduction, Oxford University Press, 2010. [6] A.-L. Barabási, et al., Network science, Cambridge university press, 2016. [7] D. J. Watts, S. H. Strogatz, Collective dynamics of ‘small-world’networks, nature 393 (1998) 440. [8] A.-L. Barabási, R. Albert, Emergence of Scaling in Random Networks, Science 286 (1999) 509. doi:10.1126/science.286.5439.509. [9] P. Erdős, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci 5 (1960) 17–60. [10] J. M. Kleinberg, R. Kumar, P. Raghavan, S. Rajagopalan, A. S. Tomkins, The web as a graph: measurements, models, and methods, in: International Computing and Combinatorics Conference, Springer, 1999, pp. 1–17. [11] X. Li, G. Chen, A local-world evolving network model, Physica A: Statistical Mechanics and its Applications 328 (2003) 274–286.

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[12] A. Santiago, R. Benito, An extended formalism for preferential attachment in heterogeneous complex networks, EPL (Europhysics Letters) 82 (2008) 58004. [13] A. Bonato, N. Hadi, P. Horn, P. Prałat, C. Wang, Models of online social networks, Internet Mathematics 6 (2009) 285–313. doi:10.1080/15427951.2009.10390642. [14] Z.-G. Shao, T. Chen, B.-q. Ai, Growing networks with temporal effect and mixed attachment mechanisms, Physica A: Statistical Mechanics and its Applications 413 (2014) 147–152. [15] T. Carletti, F. Gargiulo, R. Lambiotte, Preferential attachment with partial information, The European Physical Journal B 88 (2015) 18. [16] P. K. Pandey, B. Adhikari, Context dependent preferential attachment model for complex networks, Physica A: Statistical Mechanics and its Applications 436 (2015) 499–508. [17] P. K. Pandey, B. Adhikari, A parametric model approach for structural reconstruction of scale-free networks, IEEE Transactions on Knowledge and Data Engineering 29 (2017) 2072–2085. [18] S. Chattopadhyay, C. Murthy, Generation of power-law networks by employing various attachment schemes: Structural properties emulating real world networks, Information Sciences 397 (2017) 219–242. [19] A.-L. Barabási, R. Albert, H. Jeong, Mean-field theory for scale-free random networks, Physica A: Statistical Mechanics and its Applications 272 (1999) 173–187. [20] A. Santiago, R. Benito, Kernel nonlinearity in heterogeneous evolving networks, The European Physical Journal B 76 (2010) 557–564. [21] S. N. Dorogovtsev, J. F. F. Mendes, Evolution of networks with aging of sites, Physical Review E 62 (2000) 1842. [22] S. Mossa, M. Barthélémy, H. Eugene Stanley, L. A. Nunes Amaral, Truncation of power law behavior in “scale-free” network models due to information filtering, Phys. Rev. Lett. 88 (2002) 138701. URL: https://link.aps.org/doi/10.1103/PhysRevLett.88.138701. doi:10.1103/PhysRevLett.88.138701. [23] J. Gomez-Gardenes, Y. Moreno, Local versus global knowledge in the barabási-albert scale-free network model, Physical Review E 69 (2004) 037103. [24] H. Štefančić, V. Zlatić, Preferential attachment with information filtering—node degree probability distribution properties, Physica A: Statistical Mechanics and its Applications 350 (2005) 657–670. [25] A. Clauset, C. R. Shalizi, M. E. Newman, Power-law distributions in empirical data, SIAM review 51 (2009) 661–703. [26] A. Hagberg, P. Swart, D. S Chult, Exploring network structure, dynamics, and function using NetworkX, Technical Report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2008. [27] L. d. F. Costa, F. A. Rodrigues, G. Travieso, P. R. Villas Boas, Characterization of complex networks: A survey of measurements, Advances in physics 56 (2007) 167–242. [28] P. Holme, B. J. Kim, Growing scale-free networks with tunable clustering, Physical review E 65 (2002) 026107. [29] A. Santiago, R. M. Benito, Improved clustering through heterogeneity in preferential attachment networks, International Journal of Bifurcation and Chaos 19 (2009) 1029–1036. [30] A.-L. Barabási, Z. Dezső, E. Ravasz, S.-H. Yook, Z. Oltvai, Scale-free and hierarchical structures in complex networks, in: AIP Conference Proceedings, volume 661, AIP, 2003, pp. 1–16. [31] P. Van Mieghem, J. Omic, R. Kooij, Virus spread in networks, IEEE/ACM Transactions on Networking (TON) 17 (2009) 1–14. [32] M. E. Newman, Mixing patterns in networks, Physical Review E 67 (2003) 026126.

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HIGHLIGHTS

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 We propose an evolving network model which combines information filtering, preferential attachment, and copying mechanisms.  The in-degree distribution of the network generated by the proposed model is explored theoretically and numerically.  We analyze how model parameters influence the topology of the network by numerical simulations.  The proposed model can generate complex networks with diverse structural properties.