A Synthetical Weights' Dynamic Mechanism for Weighted Networks
aa r X i v : . [ phy s i c s . s o c - ph ] S e p A Synthetical Weights’ Dynamic Mechanismfor Weighted Networks
Lujun Fang a , b Zhongzhi Zhang a , b Shuigeng Zhou a , b , ∗ Jihong Guan c a Department of Computer Science and Engineering, Fudan University,Shanghai 200433, China b Shanghai Key Lab of Intelligent Information Processing, Fudan University,Shanghai 200433, China c Department of Computer Science and Technology, Tongji University,4800 Cao’an Road, Shanghai 201804, China
Abstract
We propose a synthetical weights’ dynamic mechanism for weighted networks whichtakes into account the influences of strengths of nodes, weights of links and incom-ing new vertices. Strength/Weight preferential strategies are used in these weights’dynamic mechanisms, which depict the evolving strategies of many real-world net-works. We give insight analysis to the synthetical weights’ dynamic mechanism andstudy how individual weights’ dynamic strategies interact and cooperate with eachother in the networks’ evolving process. Power-law distributions of strength, degreeand weight, nontrivial strength-degree correlation, clustering coefficients and assor-tativeness are found in the model with tunable parameters representing each model.Several homogenous functionalities of these independent weights’ dynamic strategyare generalized and their synergy are studied.
Key words:
Complex networks, Weighted networks, Networks
PACS: ∗ Corresponding author.
Email addresses: [email protected] (Lujun Fang), [email protected] (Zhongzhi Zhang), [email protected] (ShuigengZhou), [email protected] (Jihong Guan).
Preprint submitted to Elsevier
Introduction
Complex networks [1,2,3,4,5] depict a great many real-world networks like thescientific collaboration networks (SCN) [6,7,8,9], World Wide Web (WWW) [10],world-wide airport networks (WAN) [11,12] and so on. Simple binary net-works [1,2,3] are used to depict the topological aspects of these real-worldnetworks. Degree distribution and degree related clustering coefficients can beanalyzed from the model. Typically, Barab´asi and Albert proposed a linearpreferential attachment model on which most other binary models are basedon (BA model[13]). However, real-world networks often contains far more in-formation than that binary networks can express, since relations in the net-works are not necessarily binary. Therefore links with weights are introducedto emphasize the importance of heterogenous relations between nodes in net-works. Barrat, Barth´elemy and Vespignani first build a model for weightednetworks based on preferential attachment mechanism (BBV model [14,15]).Since then various models have been derived to mimic diverse real-world net-works [4,5,16,17,18,19,20,21,22], all of which emphasize a different kind ofweights’ dynamic mechanism.The flourishing research on various weights’ dynamics mechanisms roots in thethe heterogenous behaviors of real-life networks. Current research has alreadycovered most part of typical behaviors of weights’ dynamics. However, theinteraction and operations of these scattered weights’ dynamics are poorlystudied and their underlying homogeneity are still not know. In our paper, wegeneralize weights’ dynamics with monotonous weights’ growth and focus oninteractions and cooperations among them.With a close scrutiny into prevailing weights’ dynamic models depicting real-world networks we find that there are three main sources of weights’ incrementdynamics: the variation of traffic caused by introducing of new nodes [4,5,16,17,18,19],the increment of links’ weights based on links’ weights themselves [23], and theincrement of weights based on strength of two ends of the link [20,21,22]. Mostweights’ increment dynamics can be grouped into these three sources. Manyworks have done to empirically validate the classification, among them New-man gives the most comprehensive and convincing experiments [24]. In thispaper we will give numerical and analytical study to the synthetical weights’dynamic model which comprises the three above-mentioned mechanisms, andreveal the scale-free characteristics and nontrivial clustering coefficients andassortativeness of the model.Our paper is organized as follows. In Sec. 2 we define basic definitions andterms to represent the weighted network and gives a brief review of relatedworks done. In Sec. 3 we depict our model and give clear definition of threeweights’ dynamic mechanisms. In Sec. 4 we analytically calculate the mathe-2atical expression for probability distributions of strength and weight, degree-strength correlations and related attributes. In Sec. 5 we preform simulationsto mimic the proposed mechanisms and analyze the experiment results indetail. In Sec. 6 we give a conclusion to the paper.
Weighted networks can be represented by a adjacent matrix W where w ij defines the weight of link between vertices i and j . w ij = 0 indicates thatthere is no link between vertices i and j . Therefore topological and weightedinformation can both be revealed from W . Matrix W is symmetrical therefore w ij = w ji . Degree k i defines the number of vertices vertex i is linked with, andstrength s i defines the total weights of links that ends in the particular vertex i . s i can be written as s i = P j w ij , and k i can be written as k i = P j sgn ( w ij ),where sgn () is the signum function. P ( s ), P ( k ), P ( w ) defines the probabilitydistribution of strength, degree and weight. Previous studies have revealedthat in many weighted networks, P ( s ), as well as P ( k ) and P ( w ), displaysa power-law distribution as P ( s ) = s − γ s , P ( k ) = k − γ k and P ( w ) = w − γ w .There is also a power-law correlation between s and k that s = k α . Clusteringcoefficients and assortativeness of weighted network is also studied and thedetails will be discussed later. The model proposed in this paper starts from an initial configuration of N vertices fully connected by links with weight w = 1 ( N -clique). At each timestep, the network evolves under two coupled mechanisms: topological growthand weights’ dynamics. Weights’ dynamics are discussed in detail in this paper,where all three sources of weights’ dynamics are taken into account. At each time step, a new vertex n is introduced into the network and con-nected to p existing vertices i . Vertices are chosen according to the strengthpreferential probability Π n → i = s i P j s j , (1)and the weight of this new link is set to w = 1.3 .2 Weights’ Dynamics There are three sources of weights’ dynamics: the local increment of weightstriggered by the introduction of the new vertex, the self-increment of weightsbased on the weight of each link, and the mutual selection dynamics focusingon creation and reinforcement of links between existing vertices based on theirstrengths. These three weights’ dynamics mechanisms interact and cooperateduring the evolution of the network. There are several suggestive independentworks for these three sources weights’ dynamics: Barrat, Barth´elemy, Vespig-nani first suggest the local rearrangement model considering the impact of in-coming vertices. Dorogovtsev, Mendes’s work Wen-Xu Wang’s works suggestthe mutual selection model which well depict the second source. ********’swork initializes the idea of weights’ self-increment although the idea is notcomprehensively studied yet. There are lots of works done that empiricallyproving the validity of the three sources. Newman in his work by empiricallystudying the scientific collaboration network suggests that two scientists wouldhave better chance to enforce their collaboration if they already have a lot ofworks together, and scientists are more likely to develop new collaborative re-lationships if they already have relatively large numbers of collaborators [24].Collaborative relationships of scientists and their collaborators are analogousto weights and degree in a network. Degree can be generalized to strengthif we take into account the amount of collaborations between each pairs ofcollaborators in stead of a binary expression.The introduction of new vertex brings variation in traffic across the network.For simplicity, we restrict the variation to the neighborhood of vertex i whichhas just been chosen to link with the new vertex. An overall increment of δ is introduction at each time step. The increment is distributed among theneighborhood of Γ( i ) according to weight preferential mechanism: w ij → w ij + δ w ij s i . (2)The strengths of i and all j ∈ Γ( i ) are also increased as a result of the incrementof weights in the neighborhood of i . Considering the probability of vertices i been chosen, the increment of w ij can be rewritten as∆ w ij = p s i P k s k δ w ij s i + p s j P k s k δ w ij s j = 2 pδ w ij P k s k . (3)At each time step, n existing links are chosen to increase according to the4eight preferential probability: w ij → w ij + n w ij P k,l w kl (4)Each chosen link is increased by w = 1. The links with larger weights alwayshave more chance to reinforcement.At each time step, each existing vertex i selects m vertices according to thestrength preferential mechanism:Π i → j = m s j P k s k − s i . (5)There would be a alteration in links between i and j if and only if i and j have mutually selected each other. The probability that the linking conditionbetween i and j changes can be defined to be:Π i,j = m s j P k s k − s i m s i P k s k − s j = m s i s j ( P k s k − s i )( P k s k − s j ) ≈ m s i s j ( P k s k ) . (6)If there is not a link between i and j , a new link with assigned weight w = 1will be added. If there is already a link between i and j , the link will beincreased by w = 1.These three weights’ dynamics mechanisms interact and cooperate during thethe process of network development. Synthesize all the these three mecha-nisms, the increment of weights can be represented to be w ij → w ij + 2 pδ w ij P k s k + n w ij P k,l w kl + m s i s j ( P k s k ) . (7)Noticing the fact that P k,l w kl = P k s k , we can rewrite the above equationas w ij → w ij + (2 pδ + 2 n ) w ij P k s k + m s i s j ( P k s k ) . (8) Using the continuous approximation, we can assume that s , k , w , t are allcontinuous. Therefore we get dw ij dt = (2 pδ + 2 n ) w ij P k s k + m s i s j ( P k s k ) . (9)5here are two sources contributing the increment of strength s i , one is theweights’ dynamic and the other is linking with the new added node. Thereforethe increment of s i can be written as ds i dt = X j dw ij dt + p s i P k s k = ( m + 2 pδ + 2 n + p ) s i P k s k . (10)The sum of strength of all nodes P k s k at time t can be calculated as X k s k ( t ) = t X s i = Z t X k ds k dt + pt = ( m + 2 pδ + 2 n + 2 p ) t , (11)and using this equation we can rewrite Eq. (10) as ds i dt = m + 2 pδ + 2 n + pm + 2 pδ + 2 n + 2 p s i t . (12)With the initial condition s i ( t = i ) = 1, we can integrate the above equationto obtain s i ( t ) = (cid:18) ti (cid:19) m pδ +2 n + pm pδ +2 n +2 p (13)From the equation we can see that three parameters m , δ and n cooperativelyand interactively govern the growing speed of strength s i . Is is really amazingto find out that all three sources of weights’ dynamics influence the growingspeed of strength in similar ways. The simulation of evolution of s i is given inFig. 1. We see how m , δ , n , p contribute to the evolution of s i independently byfixing three other parameters. We also show how these four parameters interactby varying them the same time as indicated by the above equation. We see s i display a power-law distribution as t evolves, and variable m contribute largeralteration in s i with relatively small amount of increment.The knowledge of the time evolution of the various quantities allows us tocompute their statistical properties. The incoming time t i of each vertex i isuniformly distributed in [0 , t ] and the strength probability distribution can bewritten as P ( s, t ) = 1 t + N Z t δ ( s − s i ( t )) dt i , (14)where δ ( x ) is the Dirac delta function. Using Eq. (13) we can get in the infinitesize limit t → ∞ the distribution P ( s ) ∼ s − γ s with γ s = 2 + pm + 2 pδ + 2 n + p . (15)6
100 200 300 400 500-2000200400600800100012001400160018002000 p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 s t Fig. 1. Evolution of degree s i with time for vertex i = 1, the time span is from 1to 5000 and the result is average of 10 individual experiments. Various values for p , m , n , δ are chosen to display the interplay of different parameters. p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 k t Fig. 2. Evolution of degree k i with time for vertex i = 1, the time span is from 1to 5000 and the result is average of 10 individual experiments. Various values for p , m , n , δ are chosen to display the interplay of different parameters. The degree probability distribution P ( k ) ∼ k − γ k can be obtained by combining s ∼ k β with Eq. (13). From the equation of the conservation of probability Z ∞ P ( k ) dk = Z ∞ P ( s ) ds (16)7
100 200 300 400 500020406080100 p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 w t Fig. 3. Evolution of degree w i with time for vertex i = 1 and j = 2, the time spanis from 1 to 5000 and the result is average of 10 individual experiments. Variousvalues for p , m , n , δ are chosen to display the interplay of different parameters. p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 s k Fig. 4. Nontrivial correlation between degree k and strength s , the size of the networkis 5000 and the result is average of 10 individual experiments. Various values for p , m , n , δ are chosen to display the interplay of different parameters. we can get P ( k ) ∼ P ( s ) dsdk ∼ s − γ s βk β − ∼ βk − [ β ( γ s − , . (17)8 p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 P ( s ) s Fig. 5. Power-law probability distribution of strength s , the size of the network is5000 and the result is average of 10 individual experiments. Various values for p , m , n , δ are chosen to display the interplay of different parameters.
10 100 1000 100000.010.11 p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 P ( k ) k Fig. 6. Power-law probability distribution of degree k , the size of the network is5000 and the result is average of 10 individual experiments. Various values for p , m , n , δ are chosen to display the interplay of different parameters. Therefore we get γ k = β ( γ s −
1) + 1 in P ( k ) ∼ k − γ k . The simulations of k i and P ( k )are given in Fig. 2 and Fig.Fig. 6 with resemble the figures for s i and k i .The power-law correlation between s and k is reveal by Fig. 4, where we fix m = p = 1 and tune δ and n as we need. .9
10 1000.010.11 p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 P ( w ) w Fig. 7. Power-law probability distribution of weight w , the size of the network is5000 and the result is average of 10 individual experiments. Various values for p , m , n , δ are chosen to display the interplay of different parameters. The evolution and distribution of weight can be calculated similarly as we dealwith strength. Combine Eq. (9), Eq. (11) ,Eq. (13), and define a = m + 2 pδ +2 n + 2 p , we get dw ij dt = 2 pδ + 2 na w ij t + m a ij ! − a t − a (18)we can integrate the above equation and get w ij ∼ t m pδ +2 nm pδ +2 n +2 p (19)for large t . Therefore P ( w ) can be represented as P ( w ) = w − γ w with γ w =2 + pm +2 δ +2 n . The simulations of w ij and P ( w ) are given representatives inFig. 3 and Fig. 7. Clustering coefficients depict connectivity among neighborhood of given ver-tices. The local clustering coefficient c i for specific vertex i is defined to be c i = C ki P j 40 80 120 160 200 240 2800.00.10.20.30.40.50.60.70.80.9 p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 C ( k ) k p=1,m=1,n=1, =0.0 p=2,m=3,n=5, =1.0 p=2,m=5,n=5, =1.0 p=3,m=7,n=10, =2.0 p=4,m=9,n=12, =4.0 k nn ( k ) k Fig. 8. (a) Clustering coefficients for different degree k . (b) Average nearest-neighbordegree for different degree k . The size of the network is 5000 and the result is averageof 10 individual experiments. Various values for p , m , n , δ are chosen to display theinterplay of different parameters. of all vertices is denoted as C , and the average of c i for vertices with degree k is denoted as C ( k ). Average degree of nearest neighbor knn, i for vertex i isalso studied, as well as knn ( k ) for the average of knn, i of vertices with degreeof k . knn reveals the assortativeness of a network.We perform numerical experiment the simulate the growth of network andanalyze the clustering coefficient and assortativeness under the synthetical11eights’ dynamic mechanism. In Fig. 8(a) we show how clustering coefficientsvary with tunable variables. The scale-free attributes C measures the overalldensity of triangles in the network. We can see that C is consistent with thenetwork size N while varying positively when those tunable variables change.The evolution of knn is show in Fig. 8(b) which suggest the tunable assorta-tiveness of the network. In summary, we propose a new model for weighted networks which synthesizesthree sources of weights’ dynamic: local weights’ rearrangement raised by in-troduction of new vertex; self-increment of weights according to weights’ pref-erential strategy; weights’ creation and reinforcement proportional to strengthsof both ends of nodes. Three sources independently contribute to the evolutionand in the mean time also cooperatively interact. The homogenous behaviorsthat these weights’ dynamics display suggest there may be some common un-derlying mechanisms that are not yet well understood. This model would bea good start for a synthetical and general understanding of weights’ dynamicand hopefully our work would be helpful for the future study. Acknowledgements This research was supported by the National Natural Science Foundation ofChina under Grant Nos. 60496327, 60573183, and 90612007. 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