Physica A387(2008)1381–1386
/locate/physa
Complex network-bad time ries analysis
Yue Yang a,Huijie Yang b,∗
a School of Economics,Nankai University,Tianjin300071,China
b School of Management,University of Shanghai for Science and Technology,Shanghai Academy of Systems Science,Shanghai200093,China
Received6August2007;received in revid form21September2007
Available online28October2007
Abstract
关于教育的书Recent works show that complex network theory may be a powerful tool in time ries analysis.We propo in this paper a reliable procedure for constructing complex networks from the correlation matrix of a time ries.An original stock time ries, the corresponding return ries and its amplitude ries are considered.The degree distribution of the original ries can be well fitted with a power law,while that of the return ries can be wellfitted with a Gaussian function.The degree distribution of the amplitude ries contains two asymmetric Gaussian branches.Reconstruction of networks from time ries is a common problem in diver rearch.The propod strategy may be a reasonable solution to this problem.
c 2007Elvier B.V.All rights rerved.
PACS:05.45.Tp;89.75.2k
Keywords:Complex networks;Time ries analysis;Network construction
Time ries analysis[1]attracts special attention due to its practical and theoretical importance in physics, physiology,biology and society.Theoretical physics is one of the basic origins of the ideas and the methods. Applications of physical theories have led to fruitful achievements in thisfield.To cite an example,the complexity theory has led to an avalanche offindings of non-trivial features in time ries,including the long-range correlations, the scale invariance and so on[2].
One of the most important advances in statistical physics in recent years is the complex network theory[3]. Complex systems in diverfields can be described with networks.The nodes and the connections reprent the elements and their relations,respectively.Network theory provides us with a new viewpoint and an effective tool for understanding a complex system from the relations between the elements in a global way.In recent literature[4],from the perspective of complex networks the authors review their systematic works[5,6]on autocorrelations infinancial data.Network theory may be a powerful tool for revealing information embedded in time ries[7].
先就业还是先择业But how to construct a network from a time ries is still an esntial problem to be solved.In reference[4]the minimum spanning tree is extracted from the autocorrelation matrix as the network.The authors of[8]construct networks with a critical value,namely,the adjacent matrices are obtained by reassigning autocorrelation values less and larger than the critical value to0and1,respec
tively.With the increa of this critical value,the statistical properties
∗Corresponding author.
E-mail address:(H.Yang).
0378-4371/$-e front matter c 2007Elvier B.V.All rights rerved.
doi:10.1016/j.physa.2007.10.055
翼龙骑士
1382Y.Yang,H.Yang/Physica A387(2008)1381–1386
of the nearest neighbor level spacings s for the spectra of the constructed networks are found to evolve from Wigner to Poisson distributions(the two extreme conditionsβ=1andβ=0for the distribution function,∼sβ·e−sβ+1)[9–12]. The network at the transition point is regarded as physically meaningful.Rho et al.[13]consider the characteristics of the degree distribution functions for the constructed networks.At a special critical value the degree distribution will tend to obey a power law.The network at this transition point is ud to detect non-trivial characteristics embedded in the autocorrelation matrix.
This kind of characteristic transition-bad strategy may lead to different and even artificial results and conclusions. Obviously,we do not have enough evidences to support the idea that the real world systems should exist at the transition states.Robustness may be a much more reasonable benchmark for the constructed networks.That is, a constructed network capturing correctly the behaviors of a ries should behave in a critical-value-independent manner.The impacts of the critical ,the artificial characteristics,should be negligible.Hence,the transition from trivial to non-trivial characteristics cannot guarantee meaningfulness of the networks constructed with the critical value at the transition point.We should show further the existence of an interval of the critical value,in which the characteristics of the constructed networks remain unchanged.In this paper,this kind of robustness is propod as a new strategy for constructing networks from time ries.
We consider a time ries denoted as{S1,S2,...,S N}.Connecting the start and the end of this ries,we can obtain all the possible gments with length L,which read
{T m=(S m,S m+1,...,S m+L−1)|m=0,1,2,...N}.(1) For each pair of gments,T i and T j,the correlation coefficient can be written as
C i j=
L
k=1
[T i(k)− T i ]·[T j(k)− T j ]
L
k=1
[T i(k)− T i ]2·
L
k=1
[T j(k)− T j ]2
,(2)
where, T i = L
k=1
T i(k)/L.The elements C i j are restricted to the domain−1≤C i j≤1,where C i j=1,0and−1
correspond to perfect correlations,no correlations and perfect anti-correlations,respectively.
励磁线圈Regarding each gment as a point in L-dimensional space,C i j describes the state of connection between the two points T i and T j.Choosing a critical value r c,the correlation matrix C can be converted into a new matrix D,the rules of which read
D i j=
1,(|C i j|≥r c);
0,(|C i j|<r c).
(3)
All the points and their connections can form a network,and the topological structure of this network can be described with the matrix D.The conditions D i j=1and D i j=0correspond to connection and disconnection,respectively.
One of the commonly ud measurements of network structures is the degree distribution[3].The degree of a node is the number of the nodes directly connected with it.The degree distribution function(DDF)describes the heterogeneous properties and conquently can shed light on the mechanisms of the network structure’s evolution.For a completely random time ries,each pair among the points is connected randomly with a certain probability and the corresponding network should behave like a random ,the DDF obeys a binomial distribution.Correlations in time ries will induce constraints on the connections.The constraints lead to the different characteristics as compared with random networks.
快速截屏
The value of thefilter parameter r c determines the characteristics of the resulting network.If it is extremely small, the pairs with weak correlations are also connected.The physically meaningful correlations in time ries will be submerged by the nois.Increasing the value of r c,the number of connections among the points becomes smaller and smaller.More and more of the nois arefiltered out.However,if the critical point becomes extremely large,some of the physically meaningful connections are alsofiltered out and the number of connections decreas rapidly,which may induce strong statisticalfluctuations due to a smallfinite number of connections.Conquently,we can expect an interval of the critical value,r c∈(γmin,γmax),in which the network can capture the characteristics of the time
Y.Yang,H.Yang/Physica A387(2008)1381–13861383
Fig.1.Evolution of the network constructed with the original ries I.With a large value of the critical r c,the degree distribution obeys a power law.With the decrea of r c,more and more edges are added,which may induce statisticalfluctuations in the degree distribution.Andfinally,the power law is submerged in statistical nois.
ries.That is,a proper critical value r c can be found just by simulating a special dynamical process of the complex ,decreasing the number of connections while keeping the number of points unchanged.If there exists a wide interval of r c in which the DDFs of the resulting networks have almost the same characteristics,we can conclude that the dynamical process is governed by the same law.
The other adjustable parameter is the length of a gment,denoted as L.Short length will induce overestimated correlations.Increasing the length L can depress thefinite-length-induced statisticalfluctuations effectively.It should be long enough to give a reliable result.
Hence,we should adjust simultaneously the parameters r c and L,by which we canfind a considerable region where the characteristics of the DDFs for the constructed networks are independent of(r c,L).This is called the stability region in this paper.In this stability region the networks can reveal physically meaningful information embedded in the time ries.
It should be emphasized that the(r c,L)-independent characteristic in the stability region is a kind of scale invariance[14].The changes of(r c,L)can induce changes of the interval that the degree distributes in,denoted asσ.
The DDF can be reprented as1
σφF
k
σφ
.The(r c,L)-independent characteristic of the DDF is that the parameter
φremains the same in the stability region.For a power-law DDF,φis just the scaling exponent.For a Gaussian DDF,φ=0.5.
The price ries of a stock in a duration of30years are considered in the prent paper[15].This time ries is parated into four non-overlapping gments(2500length each).The original ries,the corresponding return ries and the amplitude ries of the returns are all investigated,respectively.Denoting an original ries as {s1,s1,s2,...,s N},the corresponding return ries can be constructed as{ln(s2)−ln(s1),ln(s3)−ln(s2),...,ln(s N)−ln(s N−1)}.The amplitude ries of the returns are{|ln(s2)−ln(s1)|,|ln(s3)−ln(s2)|,...,|ln(s N)−ln(s N−1)|}.For the four ts of ries we obtain similar results.In this paper,we prent only the results for thefirst t denoted as ries I.
We consider the four original riesfirstly.As a typical result,Fig.1prents the DDF for the original ries I at different critical values of r c.At r c=0.9the degree distribution function obeys a power law over a considerable scale.At r c=0.8this power law keeps its shape,but thefluctuations become distinctively large.With the decrea
1384Y.Yang,H.Yang/Physica A387(2008)1381–1386
Fig.2.(Color online)Evolution of the networks constructed with the original ries I.The parameter r c
is t in the interval[0.8,1]so that we can u the power-law exponent,α,as the index forfinding the stability region.The values of the power-law exponentαat different values of(r c,L) are shown.With the increa of the size L,thefluctuations due to thefinite length of the gments can be depresd effectively.And when L is large enough,αtends to an unchanged value.A wide stability region can be found,in which we haveα∈[0.95,1.1].The power-law behavior in this region should be physically meaningful.
of r c thefluctuations become larger and larger.For r c≤0.7this characteristic is submerged in largefluctuations. The parameters(r c,L)are adjusted,as shown in Fig.2.The parameter r c is t in the interval[0.8,1]so that we can u the power-law exponent,α,as the index forfinding the stability region.With the increa of the size L,the fluctuations due to thefinite length of the gments can be depresd effectively.And when L is large enough,αtends to an unchanged value.We canfind a wide stability region of(r c,L),in which we haveα∈[0.95,1.1].That is,φ=1.02±0.07.Hence,we canfind a physically meaningful power-law behavior.The pattern of correlation between the gments can be described with a scale-free network.
The right-skewed power law tells us that there exist some hubs in the constructed networks,that is,there are some nodes with special large degrees.The gments corresponding to the hubs and th
e cloly related gments (corresponding to the hub’s neighbors)will occur along the time ries with a significant probability.Hence,the hub gments can be regarded as the reprentation of the time ries.
Fig.3shows the typical DDFs for the return ries.In the wide region of r c∈[0.50,0.70]and L=11–30,the degree distribution can be wellfitted with a Gaussian function,that is,the considered region of the parameters(r c,L) is covered by the stability region and the indexφ=0.5.Each pair of the gments(nodes)is connected just with a certain probability and the time ries should behave randomly.This is consistent with the argument in literature that the autocorrelation function of the returns decays exponentially with a time scale of a minute[16].That is to say,the time-dependent characteristics in the original ries are destroyed by the operation of converting the original ries into the return ries.This operation is often ud tofilter out the non-stationary effects in time ries.The results show that this may not be a good solution to the non-stationary problem in time ries analysis.
Fig.4shows the typical DDFs for the amplitude ries.The parameters are adjusted in the region r c∈[0.50,0.70], L=11–30.When the gment length L is small,the DDF can be wellfitted with the Gaussian function and the left and right branches with respect to the maximum are symmetric.With i
练气化神
ncrea of L,the two branches tend to become significantly asymmetric.The left and right branches can befitted well with Gaussian functions with different widths, denoted asσL andσR,respectively.The considered region of the parameters(r c,L)is in the stability region and the indexφ=0.5.
The asymmetric branches tell us that the gments can be catalogued into two class according to their occurrence in the left or the right branch.The nodes in the left branch have small degree and may be considered as homogeneous nois in the time ries.The right branch decreas much more slowly than the left one,which may lead to some hubs occurring with significant probability in the amplitude ries.极端的意思
Y.Yang,H.Yang/Physica A387(2008)1381–13861385
黄精作用与功效
Fig.3.Degree distribution of the networks constructed with the return ries.As a typical example,the result for the return ries I is prented. The parameters are adjusted in the region r c∈[0.50,0.70],L=11–30.Only the results for r c=0.55,L=11,15,19,23,30are prented. The degree distribution obeys a perfect Gaussian function.The considered region of the parameters(r c,L)is in the stability region and the index φ=0.5.The return ries should behave randomly,which is consistent with the results in the literature.Converting the original ries to a return
ries mayfilter out the time-dependent characteristics.
Fig.4.Degree distribution of the networks constructed with the amplitude ries.As a typical example,the result for the amplitude ries I is prented.The parameters are adjusted in the region r c∈[0.50,0.70],L=11–30.Only the results for r c=0.55,L=15,19,23,30are prented.We can parate the DDF into two branches with respect to the maximum.When the gment length L is small,the DDF can be well fitted with the Gaussian function and the left and right branches with respect to the maximum are symmetric.With the increa of L,the two branches tend to being significantly asymmetric.The left and right branches can befitted well with Gaussian functions with different widths, denoted asσL andσR,respectively.The considered region of the parameters(r c,L)is in the stability region and the indexφ=0.5.
In summary,network-bad time ries analysis attracts special attention,but how to construct a reasonable network is still an esntial problem.In this paper,we propod the scale invariance of the DDF as the criterion.It is a new kind of scale invariance describing a lf-similarity characteristic in the global correlations in the time ries.
