inverse scattering problems , originally introduced for potential scattering in quantum physics ( see for instance ref. and the references therein ) , have found their applications in a wide variety of scientific and technological fields .an important limitation to the application of inverse scattering methods to real data is related to the inherently unstable behavior of the algorithms involved .we present in this paper a novel method to solve the one - dimensional inverse scattering problem , and moreover a discussion about the stability of the obtained solutions is given .the inverse problem for the one - dimensional schrdinger equation can be solved by using the marchenko equation .burridge showed that the inverse problem of the one - dimensional wave equation can be transformed into an quantum inverse scattering representation .this result made it possible to solve electro - magnetic , optical and acoustic inverse problems by using the marchenko equation .the marchenko equation maps a reflection time series , ( hereafter to be referred to as the `` data - set '' ) , onto an integral kernel from which the quantum mechanical potential function can be recovered .the data - set is the fourier transform of the reflection coefficient .sabatier has discovered a method to solve this integral kernel in the fourier domain for reflection coefficients that are a rational function of the wave number .  in ref. stability estimatesare made for the solutions derived by sabatier .a disadvantage of the method presented in ref. is that the method is highly dimensional and therefore not easily implementable for realistic data - sets . in this paperwe re - open the discussion with respect to the solvability and stability of one - dimensional marchenko - equation on the basis of novel results which have recently become available in the field of the integrability of nonlinear evolution equations .it is shown in these publications that large classes of solutions of integrable nonlinear evolution equations can be obtained by solving the fourier expansion coefficients of its solution by using a recursion relationship .since it is well known that large classes of nonlinear evolution equations are integrable by inverse scattering transformations , it is worth investigating whether the machinery developed in ref. can also be applied in solving quantum inverse scattering problems .we show in this paper that it is possible to transform the gelfand - levitan - marchenko equation into a linear recursion relationship which determines the fourier expansion coefficients of the kernel .this result made it also possible to investigate the stability of the solutions by using lyapunov exponents which can be computed from the data - uncertainties .this implies that we have a simple tool which requires only information about the data and data - uncertainties , to investigate whether the marchenko equation is stable . secondly ,if we might conclude that the inverse scattering problem is unstable , we provide machinery to extract the largest sub - set from the scattering data which leads to a stable solution .this paper has the following structure : in section 2 we focus on the linearization method which leads to the relationship between the expansion coefficients of the data and the expansion coefficients of the kernel .we firstly investigate the principle for a data - set having one fourier component only , but the method is generalized to data - sets having a arbitrary ( but finite ) number of fourier components . in section 3 , we investigate the stability of the obtained solutions by using lyapunov exponents . a simple numerical example is given .the paper is concluded with a discussion .in this paper , solutions of the quantum inverse scattering problem are discussed .the quantum inverse scattering problem is the inverse problem of the schrdinger equation which is given by : for reasons of simplicity the following restrictions on the scattering solutions are imposed .it is assumed that , and secondly , only incoming waves from the left are taken into account .the solutions of the schrdinger equation ( [ eq : schrod.2 ] ) satisfy : where is the transmission coefficient , and is the reflection coefficient from the left . in the following, the data - set is defined by the function .the relation between the data - set and the spectral reflection coefficient of the schrdinger equation is given by : the inverse problem for the schrdinger equation can be solved using the marchenko equation : the potential is recovered from the integral kernel by : in this paper , we firstly discus a method to obtain solutions of the marchenko - equation .the method is based upon results developed in ref. . as a starting point , we assume that the data can be expanded in a fourier series : by considering a data - set which can be expressed in the form as presented in eq.([datfour ] ) , we have implicitly assumed that the reflection coefficient is a rational function of the wave - number .this is not a limitation for the application to real data , since a measured data - set can be approximated sufficiently by considering a series of the form ( [ datfour ] ) for which . as a first step, we consider the most simple case in which the data - set has only one fourier component : in this special case , we propose a kernel having the following structure : if the expressions ( [ series_dat ] ) and ( [ series_mod ] ) are substituted in eq.([eq : mar.2 ] ) , we obtain the following result : if we assume that is a complex number with positive imaginary part , the integral in eq.([mar_1 ] ) can be replaced by the primitive of an exponential function : we can solve eq.([mar_2 ] ) by comparing the exponential functions of equal powers on both sides .this is done by firstly defining a number , and hence by comparing all the exponential functions for which respectively .if we use the fact that the coefficients and are determined by the data - spectrum , the kernel can be computed by collecting all the coefficients . from the -dependence on the right - hand side of eq.([mar_2 ] ) , we can immediately conclude that all the coefficients vanish for which .we consider firstly the case that : this leads to the result that .if we put , it can easily be verified by substitution that both the coefficients and vanish . in general ,only coefficients for which are unequal to zero and follow from the following recursion relationship : by using eq.([rec_1 ] ) , we can compute the kernel as a function of and : the potential function can be recovered from the kernel by using the relationship ( [ eq : v.2 ] ) .if we substitute and into eq.([kern_2 ] ) , we obtain the following expression for : the solution of the kernel ( [ k_xx2 ] ) is in agreement with solutions obtained by using sabatiers approach .the advantage of the method presented in this paper is that we solve the inverse scattering problem by deriving a linear recursion relationship between the fourier expansion coefficients of the data and the expansion coefficients of the kernel .this implies that we have discovered a method which is easy to implement from a computational point of view , and moreover , the stability of the obtained solutions can be investigated by using lyapunov exponents .  in order to generalize this method to the data - set ( [ series_dat ] ), it is also illustrating to discus a data - set consisting of two fourier components : in this special case we propose a kernel having the following structure : if we insert eq.([data_3 ] ) and eq.([kern_3 ] ) in the marchenko equation , we obtain the following result : k_{1 } + s k_{2 } ) } e^{i(p+q+1)k_{1}x } e^{i(r+s)k_{2}x } e^{i k_{1 } y } \right . \\ & + \left . \ ! \ ! \ ! \ ! \sum_{p , q , r , s=0}^{\infty }  \ ! \ !  \frac { a_{2 } b(p , q , r , s ) } { i ( [ s+1 ] k_{2 } + q k_{1 } ) } e^{i(p+q)k_{1}x } e^{i(r+s+1)k_{2}x } e^{i k_{2 } y } \right ] \end{split } \label{march_3}\ ] ] if we use the fact that the coefficients and are determined by the data , we can compute the non - zero coefficients .this is done , by firstly defining and comparing the exponential functions for which respectively .if we consider the case that , we obtain the following result : from eq.([step_3 ] ) , we obtain that and that . in general ,it follows from eq.([march_3 ] ) that all the expansion coefficients can be computed by solving the following linear iteration relationship : {1 } + s k_{2 } ) }  \\b(p+q,0,r+s+1,1 ) & =  a_{2 } \frac{b(p , q , r , s ) } { i([s+1]k_{2 } + q k_{1 } ) } \end{split } \label{it_ser}\ ] ] it can be witnessed from eq.([it_ser ] ) that the number of expansion coefficients double after each iteration . as an example , we compute the coefficients arising from the first three iterations : it can be witnessed from eq.([bla ] ) , that some of the coefficients occur more than once .since these coefficients all belong to the same exponential basis function , we can simply add them together .similarly as for the case that the data - spectrum has only one fourier component , we have a iteration relationship to solve the coefficients .if we collect all the coefficients , we find that the kernel is given by the following relationship : this result can be made general by considering the full data - set as presented in eq.([datfour ] ) . in this casewe propose a kernel having the following structure : if we substitute eq.([kern_gen ] ) and eq.([datfour ] ) into the marchenko equation we obtain the following expression : k_{1 } + \hat{n}_{2 } k_{2 }  + \cdots \hat{n}_{n } k_{n } ) } e^ { i ( n_{1 } + \hat{n}_{1 } + 1 ) k_{1 } x } e^ { i ( n_{2 } + \hat{n}_{2 } ) k_{2 } x } \cdots e^ { i ( n_{n } + \hat{n}_{n } ) k_{n } x } e^{ik_{1 } y } \right .\\ & +  \cdots \left .\frac { a_{n } b(n_{1},\hat{n}_{1 } \cdots n_{n},\hat{n}_{n } ) } { i ( \hat{n}_{1 } k_{1 } + \hat{n}_{2 } k_{2 }  + \cdots [ \hat{n}_{n } + 1 ] k_{n } ) } e^ { i ( n_{1 } + \hat{n}_{1 } ) k_{1 } x } e^ { i ( n_{2 } + \hat{n}_{2 } ) k_{2 } x } \cdots e^ { i ( n_{n } + \hat{n}_{n } + 1 ) k_{n } x } e^{ik_{n } y } \right ] \end{split}\ ] ] from this result , we can immediately derive that : moreover , we can derive by using a similar argumentation as used in the case that that the recursion relations are given by : k_{1 } + \hat{n}_{2 } k_{2 } + \cdots \hat{n}_{n } k_{n } ) }   \\b(n_{1}+\hat{n}_{1},0,n_{2}+\hat{n}_{2}+1,1 , \cdots , n_{n}+\hat{n}_{n},0 ) & =  a_{2 } \frac { b(n_{1},\hat{n}_{1},n_{2},\hat{n}_{2 } , \cdots , n_{n},\hat{n}_{n } )  } {  i ( \hat{n}_{1 } k_{1 } + [ \hat{n}_{2 } + 1 ] k_{2 } + \cdots \hat{n}_{n } k_{n } ) }   \\ & \vdots \\ b(n_{1}+\hat{n}_{1},0,n_{2}+\hat{n}_{2},0 , \cdots , n_{n}+\hat{n}_{n}+1,1 ) & =  a_{n } \frac { b(n_{1},\hat{n}_{1},n_{2},\hat{n}_{2 } , \cdots , n_{n},\hat{n}_{n } )  } {  i(\hat{n}_{1 }  k_{1 } + \hat{n}_{2 } k_{2 } + \cdots [ \hat{n}_{n } + 1 ] k_{n } ) }   \\ \end{split } \label{it_fin}\ ] ] once the coefficients are determined , the kernel can be uniquely recovered .hence the kernel can be represented in the following form : the final result which is presented by eq.([ker_fin ] ) is in agreement with results obtained in ref. .the method used for deriving eq.([ker_fin ] ) enables us to investigate the sensitivity of the method for small data - errors by using lyapunov exponents .before we proceed , we place a few remarks .firstly , it is interesting that large classes of solutions of the marchenko equation can be obtained by solving a linear recursion relationship .the instabilities in the obtained solutions can be explained by the process of repeated iteration which has to be applied to compute the full kernel .this also implies that the stability of the obtained solutions depends strongly on the the numerical value of the parameters and .the second remark is that from a conceptual point of view there is no difference in solving inverse scattering problems or solving nonlinear evolution equations .it is shown this publication that both integrable nonlinear evolution equations can be linearized using a similar approach as used in this paper .this explains why the inverse scattering method works so well in the field of integrability .in the previous section solutions of the marchenko equation are discussed .we have shown that we can compute solutions of the marchenko equation by solving the linear recursion relation ( [ it_fin ] ) .solutions of the marchenko equation can be unstable due to process of repeated iteration which is used to compute the expansion coefficients of the obtained solutions .the fact that the marchenko equation can be solved by using a recursion relationship enables us to perform the stability analysis of the inverse scattering problem by using lyapunov exponents . at a first sight the bookkeeping for the process of error propagation of the coefficients is quite tedious , but the analysis can be simplified by considering only the coefficients which are obtained repeatedly inserting in the same recursion relation of eq.([it_fin ] ) : }   \\b(0,0,n_{2}+2,1 , \cdots , 0,0 ) & =  a_{2 } \frac { b(0,0,n_{2},1,\cdots , 0,0 )  } { i k_{2 } [ n+1 ] }    \\ & \vdots \\ b(0,0,0,0 , \cdots , n_{n}+2,1 ) & =  a_{n } \frac { b(0,0,0,0,\cdots , n_{n},1 )  } { i k_{n } [ n+1 ] }     \\ \end{split } \label{w_prop}\ ] ] by discussing the error - propagation in eq.([w_prop ] ) instead of eq.([it_fin ] ) has the advantage that we can investigate the stability of every pair ( ) in the data - spectrum .suppose that all the parameters and in the data lead to coefficients in eq.([w_prop ] ) which are stable for error - propagation .this implies that automatically all the other coefficients ( which are computed by solving eq.([it_fin ] ) ) are also stable for error - propagation .if there are pairs ( ) leading to instabilities , we can simply remove them from the data - set and hence a stable inverse scattering problem remains . before computing the lyapunov exponents , it is convenient to introduce a more efficient formulation .we firstly define coefficients in the following way : due to imperfections in the data - set , we have to investigate the stability of the iteration series ( [ it_fin ] ) under the variations , and respectively .if we insert these perturbations in eq.([it_fin ] ) , we obtain the following result for the perturbation : \label{stab_1}\ ] ] where the operator $ ] which expresses the error made per iteration is given by : = \frac { a_{i } } { i[n+1]k_{i } }  [ \ : \cdot \ : ] -  \frac {  b_{i}^{(n ) } \delta a_{i}- a_{i } b_{i}^{(n ) } [ n+1 ] \delta k_{i } } {  [ n+1]^{2}k_{i}^{2 }  }  \label{stab_2}\ ] ] in principle , after iterations , we obtain an error which yields : \label{stab_4}\ ] ] if the number of iterations is large we can re - express eq.([stab_4 ] ) in the following from : where the averaged growth of the error per iteration is given by : eq.([stab_5 ] ) can be reformulated as : where we have obtained a closed expression for the lyapunov exponent which describes the average growth of errors per iteration .although the the expression for the lyapunov exponents may look complicated , the method can easily be implemented numerically .if the lyapunov exponent is positive the errors in the coefficients grow .if the lyapunov negative the errors damp out . in the latter casethe inverse scattering problem is not sensitive for small errors in the data . in principle , we have obtained a practical approach to stabilize the inverse scattering problem since each of the lyapunov exponents associated with eq.([stab_8 ] ) corresponds to a single fourier component of the data - spectrum . by removing the fourier components associated with a positive lyapunov exponent , we can find a stable sub - set of the data - spectrum .we illustrate this principle numerically for the case that that we have only one fourier component .if we assume that there are only uncertainties in , we obtain that the lyapunov exponent is given by : in fig.1 , the result of the reconstruction is presented for and .it follows from eq.([cond1 ] ) that for this choice of the coefficients and , the lyapunov exponent is negative and the inverse scattering problem is stable .if the experiment is repeated with the same value of but with , the curve corresponding to the reconstruction can not be distinguished from the unperturbed construction .this situation changes if we choose and keep equal to unity .the reconstruction is for this case presented by the solid line in fig.2 . in this casethe reconstruction is according to eq.([cond1 ] ) sensitive to small changes in . indeed , if we solve the inverse scattering problem for , we find that the reconstruction given by the broken curve in fig.2 differs from the unperturbed solution .this simple example indicates that the ratio between and determines whether the inverse scattering problem is stable or not .the results given in eq.([w_prop ] ) indicate that a similar result holds in more general cases and forms in principle a tool to investigate which part of the data leads to stable inversion results . in principleif , instabilities in the obtained solutions can be expected .however in contrast to the results presented in ref. , we show in this paper that there is a delicate balance between the amplitude and frequency of each fourier component to provide stable inversion results .in this paper , we have presented a method to solve the one - dimensional marchenko equation for finite data - sets and to investigate the stability of the obtained solutions .we have established a recursion relationship between the fourier expansion coefficients of the data and the fourier expansion coefficients of the kernel .we have also shown that we can estimate the stability of the obtained solutions from the data - uncertainties by using lyapunov exponents .we want to remark that investigating the stability of the marchenko has become an easy task from computational point of view since we have presented the lyapunov exponents in an analytically closed form .a practical approach to find a stabilizing transformation for a potentially unstable data - set consists of simply removing all the unstable pairs from the data .the low frequency components in the spectrum are responsible for the instabilities . instead of bandpass filtering the data - set ( to remove the low - frequency components of the data ) , the method presented in this publication enables us to identify the pairs , and their removal leads to a more accurate regularization method for the inverse scattering problem .  as a concluding remark we have a renewed look on the examples of data - contaminations given in ref. .the first example discussed in this paper is a small dc - error on the data - set which leads unstable inversion results .it is shown that a dc - error corresponds with a fourier component with vanishing . as a result of the discussion given in this paper ,it can be understood easily that a vanishing leads to a positive lyapunov exponent .the other examples discussed in ref. deal with an amplitude error and a timing error .it is shown for both examples that if the data - set contains fourier components with small , that inverse scattering problem is unstable .the results obtained in this paper are in agreement with ref. , but additionally quantitative results are given .faddeev , _ the inverse problem in the quantum theory of scattering _ , j. math. phys . * 4 * , 1963 , 72 - 104  p.m. carrion , _ on the stability of 1d exact inverse methods _ , inverse problems * 2 * , 1986 , 1 - 22 .f. koehler and m. taner , _ direct and inverse problems relating reflection coefficients and response for horizontally layered media _ , geophysics * 42 * , 1977 , 1199 - 1206 .r. burridge , _ the gelfand - levitan , the marchenko , and the gopinath - sondhi integral equations of inverse scattering theory , regarded in the context of inverse impulse - response problems _ , wave motion * 2 * , 1980 , 305 - 323 .sabatier , _ rational reflection coefficients and inverse scattering on the line _ , nuovo cinemento b * 78 * , 1983 , 235 - 248 .dorren , e.j .muyzert and r.k .snieder , _ the stability of one dimensional inverse scattering _ , inverse problems * 10 * , 865 - 880 , 1994  h.j.s .dorren and r.k.snieder , _ one - dimensional scattering using data contaminated with errors _ , in lecture notes on physics , quantum inversion theory and applications , h.v .von geramb ( ed . ) , 405 - 411 , springer - verlag , 1994 .dorren , and r.k .snieder , _ on the stability of inverse problems _ , inverse problems * 11 * , 889 - 911 , 1995 .dorren , _ a linearization method for the korteweg - de vries equation ; generalizations to higher dimensional s - integrable differential equations _ , j. math. phys . * 39 * , 3711 - 3729 , 1998 .dorren , _ on the integrability of nonlinear partial differential equations _ , preprint ( solv - int/9807007 ) .k.chadan and p.c .sabatier , _ inverse problems in quantum scattering theory _ , springer - verlag , 1989 .* figure 1 * : kernel of is case of one fourier component as given by eq.([k_xx2 ] ) for and ( solid line ) and for and ( broken curve ) . in the stable caseboth curves can not be distinguished .* figure 2 * : kernel of is case of one fourier component as given by eq.([k_xx2 ] ) for and ( solid line ) and for and ( broken curve ) .both curves are scaled with a factor .

Abstract: we present a novel method to estimate the stability of the marchenko equation for finite data - sets . we show that we can derive a recursion relationship for the fourier expansion coefficients of the kernel which is solved by the marchenko equation . the method can easily be implemented numerically . moreover , we discus the stability of the one - dimensional inverse scattering problem by using lyapunov exponents . we give conditions on the scattering data to provide stable inversion results . a numerical example is given .   e - mail : h.j.s.dorren.tue.nl


in the past decade , there has been a considerable interest in characterizing and understanding the structural properties of networked systems  .it has been established that scale - free behavior  is one of the most fundamental concepts for a basic understanding of the organization of many real - world systems in nature and society .this scale - free property has a profound effect on almost every aspect on dynamic processes taking place on networks  , including robustness  , percolation  , synchronization  , games  , epidemic spreading  , to name just a few .for instance , for a wide range of scale - free networks an epidemic threshold does not exist , and even infections with a low spreading rate will prevail over the entire population in these networks  . this is a radical change from the conclusions drawn from classical disease modeling  .in addition to the above - mentioned dynamics , some authors have focused their attention on the trapping problem occurring on complex networks  , which is one of the main topic of interest for random walks ( diffusion )  .the classical trapping problem first introduced in  is a random - walk issue , where a trap is located at a fixed position , absorbing all particles that visit it .an interesting quantity closely related to the trapping problem is the mean first - passage time ( mfpt ) , which is very important in the study of transport - limited reactions  , and target search  , amongst other physical problems .a result from previous research is that a power - law property can improve the efficiency of transport by diffusion on scale - free networks  : the mfpt , , scales linearly or sublinearly with the number of network nodes as with or , which shows that the efficiency of trapping processes on scale - free networks is even better than ( at least not worse than ) that on complete graphs  , the best possible structure for a fast diffusion ( with ) .although the scale - free topology has a direct effect on other structural characteristics ( e.g. , average path length  ) of networks and dynamics running on them , it can not reflect all the information of the network structure .recently , it has been discovered that many real - life networks , such as the www , metabolic networks , and yeast protein interaction networks have self - similar properties and exhibit fractal scaling  .this fractal topology is often characterized through the fractal dimension , which can be obtained by the box - counting algorithm  .it is now commonly accepted that fractal scaling  must be considered in an integral basic understanding of the organization of real - life complex systems .  as a fundamental property ,topological fractality is related to many respects of network structure and function .recently , several authors have shown that the correlation between degree and betweenness centrality of nodes is much weaker in fractal network models in comparison with non - fractal models  .it has been also shown  that fractal scale - free networks are disassortative  , and this feature , together with fractality , makes such scale - free networks more robust against intentional attacks on hub nodes , as compared to the highly vulnerable non - fractal scale - free networks  .in addition to the distinction in the robustness , fractal networks exhibit lower synchronizability than their non - fractal counterparts  .although a lot of efforts have been devoted to fractal scale - free networks  , it is still of current interest to look for a better understanding of the consequences of a fractal topology on different dynamic processes .  in this paper , we study the trapping dynamics on a fractal scale - free network in the presence of a perfect absorber located at a fixed node .we obtain a rigorous solution for the mfpt of the unbiased random walks , which is computed through the recurrence relations derived from the network structure .the resulting formula shows that for a large network , the mfpt , , scales with the network order as .this superlinear growth is significantly different from the linear or sublinear scaling previously found for nonfractal scale - free networks .in this section we introduce a network model defined in an iterative way  , which has attracted a great amount of attention  .we call this model _ iterative fractal scale - free tree _ ( ifsft ) .we study the ifsft because of its intrinsic interest and its relevance to real - world systems .for example , the so - called border tree motifs have been shown to be present , in a significant way , in real - life networks  . moreover , the ifsft is deterministic , which allows us to study analytically its topological properties and dynamical processes taking place on it .it is thus a good test - bed and an ideal substrate network.   stands for a new external node , while represents an internal node . ]the ifsft , denoted by after ( ) iterations ( the number of iterations is also called generation hereafter ) , is constructed as follows . for , is an edge connecting two nodes . for , obtained from : for each edges in , two new nodes ( called external nodes with degree of 1 ) are firstly introduced and linked respectively to both ends of the edge ; then , the edge is broken , another new node ( referred to as an internal node ) is positioned in its middle and connected to both ends ( see fig . [ construct ] ) . in other words , is obtained from by performing the following operations on every edge in : replace the edge by a path of 2 links long , with the two endpoints of the path being the same endpoints of the original edge , then attach a new node to each endpoint of the path .figure [ network ] shows the construction process for the first three iterative processes .    according to the network construction, one can see that at each generation ( ) the number of newly introduced nodes is . from this result, we can easily compute the network order ( i.e. , the total number of nodes ) at generation :   to facilitate the description in what follows , we distinguish different nodes of by labeling them as follows . for ,the newly created internal node is labeled 1 , the initial two nodes belonging to are labeled as 2 and 3 , and the two new external nodes have labels 4 and 5 , see fig .[ labeling ] . for each new iteration , we label consecutively the new nodes generated at this iteration , while we keep the labels of the old nodes unchanged .namely , new nodes are labeled sequentially as . in this way, we label each node by a unique integer : at generation all nodes are labeled from 1 to .labels of all nodes of . ]let be the degree of a node at generation , which entered the network at generation ( ) . if node was an external node when it was introduced , otherwise , if was an internal node when it entered the network , from eqs .( [ ki01 ] ) and ( [ ki02 ] ) , one can easily see that after each new iteration the degree of a node doubles , i.e. ,   the ifsft presents some interesting topological characteristics .it has a power law degree distribution with exponent  .its average path length , defined as the mean of shortest distances between all pairs of nodes , grows as a square power of the network order  .the betweenness distribution of its nodes exhibits a power - law behavior with exponent  .particulary , it is fractal with the fractal dimension .after introducing the ifsft , in what follows we will study the mfpt for random walks with a single immobile trap on the network . we will show that the fractal property has an essential influence on the mfpt , i.e. , the fractality can induce a general slowing down of diffusion .here we formulate the trapping problem of a simple random walk of a particle on the ifsft in the presence of an absorbing trap positioned at the central hub node , represented as .to this end , we first represent by its adjacency matrix of order .the entry of is either 1 or 0 : if and are adjacent and otherwise .the diagonal degree matrix of is .then , the normalized laplacian matrix of is given by , where is the identity matrix .  in the trapping problem , at each time step , a particle , starting from any node except the trap , moves from its current location to any of its nearest neighbors with equal probabilities .it is easy to see that in the end the particle will be necessarily absorbed by the trap , regardless of its starting location  .we are interested in the mean transmit time ( first - passage time , or trapping time ) for a particle , originating at node , to first reach the trap in .such a random walk can be described by a markov chain  , whose fundamental matrix is the inverse of matrix that is defined as a sub - matrix of the normalized laplacian matrix obtained by deleting from it the first row and column , corresponding to the absorbing node .the entry of the fundamental matrix expresses the mean residence time , which is the mean number of visitations of node by the particle , starting from node , before trapping occurs .thus , we have then , the mfpt , , which is the mean of over all nodes distributed uniformly over nodes in other than the trap , is given by   equation ( [ mfpt5 ] ) shows that the problem of finding is reduced to calculating the sum of all entries of the fundamental matrix .although the expression of eq .( [ mfpt5 ] ) seems compact , since the order , , of increases exponentially with , for large , it becomes impossible to get through direct calculation from eq .( [ mfpt5 ] ) as we are restricted by time and computer memory , and one can calculate directly the mfpt only for the first iterations , see fig .however , the particular construction of the ifsft and the special choice of the trap location allow to calculate analytically mfpt to obtain a closed - form formula . the derivation details of which will be provided in the following section.   mean first - passage time versus network order on a log - log scale .the numerical results are obtained by direct calculation from eq .( [ mfpt5 ] ) , while the analytical results are from eq .( [ mfpt19 ] ) .the solid line is the guide to the eye . ]before giving a general formula for mfpt , , we first establish the dependence of on .   for each ,the values of can be obtained straightforwardly via eq .( [ mfpt4 ] ) .table [ tab : amta2 ] lists the numerical values of for nodes for the first several generations up to .the numerical values quoted in table [ tab : amta2 ] show that for a given node we have .that is to say , upon growth of the ifsft from generation to , the mean time to first reach the trap increases by a factor of 8 .this is a basic characteristic of random walks on the ifsft , which can be established from the arguments below .consider an arbitrary node in the ifsft . from eq .( [ ki03 ] ) , we know that at iteration , the degree of node of ifsft doubles , i.e. , it grows from ( degree at iteration ) to . moreover , all these neighbors are new nodes created at iteration , among which neighbors are external nodes , and the rest neighbors are internal nodes .we now examine the standard random walk in : let be the mean transmit time for a particle starting from node to any of its old neighbors , i.e. , those nodes directly linked to node at iteration ; and let ( resp . ) be the mean transmit time for going from any of the internal ( resp .external ) neighbors of to one of the old neighbors .then the mean transmit times follow the relations : equation ( [ mfpt6 ] ) has a solution , found by eliminating and .that is to say , when the network grows from iteration to iteration , the first - passage time from any node to any node ( both and belong to ) increases by a factor of 8 .thus , we have , which will be useful for the derivation of the exact formula for the mfpt below .  [ cols="<,^,^,^,^,^,^,^,^,^,^",options="header " , ]   after obtaining the scaling of mean trapping time for old nodes , we now derive the analytical rigorous expression for the mfpt . before proceeding further ,we first introduce some notation used in the rest of this section .let denote the set of nodes in , and let stand for the set of those nodes introduced at generation . to facilitate the computation, we also define the following quantities for : and   by definition , it follows that .thus , we have where the relation of has been made use of . hence , to determine , one should first explicitly determine the quantity .for this purpose , we further separate into two sets : one set of external nodes and the other set of internal nodes as defined in the second section , which are denoted as and , respectively .clearly , . on the other hand , we have shown that the cardinality of set is , and that .thus , we can obtain and .then , two corresponding quantities can be defined :  it is obvious that     in order to obtain , one may alternatively get and . to this end , we first establish the relationship between the two quantities and . by construction , at a given generation , each edge connecting two nodes and ( see fig . [ iter ] ) will generate three new nodes in the next generation : two external nodes ( and ) and one internal node ( ) , and the mean transmit times for these three new nodes satisfy the following relations : hence , we have summing eq .( [ eq : intext1 ] ) over all old edges at the generation before growth , we can easily obtain that for all , equation ( [ eq : intext2 ] ) gives the relationship between and , which is very significant since it is useful for the computation in the following text .therefore , the issue of determining is reduced to finding the quantity that can be obtained as follows .for an arbitrary external node in , which is created at generation and attached to an old node , we have since a particle starting from node will be on node after one jump .note that eq .( [ eq : intextx2 ] ) holds for any node pair consisting of an old node and any one of its new external adjacent nodes . by applying eq .( [ eq : intextx2 ] ) to two sum ( the first one is over a given old node and all its new external nodes , the other is summing the first one over all old nodes ) , we get where eqs .( [ eq : ttotr ] ) and ( [ eq : intext2 ] ) were used .now , we can determine through a recurrence relation , which can be obtained easily .from eq .( [ eq : rec1 ] ) , it is not difficult to write out as   equation ( [ eq : rec2 ] ) minus eq .( [ eq : rec1 ] ) times 16 and making use of the relation , one gets the following recurrence relation considering the initial condition , this recurrence relation is solved to obtain inserting eq .( [ eq : tx1 ] ) into eq .( [ eq : ttotr ] ) and considering the relation , we have substituting the last expression into eq .( [ eq : ttot ] ) yields using , eq .( [ eq : t2 ] ) is solved inductively   inserting eq .( [ eq : tnew1 ] ) into eq .( [ mfpt5 ] ) , we obtain the rigorous expression for the mfpt for the trapping problem on the -th generation of the ifsft : we have checked this exact solution for the mfpt against numerical values given by eq .( [ mfpt5 ] ) , see fig .[ trap ] . for all , the analytical values obtained from eq .( [ mfpt19 ] ) are perfectly consistent with the numerical results .this agreement is an independent test of our theoretical formula .we show next how to represent mfpt as a function of the network order , with the aim of obtaining the relation between these two quantities .recalling eq .( [ nt ] ) , we have and .hence , eq . ( [ mfpt19 ] ) can be recast as for a large network , i.e. , , with the exponent much larger than 1 .thus , in the limit of large network order , the mfpt grows superlinearly with the number of network nodes .recently , it has been shown that for non - fractal scale - free networks with a large network order , their mfpt behaves linearly or sublinearly with as with  .however , we have seen that for the ifsfn , the mfpt , , increases superlinearly with ( i.e. , ) irrespective of its scale - free property , presenting an obvious difference from the results previously obtained for its non - fractal scale - free counterparts .why is the mfpt for the ifsft far larger than that for nonfractal scale - free networks such as the apollonian network  ? the reasons behind this discrepancy may be explained as follows . for the apollonian network , the large - degree nodes , including the trap node , are directly connected to one another and compose a core group sharing more neighbors , which make the apollonian network be a very compact system .so , these large - degree nodes can be easily visited by a particle in spite of its starting location .the interconnection within nodes with large degrees makes the particle spend a short time to find the trap . on the contrary , in the ifsft, the large - degree nodes are not linked to each other , they are exclusively connected to low - degree nodes , as a result from the fractal property of ifsft . in other words , there is an effective ` repulsion ' between the large - degree nodes in the ifsft  , which seems to be a main feature that distinguishes the ifsft from the apollonian network .hence , for the trapping problem in the ifsft , because of the isolation of the large - degree nodes from each other , the particle will first reach a hub node , and before being absorbed it will spend a lot of time in the intermediate regions constituted by small - degree nodes , which connect indirectly the large - degree nodes to one another .therefore , it takes a longer time for the particle to arrive at the trap .we have studied a classical trapping problem performed on a deterministically growing scale - free network with fractal topology .the self - similarity of the network allows us to derive the exact expression for the mfpt .we have shown that , for large networks , the mfpt increases as a power - law function of the network order , with an exponent much larger than 1 , in contrast with previous results found for non - fractal scale - free networks .we see that this slow transport efficiency lies with the inherent fractality and its associated disassortativity .thus , we can infer that scale - free networks do not always tend to accelerate the diffusion processes occurring on them .our research may be helpful for a better understanding of the role that network structure plays in a trapping process .we thank f. comellas for his help in revising the manuscript .this research was supported by the national basic research program of china under grant no .2007cb310806 , the national natural science foundation of china under grant nos .60704044 , 60873040 and 60873070 , shanghai leading academic discipline project no .b114 , and the program for new century excellent talents in university of china ( ncet-06 - 0376 ) .w l xie also acknowledges the support provided by hui - chun chin and tsung - dao lee chinese undergraduate research endowment ( cure ) .

Abstract: it is known that the heterogeneity of scale - free networks helps enhancing the efficiency of trapping processes performed on them . in this paper , we show that transport efficiency is much lower in a fractal scale - free network than in non - fractal networks . to this end , we examine a simple random walk with a fixed trap at a given position on a fractal scale - free network . we calculate analytically the mean first - passage time ( mfpt ) as a measure of the efficiency for the trapping process , and obtain a closed - form expression for mfpt , which agrees with direct numerical calculations . we find that , in the limit of a large network order , the mfpt behaves superlinearly as with an exponent much larger than 1 , which is in sharp contrast to the scaling with , previously obtained for non - fractal scale - free networks . our results indicate that the degree distribution of scale - free networks is not sufficient to characterize trapping processes taking place on them . since various real - world networks are simultaneously scale - free and fractal , our results may shed light on the understanding of trapping processes running on real - life systems .


a number of papers have recently considered the problem of constructing a coherent quantum observer for a quantum system ; see . in the coherent quantum observer problem , a quantum plantis coupled to a quantum observer which is also a quantum system .the quantum observer is constructed to be a physically realizable quantum system so that the system variables of the quantum observer converge in some suitable sense to the system variables of the quantum plant .  in the papers ,the quantum plant under consideration is a linear quantum system . in recent years, there has been considerable interest in the modeling and feedback control of linear quantum systems ; e.g. , see .such linear quantum systems commonly arise in the area of quantum optics ; e.g. , see . for such linear quantum system models an important class of quantum control problemsare referred to as coherent quantum feedback control problems ; e.g. , see . in these coherent quantum feedback control problems , both the plant and the controllerare quantum systems and the controller is typically to be designed to optimize some performance index .the coherent quantum observer problem can be regarded as a special case of the coherent quantum feedback control problem in which the objective of the observer is to estimate the system variables of the quantum plant .  in the previous papers on quantum observers such as ,the coupling between the plant and the observer is via a field coupling .this leads to an observer structure of the form shown in figure [ f1 ] .this enables a one way connection between the quantum plant and the quantum observer .also , since both the quantum plant and the quantum observer are open quantum systems , they are both subject to quantum noise .however in the paper , a coherent quantum control problem is considered in which both field coupling and direct coupling is considered between the quantum plant and the quantum controller . in this paper , we explore the construction of a coherent quantum observer in which there is only direct coupling between quantum plant and the quantum observer . furthermore ,both the quantum plant and the quantum observer are assumed to be closed quantum systems which means that they are not subject to quantum noise and are purely deterministic systems .this leads to an observer structure of the form shown in figure [ f2 ] .it is shown that for the case being considered , a quantum observer can be constructed to estimate some but not all of the system variables of the quantum plant .also , the observer variables converge to the plant variables in a time averaged sense rather than a quantum expectation sense such as considered in the papers .in this section , we describe the class of closed linear quantum systems under consideration ; see also .we consider linear non - commutative systems of the form where is a real matrix in , and \trp ] . in the case of a single degree of freedom quantum particle , where is the position operator , and is the momentum operator .the commutation relations are =2 i ] and  ] where and . then ,  ] .hence , the augmented system equations ( [ augmented_system ] ) describing the combined plant - observer system become we now use laplace transforms to solve these equations .it follows that and hence , however , = 0\ ] ] since is skew - symmetric .therefore , taking the inverse laplace transform of this equation , we obtain   also , if we substitute ( [ xos ] ) into ( [ laplace_aug_sys ] ) , we obtain taking the inverse laplace transform of this equation , we obtain we now choose the parameters of the quantum observer so that .it follows from ( [ xot ] ) and ( [ xpt ] ) that the quantities and are given by and where we have used the fact that .that is , the quantity remains constant and is not affected by the coupling to the coherent quantum observer .note that the equation ( [ zop ] ) can be derived directly since  & = & \left[\begin{array}{ll}\beta^t & 0\end{array}\right ] \left[\begin{array}{ll}0 & 2\theta_1 \beta \alpha^t\\2 \theta_2\alpha \beta^t & 2 \theta_2 r_o\end{array}\right ] \nonumber \\ & = & \left[\begin{array}{ll}0 & 2\beta^t\theta_1 \beta \alpha^t\end{array}\right ] \nonumber \\ & = & 0\end{aligned}\ ] ] since .hence , ^{a_at}x_a(0 ) = \left[\begin{array}{ll}c_p & 0\end{array}\right]x_a(0 ) = z_p(0)\ ] ] for all .note that the matrix will have all purely imaginary eigenvalues . to see this, we first observe that the matrix has purely real eigenvalues since is a hermitian matrix and is assumed to be a positive definite matrix .indeed , and thus is similar to the hermitian matrix which has purely real eigenvalues .hence , must have purely imaginary eigenvalues .now suppose the vector  ] is an eigenvector of .thus , we must have .thus , we can now conclude that is an eigenvalue of which we have already established has only purely imaginary eigenvalues .thus , must be purely imaginary in this case as well .we now verify that the condition ( [ average_convergence ] ) is satisfied for this quantum observer .we recall from remark [ r1 ] that the quantity remains constant in time for the linear system : that is however , and .therefore , it follows from ( [ roconst ] ) that for all and .hence , for all .now since and are non - singular , and therefore , it follows from ( [ exp_bound ] ) that as .hence , ( [ zot ] ) implies also , ( [ zop ] ) implies therefore , condition ( [ average_convergence ] ) is satisfied .thus , we have established the following theorem .[ t2 ] consider a quantum plant of the form ( [ plant ] ) where , and is as defined in ( [ beta ] ) .then the matrices , , will define direct coupled quantum observer for this quantum plant if is of the form where and .[ r2 ] we consider the above result for the single mode case with , , in which  ] . by choosing , , ] and  ] . before the quantum observer is connected to the quantum plant, the quantities and will remain constant since we have assumed that .now suppose that the quantum observer is connected to the quantum plant at time . according to ( [ zop ] ), the plant position operator will remain constant at its initial value but the plant momentum operator will evolve in an time varying and oscillatory way as defined by ( [ xpt ] ) .in addition , the observer position operator will evolve in an oscillatory way as defined by ( [ zot ] ) but its time average will converge to according to ( [ average_convergence ] ) .  now suppose that after a sufficiently long time such that the time average of has essentially converged to , the observer is disconnected from the quantum plant. then , the plant position operator will remain constant at and the plant momentum operator will remain constant at a value which is determined by the formula ( [ xpt ] ) in terms of , and the time .this will be an essentially random value .if at a later time an observer with the same parameters as above is connected to the quantum plant , then time average of its output will again converge to and will remain constant at .however , suppose that instead an observer with different parameters ,  ] is used .this observer is designed so that the time average of the observer output converges to the momentum operator of the quantum plant .this quantity is the essentially random value mentioned above . in addition , the previously constant value of will now be destroyed and will evolve to another essential random value . this behavior of the quantum observer is similar to the behavior of quantum measurements ; e.g. , see .this is not surprising since the behavior of the direct coupled quantum observers considered in this paper and the behavior of quantum measurements are both determined by the quantum commutation relations which are fundamental to the theory of quantum mechanics .we now present some numerical simulations to illustrate the direct coupled quantum observer described in the previous section .we consider the quantum observer considered in remark [ r2 ] above where , , , ,  ] ,  ] .as described in remark [ r2 ] , the variable to be estimated by the quantum observer is the position operator of the quantum plant ; i.e. , where  ] .then the augmented plant - observer system is described by the equations = a_a \left[\begin{array}{l } q_p(t)\\  p_p(t)\\  q_o(t)\\ p_o(t)\end{array}\right]\ ] ] where = \left[\begin{array}{llll }    0  &  0  &  0  &  0 \\    0 &   0  &  2  &  0 \\   0 &   0  &  0  &  2 \\    2 &   0  &  -2  &  0 \end{array}\right].\ ] ] then , we can write  = \phi(t ) \left[\begin{array}{l } q_p(0)\\  p_p(0)\\  q_o(0)\\ p_o(0)\end{array}\right]\ ] ] where = e^{a_a t}.\ ] ] thus , the plant variable to be estimated is given by and we plot the functions , , , in figure [ f3 ] ..,width=264 ]  from this figure , we can see that , , , , and will remain constant at for all .also , the other plant variable is given by and we plot the functions , , , in figure [ f4 ] ..,width=264 ]  from this figure , we can see that evolves in a time - varying and oscillatory way when the quantum plant is connected to the quantum observer .we now consider the output variable of the quantum observer which is given by and we plot the functions , , , in figure [ f5 ] ..,width=264 ]  to illustrate the time average convergence property of the quantum observer ( [ average_convergence ] ) , we now plot the quantities in figure [ f6 ] ..,width=264 ]  from this figure , we can see that the time average of converges to as .note that the effect of time averaging can be regarded as a low pass filtering effect which removes the sinusoidal oscillations but retains the dc component which represents the estimate of the plant variable .we now consider the other variable of the quantum observer which is given by and we plot the functions , , , in figure [ f6a ] ..,width=264 ]  to investigate the time average property of the other quantum observer variable , we now plot the quantities in figure [ f6b ] ..,width=264 ]  we now illustrate the comments in remark [ r2 ] by supposing that the above quantum observer is applied to the quantum plant for the time interval  ] . during this time internal, the quantum plant can be regarded to be connected to a null quantum observer so that in this time interval . at time , the quantum plant is then connected to a different quantum observer defined by the parameters ,  ] and  ] , the matrix defines the dynamics of the augmented plant - observer system in the time interval $ ] , and the matrix defines the dynamics of the augmented plant - observer system for .then , we can write where ,  \\  e^{a_{a2 } ( t-20)}e^{a_{a1 } 20}= e^{a_{a1 } 20}\mbox { for } t \in [ 20,25 ] , \\  e^{a_{a3 } ( t-25)}e^{a_{a1 } 20}\mbox { for } t \geq 25 \end{array } \right.\\ & = & \left[\begin{array}{llll }    \tilde\phi_{11}(t )  &  \tilde\phi_{12}(t )  &  \tilde\phi_{13}(t )  &  \tilde\phi_{14}(t ) \\    \tilde\phi_{21}(t )  &  \tilde\phi_{22}(t )  &  \tilde\phi_{23}(t )  &  \tilde\phi_{24}(t ) \\ \tilde\phi_{31}(t )  &\tilde\phi_{32}(t )  &  \tilde\phi_{33}(t )  &  \tilde\phi_{34}(t ) \\\tilde\phi_{41}(t )  &  \tilde\phi_{42}(t )  &  \tilde\phi_{43}(t )  &  \tilde\phi_{44}(t ) \end{array}\right ] .  \end{aligned}\ ] ] now in a similar fashion to figure [ f3 ] , we plot the quantities , , , and in figure [ f7 ].  .,width=264 ]  also , in a similar fashion to figure [ f4 ] we plot the quantities , , , and in figure [ f8 ] ..,width=264 ]  moreover , in a similar fashion to figure [ f5 ] we plot the quantities , , , and in figure [ f9 ] ..,width=264 ]  in addition , in a similar fashion to figure [ f6 ] , we now plot the quantities in figure [ f6 ] ..,width=264 ]  also , in a similar fashion to figure [ f6a ] , we plot the quantities , , , and in figure [ f11 ] ..,width=264 ]  in addition , in a similar fashion to figure [ f6b ] , we now plot the quantities in figure [ f12 ] .in this paper we have introduced a notion of a direct coupling observer for closed quantum linear systems and given a result which shows how such an observer can be constructed .the main result shows the time average convergence properties of the direct coupling observer .we have also presented an illustrative example along with simulations to investigate the behavior of a direct coupling observer when applied to a simple one mode quantum linear system .future research in this area might involve extending the class of quantum linear systems for which a direct coupling observer can be designed and also considering the problem of constructing an observer which is optimal in some sense .also , future research could investigate the role of direct coupling observers in the design of direct coupling coherent quantum feedback control systems .the author would like to thank dr .igor vladimirov for his useful comments .i. vladimirov and i. r. petersen , `` coherent quantum filtering for physically realizable linear quantum plants , '' in _ proceedings of the 2013 european control conference _ ,zurich , switzerland , july 2013 , arxiv:1301.3154 .z. miao , l. a. d. espinosa , i. r. petersen , v. ugrinovskii , and m. r. james , `` coherent quantum observers for finite level quantum systems , '' in _australian control conference _ , perth , australia , november 2013 .m. r. james , h. i. nurdin , and i. r. petersen , `` control of linear quantum stochastic systems , '' _ ieee transactions on automatic control _ , vol .53 , no . 8 , pp . 17871803 , 2008 , arxiv : quant - ph/0703150 .a. j. shaiju and i. r. petersen , `` a frequency domain condition for the physical realizability of linear quantum systems , '' _ ieee transactions on automatic control _ ,57 , no . 8 , pp . 2033 2044 , 2012 .i. g. vladimirov and i. r. petersen , `` a quasi - separation principle and newton - like scheme for coherent quantum lqg control , '' _ systems & control letters _62 , no . 7 , pp . 550559 , 2013 , arxiv:1010.3125 .i. vladimirov and i. r. petersen , `` a dynamic programming approach to finite - horizon coherent quantum lqg control , '' in _ proceedings of the 2011 australian control conference _ , melbourne ,november 2011 , arxiv:1105.1574 .

Abstract: this paper considers the problem of constructing a direct coupling quantum observer for a closed linear quantum system . the proposed observer is shown to be able to estimate some but not all of the plant variables in a time averaged sense . a simple example and simulations are included to illustrate the properties of the observer .


recent developments in digital technology have made possible the collection and analysis of massive amount of human social data and the ensuing discovery of a number of strong online behavioral patterns  .these patterns are important for two reasons .first , they yield predictions about future behavior that can be incorporated into the design of useful social media and services , and second , they provide an empirical test of the many social theoretical models that have been proposed in the literature . as an example , the assumption that events in web traffic data are described by a series of poisson process  was shown to be contradicted by measurements of the the waiting time between two consecutive events , which display power law scaling .these power laws are ubiquitous and appear in the analysis of email exchanges  and web browsing  . on the other hand , regular behavioral patterns in real life are a well known phenomenon , as exemplified by vehicular traffic patterns , daily routines , work schedules and the seasonality of economic transactions . at the aggregate level , these regularities are often induced by spatial and temporal constraints , such as the disposition of roads and streets in urban settings or the timing of daily routines .other examples are provided by the existence of deterministic patterns in human daily communication  and phone call location sequences  .when it comes to human online activities many theoretical studies curiously assume uncorrelated random events on the part of the users  which makes their behavior rather unpredictable . moreover , that literature assumes that a user s future partners in comments and reviews , or how web pages are visited are independent of the history of the process or at best on the previous time step . while these assumptions work well for page ranking in web searching  , online recommendation systems  , link prediction  , and advertising  , it is not clear that they apply to more interactive processes such as contacting friends within online social networks , participating in online discourse and exchanges of email and text messages . even in cases where a markovian assumption seems to yield good results ,the discovery of deterministic components to online browsing and searching can improve existing algorithms  .in this paper we study the predictability of online interactions both at the group and individual levels . to this end, we measure the predictability of online user behavior by using information - theoretic methods applied to real time data of online user activities .this is in the same spirit as a recent study of offline conversations within an organization  . using ideas first articulated in studies of gene expressions  , predictability is here defined as the degree to which one can forecast a user s interacions based on observations of his previous activity .the main focus of this study is to be contrasted to existing studies of online social behavior , such as recommender systems  and link prediction  , which use statistical learning models to improve the prediction accuracy of novel links and recommendations . by examining datasets from user commenting activities and place visiting logs, we found that the observed activity sequences deviate from a random walk model with deterministic components .furthermore , we also compared the predictability of activities when individuals act alone as opposed to as members of a group .in contrast to many model assumptions in studies of online communites and group behavior  , we observed that individuals are less predictable when attending group or social activities than when acting on their own .we examined the predictability of online user behavior using datasets from two different websites : epinions and whrrl .epinions is a who - trust - who consumer review site , where users write their personal reviews of a wide variety of products , ranging from automobiles to media ( including music , books and movies ) .each user can comment on other users reviews or comments .the thread of comments forms a conversation of two or more users . to trace the predictability of commenting partners, we collected unique users comments from the website . for each user, we used the website s api to collect all of their comments with a time stamp for each comment . in total , we gathered threads of comments from different categories containing user comments .the other dataset that we used is from whrrl.com .whrrl is a popular lbsn ( location based social network ) that people use to explore , rate and share points - of - interest .it also allows users to declare friendships with each other and to interact through visits and check - ins at physical places .users can check in by using a mobile application on a gps - equipped smart phone .the types of places that are often visited include restaurants , hotels and bookstores .a distinctive feature of this dataset is that a user can check - in by herself or with a group of other people , thus providing a forum for social activities .users of the site are identified by unique user - ids . in our study, we crawled a friendship network consisting of users and social ties and collected the check - in records of these users activities from january 2009 to january 2011 .the resulting undirected graph had an average degree of and an average shortest - path length of , which is typical of a small - world social network . in our observational period of 2 years , there were check - in records over different places associated with these users .for each check - in record , we also collected information such as the exact location ( i.e. , longitude and latitude ) , time of check - in and the users involved ( i.e. , there may be more than one user - id for group check - ins ) .we were thus able to obtain a series of places the users visited in chronological order .the activity sequence is obtained by neglecting the absolute timing of events in the raw dataset . to generate the activity sequence of a certain user , we first sifted out all the events that are associated with the user and we then listed the chronologically ordered sequence of states identified by a unique number . for the whrrl dataset , we labeled each activity as a group one if the user was checking in with others . to determine the extent towhich user behavior is predictable we used standard information - theoretic methods similar to those used in the analysis of gene expression  .for instance , we consider a user as having possible states , where each state in the sequence can correspond to either an online conversation partner or a check - in location .an example of a user s activity sequence is shown in figure [ fig : activity_sequence_sample ] , where two states , and , form the sequence .we then used the observed sequences to examine the degree of second order dependences , which signal the extend to which activities depart from random interactions .we used entropy to measure the randomness of a user s activities .the estimated probabilities for all states have the property that .if we assume that these probabilities do not change with time , the randomness of user s possible states can be measured by the uncorrelated entropy , defined as notice that if each state is equally probable , this uncorrelated entropy is maximal and equal to to measure the randomness of the sequence from knowledge of the previous states we introduce the conditional entropy , defined as and we quantify the predictability of the user s activity sequence by using the mutual information for each user , the inequalities are satisfied . is equal to the amount of information one can gain about the next state by knowing the current state . if there is no second order correlation between state sequences , is equal to , and takes the minimum value of .if the next state is completely determined by the previous state , or in other words the user activity is completely predictable , takes the maximum value of .the calculations of these quantities require an accurate estimation of the probabilities and .however , in the absence of unlimited data , estimating these probabilities with finite sampling renders a biased estimation of the entropy , since the finite sampling makes the user activity less variable than it is , resulting in a downward bias of the entropy , and a upward bias of the the mutual information .the problems associated with estimating entropies for sparse data have been extensively explored in the literature and a variety of remedies proposed  .the most common solution is to restrict the measurements to situations where one has an adequate amount of user activity data  . in what follows we filter out users who are below a certain activity level , in our observational period .since both and generally decrease by different amounts when taking into account finite size effects , we also performed a through bootstrap test to confirm that the empirical values of mutual information are significantly different from zero .another widely accepted method is to estimate the magnitude of the systematic bias that originates from finite size effects and then subtract this bias from the estimated entropy .to do so , we used the panzeri - treves bias correction method  in our calculations .the lead terms in the bias are , respectively =  - \frac{1}{{2n\ln ( 2)}}[\overline m  - 1],\ ] ] and =  - \frac{1}{{2n\ln ( 2)}}\sum\limits_j { [ \overline { { m_j } }  - 1]},\ ] ] where denotes the estimated number of outcome states , denotes the number of different states with nonzero probability of being observed given that the previous state is , and is the total number of observations .thus , the leading term of the mutual information bias equals = \frac{1}{{2n\ln ( 2)}}\{\sum\limits_j { [ \overline { { m_j } }  - 1 ] - [ \overline m  - 1]}\}.\ ] ] in what follows , we include the above adjustments to eliminate the impact of the finite size amount of data .we start by looking at the predictability of individual activities as measured by both the entropy and the mutual information extracted from sequences in the whrrl and epinion datasets , respectively .the histograms of , and calculated from user activity sequences are shown in figure [ fig : indv_entropy ] .the gray solid line in the plot shows a normal fit to the frequency count .the gap between and suggests a preference for certain activities , while the difference between the values of and in the figure indicates the existence of second order correlations between states .values of and for each individual in the online conversation network and the location check - in one are shown in figure [ fig : h1_h2 ] .the straight line corresponds to equal to .one interesting fact is that all the dots are below the straight line , which confirms that there is a positive difference between and for all individuals .this difference , which is the mutual information conditioned on previous states of user activity , is plotted in increasing order as the red line in figure [ fig : mutual_info ] for ( a ) conversations and ( b ) location check - ins .the positive values of the mutual information indicate information gain , or predictability , conditioned on historical states .we now examine the validity of the positive mutual information values in greater detail .there are usually two limitations when performing mutual information measurements .the first one is the potential bias resulting from the finite data size .the second one is the possibility of missing data points in the observation process . to make sure that our results are significant andare not impacted by these two limitations , we performed the following analysis . to establish that the observed positive value of the mutual information is not due to the finite size of our data sets, we performed a bootstrap test similar to that used in human conversation studies  .the null hypothesis of this test is that the mutual information has a positive value because of the finite size of the dataset . for this testwe set the significance level to .we first shuffled the true activity sequence and constructed a new sequence by drawing elements randomly one by one from the original sequence without replacement .if there is a second order correlation in the original sequence the shuffled sequence breaks the order and will thus have a higher value , while the value of would be the same before and after the bootstrap .this would result in a mutual information value smaller than that of the true sequence .the test checks if the value of obtained from the original activity sequence is significantly different from the shuffled one . to obtain an estimate of the distribution for the shuffled sequence we performed the shuffling procedure a times for each user and calculatedeach individual s shuffled mutual information .the value of the simulated sequence ranging from to is shown by the blue column in figure [ fig : mutual_info ] . as can be seen, the red line ( mutual information for true activity sequence ) lies well above the upper end of the error bar , which suggests that the value of the original sequence is significantly different from that generated by the simulated sequences .we can then reject the null hypothesis at the significance level and conclude that the positive mutual information we obtained is not due to the limited size of the data .furthermore , the fact that the mutual information is significantly different from zero suggests that a user s current online activity predicts his next interactions .next , we assessed the impact of potential loss of data points in the observation period by marking off a percentage of data points from the observed location check - in sequence from whrrl dataset . in real applications of predicting user behavior , a key question to apply maximum likelihood estimationdepends on the size of observations and the ratio of missing points . to examine the impact of ratio , we perform a mark - off on the bootstrap test of mutual information .we hide data points randomly from the true sequence while keeping the chronological order in the remaining sequence .for example if we have a mark - off rate of 0.5 , then of the states from the true sequence is marked off .the result of the bootstrap test after mark - off is shown in figure [ fig : whrrl_markoff ] .in the plot , the red dot shows the mutual information of the true sequence after performing mark - off procedure .the thick blue bar in the plot demonstrates the mutual information of the exact same sequence with shuffling .the mutual information is significantly different from that of the random shuffled sequence until the mark - off rate reaches . for values of the mark - off rate larger than , the difference between the two is broken when we fail to reject the null hypothesis that the sequence is significantly different from randomly shuffled .it is thus a confirmation that the deterministic pattern we observed is a robust one .this test also suggests the existence of a higher order correlations , larger than two , in human social online behavior .thus , the deterministic pattern discussed in this study is a robust phenomenon which can be applied to the general situations with missing or incomplete observations .as mentioned earlier , we also explored whether individuals acting alone are less predictable than when becoming members of a group .specifically , we investigated how predictable each user s is when engaged in group activities as compared with the predictability of individual ones . in the whrrl dataset userscan expose their position with a group of other users thus providing a sequence of group attendances by users and filtering out the places that were checked in by the user alone .we then calculated the information entropies and performed the same bootstrap test as before .the calculated mutual information of the activity sequences and shuffled sequences are shown in figure [ fig : whrrl_group_mutual_info ] .interestingly , the gap between the red line of true observation and the upper end of the error bar is is smaller than the one we obtained for the individual activities .in contrast with figure [ fig : mutual_info](b ) , the differences between the randomly shuffled sequences and the true observations are smaller .to quantify the observed difference , we calculated the gap between the mutual information from the true activity sequence and the percentile value of the shuffled sequences , defined by and .this allows for a comparison of sequences with different lengths .the relative frequency plot of this and is plotted in figure [ fig : whrrl_indv_group_unbias ] .the upper plot in figure [ fig : whrrl_indv_group_unbias ] shows the density plot of the gap for individual activity sequences while the lower plot shows the gap for group activities .as can be seen , the gap for individual activities has a larger of the mode compared with that of the group activities . under the assumption that both populations from and are random , independent , and arising from a normally distributed population with equal variances , the two sample t - test rejects the null hypothesis of an equal mean with a p - value of under significance level .this implies that it is harder to predict the a user s group activities than his individual ones .the values of versus for each individual are plotted in figure [ fig : whrrl_indv_group_unbias ] .the mean of is larger than .one possible explanation for this observation is that when individuals attend group activities , the decision as to what to do next is not usually made by the individual himself .thus , the tendency to follow others in their decisions tends to break one s regular patterns .this extra randomness would result in a larger value of and thus become less predictable .in summary , we have shown that sequences of user online activities have deterministic components that can be used for predicting future activities . using methods from information theory, we experimentally measured how much additional information can be gained from knowledge of previous states within a users activity sequences .while the degree of predictability varies from person to person , we also established that it is different when individuals join a group . besides the intrinsic interest of these findings, the fact that one can predict online social interactions should be helpful in improving the design of algorithms and applications for online social sites .c. w. would like to thank hp labs for financial support .c.w . and b.h .conceived and designed the study . c.w . collected the data .c.w . and b.h .discussed the results and wrote the manuscript .

Abstract: the massive amounts of data that social media generates has facilitated the study of online human behavior on a scale unimaginable a few years ago . at the same time , the much discussed apparent randomness with which people interact online makes it appear as if these studies can not reveal predictive social behaviors that could be used for developing better platforms and services . we use two large social databases to measure the mutual information entropy that both individual and group actions generate as they evolve over time . we show that user s interaction sequences have strong deterministic components , in contrast with existing assumptions and models . in addition , we show that individual interactions are more predictable when users act on their own rather than when attending group activities .